Multiscale Numerical Modelling ofMicrostructured Reinforced Composites

Fermin Otero Gruer

Advisors:

Prof. S. Oller MartınezProf. X. Martınez Garcia

Doctoral ThesisStructural Analysis Programme

Escola Tecnica Superior d’Enginyers deCamins, Canals i Ports de Barcelona

UNIVERSITAT POLITECNICADE CATALUNYABARCELONATECH

November 2015

Multiscale Numerical Modelling of MicrostructuredReinforced Composites

November 2015Universitat Politecnica de Catalunyac/ Jordi Girona 1-3, 08034 Barcelonawww.upc.edu

A mis Padres

Acknowledgements

Este trabajo no hubiera sido posible sin la ayuda, el consejo, la colaboracion, lacompanıa, el carino y sobre todo, la paciencia de personas que me han acom-panado en este camino hasta llegar aquı.

En primer lugar, un gracias especial a mis directores Sergio y Xavier por laconfianza, la paciencia y mas paciencia, la sincera dedicacion y mas que nada,los consejos en lo academico primero y por los otros, los formativos de vida, losbien intencionados, aquellos orientados solo al mejor desarrollo personal sin nadamas. A ellos, Gracias.

A mis otros “co” directores, y personas que dedicaron tiempo y esfuerzo enguiar, aconsejar y acompanar durante mis estancias de investigacion. Ever, milgracias por la predisposicion, la constante ayuda y colaboracion, por hacer de miestancia en West Virginia una experiencia inolvidable. Eduardo y Ana graciaspor hacernos sentir como en casa estando tan lejos, por la amabilidad y cercanıa.A Salvador y Miguel, gracias por la colaboracion durante mis dıas en Guanajuato,por toda la ayuda, y tiempo, para hacer que la cosa “jale”.

Mas que un gracias, un sincero reconocimiento a mi familia, mis padres yhermanos, por su incondicional apoyo, por el carino permanente entregado, porsu confianza y su creencia en mı (muchas veces desmedida). Todo eso ha influidopositivamente en mi persona empujandome hacia adelante, dando ese extra deenergıa que se necesita en esos momentos de incertidumbre, haciendo creıble esode “vamos que se puede. . . ”.

A mis companeros de oficina y pasillo, los presentes y pasados, por compartir,escuchar, aconsejar y mas que nada, soportar. Al cafe/infusion post almuerzo,en los que tantas veces arreglamos el mundo y mas importante, por escucharmey ayudarme en arreglar mi propio mundo. A todos esos asados en los que entrecorte y corte compartimos vida, por hacer de ellos un lugar de encuentro.

Muchas gracias Manuel y Mari Carmen por adoptarme como a un hijo, porla ayuda constante, la amable companıa, carino y el buen comer. Gracias porhacerme parte de su familia y por hacermelo sentir.

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Un especial GRACIAS “en mayusculas” a Raquel, mi companera, mi confi-dente, mi amiga y amante. Por su confianza y paciencia, por escuchar. . . yvolver a escuchar. . . por dejarse aburrir. Por mostrarme y recordarme lo que esimportante y lo que no. Por estar conmigo y sobre todo, por amarme libremente.

A Martina. Gracias por existir, gracias por tu increıble comprension. Crecepero intenta no cambiar tu esencia, eres genial. Te amo por sobre todo lo demas.

Y por supuesto, a todos mis amigos de vida, aquellas personas que se hancruzado en mi camino y que de una forma u otra han influido para ser el que soyhoy. A todos ellos, muchas gracias.

This work was financially supported by CIMNE together with the EuropeanCommunity under grant:NMP-2009-2.5-1 246067 M RECT “Multiscale Reinforcement of Semi-crystallineThermoplastic Sheets and Honeycombs”,FP7-PEOPLE-2013-IRSES 612607 TCAiNMaND “Tri Continental Alliance inNumerical Methods applied to Natural Disasters”, by European Research Coun-cil through of Advanced Grant:ERC-2012-AdG 320815 COMP-DES-MAT “Advanced tools for computationaldesign of engineering materials”, by Direccion General de Investigacion Cientıficay Tecnica:MAT2014-60647-R OMMC “Optimizacion multi-escala y multi-objetivo de es-tructuras de laminados compuestos”, by “Abengoa Research”, and by UniversitatPolitecnica de Catalunya (UPC).All this support is gratefully acknowledged.

Abstract

Most of the existing materials around us can be considered composite materials,since they are composed by several phases or components at certain spatial scale.The physical and chemical properties of composites, as occurs with structurescomposed by two or more materials, is defined by the response provided by theirconstituents. Therefore, a good characterization of the composite requires con-sidering the performance of its components. In the last decades, several methodshave been proposed with this approach to characterize composite materials, mostof them based on multiscale techniques.

Nowadays, multiscale hom*ogenization analysis is a popular topic in the simu-lation of composite materials. This is because the complexity of new compositesdemands of advanced analysis techniques for their correct characterization, andthanks to the continuous increase of computational capacity. However, the com-putational cost when multiscale procedures are taken to the non-linear rangeand are applied to real-size structures is still excessively high. In this context,this work presents a comprehensive hom*ogenization formulation for an efficientnon-linear multiscale modeling of composite structures.

The development of a composite multiscale constitutive model is addressedfrom two different hom*ogenization approaches. The first one corresponds to aphenomenological hom*ogenization procedure for the non-linear analysis of car-bon nanotubes reinforced composites. The second one is a general two-scalehom*ogenization procedure to analyze three-dimensional composite structures.

Carbon nanotubes (CNTs) have been regarded as ideal reinforcements forhigh-performance composites. The formulation developed takes into account ex-plicitly the performance of the interface between the matrix and the CNTs. Theload is transferred to the nanotubes through the considered interface. The com-posite non-linear behavior results from the non-linearities of its constituents, andin case of interface damage, it also becomes non-linear the law defined to couplethe interface with the CNTs. The formulation is validated studying the elasticresponse and non-linear behavior of several composites.

In the context of multiscale hom*ogenization, a first-order and an enhanced-first-order formulation is developed. The results obtained for laminate composites

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using the first-order formulation are compared with other microscopic formula-tions, showing that the hom*ogenization method is an excellent alternative whenmicrostructural effects must be taken into account. Then, a strategy to conductnon-linear multiscale analysis in an efficient way is proposed. The procedureconserves the dissipated energy through the scales and is mesh independence.The analysis of academic examples is used to show the capacity of the non-linearstrategy. Finally, the simulation of an industrial composite component proves theperformance and benefits of the non-linear hom*ogenization procedure developed.

Contents

Abstract vii

List of Figures xv

List of Acronyms xvii

Introduction 1Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 2

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Research dissemination . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

I Phenomenological hom*ogenization 9

1 Introduction 11

1.1 Part’s outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 State of the art 15

2.1 Classical mixing theory . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Modifications to the mixing theory . . . . . . . . . . . . . . . . . . 17

2.2.1 Mixing theory using serial-parallel model . . . . . . . . . . 17

2.2.2 Generalized mixing theory . . . . . . . . . . . . . . . . . . . 17

2.2.3 Mixing theory expressed in finite strains . . . . . . . . . . . 18

2.2.4 Generalized mixing theory expressed in finite strains . . . . 19

2.2.5 Mixing Theory by short fiber reinforcements . . . . . . . . 20

2.2.6 Serial-Parallel (SP) continuum approach . . . . . . . . . . . 21

3 Carbon nanotubes reinforced polymers 23

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4 Phenomenological hom*ogenization of CNTs reinforced polymers 274.1 Description of the composite constitutive model . . . . . . . . . . . 274.2 Formulation of the composite constitutive model . . . . . . . . . . 30

4.2.1 Definition of the parallel factor . . . . . . . . . . . . . . . . 314.2.2 Definition of the volume fraction of the interface region . . 324.2.3 Equivalent properties for MWCNTs . . . . . . . . . . . . . 334.2.4 Material non-linearity of the proposed model . . . . . . . . 34

4.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . 35

5 Validation and numerical results 395.1 Validation of the elastic response . . . . . . . . . . . . . . . . . . . 395.2 Validation of the non-linear performance . . . . . . . . . . . . . . . 415.3 Simulation of a four points bending beam . . . . . . . . . . . . . . 45

5.3.1 Materials properties, geometry and FE model of the ana-lyzed structure . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3.2 Linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . 495.3.3 Non-linear analysis . . . . . . . . . . . . . . . . . . . . . . . 505.3.4 Visco-elastic analysis . . . . . . . . . . . . . . . . . . . . . . 54

5.4 Effect of the CNTs angle on the elastic properties . . . . . . . . . . 55

6 Phenomenological hom*ogenization. Concluding remarks 59

II Multiscale hom*ogenization 61

1 Introduction 631.1 Part’s outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2 State of the art 672.1 The effective medium approximation . . . . . . . . . . . . . . . . . 672.2 The self-consistent method . . . . . . . . . . . . . . . . . . . . . . 682.3 Bounding methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.3.1 The classical bounds of Voigt and Reuss . . . . . . . . . . . 692.3.2 The variational bounding method . . . . . . . . . . . . . . . 70

2.4 The asymptotic hom*ogenization theory . . . . . . . . . . . . . . . . 712.5 hom*ogenization using the RVE concept . . . . . . . . . . . . . . . 72

2.5.1 Multiscale computational hom*ogenization . . . . . . . . . . 73

3 Multiscale hom*ogenization formulations 793.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.2 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . 793.3 First-order hom*ogenization approach . . . . . . . . . . . . . . . . . 80

3.3.1 Displacement field on the RVE . . . . . . . . . . . . . . . . 81

3.3.2 Kinematically admissible displacement fields and boundaryconditions in the RVE . . . . . . . . . . . . . . . . . . . . . 82

3.3.3 Microscopic and macroscopic strain tensor . . . . . . . . . . 853.3.4 Hill-Mandel principle and RVE equilibrium . . . . . . . . . 853.3.5 Microscopic and macroscopic stress tensor . . . . . . . . . . 86

3.4 Enhanced-first-order hom*ogenization approach . . . . . . . . . . . 873.4.1 Kinematically admissible displacement fields and boundary

conditions in the RVE . . . . . . . . . . . . . . . . . . . . . 883.4.2 Microscopic and macroscopic strain tensor . . . . . . . . . . 943.4.3 Hill-Mandel principle and RVE equilibrium . . . . . . . . . 953.4.4 hom*ogenized stress and second-order stress tensor . . . . . 963.4.5 Final remarks on the enhanced-first-order hom*ogenization

approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.5 Macroscopic and microscopic formulation . . . . . . . . . . . . . . 99

3.5.1 Macroscopic BVP . . . . . . . . . . . . . . . . . . . . . . . 993.5.2 Microscopic BVP . . . . . . . . . . . . . . . . . . . . . . . . 1003.5.3 Consequence of the boundary condition considered . . . . . 103

3.6 Finite element implementation . . . . . . . . . . . . . . . . . . . . 1053.6.1 Microscopic numerical implementation . . . . . . . . . . . . 106

3.7 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.7.1 Linear FE in the macroscopic mesh . . . . . . . . . . . . . . 1093.7.2 High-order FE in the macroscopic mesh . . . . . . . . . . . 110

3.8 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.8.1 Geometry, support scheme and mesh information . . . . . . 1113.8.2 Results and analysis . . . . . . . . . . . . . . . . . . . . . . 114

4 Numerical comparison with other formulations 1294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2 Geometry and numerical models . . . . . . . . . . . . . . . . . . . 130

4.2.1 Macro and micro numerical models . . . . . . . . . . . . . . 1304.2.2 Simple materials and composite description . . . . . . . . . 131

4.3 Comparison for several material configurations . . . . . . . . . . . 1324.3.1 Undamaged case . . . . . . . . . . . . . . . . . . . . . . . . 1324.3.2 Global damage case . . . . . . . . . . . . . . . . . . . . . . 1334.3.3 Local damage case . . . . . . . . . . . . . . . . . . . . . . . 1354.3.4 Local damage case in a localized region of the beam . . . . 137

4.4 Run times and memory used . . . . . . . . . . . . . . . . . . . . . 137

5 Non-linear extension proposed for multiscale methods 1395.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.2 General concepts of the proposed approach . . . . . . . . . . . . . 140

5.2.1 Non-linear activation function . . . . . . . . . . . . . . . . . 1415.2.2 Smart first step . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.2.3 Numerical hom*ogenized tangent constitutive tensor . . . . . 1435.3 Energy dissipation in a multiscale analysis . . . . . . . . . . . . . . 145

5.3.1 Fracture energy . . . . . . . . . . . . . . . . . . . . . . . . . 1455.3.2 Localization at the microstructural scale . . . . . . . . . . . 1465.3.3 Validation example . . . . . . . . . . . . . . . . . . . . . . . 147

5.4 Numerical examples of non-linear analyses . . . . . . . . . . . . . . 1515.4.1 Tensile test of a plate with a hole. . . . . . . . . . . . . . . 1515.4.2 Industrial Component . . . . . . . . . . . . . . . . . . . . . 157

6 Numerical hom*ogenization. Concluding remarks 165

Conclusions 167Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Appendices

A Constitutive models 175A.1 Elastic constitutive model . . . . . . . . . . . . . . . . . . . . . . . 175A.2 Elasto-plastic constitutive model . . . . . . . . . . . . . . . . . . . 176A.3 Elasto-damage constitutive model . . . . . . . . . . . . . . . . . . . 176A.4 Visco-elastic constitutive model . . . . . . . . . . . . . . . . . . . . 177

B Computational implementation 181B.1 Microscopic Kinematic relationships . . . . . . . . . . . . . . . . . 181

B.1.1 Master-slave kinematic relationships . . . . . . . . . . . . . 183B.2 Elimination of the slave degrees of freedom . . . . . . . . . . . . . 188

B.2.1 Linear implementation . . . . . . . . . . . . . . . . . . . . . 188B.2.2 Non-linear implementation . . . . . . . . . . . . . . . . . . 190

B.3 Derivatives of the shape functions . . . . . . . . . . . . . . . . . . . 192B.4 Derivation of displacement field, deformation gradient and gradi-

ent of the deformation gradient tensor . . . . . . . . . . . . . . . . 194B.5 FEM implementation in PLCd code . . . . . . . . . . . . . . . . . 195

B.5.1 PLCd FE2 implementation . . . . . . . . . . . . . . . . . . 197B.5.2 Numerical tangent constitutive tensor . . . . . . . . . . . . 202B.5.3 PLCd parallelization and memory improvements . . . . . . 204

References 207

List of Figures

Figures in Part I 11

4.1 SEM image of nanomanipulation and fracture surface of compos-ites [26]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Representation of formation for reinforced composite. . . . . . . . 29

4.3 Different regions in the new material CNT-interface. . . . . . . . . 29

4.4 Flow chart of the proposed model in a FEM code. . . . . . . . . . 37

5.1 Comparison of numerical and experimental results [16, 15]. . . . . 41

5.2 PA6 stress-strain relations for static tests [85]. . . . . . . . . . . . 42

5.3 PA6/A-MWCNT stress-strain relations for static tests [85]. . . . . 44

5.4 PA6/U-MWCNT stress-strain relations for static tests [85]. . . . . 45

5.5 Comparison of the experimental data with the numerical resultsfor PEEK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.6 Experimental data and numerical response with the calibrated in-terface component model. . . . . . . . . . . . . . . . . . . . . . . . 47

5.7 MWCNTs orientation distribution in the composite. . . . . . . . . 48

5.8 Geometry and extra information of the analyzed structure. . . . . 49

5.9 FE mesh used in the reduce model. . . . . . . . . . . . . . . . . . . 50

5.10 No-linear structural response for PEEK-CNT. . . . . . . . . . . . . 52

5.11 No-linear structural response for PEEK-CNT up to 50 [mm] ofvertical displacement. . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.12 No-linear results obtained at 1 [mm] of vertical displacement forPEEK-0.5%CNT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.13 Scheme of the generalized visco-elastic Maxwell model [99]. . . . . 54

5.14 Structural response for a sinusoidal load-unload of 1 Hz. . . . . . . 55

5.15 Change of elastic properties of CNTs reinforced PEEK with theCNTs angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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Figures in Part II 63

2.1 Ellipsoidal region of the Eshelby inclusion [1]. . . . . . . . . . . . . 682.2 Representative volume element models: (a) unit cell approach; (b)

statistical and ergodic approach [1]. . . . . . . . . . . . . . . . . . . 722.3 Representative hom*ogenization scheme [1]. . . . . . . . . . . . . . . 732.4 Schematic representation of a multiscale finite element program

[124]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.1 Macrostructure and microstructure around of the point Xo. . . . . 803.2 Reference and current configuration of the RVE. . . . . . . . . . . 813.3 Normal vectors to the surfaces in the reference configuration of a

Cubic RVE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.4 Normal vectors to the lines in the YZ surface of the Cubic RVE. . 903.5 Macro volume ΩM around point Xo and its micro structure. . . . . 953.6 Dimensions of the geometry and support scheme of the structure

simulated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.7 Different mesh sizes used in the macroscopic numerical model. . . . 1123.8 RVE models for the two different numerical simulation cases. . . . 1133.9 Mesh used on the RVE models for the different numerical simulations.1143.10 Reaction force vs number of elements in Z direction for the different

approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.11 Longitudinal stress obtained close to point A for the different

meshes and approaches used. . . . . . . . . . . . . . . . . . . . . . 1183.12 Macroscopic and Microscopic (RVE closest to point A) SXX field

for the mesh Macro3. . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.13 Shear stress obtained close to point A for the different meshes and

approaches used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.14 Reaction force vs number of elements in Z direction for the different

approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.15 Composite SXX obtained close to point A for the different meshes

and approaches used. . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.16 Composite SXZ obtained close to point A for the different meshes

and approaches used. . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.17 Fiber and matrix longitudinal stress field in the RVE of the Gauss

point closest to point A for the mesh Macro3. . . . . . . . . . . . . 1263.18 Longitudinal stress of the fiber and matrix components in the RVE

close to point A for the different meshes and approaches used. . . . 127

4.1 Geometric of the beam studied. . . . . . . . . . . . . . . . . . . . . 1314.2 Beam numerical model . . . . . . . . . . . . . . . . . . . . . . . . . 1314.3 RVE used for the undamaged case. . . . . . . . . . . . . . . . . . . 1334.4 Convergence analysis results. . . . . . . . . . . . . . . . . . . . . . 133

4.5 Reaction force obtained in the global damage case. . . . . . . . . . 1344.6 RVEs containing 50%, 25% and 12.5% of damaged layers. . . . . . 1354.7 Reaction force obtained in the local damage case. . . . . . . . . . . 1364.8 RVEs with 50%, 25% and 12.5% of damaged layers under load. . . 1364.9 FE mesh of the macro-model of the beam with two laminates. . . . 137

5.1 Non-linear activation function scheme. . . . . . . . . . . . . . . . . 1425.2 Non-linear strategy algorithm scheme. . . . . . . . . . . . . . . . . 1435.3 Smart first step algorithm scheme. . . . . . . . . . . . . . . . . . . 1445.4 Representative volume of the subscale. . . . . . . . . . . . . . . . . 1465.5 Structure simulated in the tensile test. . . . . . . . . . . . . . . . . 1475.6 Different meshes used in the tensile test. . . . . . . . . . . . . . . . 1485.7 Traction force vs displacement curves obtained in the tensile test. . 1495.8 Macrostructural results obtained at the end of the tensile test. . . 1505.9 Microstructural results obtained at the end of the tensile test. . . . 1505.10 Simulated structure of the plate with a hole. . . . . . . . . . . . . . 1515.11 Different meshes used in the plate with a hole. . . . . . . . . . . . 1525.12 Forve vs Displacement for the plate with a hole. . . . . . . . . . . 1525.13 Results obtained in the plate with a hole to Mesh1. . . . . . . . . . 1545.14 Results obtained in the plate with a hole to Mesh2. . . . . . . . . . 1555.15 Engine stiffener part. . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.16 Geometry and mesh of the RVE used in the external sheets. . . . . 1585.17 Mesh and boundary conditions used in the engine stiffener. . . . . 1605.18 Results obtained in the engine stiffener. . . . . . . . . . . . . . . . 1615.19 Results obtained in the RVE with maximum hom*ogenized damage. 162

Figures in Appendices 175

A.1 Scheme of the generalized visco-elastic Maxwell model [99]. . . . . 177

B.1 Master and slaves nodes in a general hexagonal RVE. . . . . . . . 182B.2 Master and slaves nodes on the negative faces of the RVE. . . . . . 185B.3 PLCd Flow diagram of the FEM implementation at the macro-scale.196B.4 Flow diagram of the hom*ogenized elastic constitutive tensor cal-

culation in PLCd code. . . . . . . . . . . . . . . . . . . . . . . . . . 198B.5 PLCd flow diagram of the FE2 implementation. . . . . . . . . . . . 200B.6 PLCd flow diagram of the FE2 implementation at the micro-scale. 201B.7 Flow diagram of the perturbation method implemented at PLCd

to obtain the macroscopic tangent constitutive tensor. . . . . . . . 203B.8 PLCd flow diagram of the parallel element loop implementation

at the macro-scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

List of Acronyms

BVP Boundary Value ProblemCNTs Carbon NanotubesCVD Chemical Vapor DepositionDDM Discrete Damage MechanicsEFO Enhanced-First-OrderFE Finite ElementFEM Finite Element MethodFE2 Finite Element Two-ScaleFO First-OrderIFSS Interfacial Shear StrengthLE Linear ElementMWCNTs Multiwall Carbon NanotubesNLAF Non-Linear Activation FunctionNLS Non-Linear StrategyOpenMP Open Multi-ProcessingPOD Proper Orthogonal DecompositionQE Quadratic ElementRHS Right-Hand SideRVE Representative Volume ElementSFS Smart First StepSP Serial-ParallelSWCNTs Single Wall Carbon Nanotubes

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Introduction

The continuum mechanics theory has made a great effort to obtain the behaviorof hom*ogeneous materials using physical and mathematical concepts showing agood agreement with reality. Furthermore, the constant improvements on com-puter technology and computer architecture have allowed to improve the numer-ical tools used to simulate mechanical structures. In the numerical simulationfield, one of the most extended methods used for several applications is the FiniteElement Method (FEM) [145]. In a FEM analysis the behavior of the hom*oge-neous materials in the structure is simulated by a specific constitutive law orconstitutive model with some calibrated parameters.

Composites are non-hom*ogeneous materials formed by two or more differentcomponents which can be hom*ogeneous materials or even micro-heterogeneousmaterials. The hom*ogenized behavior of the composites depends strongly on theinternal spatial distribution, the size and the properties of the material compo-nents and their respective interfaces. Therefore, composites require more complexand advanced constitutive models than the ones use in single materials.

For a linear analysis or a structural analysis to failure study it is enoughto simulate the composite with one orthotropic hom*ogenized characterizationand a constitutive law with some complexity. However, more realistic compositeconstitutive models are necessary to simulate the structures beyond their elasticlimit, to obtain the post critic behavior of these or to estimate their tenacity andstructural integrity.

The direct application of the FEM is not the most appropriate or effectivemanner to face the described problem. In a classical FEM analysis each com-ponent material has its own constitutive model. Therefore, the numerical modelof the structure must to be discretized with a Finite Element (FE) size of atleast the size of the components in the composite. In general, this restrictiongives as result FE meshes with large number of finite elements which demandan extremely expensive computational cost, and in some cases this analysis isunfeasible to perform. Consequently, to analyze composite structures and tocharacterize their behavior or fracture modes more suitable strategies must bedeveloped.

1

2 Introduction

Background and motivation

The complexity of the composite materials has promoted that several formula-tions appeared to predict their behavior, which are more o less suitable accordingto the computational cost required, the accuracy in the results desired or eventhe expected failure type. Further, the development of a new generation of com-posites with improved properties, more reliable and cheap has extended its useto many industrial applications.

The phenomenological hom*ogenization methods are a possibility to analyzecomposite materials with a heterogeneous internal structure. In this context, themost usual method is the classical mixing theory proposed initially by Truesdelland Toupin [131]. The formulation obtains the hom*ogenized behavior of the com-posite through the compatibility equation and from the mechanical performanceof the component materials, which are simulated with their own constitutive laws.Later, several modifications and extensions of this classical theory of mixtureshave enabled the resolution of any composite with reinforced matrix, without thelimitation required by the compatibility equation [99].

One of the most significant modifications of the mixing theory is the Serial-Parallel (SP) continuum approach. In the SP formulation the mechanic charac-teristic of the composite is obtained using not only the properties and constitutivemodel of the material components but also taking into account their topologicaldistribution [114]. The SP mixing theory assumes a serial-parallel self-adjustingbehavior to the topological distribution of fiber embedded in the matrix of thecomposite material. This approach imposes the iso-strain condition in the fiberalignment direction on the components of the composite (as parallel materials)and the iso-stress condition in the orthogonal direction (as serial materials).

The reinforcements developed today, as the nanofibers or Carbon Nanotubes(CNTs) and the renewed composites as the reinforced concrete with short fibers,require sophisticated formulations for their simulation. The interface zone be-tween matrix and reinforcement has a meaningful effect in the final propertiesand response of the composite structures made with these current composites.Many effort have been made to consider the debonding phenomenon in lami-nated composites but it is not enough to totally characterize the behavior ofthese micro-heterogeneous composites. Therefore, renovated formulations or re-newed modifications of existing theories should be developed to face with thechallenge of predict the behavior of these new composites.

The hom*ogenization techniques are another option to analyze composite ma-terials. In these methods the characterization of the entire composite is obtainedthrough the analysis of its internal structure or microscopic structure. In thiscontext, an approach extensively used is the called multiscale hom*ogenizationmethod. In general, the formulation is based on the use of the concept of unitcell or Representative Volume Element (RVE) [46]. The definition of the RVEcorresponds with a microscopic subregion which is representative of the entire

Introduction 3

micro-structure (referred as micro-scale) level of the composite. This is employedto determine the hom*ogenized properties and behavior of the composite level(also known as macro-scale). It is assumed that the RVE must contain a suf-ficient number of inclusions to make the hom*ogenized moduli independent ofhom*ogeneous forces or displacements on the RVE boundary.

Within this context, one of the most extended and popular method is theknown first-order hom*ogenization approach [125]. This multiscale method usesthe macro-scale deformation gradient tensor (or the strain tensor) to solve themicro-scale problem. The composite behavior (the macro-scale stress-strain rela-tionship) is obtained by a detailed modeling of the internal heterogeneous struc-ture of the composite in the RVE. Therefore, the approach does not requireany composite constitutive assumption or compatibility equation to address thecomposite response. Moreover, there are not restriction about the constitutivelaw of the component materials, even non-linear materials and time-dependencymodels can be taken into account. The benefits of the method becomes in achallenge when a non-linear analysis of a three-dimensional structure is studied.Considering a FE2 hom*ogenization technique [124], it is required for each timestep to solve one RVE at each point of integration at the macro-scale becausethe non-linear threshold and non-linear behavior of the hom*ogenized compositeare unknown. Therefore, the computational cost in the non-linear analysis ofan industrial component by using multiscale FE2 hom*ogenization is extremelyexpensive, and in many cases, is unsuitable to perform.

In addition to the computational cost to address the non-linear problem withmultiscale hom*ogenization methods, the softening issue must be considered too.The non-linear constitutive law of the component materials are defined in theRVE problem. Consequently, the non-linear behavior starts in the micro-scaleand then, it moves up to the macro-scale. Because of this, novel computationallyefficient multiscale strategies dealing with non-linear problem should be developedtaking into account also the conservation of the dissipated energy through thescales. Besides, they must be macro and micro mesh independence for the caseof FE2 hom*ogenization.

In the last decade, a second-order computational hom*ogenization was pro-posed as a natural extension of the first-order hom*ogenization method [60]. Itwas developed to be applied in critical regions of intense deformation, where thecharacteristic wave length of the macro-scale deformation field is of the order ofthe size of the micro-scale. Therefore, in this approach the macroscopic gradientof the deformation gradient is also incorporated in the microscopic scale prob-lem. The first-order equilibrium problem is conserved in the micro-scale thougha higher-order equilibrium problem appears in the macro-scale. The solutionof the proposed multiscale approach is made through a complex finite elementimplementation, which restricted its popular application.

The main advantage of the described second-order hom*ogenization is that it

4 Introduction

can consider intense localization phenomena, then it is a desirable approach fornon-linear analysis. On the other hand, the benefit of the first-order hom*ogeniza-tion is that it considers first-order equilibrium equations at both scales, whichrepresents an advantage from a point of view of computational implementation.Therefore, an enhanced-first-order approach could be an interesting option toaccount second-order effects of the macro-scale from the micro-scale by the in-corporation of macroscopic second-order deformation measure in the microscopicboundary value problem.

Objectives

The main objective of this study is to develop a comprehensive formulation for theanalysis of three-dimensional composite structures in linear and non-linear range.In this context, the partial targets to address the global aim of this dissertationcan be written in a synthesized form as:

- Development of a phenomenological hom*ogenization formulation based on themixing theory for the analysis of CNTs reinforced composites. The formula-tion should consider the effect of the CNTs-matrix interface in the compositebehavior.

- Extension to three dimensions of the first-order multiscale hom*ogenization forthe numerical analysis of composite structures. Implementation of an eliminationof redundant unknowns method to solve the microscopic boundary value problemconsidering the constraint conditions on the boundary domain.

- Improvement of the first-order multiscale approach implemented to considersecond-order effects in the microscopic scale from the macroscopic scale.

- Comparison of numerical simulations with other microscopic formulations toshow the advantages and drawbacks of the developed first-order multiscale pro-cedure.

- Development of a non-linear strategy to optimize the computational cost of theanalysis of real-size composites structures using a multiscale hom*ogenizationapproach.

In order to achieve the objectives described previously, there are parallel tasksthat must be addressed. These are important milestones of this work that areworth to be mentioned. Among them:

- Parallel numerical implementation of the developed approaches through an OpenMulti-Processing (OpenMP) philosophy in the finite element code PLCd [111].

- Development of a preprocessor manager to deal with the numerical models ofthe microstructure (RVE) using a GID problem type [84, 18].

Introduction 5

- Numerical validation of the different formulations implemented in PLCd throughthe simulation of several composites and by the analysis of real-structural com-ponents.

Outline

In the present dissertation is possible to observe that from a theoretical point ofview the main goal can be divided in two parts. For this reason, the documentis arranged in two major self-contained parts.

In Part I a phenomenological hom*ogenization model for the analysis of com-posites material using CNTs as reinforcement is presented. The formulationdeveloped is based on the mixing theory. In this context, Chapter 2 shows areview of the state of the art of the classical mixing theory and its subsequentmodifications while Chapter 3 introduces general considerations about the CNTsand a state of the art of the production methods and of the measured mechanicalproperties. Then, Chapter 4 presents the formulation and numerical implemen-tation of the “ad hoc” hom*ogenization model developed in this study. Chapter5 shows the validation and numerical examples analyzed using the formulationproposed in the above chapter. Finally, in Chapter 6 the conclusions and futurework about the model developed in this part of the dissertation is approached.

In Part II the developed multiscale hom*ogenization approach for compositestructures is described. The state of the art is addressed in Chapter 2, whichshows the fundamental theories and the latest developments about this researchtopic. Chapter 3 presents the formulations and implementations of the first-order hom*ogenization and the proposed enhanced-first-order extension to con-sider second-order effects. In Chapter 4 the implemented two-scale hom*ogeniza-tion procedure is compared with other micro-structural formulations. Chapter5 describes the non-linear strategy proposed for multiscale approaches, also itsvalidation and numerical applications are shown. The conclusions and futurework of this second part are addressed in Chapter 6.

In the final conclusions chapter the achievements of the present study areexposed along with concluding remarks and future works derived from this dis-sertation.

Research dissemination

The work included in this dissertation resulted in the following scientific publi-cations:

6 Introduction

Part I

F. Otero, S. Oller, X. Martinez and O. Salomon. Modelling of behaviour of carbon

nanotube-reinforced composites. In: MATCOMP’11 - IX Congreso Nacional de Ma-

teriales Compuestos. Girona, Espana 2011.

F. Otero, S. Oller, X. Martinez and O. Salomon. Numerical modelling of behaviour

of carbon nanotube-reinforced composites. In: COMPLAS XI - XI International

Conference on Computational Plasticity. Fundamentals and Applications. Barcelona,

Espana, 2011.

F. Otero, S. Oller, X. Martinez and O. Salomon. Modelling the elastic behavior

of carbon nanotube-reinforced composites. In: Composites 2001 - 3rd ECCOMAS

Thematic Conference on the Mechanical Response of Composites. Hannover, Germany,

2011.

F. Otero, S. Oller, X. Martinez and O. Salomon. Modelling viscoelastic behaviour

of carbon nanotube-reinforced thermo-plastics. In: MECOM 2012 - X Congreso Ar-

gentino de Mecanica Computacional. Salta, Argentina, 2012.

F. Otero, X. Martinez, S. Oller and O. Salomon. Study and prediction of the mechan-

ical performance of a nanotube-reinforced composite. Composite Structures. 2012,

94(9):2920-2930. doi: 10.1016/j.compstruct.2012.04.001.

Part II

X. Martinez, F. Otero and S. Oller. Strategy for an efficient material non-linear

multiscale analysis In: COMAT 2015 - VI International Conference on Science and

Technology of Composite Materials. Buenos Aires, Argentina, 2015.

F. Otero, S. Oller, X. Martinez and O. Salomon. Numerical hom*ogenization for

composite materials analysis. Comparison with other micro mechanical formulations.

Composite Structures. 2015, 122:405-416. doi: 10.1016/j.compstruct.2014.11.041.

F. Otero, X. Martinez, S. Oller and O. Salomon. An efficient multi-scale method for

non-linear analysis of composite structures. Composite Structures. 2015, 131:707-719.

doi: 10.1016/j.compstruct.2015.06.006.

In addition, part of the work was presented at the following unpublishedconferences:

F. Otero, S. Oller, X. Martinez and O. Salomon. Numerical hom*ogenization for the

simulation of composites materials. Comparison with other micro mechanical formu-

lations. Mechanics of Composites (MECHCOMP2014). Long Island, NY State, USA,

8-12 June 2014.

Introduction 7

F. Otero, S. Oller and X. Martinez. Non-linear multiscale strategy to analyze com-

posite materials efficiently. 18th International Conference on Composites Structures

(ICCS18). Lisbon, Portugal 15-18 June, 2015.

Finally, part of the work presented in this document is the result of the col-laborating with external researchers during the following research stays:

WVU-(West Virginia University), 6-month doctoral research stay. Worked underthe direct supervision of Prof. Ever J. Barbero in the Mechanical and AerospaceEngineering of the Benjamin M. Statler College of Engineering and Mineral Resources,WVU. Morgantown, USA. January - June 2014. The following article resulted fromthe work developed during the stay:

M.M. Moure, F. Otero, S.K. Garcıa-Castillo, S. Sanchez-Saez, E. Barbero and E.J.

Barbero. Damage evolution in open-hole laminated composite plates subjected to in-

plane loads. Composites Structures. 2015, 133:1048-1057. doi:

10.1016/j.compstruct.2015.08.045.

CIMAT-(Centro de Investigacion en MATematicas), 2-month research stay in the

framework of the TCAiNMaND project, a Marie Curie International Research Staff

Exchange Scheme (IRSES) under grant agreement FP7-612607. Worked under the

direct supervision of Dr. Salvador Botello in the Computational Sciences Department

of CIMAT. Guanajuato, Mexico. November - Dicember 2015.

Part I

Phenomenologicalhom*ogenization

9

Chapter 1

Introduction

Composites are materials made of at least two different components. Generally,are constituted by a matrix that surrounds the reinforcing elements, which maybe in the form of particles, nanotubes, short fibers, fibers, etc [4]. The mainfunction of matrix component is to give cohesion, support the reinforcement andtransfer the external actions to the reinforcements. While the main task of thereinforcement component is to improve the matrix properties. The appropriatedesign of structural elements made of this type of composite material requiresthe use of composite constitutive models capable of estimating their stiffness,strength and different failure modes.

In case of using fibers or nanotubes as reinforcement components, the perfor-mance of the composite depends on the achievement of the following four maincharacteristics:

Aspect ratio of the fibers. The fiber aspect ratio is a dimensionless geometricmeasurement that results from dividing the length of the reinforcement by itsdiameter. This parameter is important because the stress distribution in thereinforcement depends on it [52]. In fibers with high aspect ratios the fiber endeffect is less important. The reinforcement is considered to behave as a longfiber when its aspect ratio is larger than 1000.

Dispersion of fibers in the composite. A uniform distribution of reinforce-ment in the composite is fundamental to ensure that it is completely surroundedby matrix. This is necessary to obtain an effective stress transfer. A good dis-persion of the reinforcement in the composite also helps to have a more uniformstress distribution in it, reducing the regions susceptible having stress concen-trations.

Fiber alignment. It has been shown that the difference between randomdistribution and perfect alignment may represent a factor of five in the composite

11

12 Chapter 1. Introduction

Young’s modulus [42]. Fiber alignment also affects the isotropy of the composite,as perfect alignments increase its anisotropy.

Load transmission from the matrix to the fiber. The last and probablymost important factor is the interfacial tension between matrix and reinforce-ment. In general, the loads in a composite structure are introduced throughthe matrix and are transferred to the reinforcement through the interface [52].Therefore, the interface can be defined as the region surrounding the reinforce-ment where this stress transfer takes place. The properties of the compositedepend on the properties of this region, and on its ability to transfer the loadefficiently.

On the other hand, the external load applied to a composite is shared dispro-portionately by the different components, as their elastic properties are different.In case of considering an iso-strain hypothesis [131] the stresses on the reinforce-ment will be larger than in the matrix, as the reinforcement is stiffer than thematrix. This unequal stress distribution generates shear stresses between bothmaterials in a region that can is usually called the interface. The load transferfrom the matrix to the reinforcement is produced in this region. Shear stressesin the interface increase proportionally to the external load until a critical value,beyond which the interface breaks. This critical value is known as InterfacialShear Strength (IFSS) and it limits the stress transfer capacity.

In this context, the classical rule of mixtures was one of the first theoriesused to address a composite constitutive model from a phenomenological pointof view [131]. The theory defines the manner in which all components interactto provide the material performance. The iso-strain hypothesis defined in themixing theory implies a parallel distribution on the components in the composite.It is possible to think in an inverse mixing theory which replaces the iso-strainassumption by an iso-stress assumption, therefore it means a serial distribution onthe components in the composite. The characterization of the composite dependsof the hypothesis used in the formulation. Then, modifications of the mixingtheory capable to consider all possible behaviors of the composite: parallel, serialand mixed were proposed [95]. Finally, the serial/parallel concept was introducedin the theory, which replaced the iso-strain hypothesis by an iso-strain conditionin the fiber direction and iso-stress condition in the transversal directions [114].

The mixing theory began to be considered as a constitutive equation managerwhen its hypothesis were coupled with a thermodynamical description of thecomposite components [103]. Therefore, the formulation obtains the relationbetween the components even when they have reached their elastic limit. Withthis at hand, the different failure phenomenons present in the composites such asdebonding or delamination were modeled through of the constitutive law of thematerial components.

1.1. Part’s outline 13

1.1 Part’s outline

In this part of the dissertation a renewed modification of the mixing theory isproposed to consider the effect of the reinforcement-matrix interface zone in thefinal response of composite. The present formulation is developed for compositesthat use CNTs as reinforcement.

Following this aim, Chapter 2 shows an state of the art of the classical mixingtheory and its modifications since the early inclusion of the SP concept until thecurrent sophisticated modifications. Chapter 3 presents a review of the stateof the art regarding the different production methods and measured mechanicalproperties found in the literature regarding CNTs is presented. In this chapteris also addressed the different issues that should be considered in a constitutiveformulation for reinforced composites with CNTs.

In Chapter 4 the phenomenological hom*ogenization based in the on the mixingtheory is developed. The insertion of the concepts of serial and parallel behaviorin the CNTs-matrix bonding through of the definition of a parallel factor is shown.The CNTs debonding phenomena is also considered by a material non-linearityproposed. At the end of the chapter, the implementation in the FEM code PLCdis presented.

Chapter 5 shows the results of validation of the implemented composite con-stitutive model using information data from literature. The validation of themodel is made for linear and non-linear behavior using experimental data ofseveral composites. Then, numerical examples are developed showing the goodbehavior of the phenomenological hom*ogenization model developed.

Finally, Chapter 6 the conclusions about the composite constitutive formula-tion developed in this part of the study are addressed in detail.

Chapter 2

State of the art

The first part of the present study proposes a phenomenological composite con-stitutive model which is based on the classical rule of mixtures. Therefore, in thefollowing a literature review is dedicated to explain this mixing theory and themodifications which have been developed over the years.

2.1 Classical mixing theory

The mixing theory was originally proposed by Truesdell and Toupin [131]. Later,Truesdell [130] extended the theory to linear systems and Green and Adkins [37]presented a general non-lineal constitutive equations. Finally, Ortiz and Popov[104, 105] proposed a general constitutive equations for unreinforced concreteidealized as a composite material.

The classical mixing theory is based on the mechanical of local continuoussolid and it is appropriated to explain the behavior of a point in a solid com-pound. It is based on the principle of interaction of the component substances inthe composite material, assuming the following hypothesis: i) on each infinitesi-mal volume of the composite are involved a set of component substances,ii) eachcomponent contributes to the behavior of the compound material in the sameproportion as their volumetric fraction, iii) all component materials have thesame deformation (compatibility equation or closure equation) and iv) the oc-cupied volume of each component is much smaller than the total volume of thecomposite.

The second hypothesis implies a hom*ogeneous distribution of all substancesin the compound material. The volume fraction, the internal distribution andthe interaction between the different substances components, which have its re-spective constitutive law, determine the behavior of the composite material. Thisallows combining materials with different behavior (elastic, elasto-plastic, elasto-

15

16 Chapter 2. State of the art

damage, etc.), which have an evolutionary behavior governed by its own law andinternal variables [105, 103].

The third hypothesis demands that the following condition of compatibilitymust be fulfilled

εij = (εij)1 = (εij)2 = · · · = (εij)n , (2.1)

where the assumption of infinitesimal deformations on each components are con-sidered and where, εij and (εij)n are the strain tensors of the composite and ofthe n− th component of the compound material, respectively.

The specific Helmholtz free energy of the composite is given by the sum of thespecific Helmholtz free energies of each components of the composite multipliedby its volume fraction, that is

Ψ (εe, θ,α) =

n∑c=1

kcΨc (εc, εpc , θ,αc) , (2.2)

where Ψc is the specific Helmholtz free energy, kc is the volume fraction, εpc isthe plastic strain tensor and αc are the inner variables of each one of the n− thcomponents in the composite.

The volume fraction coefficient allows to consider the contribution of eachmaterial to the composite and it is obtained with the following equation as

kc =dVcdV0

, (2.3)

where dVc is the volume of the c − th component and dV0 is the total volumeof the composite. The volume fractions of the components must to satisfied thefollowing condition:

n∑c=1

kc = 1. (2.4)

Equation (2.4) guarantees the conservation of mass. Following with the pro-cedure used for a simple material [98, 71, 70, 102], from the Clausius-Duheminequality and applying the Coleman method, the constitutive equation of thecomposite is obtained as

σij =∂Ψ (εij , θ, αi)

∂εij=

n∑c=1

kc∂Ψc (εij , θ, αi)

∂εij=

n∑c=1

kc (σij)c , (2.5)

where, σij and (σij)c are the stress tensors of the composite and of the c − thcomponent of the compound material, respectively. The composite constitutivetensor is obtained considering the variation of the composite stress tensor respectto the strain tensor, therefore

Cijkl =∂σij∂εkl

=∂2Ψ (εij , θ, αi)

∂εij∂εkl=

n∑c=1

kc (Cijkl)c . (2.6)

2.2. Modifications to the mixing theory 17

However, the closure equation given by (2.1) imposes a strong limitation ofthe classical theory of mixtures because it is strictly valid only for compositeswith parallel behavior. Moreover, this limitation is extended to non-linear rangebecause each component can have different deformation for a given load step.Because of this, several formulations have been proposed from the classical mixingtheory in order to consider different internal behavior (not only parallel behavior)and the nonlinearity of each components too.

2.2 Modifications to the mixing theory

Over the years the classical mixing theory has had many modifications and im-provements with the objective of expanding its scope. Some of the most relevantdevelopments are presented in this section.

2.2.1 Mixing theory using serial-parallel model

The classical theory of mixtures was modified by Oller et al. [101] and Neamtuet al. [95] introducing the serial-parallel concept. The model allows to representcomposites for various possible combinations of serial and/or parallel behaviorof their components. The properties of the composite are obtained using theproperties of each component and taking into account its topological distribution.The modification is based on the definition of the total strain field as a weightedsum of the contributions of the deformation components in series and parallel.Therefore

εij = (1− ℵ) εparij + ℵ εserij , (2.7)

where εij is the total strain tensor of the composite, εparij and εserij represent theparallel and serial strain tensor, respectively. And ℵ is the coupling parameterthat relates in weighted form the serial-parallel behavior, it has a value rangefrom 0 to 1. The deformation components in parallel and serial behavior areapproximated by

εparij u1

n

n∑c=1

(εij)c , εserij =

n∑c=1

kc (εij)c . (2.8)

This modification of the classical mixing theory has the disadvantage that thecoupling parameter, in general, must be calibrated with experimental tests of thecomposite.

2.2.2 Generalized mixing theory

The proposed modification of the classical theory of mixtures by Oller [99] is ageneralization of this theory. The new proposed is enabled to solve any reinforced

18 Chapter 2. State of the art

matrix composite, without the limitation of the compatibility equation (see (2.1))required by the classical theory. The closure equation is satisfied automatically bythe proposed modification. The fundamental hypothesis of this theory’s general-ization is a new definition of the third hypothesis of the classical theory. The newthird hypothesis is: iii) the components must satisfy a generalized compatibilityequation to fit the topology of the serial-parallel composite. The new hypothesisallows establishing the relationship between the composite deformation and thedeformation of each component. The new compatibility equation provides thelink between the parallel behavior and the serial behavior and can be expressedas

(εij)c = (1− χc) · Iijkl εkl︸ ︷︷ ︸(εparij )

c

+χc · [(φijkl)c · (εkl − εpkl) + (εpkl)c]︸ ︷︷ ︸(εserij )

c

, (2.9)

where (εij)c is the strain tensor of the c− th component, which can be separatedin its parallel (εparij )c and serial (εserij )c component, respectively, and εij is thetotal strain tensor in the composite. Equation (2.9) can be rewritten as

(εij)c = [(1− χc) · Iijkl + χc · (φijkl)c] : εkl − χc(εpkl)c, (2.10)

where (εpkl)c is a plastic strain tensor without physical meaning, which is definedonly for operating purpose and it is obtained from the plastic strain tensor of thecomposite distributed among its components according to (φijkl)c ε

pkl and the

plastic strain tensor of the current component (εpkl)c. The serial-parallel couplingparameter is defined as 0 ≤ χc = sinαχ ≤ 1, where αχ corresponds to theexisting angle between the reinforcement orientation and the orientation of thehigher principal stress.

2.2.3 Mixing theory expressed in finite strains

The extension to finite strains of the classical theory of mixtures considers that thethird hypothesis (original closure equation given by (2.1)) must be verified on thereferential configuration and on the spatial configuration for each component[11]

Eij = (Eij)1 = (Eij)2 = · · · = (Eij)n

eij = (eij)1 = (eij)2 = · · · = (eij)n,(2.11)

where Eij is the Green-Lagrange strain tensor and eij is the Almansi strain ten-sor. Considering the definition of the right Gauchy-Green tensor and (2.11) thecompatibility equation can be written as a function of the deformation gradienttensor as

Fij = (Fij)1 = (Fij)2 = · · · = (Fij)n. (2.12)

The others hypothesis of the classical theory must be also verified. The re-lationship between the volume of a component in the spatial configuration and

2.2. Modifications to the mixing theory 19

in the referential configuration is given by the determinant of the deformationgradient tensor and it is

dVc =1

Jdvc. (2.13)

With (2.13) it is possible to demonstrate that the volume fraction of the compo-nents do not change in both configurations.

The solution process starts by estimating the strain increments at the refer-ence configuration and then through tensor transport operations (“push-forward”)the strain tensor in the updated configuration can be obtained. The constitu-tive equation of each components of the composite is integrated in the updatedconfiguration. Each of these components may have different kinds of constitu-tive behavior (plasticity, damage, etc.) and also, these constitutive models maybe isotropic or anisotropic. Then, with the integrated stress state of the com-ponents it is possible to obtain the stress state and the constitutive tensor ofthe composite. Finally, the obtained composite informations are transported(“back-forward”) to the reference configuration and then, the internal forces arecomputed. The balance between the internal forces and applied external forcesis verified in an iterative procedure until convergence.

2.2.4 Generalized mixing theory expressed in finite strains

The procedure to extend the generalized theory of mixtures to finite strains isthe same than the one used to extend the classical mixing theory in Section2.2.3. This generalized theory starts with the non-compliance of the classicalcompatibility equation. Therefore, the proposed new closure equation given by(2.10) must be written now in the reference and the updated configuration, thatis

(Eij)c = [(1− χc) · Iijkl + χc · (Φijkl)c] : Ekl − χc(Epkl)c

(eij)c = [(1− χc) · Iijkl + χc · (φijkl)c] : ekl − χc(epkl)c.(2.14)

Equations (2.14) give the strain tensors for each component in both configu-rations. The constitutive equations are obtained following a similar formulationused in the classical theory of mixtures in finite strains. Finally, the stress tensorsof the composite in the reference configuration S and in the current configuration,the Kirchoff stress tensor τ are, respectively

S =

n∑c=1

kc[(1−χc)·I4+χc·(Φ)c]T :[(CS)c:[(1−χc)·I4+χc·(Φ)c]:(E

e)c] (2.15)

τ =

n∑c=1

kc[(1−χc)·I4+χc·(Φ)c]T :[(cτ )c:[(1−χc)·I4+χc·(Φ)c]:(e

e)c] = Jσ, (2.16)

20 Chapter 2. State of the art

where I4 is the fourth order identify tensor, (CS)c and (cτ )c are the tangentconstitutive tensors for the c − th component in each configuration, (Ee)c and(ee)c are the elastic strain tensors and σ is the Cauchy stress tensor.

2.2.5 Mixing Theory by short fiber reinforcements

The formulation of the mixing theory is oriented to a composite where the re-inforcements are long fibers, and the condition of the compatibility equations isverified. However, when the aspect ratio of the fiber decreases, the condition offiber-matrix compatibility is not satisfied. This is because the effect of slip andthe limit transmission of forces between fiber and matrix at the ends of the fibertake increasingly significant. This situation creates conditions of stress concen-tration and distortion in the fiber and the surrounding matrix because of thediscontinuity. The effectiveness of the fibers in the composite stiffness decreaseswhen the length of the fiber decreases.

Equation (2.17) shows the axial stress distribution along the fiber [52] as

σf (x) = CσfEm

[1− cosh

(β(l2 − x

))cosh

(β l2) ]

∀ 0 ≤ x ≤ l

2, (2.17)

where Cσf is the Young’s modulus, l is the length of the reinforcement, Em is thelongitudinal strain of the matrix and the parameter β is defined as

β =

√GcCσf

2π

Af lnr′

r

, (2.18)

where Af is the cross section of the fiber, Gc is the shear modulus of the composite

and r′

is the mean distance between the reinforcing fibers.One way to consider the contribution of the short fiber reinforcement in the

classical mixing theory is through the average stress along the fiber, then

σf =1

l

∫ l

σf (x) dx = Cσf

[1− tanh

(β l2)(

β l2) ]

Em = CσfEm. (2.19)

Here, Cσf is the average or hom*ogenized Young’s modulus of the reinforcement,which is function of the length of the fiber and of the geometric parameters ofthe composite.

The obtained short fiber hom*ogenized Young’s modulus is smaller than thereal fiber Young’s modulus, this shows that its participation on the mechanicalproperties of the composite depend not only of its mechanical properties butalso of the overall properties of the matrix-reinforcement assemblage. The sameconcept used to hom*ogenize the stress along the fiber can be extended to get the

2.2. Modifications to the mixing theory 21

three dimension hom*ogenized constitutive tensor of the short fiber reinforcementas

CSf = CSf

[1− tanh

(β l2)(

β l2) ]

, (2.20)

where CSf is the orthotropic constitutive tensor in the referential configuration ofthe reinforcement. Using the previously described concept, the incorporation ofthe short fiber in the theory of mixtures can be extend to finite strains too [99].

2.2.6 Serial-Parallel (SP) continuum approach

The SP continuum approach has been proposed by Rastellini et al. [114], andit is a natural evolution of the parallel mixing theory developed by Car et al.[11, 99]. The theory is based on the compatibility conditions defined by Trusdelland Toupin [131], but introduces a modification in the iso-strain hypothesis. Theiso-strain condition is imposed in the reinforcement direction (normally fiber) anda new iso-stress condition is imposed in the transversal directions. The theory isbased on the following hypotheses:

• The constituent materials of the composite are subjected to the same strainin the parallel (fiber) direction.

• Constituent materials are subjected to the same stress in the serial direc-tion.

• The response of the composite material is directly related to the volumefractions of its constituent materials.

• The phases in the composite are considered to be hom*ogeneously ditributed.

• The constituent materials are considered to be perfectly bonded.

In this formulation the definition of the constitutive model for the compositeneeds the introduction of additional equations that specify somehow the inter-action between the component phases. Then, the resulting composite materialmodel will depend crucially on the adopted specific additional equations thatcharacterizes the mechanical interaction at the micro-scale. These additionalsets of equations are referred to as “closure equations” and are obtained takingthe iso-strain hypothesis in the reinforcement direction and iso-stress hypothe-sis in transversal directions. Considering only two composite components, theequations that define the stress (σ) equilibrium and setting up the strain (ε)compatibility between the individual components follow the hypothesis previ-ously described are:

22 Chapter 2. State of the art

• Parallel behaviorcεp = mεp = fεpcσp = mkmσp + fkfσp

(2.21)

• Serial behaviorcεs = mkmεs + fkfεscσs = mσs = fσs

(2.22)

where, the superscripts c, m and f stand for composite, matrix and fiber, respec-tively, the subscripts s and p correspond to the serial and parallel behavior andik is the volume fraction of each constituent in the composite.

Composite materials that can be modeled with this formulation are thoseformed of long fibers embedded in a matrix. The theory predicts the differentbehavior of the composite, depending on the load direction. This formulation canobtain the linear and non linear behavior of structural elements made of compos-ite materials as has been proved in several papers [79, 76, 75, 108, 78, 109]. TheSP theory is able to simulate the delamination problem naturally, without havingto define specific elements or predefine the path of fracture. The approach hasbeen also extended to tri-dimensional framework by Martinez et al. [77] and ap-plied for the numerical simulation of structures of reinforced concrete retrofittedwith carbon fiber reinforced polymers. In this study the tangent constitutive ten-sor of each component of the composite is obtained by means of a perturbationmethod.

Chapter 3

Carbon nanotubesreinforced polymers

Since their discovery by Lijima in 1991 [50], CNTs are considered a new gener-ation of reinforcements [17]. Their “nano” size structure makes them potentiallyfree of defects, which gives them excellent physical properties [116, 117]. A nan-otube is a tubular cylinder formed by sp2 bonds between the carbon atoms alongits length. There are two main nanotube types: Single Wall Carbon Nanotubes(SWCNTs), which are made of a single wall tube with an outer diameter in theorder of 1 nm; and Multiwall Carbon Nanotubes (MWCNTs), which consist inseveral concentric walls, one inside the other, separated by a distance of 0.34 nm[50]. The diameters range of MWCNT varies from 2 to 100 nm. MWCNT canhave lengths up to 100 µm.

Carbon nanotubes can be obtained by several procedures. The first methodused was the arc-discharge [10], which consists in generating an arc dischargebetween two graphite electrodes in an inert gas atmosphere at low pressure. Thecontinuous electric discharge sublimates the carbon atoms of the electrodes andforms a plasma around them. This method produces free defect nanotubes alongtheir length. The length of these nanotubes can reach 50 µm. Another proce-dure is the laser ablation. This consists in vaporizing the graphite by radiationwith a laser pulse, in an inert gas atmosphere, inside a high temperature reac-tor. The nanotubes are formed when the graphite vapor touches the cold wallsof the reactor. Finally, the most common procedure used for commercial pro-duction of carbon nanotubes is the deposition of Catalytic Vapour Phase (alsonamed, Chemical Vapor Deposition (CVD)). This procedure allows producinglarge amounts of nanotubes at a low cost. This method prepares a substratewith a metal layer. The nanotube diameter depends on the size of the metalparticles. The process starts by mixing two gases; one of them is used as a source

23

24 Chapter 3. Carbon nanotubes reinforced polymers

of carbon, and the other for the process itself. The nanotubes grow on the sideof the metal catalyst. The generated nanotubes have defects on its surface. Thismethod can provide oriented nanotubes if there is plasma during their growth.

Nanotubes obtained by arc-discharge have Young’s modulus values in theorder of 1TPa. Recent measurements carried out in arc-MWCNTs (multiwallnanotubes made by arc-discharge) have provide Young’s modulus values withvalues varying from 0.27 to 0.95 TPa, ultimate strain values higher than 12%, andultimate tensile stresses in the range of 11 to 63 GPa [141]. In these measurementsit was also obtained the stress-strain curve of the MWCNTs with help an electricmicroscope.

The properties obtained for CVD-MWCNTs (multiwall carbon nanotubesobtained by CVD) are low due to the defects in the nanotubes surface. The firstsYoung modulus measurement known was made with an atomic force microscope[118] and the values obtained were in the range of 12 to 50 GPa. Later on,new measurements have shown Young modulus values in order of 0.45 TPA,and ultimate tensile stresses of 3.6 GPa [139]. The lower measured values wereassociated with defects in the nanotube and with the slipping of the inner tubes inMWCNTs. The difference in measured values between CVD-MWCNTs and arc-MWCNTs shows the influence of defects on the properties of these new materials.

It is not entirely clear which nanotube type performs better as a reinforcement.A recent study made by Cadeck et al. [9] comparing the properties of a polyviny-lalcohol (PVA) matrix reinforced with different types of CNTs nanotubes (dou-ble wall nanotubes (DWCNT), SWCNTs, arc-MWCNTs and CVD-MWCNTs)showed that the effectiveness of reinforcement is inversely proportional to its di-ameter, except when using SWCNTs. The study also proved that the compositeproperties are proportional to the total interface area. The composite reinforcedwith SWCNTs had the lowest properties; this result is associated with slippingof SWCNTs inside the bundles. Finally, the study states that the best propertiesare obtained with the CVD-MWCNTs with smaller diameter.

Currently, there are several methods that can be used to produce nanotube-reinforced composites. The choice of the most appropriate method depends ofnature of the involved components [17]. All methods seek to produce a compos-ite with a good dispersion of the CNT reinforcement and to create an interfacecapable of transmitting the external load to the nanotubes. The manufactur-ing process has to be selected taking into account that it must not affect theproperties of the composite components.

Several studies have shown that the composite formation generates an inter-face zone around the carbon nanotubes. This interface has a different morphologyand properties than the original matrix [83, 82]. The size, shape and properties ofthe interface have a strong dependence on the matrix type [140] and the formationprocess. Pull out experimental tests indicate that IFSS values are much higherthan the theoretical ones [132], which are calculated using the shear strength of

25

the matrix. This result suggests that the interface region around the nanotubehas better properties than the rest of the matrix [19]. Some studies estimate thatin this region matrix properties may improve by an order of magnitude [3]. Frac-ture surface images obtained from composites with strong nanotube-matrix bondshow that the interface zone has a thickness several times larger than the nan-otube diameter [26, 122]. In the case of semi-crystalline matrices, the interfacezone is associated with crystal nucleation around the nanotubes [121].

All manufacture processes seek to obtain a composite with a strong bondbetween the nanotube and the matrix, in order to transfer effectively the loads.The IFSS defines the capacity of the bond. Values of 500 MPa have been ob-tained for the IFSS when observing the stresses induced to a broken nanotube,these values where obtained using a Transmission Electron Microscope (TEM).The study attributes this value to the presence of covalent bonds between thematrix and the nanotube [134]. Molecular Dynamics (MD) simulations carriedout confirm that strong bonds are obtained when these are covalent. In fact, thetransfer load of the interface increases by an order of magnitude with just a 1%of covalent bonds in its surface [29]. On the other hand, the generation of manycovalent bonds in the interface is detrimental to the intrinsic properties of thenanotube [29, 30].

When there are not covalent bonds, the interaction between matrix and nan-otube is made with Van der Waals forces. Several studies show that this unionis weaker. Molecular Dynamics simulations made by [29] predicted values of theIFSS that do not exceed 2.8 MPa. Another study made by [67] predicted valuesup to 160 MPa. According to [68], the differences in the results depend on thepolymer type and they can be in the range of 80 to 135 Mpa. The differencein the results, and the good values of IFSS, were attributed to the morphologyand the capacity of the matrix to generate helical chains around the nanotube.On the other hand, nanotubes have a smoother outer surface and therefore, thecontribution of the frictional forces to the IFSS are an order of magnitude lower[2].

Experimental results of pull-out tests show values of IFSS between 20-90 MPa[3, 2]. Other experiments using the drag-out technique have shown values between35-376 MPa [19]. The disparity of the results suggests that is not always possibleto generate covalent bonds. The maximum values obtained experimentally areassociated to covalent bonds and consider that the interface zone has betterproperties than the rest of the matrix.

Carbon nanotubes, mainly SWCNTs, tend to agglomerate. This makes verydifficult to obtain a good dispersion of those in the polymer. Besides, the smoothsurface of the nanotubes leads to a possible lack of bond between the nanotubeand the matrix. Currently these problems are solved with a chemical function-alization of the CNTs. The covalent functionalization can be done by modifyingthe carboxylic acid groups on the nanotube surface and or by direct addition

26 Chapter 3. Carbon nanotubes reinforced polymers

of reagents. The drawback of functionalizing the nanotubes is that there is anintrinsic degradation of their properties [30]. In general, two different methodshave been used for the functionalization: “grafting from” and “grafting to”.

The “grafting from” method is based on the initial immobilization of initiatorson the nanotube surface, followed by an in situ polymerization of the suitablematrix for the formation of polymer molecules around the nanotube [49, 138].The advantage of this method is that it allows the formation of composites witha high density of nanotubes. The disadvantage is that this method requires strictcontrol of the quantities and the conditions in which the polymerization reactiontakes place.

The “grafting to” method makes the union of preformed polymer moleculesto functional groups on the surface of the nanotube through chemical reactions[69, 7]. The advantage of this method is that it can be used with commercialpolymers. However, it has as a limitation that the initial union of the polymerchains inhibits the diffusion of macromolecules to the surface. Therefore, thedensity of functionalization is low.

The above description shows that the final properties of the composite de-pend on many parameters. Together with these, there are others aspects thatmay also condition the final properties of the composite, such as the undulationand misalignment of the nanotubes inside the matrix. All this variability can beconsidered the responsible of not having yet an accepted theory capable of de-scribing correctly the performance of nanotube-reinforced composites. It is alsothe reason because the existing theories fail in their predictions. Comparisonsbetween measured mechanical properties and theoretical results, show that thetheoretical predictions are generally three times higher than measured results[122, 23].

Chapter 4

Phenomenologicalhom*ogenization of CNTsreinforced polymers

Carbon nanotubes have been regarded as ideal reinforcements of high perfor-mance composites. A key factor for the reinforcement efficiency is the interfacebonding between the CNTs and the matrix. In this chapter the formulationand numerical implementation of a new constitutive model to predict the per-formance of composites made of CNTs is presented. The composite constitutivemodel takes intro account explicitly the mechanical performance of the interfacebetween the matrix and the CNTs. The proposed model is based in the clas-sical mixing theory. As it is written, the mixing theory can be understood asa constitutive model manager. Therefore, the mechanical performances of thecomposite are obtained from the behavior of the composite components, eachone simulated with its own constitutive law [11]. The present new compositeconstitutive model is formulated with the same philosophy, which increases itsversatility and simulation capability.

4.1 Description of the composite constitutivemodel

The proposed composite constitutive model assumes that the composite is acombination of three different materials: matrix, CNTs and an interface [16].The interface component corresponds to the matrix that surrounds the CNTs. Itis considered as an independent component, with its own constitutive law. Theinterface is used to define the capacity of the matrix to transfer the loads to the

27

28 Chapter 4. Phenomenological hom*ogenization

reinforcement.

Although the phenomenological performance of the composite already justifiesthe definition of an interface material; images obtained with Scanning ElectronMicroscope (SEM) of CNTs reinforced composites, such the ones shown in Figure4.1, prove its actual existence. These images reveal that the structures protrudingfrom the fractured surface have larger diameters than the original MWCNTs usedin the sample preparation [26]. The material surrounding the CNTs correspondsto the interface. The presence of an interface, as a differentiable material, is alsoproved by Differential Scanning Calorimetry (DSC) measurements carried out incomposites with a semi-crystalline polymer as matrix. These measurements showa linear increase of crystalline matrix as the nanotube volume fraction increases,suggesting that each nanotube has a crystalline coating [8].

Figure 4.1: SEM image of nanomanipulation and fracture surface of composites [26].

Once having conveyed the necessity of including the interface material in theformulation to simulate the mechanical performance of CNTs reinforced compos-ites, in the following is described the new procedure proposed, which is summa-rized in Figure 4.2. This figure shows that the composite is divided in severallayers, each one containing carbon nanotubes with a different orientation. Alllayers are coupled together using the parallel mixing theory. This is, assumingthat all layers have the same deformation. The new formulation developed pro-vides the mechanical performance of each layer by combining the response ofthe three coexisting materials: matrix, interface and CNTs. The layer responsedepends on the materials and on their volumetric participation in the composite.

First, the layer is split into matrix and a new material that results of couplingthe CNTs with the interface. The relation between the matrix and the CNT-interface material is established in terms of the parallel mixing theory (they areassumed to have an iso-strain behavior). On the other hand, CNTs and theinterface are coupled together with a combination of parallel and serial mixingtheories. The serial mixing theory assumes that all components have the samestresses.

4.1. Description of the composite constitutive model 29

Figure 4.2: Representation of formation for reinforced composite.

Figure 4.3 shows scheme used to obtain the performance of the CNT-interfacematerial. This is based in the short-fiber model developed by Jayatilaka [52].According to this model, the load is transferred from the interface to the nanotubeat the ends of the reinforcement, through shear stresses. In this region normalstresses in the fiber increase from zero to their maximum value, which is reachedin the central part of the reinforcement. In this region there is not load transferand shear stresses are null. This whole stress transfer scheme can be simplifiedassuming a CNT-interface performance defined by a serial mixing theory at theends of the reinforcement and a parallel mixing theory at the center of it.

Figure 4.3: Different regions in the new material CNT-interface.

A parallel factor named Npar is defined to differentiate these two regions.This parameter, multiplied by the nanotube length, provides the length of thenanotube-interface element with a parallel behavior. The length with a serialperformance is defined by the complementary factor.

30 Chapter 4. Phenomenological hom*ogenization

4.2 Formulation of the composite constitutivemodel

The Helmholtz free energy [73] of a material point subjected to infinitesimaldeformations can be described with the following thermodynamic formulation[103, 71]

Ψ = Ψ (ε, θ,α) , (4.1)

where ε is the strain tensor, θ a measure of temperature and α = εp, d, s a setof inner variables, for example: εp is the plastic strain tensor, d damage innervariable and s any other material internal variables.

The proposed model simulates the composite combining the different com-ponents using the serial and parallel mixing theories. If this combination isperformed according to what has been described in previous Section 4.1, theexpression of the Helmholtz free energy may be written as

Ψ = kmΨm+

+(knt + kiz)[Npar(kntΨnt+kizΨiz)︸ ︷︷ ︸

Ψparntiz

+ (1−Npar) (kntΨnt+kizΨiz)︸ ︷︷ ︸Ψserntiz

], (4.2)

where Ψm, Ψnt and Ψiz are the specific Helmholtz free energy for the matrix,the nanotube and the interface components, respectively; km, knt and kiz are thevolume fraction of each component, Npar is the parallel factor and,

knt =knt

knt + kizkiz =

kizknt + kiz

(4.3)

are the volume fractions of the carbon nanotubes and the interface in the newCNT-interface material. These volume fractions must verify

km + knt + kiz = 1 knt + kiz = 1. (4.4)

The relation among the strain tensors of the different components is

ε = εm = εparntiz = εserntiz, (4.5)

being ε and εm the composite and matrix strain tensor, respectively; εparntiz thestrain tensor of the new CNT-interface material with a parallel behavior; andεserntiz the strain tensor of the CNT-interface material with a serial behavior.

The tangent constitutive tensor of the composite material may be derivedfrom (4.2) as

C =∂2Ψ

∂ε⊗ ∂ε = km∂2Ψm

∂εm ⊗ ∂εm+

∂2Ψparntiz

∂εparntiz ⊗ ∂εparntiz

+∂2Ψser

ntiz

∂εserntiz ⊗ ∂εserntiz. (4.6)

4.2. Formulation of the composite constitutive model 31

A parallel behavior means that all composite constituents have the same strainvalue. Therefore:

εparntiz = εnt = εiz (4.7)

∂2Ψparntiz

∂εparntiz ⊗ ∂εparntiz

= Npar[kntCnt + kizCiz

]= NparCpar

ntiz. (4.8)

And, a serial behavior means that all composite constituents have the samestress value. Thus:

σserntiz = σnt = σiz

εnt = C−1nt : Cser

ntiz : εserntiz ; εiz = C−1iz : Cser

ntiz : εserntiz

(4.9)

∂2Ψserntiz

∂εserntiz ⊗ ∂εserntiz= (1−Npar)

[kntC

−1nt + kizC

−1iz

]−1= (1−Npar) Cser

ntiz.

(4.10)Replacing (4.8) and (4.10) in (4.6) it is possible to obtain a simplified expres-

sion of the tangent constitutive tensor as

C = kmCm + (knt + kiz) [NparCparntiz + (1−Npar) Cser

ntiz] . (4.11)

The formulation developed require all composite components to fulfill theexpression given by (4.1). Therefore, it is possible to use any constitutive law todescribe the mechanical performance of the different components.

4.2.1 Definition of the parallel factor

The parallel factor is defined as

Npar =lparlnt

, 0 ≤ Npar ≤ 1, (4.12)

where lnt is the length of the nanotube and lpar is function of geometry andmechanical properties of the nanotube and the interface. The value of this lengthcan be obtained from the equation of tension distribution in a reinforcementconsidering perfect bond with the matrix, which is [52]

σnt (x) = Ent

[1− cosh (β (lnt − 2x))

cosh (βlnt)

]εm (4.13)

β =

√√√√ 2Giz

Entd2nt ln

(1 + b

rnt

) , (4.14)

32 Chapter 4. Phenomenological hom*ogenization

where x represents the longitudinal positions in the reinforcement, and the sub-scripts “nt” and “iz” refers to the properties of nanotube and interface zone,respectively. E and G are the Young’s modulus and the shear modulus, and b isthe thickness material around of the CNTs associated with the interface zone.

Defining lpar = lnt − 2x, its value can be obtained by finding the position“x” for which the effective modulus obtained from the integration of the tensiondistribution becomes

Eeff =lparlnt

Eparntiz +

(1− lpar

lnt

)Eserntiz. (4.15)

This procedure provides a value of the parallel length of

lpar =1

βcosh−1

[1

3cosh (βlnt)

]. (4.16)

4.2.2 Definition of the volume fraction of the interface re-gion

Based on the results reported in [8], the interface zone can be considered theregion surrounding the carbon nanotube in which an amorphous matrix becomescrystalline. The volume fraction of the interface zone can be obtained as

χc = χo + kiz, (4.17)

where χc, χo are the volume fractions of crystalline matrix with and withoutCNTs, respectively. Assuming that the interface zone is a cylinder around theCNTs, it is possible to relate the volume fraction of the interface zone with theparameter b

rntas

kiz =N(πr2lnt−πr2

ntlnt)V =

Nπr2ntlntV

[(rrnt

)2

− 1

]= knt

[(rrnt

)2

− 1

], (4.18)

where V is the total composite volume, r is the radius of interface zone and N isthe total number of nanotubes in the composite.

The relation between the radius of the nanotube and the interface is obtainedreplacing (4.18) in (4.17) as

r

rnt= +

√(χc − χo)

knt+ 1 , χc ≥ χo, (4.19)

and therefore

r

rnt= 1 +

b

rnt⇒ b

rnt= +

√kizknt

+ 1− 1 , kiz ≥ 0. (4.20)

4.2. Formulation of the composite constitutive model 33

4.2.3 Equivalent properties for MWCNTs

MWCNTs consist of concentric SWCNTs joined together with relatively weakvan der Waals forces. For this reason, the capacity to transfer the load fromthe external wall to the internal walls is low. Some papers [129, 144] propose tosimulate the CNTs like a solid cylinder with same exterior diameter and length,but with effective properties. The effective properties are obtained assumingthat the outer wall takes the total load. In this approach it is assumed that theproperties of the outer wall correspond to those of a graphite sheet. The effectivestiffness of the MWCNT is calculated by imposing that for a same applied force,the deformation must be the same

εnt = εnt ⇒ Ent =AowAnt

Eg, (4.21)

where Ent and Eg are the Young’s modulus of the effective solid nanotube andgraphite sheet, respectively, and Ant and Aow are the areas of the effective solidnanotube and outer wall, respectively. Equation (4.21) can be also read as

Ent =

[1−

(1− 2t

dnt

)2]Eg ,

t

dnt≤ 0.5, (4.22)

being t the thickness of one wall in the MWCNT and dnt is the external diameterof the MWCNT.

Using the same procedure it is possible to obtain the shear modulus of thesolid cylinder, by forcing the same twist when applying the same torque (T).

φnt = φnt ⇒ T lntGntJnt

=T lntGgJow

⇒ Gnt =JowJnt

Gg, (4.23)

where Gnt and Gg are the shear modulus of the effective solid CNTs and graphitesheet, respectively, and Jnt and Jow are the polar moment of inertia of the effec-tive solid CNTs and outer wall, respectively. They are

Jnt =πd4

nt

32, Jow =

π(d4nt − (dnt − 2t)

4)

32. (4.24)

Replacing the expressions given by (4.24) in (4.23), the equivalent shear moduluscan be written as

Gnt =

[1−

(1− 2t

dnt

)4]Gg. (4.25)

Finally, it is necessary to obtain the new density of the effective solid CNTs,as the total weight of the MWCNTs can not change in the composite when they

34 Chapter 4. Phenomenological hom*ogenization

are considered a solid cylinder, then

ρnt =AntAnt

ρg ⇒ ρnt =

[1−

(didnt

)2]ρg, (4.26)

being ρg the density of the graphite sheet (ρg = 2.25[g cm−3]) and di the internaldiameter of the MWCNTs.

The most common parameter used to define the amount of CNTs added toa composite is their weight fraction. However, the composite constitutive modeldeveloped requires knowing the volume fraction. The volume fraction of CNTs inthe composite is the volume that occupies a solid cylinder with the same externaldiameter. This parameter can be calculated with the following expression [129]as

knt =wnt

wnt + ¯ρntρm− ¯ρnt

ρmwnt

, (4.27)

where wnt is the weight fraction and ρm is the density of the matrix.

4.2.4 Material non-linearity of the proposed model

In the proposed model, the composite performance is obtained from the mechan-ical response of its constituent materials, and each component is simulated withits own constitutive law. Therefore, if a constituent (i.e. the interface) is sim-ulated with a non-linear law, the whole composite will become non-linear. Asit has been already explained, with the present model it is possible to use anynon-linear formulation to simulate the component behavior, such as plasticity,damage, viscosity, etc.

Besides the non-linear performance provided by each constituent, the loadtransfer capacity of the interface region is also affected if the interface is damaged.This effect must be included in the formulation.

According to Figure 4.3, the load is transferred from the interface to theCNTs reinforcement at their ends. Interface damage is expected to occur at theends of the reinforcement, where there is larger stress concentrations. Assumingthat the damaged region is unable to transfer loads and that the length requiredto transfer loads must remain constant, interface damage ends up affecting theparallel length of the nanotube, which can be calculated as

lpar = lopar (1− d) . (4.28)

Here, lopar is the initial length of the nanotube working in parallel and d is theinterface damage inner variable.

The dependence of the parallel length on the interface material damage pro-vides a non-linear response of the composite, even when matrix and the carbonnanotube reinforcement are in their linear range.

4.3. Numerical Implementation 35

4.3 Numerical Implementation

The proposed composite constitutive model has been implemented in PLCd [111],a finite element code that works with 3D solid geometries. The algorithm devel-oped is described in Figure 4.4. PLCd has already implemented the constitutivelaws that will be used to predict the performance of the composite components(elasto-plastic, elasto-damage and elastic). The formulation proposed has beenwritten so that the constitutive laws of the constituents are seen as “black boxes”,following the recommendations of [77] and [114].

The FEM code enters into the new formulation with the prediction of thestrain tensor of the composite material in the actual time step. Layers are as-sumed to have all the same strain; therefore the strain tensor of each layer isobtained rotating the composite strain to the direction in which the CNTs areoriented. In each layer, the strain of the matrix and the CNTs-interface are thesame, as they work in parallel, (see (4.5)). Knowing the strains for matrix ma-terial it is possible to obtain its stresses straightforward. On the other hand, toobtain the stresses for the CNT-interface material, it is necessary to separate itin two regions. In the flow chart shown in Figure 4.4, these two regions are rep-resented as “Parallel Block” and “Serial Block”. This division is performed basedon the value of Npar (defined in (4.12)). This value depends on the damageevolution of the interface, as has been explained in section 4.2.4.

The Parallel Block corresponds to the central region, where the CNTs and theinterface work in parallel behavior and, therefore, they have the same strains. Inthis region the stresses for each component are obtained from the strain tensor,using their constitutive equation. Finally, the stress tensor of the CNT-interfacematerial in the “Parallel Block” at time t+ ∆t is

[σparntiz]t+∆t

= knt [σparnt ]t+∆t

+ kiz [σpariz ]t+∆t

. (4.29)

On the other hand, at the ends of the CNTs, the interface-CNTs material hasa serial behavior and it is necessary an initial prediction of the CNT or of theinterface strains, in order to integrate the local stress in both components. If thisinitial prediction is made on the interface, its strains can be computed as

∆εserntiz = [εserntiz]t+∆t − [εserntiz]

t(4.30)

[∆εseriz ]o = [Cseriz ]−1

: Cserntiz : ∆εserntiz (4.31)

[εseriz ]t+∆to = [εseriz ]

t+ [∆εseriz ]o . (4.32)

And, the strain tensor of the interface in the iteration step n is used to calculatethe strain tensor of the CNT as

εserntiz = kntεsernt + kizε

seriz (4.33)

36 Chapter 4. Phenomenological hom*ogenization

[εsernt ]n =1

knt[εserntiz]−

kizknt

[εseriz ]n . (4.34)

Once knowing the strain tensor of both component materials, the constitutivelaw of each one is used to calculate their stress tensor. Afterwards it is necessaryto verify that the iso-stress condition is indeed fulfilled. Therefore

[∆σser]n = [σseriz ]n − [σsernt ]n ≤ tolerance. (4.35)

If the residual stress is greater than the tolerance, the prediction of the inter-face strain must be corrected. A Newton-Raphson scheme is adopted to do thiscorrection. The method uses the Jacobian to update the unknown variable, inthis case, the interface strain, then

Jn =∂[∆σser]n∂εseriz

∣∣∣εseriz =[εseriz ]

n

=∂[σseriz ]

n

∂εseriz− ∂[σsernt ]

n

∂εsernt:∂εsernt∂εseriz

= [Cseriz ]n − [Csernt ]n

(− kizknt

),

(4.36)

and, finally

Jn = [Cseriz ]n + [Csernt ]n

(kizknt

). (4.37)

Therefore, the strain tensor of the interface for the next step n+1 is estimated as

[εseriz ]n+1 = [εseriz ]n − J−1n : [∆σser]n . (4.38)

This iterative process continues until the residual stress is smaller than the re-quired tolerance.

The final stresses in the serial region “Serial Block” of the CNTs-interface are

[σserntiz]t+∆t

= [σsernt ]t+∆t

= [σseriz ]t+∆t

. (4.39)

And at the end, the final stress tensor for a specific layer is obtained as

[σ]t+∆t

= km [σm]t+∆t

+

+ (knt + kiz)

[Npar]t+∆t[σparntiz]t+∆t

+[1−Npar]t+∆t[σserntiz ]t+∆t.

(4.40)

4.3. Numerical Implementation 37

[εc]t+4t = [εL1

]t+4t = · · · = [εLi

]t+4t = · · · = [εLn

]t+4t

Rotation by the CNTs Orientation

[εLi

]t+4t = [εm]t+4t = [εparntiz]t+4t = [εserntiz]t+4t

Calculation of Npar

[σLi

]t+4t = km[σm]t+4t + (knt +kiz)Npar[σparntiz]t+4t + (1 − Npar)[σserntiz]t+4t

Rotation by the CNTs Orientation

[σc]t+4t =n∑

i=1k

Li[σ

Li]t+4t

OTHER

LAYERS

Parallel

Theory

Integration

[εparntiz

]t+4t

[σparntiz

]t+4t

PARALLELBLOCK

Con

stitutive

Model

[εm]t+4t

[σm]t+4t

MATRIX

SerialTheory

Integration

[εserntiz ]t+4t

[σserntiz ]t+4t

SERIA

LBLOCK

Figure 4.4: Flow chart of the proposed model in a FEM code.

Chapter 5

Validation and numericalresults

In this chapter, the validation of the proposed constitutive model using data fromthe literature is presented. Then, a numerical example is shown using the modelcalibrated. The basic formulation of the different constitutive models used forthe simple materials can be seen in the Appendix A.

5.1 Validation of the elastic response

In the following section are compared the composite stiffness predicted by thecomposite constitutive model (see Section 4.2) with experimental data obtainedfrom the literature. For this elastic properties validation the experimental datapresented in the papers of Coleman et al. [16, 15] is used. In these works severalcomposites made of the same matrix with different MWCNTs are experimentaltested.

Materials description

In the following, it will present the mechanical properties of the material compo-nents used and the composites data.

Matrix component: The matrix material is polyvinyl alcohol (PVA) and itsYoung’s modulus is given by the authors as Em = 1.9± 0.3 [GPa] [16].

Interface component: The authors found that the Young’s modulus of thecrystalline polymer phase is of Eiz = 46 [GPa]. On the other hand, the parameterbrnt

is estimated following the procedure described in Section 4.2.2.

39

40 Chapter 5. Validation and numerical results

MWNTs component: The nanotubes used in [16] are an arc grown MWCNT(Arc-MWCNT), two types of catalytic MWCNT from Nanocyl S.A. (CVD-1,CVD-2), a catalytic MWCNT produced in Orleans (France) (CVD-3), and adouble walled nanotube (Dwnt). While in [15] the nanotube used is MWCNTfrom Nanocyl S.A. (MWCNT).

The maximum Young’s modulus of the CNTs is ∼ 1 [TPa][16], which corre-sponds to the stiffness of a perfect graphite sheet. The equivalent stiffness (seeSection 4.2.3) of the nanotubes are calculated using this perfect stiffness valueand considering a thickness of the outer layer of t = 0.34 [nm][50, 129].

The most important collected data of the nanotubes used are presented intable 5.1:

Type dnt (nm) lnt (µm) lnt/dnt b/rnt Ent (GPa) Npar

Arc-MWCNT 24 1 42 0.81 56 0.97CVD-3 16 3.8 238 1.47 83 0.99CVD-2 14 2.1 150 2.27 95 0.99CVD-1 15 1.8 120 2.83 89 0.98Dwnt 2.5 2.2 880 4.87 470 0.99MWCNT 15 1.72 115 3.30 89 0.98

Table 5.1: Relevant data of the nanotubes used by Coleman el al. [16, 15].

Composites: A parameter missing in Table 5.1 is the direction distributionsof the CNT. In general, obtaining this information from the composite is verycomplicated. To outstep this impediment it is possible to rewrite equation givenby (4.11) for one layer as

Clayer = kmCm + kntizCeffntiz, (5.1)

where

kntiz = knt + kiz Ceffntiz = NparCpar

ntiz + (1−Npar) Cserntiz. (5.2)

Cox [20] and Krenchel [61] modified the rule of mixtures proposing the fol-lowing equation to calculate the composite Young’s modulus

E = kmEm + kfηoEeff , (5.3)

where Em and Eeff , are the Young’s modulus of the matrix and effective re-inforcement, respectively. The volume fraction for each component is k and ηois a fiber orientation efficiency factor. For the present validation (5.3) will bemodified, adapting it to the developed formulation. Therefore

Ccomposite = kmCm + kntizηoCeffntiz. (5.4)

5.2. Validation of the non-linear performance 41

The value of the efficiency factor related to fiber orientation was taken fromliterature. In composites with a random distribution, ηo = 0.38.

Results

Figure 5.1: Comparison of numerical and experimental results [16, 15].

Figure 5.1 shows the values of dC/dknt, this is: the slope of the curves ofYoung’s modulus (C) divided by volume fractions of nanotubes (knt), for thedifferent composites considered. In the figure the short lines represent the limits ofthe range experimental results presented in [16, 15] and the red points correspondto the numerical result for each CNT type, obtained with the proposed compositemodel.

This figure shows that the formulation is capable of predicting the elasticstiffness of the composite, as most of the values obtained are comprehendedbetween the limits defined by the experimental tests. There is only one case inwhich the value obtained exceeds the limits of the experimental test. This isbecause the effective Young’s modulus of the Dwnt is highest since its diameteris really low.

5.2 Validation of the non-linear performance

The non-linear behavior of the composite constitutive model has been validatedcomparing the results provided by the model with the experimental data obtainedfrom the paper of Meng et al. [85]. In that article the matrix used is Polyamide6 (PA6) and all composites contained a 1 wt% of MWCNTs reinforcement.

The MWCNTs used in the experimental tests were purchased from ChengduOrganic Chemistry Co. Ltd. Two different composites where manufacturedwith these nanotubes. One of them contains the nanotubes “as is”, without

42 Chapter 5. Validation and numerical results

any previous treatments. These nanotubes are called U-MWCNT. The othercomposite uses nanotubes that where treated with a mixture of concentratedsulfuric and nitric acids. These are called A-MWCNT.

Materials description

In the following, it will present the properties of the material components usedand the information of the composites.

Matrix component: The matrix material is characterized with an isotropic,elasto-plastic model using a Von-Mises yield criterion. The mechanical parame-ters of the model were calibrated using the experimental data described in [85],obtaining a Young’s modulus of Em = 2.67 [GPa], a Poisson ratio of νm = 0.4and an elastic threshold of 35 [MPa]. The parameters used to simulate matrixmaterial are validated comparing the stress-strain graph obtained with the nu-merical model with the experimental one. This comparison is shown in Figure5.2.

Figure 5.2: PA6 stress-strain relations for static tests [85].

Interface component: The interface zone is associated with the crystallinematrix around of MWCNTs. The properties of this material are better than thoseof the amorphous matrix. The volume fraction of the interface zone has beenestimated with the data presented in the paper of Meng [85] and the equations

5.2. Validation of the non-linear performance 43

developed in Section 4.2.2. On the other hand, the mechanical properties of theinterface are used to calibrate the model. In current simulation, the interfacehas been defined with a isotropic, elasto-damage model with linear softeningand Tresca yield surface. The mechanical parameters used are Eiz = 5 [GPa],νiz = 0.4 and Giz = 1.8 [GPa]. Damage in the interface starts for a stressthreshold of 120 [MPa]. This value is in the range of theoretical and experimentaltests value obtained in [132].

MWCNTs component: Numerical simulations of molecular structural me-chanics of CNTs show that the Young’s moduli are in the range of 1.05 ± 0.05[TPa] and the shear moduli is about 0.4±0.05 [TPa] [66]. It has been also shownthat these values do not change significantly for CNTs with two, tree or fourwalls.

Regarding the transverse modulus of CNTs, it has been assessed from nu-merical and experimental results that there is an inverse relationship betweenaxial and transverse modulus for carbon fibers [81]. Higher axial stiffness is as-sociated to a longer and more aligned crystalline structure of the nanotube inthis direction, which reduces properties in the transverse direction. Followingthis approach, in current simulation the transverse moduli of the MWCNTs aredefined with the same values of the interface component.

Therefore, the equivalent properties of the MWCNTs were obtained using theequations described in Section 4.2.3. The diameter of MWCNT is dnt = 50 [nm].The measurement of several MWCNTs provided an estimation of the internaldiameter of di = 8.2 [nm] [129]. The effective density of MWCNTs has a valueof ρnt = 2.2 [g cm−3]; and the volume fraction of MWCNTs in the composite is0.51 %. The MWCNTs have been simulated using an elastic orthotropic materialwith the following properties:

E1nt = Ent = 56 [GPa] , E2nt = E3nt = Eiz = 5 [GPa]

G12nt = G13nt = Gnt = 41 [GPa] , G23nt = Giz = 1.8 [GPa]

ν12nt = ν13nt = ν23nt = νnt = 0.2

νij = EiEjνji ⇒ ν21nt = ν31nt = 0.018 ν32nt = 0.2

Composite: The composites tested had a random distribution of the MWC-NTs. This is simulated in the numerical model by dividing the composite inseveral layers, each one containing CNTs with a different orientation. Currentsimulation divides the composite in 10 layers and CNTs angles varying from 0

to 90 . Each layer has a volume fraction of 10 %. Table 5.2 shows the volumefractions of the three composite components in each layer. This table also showssome geometry information of the MWCNTs and the interface zone, as well asthe initial value of Npar.

44 Chapter 5. Validation and numerical results

Composite knt[%] kiz[%] km[%] lnt/dnt b/rnt Npar

PA6/A-MWCNT 0.5 4.1 95.4 250 2.00 0.98PA6/U-MWCNT 0.5 5.3 94.2 250 2.35 0.98

Table 5.2: Data of the composites.

Results

In Figure 5.3 are represented the numerical and experimental results obtainedfor the composite made with A-MWCNTs. This figure shows an initial reductionof the composite stiffness, result of matrix yielding. Afterwards damage beginsin the interface zone and, consequently, the composite continues reducing itsstiffness. Interface damage leads to a reduction of the parallel length (see (4.28)).When the interface is completely damaged, the whole CNT-interface material hasa serial performance. At this stage stresses in the interface are zero, and so mustbe the stresses in the carbon nanotubes. Therefore, the final stiffness of thecomposite corresponds to a material with a volumetric participation of 95.4 % ofPA6 matrix, and the rest of the material correspond to voids.

Figure 5.3: PA6/A-MWCNT stress-strain relations for static tests [85].

Figure 5.4 shows the results for the composite made with U-MWCNTs. Thiscomposite is the same than the previous one (made with A-MWCNTs), with theonly difference that in this case the bond between U-MWCNTs and interface

5.3. Simulation of a four points bending beam 45

zone is weaker. To take into account this difference, the numerical model usedfor this composite is the same used for the previous one, varying the threshold atwhich damage starts in the interface. In current simulation this value is reducedto 70 [MPa].

This simulation provides a maximum stress in the composite lower than thevalue obtained for previous one, consequence of having a weaker interface. Thesimulation also shows some divergences between the numerical and the experi-mental values. Both graphs start to differ for a strain of 2.5 % and the maximumload reached by the numerical simulation is larger than in the experimental tests.However, it has to be noted that the experimental tests provide a maximum stresslower than having just plain matrix (see Figure 5.2). Therefore, the differencesobserved in Figure 5.4 may be justified.

Figure 5.4: PA6/U-MWCNT stress-strain relations for static tests [85].

5.3 Simulation of a four points bending beam

5.3.1 Materials properties, geometry and FE model of theanalyzed structure

This section presents the numerical simulation of a four points bending beam,which is shown in Figure 5.8. Although there is no experimental results availableto compare the results obtained with the numerical simulations, the analysisperformed is used to show the numerical performance of the proposed model and

46 Chapter 5. Validation and numerical results

the effect of reinforcing the polymer with CNTs. The validity of the results isassumed, based on the comparisons made in Section 5.1 and 5.2. The compositeused in the simulation is a reinforced matrix with MWCNTs and it has beenproposed in the framework of M RECT project. The matrix in the composite isa Polyether Ether Ketone (PEEK) thermoplastic polymer provided by Victrex R©

company. While the MWCNTs are the NanocylTMNC7000 from Nanocyl S.A.Two composite with different weight fractions percentage (0.5 and 2.0 wt%) ofMWCNTs reinforcement will be analyzed.

In the following, it will describe the properties and constitutive model of thematerial components and the composites information.

Matrix component

An elasto-plastic constitutive model with hardening is applied to characterize thebehavior of the PEEK component. The matrix material has a Young’s modulusof Em = 3.9 [GPA], a shear modulus of Gm = 1.9 [GPA], a Poisson ratio of vm =0.4. These elastic mechanical properties are obtained from M-RECT projectand of the information provided by Victrex R©(http://www.victrex.com). Theconstitutive model is calibrated with an elastic threshold of 32 [MPa] and anultimate tensile strength of 90 [MPa]. Figure 5.5 shows the comparison betweenthe experimental data from the project and numerical results obtained with theconstitutive model calibrated.

(a) In a tensile test. (b) In a shear test.

Figure 5.5: Comparison of the experimental data with the numerical results for PEEK.

Interface component

The constitutive model used to simulate the behavior of the crystalline PEEKaround of the MWCNTs is an elasto-damage model with exponential softening.The mechanic properties of this interface zone are obtained following the sameprocedure used by Coleman et al. [16] but using the information presented in

5.3. Simulation of a four points bending beam 47

the works of Diez-Pascual et al. [24, 25], which use the PEEK material as matrixtoo. Then, the properties obtained are, a Young’s modulus of Eiz = 5.07 [GPa],a shear modulus of Giz = 2.47 [GPa] and a Poisson ratio of viz = 0.4. Thevalue of the elastic threshold used in the model is of 28 [MPa]. This parameteris obtained when the composite constitutive model is calibrated to reproduce theexperimental curve shown in Figure 5.6. This experimental data is for a PEEKreinforced with 3 wt% of MWCNTs obtained in the framework of the previousreferred project.

Figure 5.6: Experimental data and numerical response with the calibrated interfacecomponent model.

MWCNTs component

The geometric characteristics of the MWCNTs are obtained from the paper ofJiang et al. [53], who obtains as average diameter and length of 10.4 [nm] and0.7 [µm], respectively. For the simulation the MWCNTs are considered as anorthotropic elastic material. The equivalent properties are obtained using theequations described in Section 4.2.3, assuming Eg = 1.05± 0.05 [TPa] and Gg =0.4 ± 0.05 [TPa][66] and a thickness of the outer layer of t = 0.34 [nm][50, 129].And taking the same consideration than before, the transverse properties aredefined with the same values of the interface.

E1nt = Ent = 131 [GPa], E2nt = E3nt = Eiz = 5.07 [GPa],

G12nt = G13nt = Gnt = 104 [GPa], G23nt = Giz = 2.47 [GPa],

ν12nt = ν13nt = ν23nt = νnt = 0.2,

48 Chapter 5. Validation and numerical results

νij = EiEjνji ⇒ ν21nt = ν31nt = 0.008 ν32nt = 0.2.

Composites

Table 5.3 shows the volume fractions of each component in the composites sim-ulated.

Composite km[%] knt[%] kiz[%]PEEK-0.5CNT 84.95 0.35 14.7PEEK-2.0CNT 91.89 1.41 6.70

Table 5.3: Volume fractions in the composites.

The orientation distribution of the MWCNTs has been defined assuming thatthe composite is formed by several layers, each one with a specific angle andvolume fraction of MWCNTs. The volume fractions of the MWCNTs for thedifferent layer in the composite are shown in the Figure 5.7. The value of thevolume fractions in the figure are relative values respect to the total volumefraction of the MWCNTs in the composite.

Figure 5.7: MWCNTs orientation distribution in the composite.

Geometry of the simulated structure

The selected structure for the numerical simulations is a simple supported beamwith two concentrated loads, which are applied at 1/3 of both beam ends. Figure5.8 shows the geometry and its dimensions, the boundary conditions and the loadposition on the analyzed structure.

5.3. Simulation of a four points bending beam 49

Figure 5.8: Geometry and extra information of the analyzed structure.

FE model used

The symmetry of the geometry, of the applied load and of the boundary conditionsof the structure allows to reduce the numerical model in the simulation. For thiscase, the reduced FE model is a quarter of the real geometry of the structure.Figure 5.9 shows the numerical model and the FE mesh used for the numericalanalysis. The more relevant data about the FE mesh is shown in the Table 5.4.

Item Nodes Elements Type Elem. OrderQuantity/Type 1953 1200 Hexahedron Quadratic

Table 5.4: Mesh information.

In order to obtain the real behavior of the structure with the reduced FEmodel it is necessary impose the restrictions on the numerical model given by thesymmetry. There are two symmetry planes: The X-axis symmetry plane thathas as normal axis the longitudinal axis (X-axis). In this symmetry plane, Xdirection displacements on the plane’s nodes are restricted to zero. The othersymmetry plane is the Y-axis symmetry plate, which has as normal axis the Y-axis in the model. For this symmetry plane, the null displacements restrictionon the plane’s nodes is Y direction. The longitudinal direction in the numericalmodel (X-axis) is taking as a reference direction for the definition of the layes’angle in the composite.

5.3.2 Linear analysis

The numerical results obtained in the simulation are presented in comparativeform, taking as reference the result obtained for the non-reinforced matrix (plainPEEK material).

In all cases, the applied load for elastic analysis is a fixed displacement of a−0.001 [mm] in Z direction at P position (see the Figure 5.8). The result consid-

50 Chapter 5. Validation and numerical results

Figure 5.9: FE mesh used in the reduce model.

ered for the comparison is the reaction force in Z direction on the support. Asthe imposed displacement is the same for all analysis, the reaction force increaseswhen the material is the PEEK reinforced. Table 5.5 shows the results obtainedfor the different composites normalized by the non-reinforced PEEK results.

PEEK PEEK-0.5%CNT PEEK-2.0%CNT1 1.20 1.52

Table 5.5: Normalized values obtained in the elastic simulation.

In the central section of the beam, between concentrated loads, there is a purebending situation. While, at both ends of the beam there are a coupling bendingand shear conditions. In Table 5.6, it is possible to observe the longitudinal (Xdirection) and shear (XZ direction) stresses obtained in the structure with thedifferent composites.

5.3.3 Non-linear analysis

In order to obtain the non-linear response of the structure the fixed displacementat P position is gradually increased in the simulation. Therefore, the reactionforce in the support in Z direction increases too until the maximum value that thestructure is able to take. The total force is four times the value obtained from thenumerical model because the symmetry. The vertical (Z direction) displacement

5.3. Simulation of a four points bending beam 51

Longitudinal Stress Shear StressP

EE

KP

EE

K-0

.5%

CN

TP

EE

K-2

.0%

CN

T

Table 5.6: Longitudinal and shear stresses distribution obtained in the beam for elasticcase.

at the middle of the beam is taking as the reference loading increase.

The fixed displacement is applied to the numerical model in 100 load steps of0.1 [mm]. Figure 5.10 shows the results obtained in the simulation for the differentcomposites. When the vertical displacement is around 1.5 [mm] the curves showthe first loss of stiffness. This is because in the middle of the beam starts theplasticity in the PEEK. Then, when the vertical deformation is between 3 [mm]to 6 [mm] there is the second loss of stiffness. In this case, it is due to the damagein the interface zone. Subsequently, it is possible to observe that the model withnon-reinforced PEEK has a higher stiffness than the ones with MWCNTs. Thisstrange phenomenon is because at this point of the simulation the interface zoneis completely damaged and, therefore, the contribution of the MWCNTs to theglobal stiffness is null. The stiffness obtained with the composites with MWCNTsare equivalent to the stiffness of a plain PEEK material but with some holes. This

52 Chapter 5. Validation and numerical results

effect is clearly observed in Figure 5.11. This Figure shows the curves obtainedfor the simulation until a vertical displacement of 50 [mm].

0.5

1

1.5

2

2.5

3

-10-8-6-4-2 0

Rea

ctio

n fo

rce

[KN

]

Vertical displacement [mm]

PEEKPEEK-0.5%CNTPEEK-2.0%CNT

Figure 5.10: No-linear structural response for PEEK-CNT.

Figure 5.11 shows a new loss of stiffness that takes place from 30 [mm] to40 [mm] of vertical displacement. This laster loss of stiffness is because theplasticity model in the PEEK arrives to the ultimate tensile strength in themiddle zone of the beam.

Figure 5.12a shows the distribution of the longitudinal stress and Figure 5.12bof the shear stress for the composite reinforced with 0.5% of MWCNTs for avertical displacement of 1 [mm]. The longitudinal plastic strain for the samecomposite and displacement state is shown in Figure 5.12c and the equivalentstress in Figure 5.12d.

5.3. Simulation of a four points bending beam 53

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

-50-40-30-20-10 0

Rea

ctio

n fo

rce

[KN

]

Vertical displacement [mm]

PEEKPEEK-0.5%CNTPEEK-2.0%CNT

Figure 5.11: No-linear structural response for PEEK-CNT up to 50 [mm] of verticaldisplacement.

(a) Longitudinal stress distribution. (b) Shear stress distribution.

(c) Longitudinal plastic strain distribution. (d) Equivalent stress distribution.

Figure 5.12: No-linear results obtained at 1 [mm] of vertical displacement for PEEK-0.5%CNT.

54 Chapter 5. Validation and numerical results

5.3.4 Visco-elastic analysis

One of the main improvements shown by CNTs reinforced composites is theirgood damping response and energy dissipation, which makes them very usefulfor impact or vibration absorption purposes [144, 58, 34]. For this reason, inthe following is analyzed the visco-elastic performance of the numerical modeldeveloped under such conditions. In order to obtain a viscous response of thecomposite it is necessary to use a visco-elastic model to characterize its compo-nents. The visco-elastic model used for the matrix and the interface zone is thegeneralized Maxwell model (see Figure 5.13) [99], which is already implementedin the PLCd code (see Section A.4 in Appendix A). To conduct the visco-elasticanalysis, the MWCNTs are considered to have a linear elastic behavior.

Figure 5.13: Scheme of the generalized visco-elastic Maxwell model [99].

Mat./Prop. C∞ [MPa] C1 [MPa] ξ1 [MPa.seg]PEEK 3900 390 39Interface 5070 1521 152

Table 5.7: Materials properties used in the visco-elastic model.

The response obtained, after calibrating the model (see Table 5.7), is shownin Figure 5.14. This figure shows the stress-strain curve of a point inside of thestructure in a complete sinusoidal load-unload cycle. The numerical simulationhas been conducted using the two composites previously described (0.5% and2.0% wt) and the plain PEEK. Figure 5.14 shows that the areas enclosed bythe curve, in the load-unload cycle, in the composites reinforced with MWCNTsare larger than the non-reinforced matrix (plain PEEK). In other words, thedissipation capacity of a composite with MWCNTs is better than the matrix

5.4. Effect of the CNTs angle on the elastic properties 55

alone. The composite reinforced with 0.5% wt of MWCNTs has higher dissipativecapacity than the other composite (2.0% wt). This is because, in this composite,the volume fraction of the interface zone is higher than in the other, as it is shownin Table 5.3. This phenomenon occurs because when the volume fraction of theMWCNTs in the composite is low, their distribution in the composite improves,and then, increasing the interface volume.

10

20

30

40

50

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

Str

ess

[MP

a]

Strain

PEEKPEEK-0.5%CNTPEEK-2.0%CNT

Figure 5.14: Structural response for a sinusoidal load-unload of 1 Hz.

5.4 Effect of the CNTs angle on the elastic prop-erties

The present study about the influence of the CNTs angle in the resulting elasticproperties using the developed composite constitutive model has been performedto asses the importance of having a good orientation of the CNTs in the com-posite.

The composite used for the analysis is a CNTs reinforced PEEK matrix andit has been also proposed in the framework of M RECT project. The PEEK ma-terial used as matrix in the composite is the same than the one used in the abovesection (see Section 5.3.1). Therefore, the elastic properties previously obtainedfor the matrix and interface components have been used in this study. However,the MWCNTs used as reinforcements in the composite are the Baytubes R© C70Pfrom Bayer, which have different geometric characteristics than the ones used inthe above section. The current CNTs have an average diameter and length of13 [nm] and 1 [µm], respectively. The CNTs are considered as an orthotropic

56 Chapter 5. Validation and numerical results

elastic material and the equivalent properties are obtained using the equationsdescribed in Section 4.2.3, assuming Eg = 1.05± 0.05 [TPa] and Gg = 0.4± 0.05[TPa][66] and a thickness of the outer layer of t = 0.34 [nm][50, 129]. Consideringthat the transverse properties are defined with the same values of the interface,the CNTs properties are

E1nt = Ent = 105 [GPa], E2nt = E3nt = Eiz = 5.07 [GPa],

G12nt = G13nt = Gnt = 85 [GPa], G23nt = Giz = 2.47 [GPa],

ν12nt = ν13nt = ν23nt = νnt = 0.2,

νij = EiEjνji ⇒ ν21nt = ν31nt = 0.008 ν32nt = 0.2.

The composite simulated in the analysis has a 3% weight of CNTs, whichrepresent a 1.94% of volume fraction. However, measurements made with X-raysshow an apparent 5% weight. This difference is obtained because the MWCNTshave a higher apparent diameter than the original one. Therefore, the param-eter b

rntis estimated assuming that this extra 2% weight in the measurement

is the coating polymer around the nanotubes and then, it is the interface com-ponent considered. Taking this into account, the interface component has anapproximate volume fraction of 1.31% and b

rnt= 0.3.

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80 90

Ela

stic

pro

pert

ies

[GP

a]

CNTs angle [º]

Long. modulusTrans. modulusShear modulus

Figure 5.15: Change of elastic properties of CNTs reinforced PEEK with the CNTsangle.

Figure 5.15 shows the variation in the longitudinal elastic modulus, thetransversal elastic modulus and the shear elastic modulus in the composite ob-tained when the CNTs angle is changed. The curves obtained, which are shown

5.4. Effect of the CNTs angle on the elastic properties 57

in the figure are symmetric with respect to the 45o vertical line. This behavioris different to the typical ones expected for reinforced matrices. In general, themaximum value of the longitudinal Young’s modulus in the composite is obtainedwhen the angle of the reinforcement is equal to 0o. For this reinforced PEEKwith CNTs, the longitudinal Young’s modulus increases from 0o until 40o andthen, it decreases until its minimum value, which is obtained for an angle of 90o.A similar behavior presents the transversal Young’s modulus but symmetricallyrespect to the 45o vertical line. The maximum value for the shear modulus ispresented for an angle of 0o and the minimum for 45o.

Chapter 6

Phenomenologicalhom*ogenization. Concludingremarks

A new phenomenological composite constitutive model, based on the mixing the-ory, capable of predicting the mechanical performance of composites reinforcedwith carbon nanotubes has been presented. The formulation presented relatesthe reinforcement and the matrix in which they are embedded, using an interfacematerial. This approach makes possible to consider non-linear phenomenons,such as debonding, by using non-linear constitutive laws to characterize the in-terface material. The formulation is written in a way in which all materials canbe defined with their own constitutive law, improving the versatility of the model.

It has been shown that the elastic properties estimated with the model arein good agreement with experimental values obtained from literature. Only thenumerical model of the composite made with the Dwnt reinforcement has givenresults in which the composite stiffness is overestimated. This is because theDwnt has a very small diameter, which leads to a very high value of its equivalentYoung’s modulus.

The validation of the non-linear response provided by the proposed compositemodel has been performed using the experimental data of two different compositesmade with MWCNTs randomly distributed. The numerical curve obtained forthe A-MWCNT is in good agreement with the experimental results. On theother hand, the numerical prediction obtained for the U-MWCNTs differs fromthe experimental results for strains larger than 2.5%. However, it has to be saidthat the experimental results are lower than expected, as this composite is weakerthan plain matrix.

Then, the formulation has used to predict and compare the mechanical prop-

59

60 Chapter 6. Concluding remarks

erties of a straight beam subjected to four-point bending, with different materialconfigurations. A non-linear analysis has also been made using the same struc-ture and composites. The non-linear response of the beam obtained from thenumerical simulation shows different points where there is a loss of structuralstiffness. This structural behavior is obtained because the component materialsin the composite reach their elastic thresholds and ultimate strength.

A visco-elastic material constitutive model is used for the polymeric matrixand the interface zone. The viscous response within the elastic range of thematerials has been studied. The good capacity of energy dissipation in compositesreinforced with MWCNTs has been proved with the simulations performed.

Finally, the composite constitutive model has been used to analyze the effectof nanotube orientation in the elastic properties of the composite, showing thatthe CNTs distribution within the reinforced matrix has a meaningful influenceon the composite properties achieved.

All these tests have proved the validity of using a phenomenological modelfor the characterization of these materials. The developed composite constitutivemodel allows an accurate characterization of this kind of composites with anaffordable computational cost, moreover taking into account the scale size ofthese reinforcements.

Part II

Multiscale hom*ogenization

61

Chapter 1

Introduction

In the last decades, several formulations have been developed to mathematicallycharacterize and model composites as heterogeneous materials. Large number ofcomposite models have been proposed to assess the global behavior of these mate-rials by fulfilling the thermodynamic laws in linear and non-linear range. Consti-tutive equations have been developed for composites with different arrangementssuch as materials with long and short fibers, nanofibers, fibers laminated with oneor more directions, and even random reinforcement distributions, etc. However,these formulations are limited because the constitutive relationships were madeon a particular composite material and in general, can not be extrapolated toother composites.

The hom*ogenization methods analyze the composite materials from an in-ternal structure point of view. Over the years, many techniques have been de-veloped, among them: the effective medium approach [27]; the self-consistentmethod [47]; the variational boundary method, which provides upper andlower limits of the total stiffness [44, 112]; and the asymptotic hom*ogenizationmethod [6, 119]. Due to the complex task that is required to represent the mi-croscopic mechanical behavior of composites, hom*ogenization approaches thatuse the RVE concept, together with stable computational methods, are very con-venient in most cases. The behavior of the whole composite is obtained by amicromechanical study of the material components and their interaction withinof the composite’s microstructure through an RVE model.

Within the context of the hom*ogenization methods, the known multiscaletechniques use the RVE concept to address the characterization of composites andstructures. As a result, these multiscale procedures do not obtain a closed form forthe general constitutive equations. The stress-strain relationship of the compositeis obtained by performing a detailed modeling of the its microstructure at themicro-scale in the RVE model. Among the main advantages of these techniquesit is found that: they do not require any composite constitute assumption at the

63

64 Chapter 1. Introduction

macro-scale; they can use any constitutive law in the simple materials, even non-linear response and time-dependency; and they can employ many computationaltechniques to find the response at the micro-scale, such as, the FEM, the Voronoicell method, or numerical methods based on the fast Fourier transforms, etc.

The first-order hom*ogenization [125] is one of the most extended and pop-ular multiscale methods used nowadays. The approach uses the macro-scaledeformation gradient tensor (or the strain tensor) to solve the micro-scale prob-lem and then, by means of the microscopic results obtain the macro-scale stresstensor. The microscopic problem is solved through a Boundary Value Problem(BVP) on the RVE with particular boundary conditions, which are obtainedusing the macroscopic input. The resolution of the micro-scale BVP can be ob-tained through any mathematical or numerical approaches. After the solutionof microstructural BVP, the microscopic displacement, deformation and stressfields are found and then, the macroscopic stress tensor is calculated as the vol-ume average of the microscopic stress field. The BVP on the macro structurecan be also approached through mathematical or numerical methods. When thesolution of the coupled macro-micro BVPs is faced by FEM at both scales theformulation/implementation is called FE2 hom*ogenization [124].

The advantages above described of the multiscale techniques become a chal-lenge when a non-linear analysis is made on a realistic three-dimensional structureusing a FE2 hom*ogenization approach. In general, the required computationalcost is extremely expensive for non-linear simulations because they require tosolve, for each integration point at the macro-scale, one RVE at each time step ofthe analysis. Furthermore, from an energetic point of view the method must beconsistent and therefore, the conservation of the dissipated energy through thescales should be guaranteed.

In the last decade, a second-order computational approach was proposed tobe applied in critical regions of intense deformation, where the characteristicwave length of the macro-scale deformation field is of the order of the size ofthe micro-scale [60]. The hom*ogenization technique has been developed as anatural extension of the first-order hom*ogenization method. In the approach,the macroscopic gradient of the deformation gradient is also incorporated in themicroscopic BVP and them, the stress tensor and a second-order stress tensor areretrieved. In the micro-scale level a first-order equilibrium problem is conservedbut extra boundary conditions in the BVP must be verified. However, a fullsecond gradient continuum theory appears in the macro-scale level, which requiressolving a higher-order equilibrium problem. The solution of the reformulatedmacroscopic BVP is made through a complex finite element implementation,which has restricted its massive application.

The use of a second-order approach for the non-linear analysis of structureshas the advantage of including higher-order effects on the micro-scale. Neverthe-less, the complex computational implementation required to solve the macro-scale

1.1. Part’s outline 65

BVP restricts its application to realistic composite structures. From a mathe-matical and computational point of view the first-order approach is simpler thanthe second-order scheme. Then, it would be interesting to develop an enhanced-first-order approach which retains the easy computational implementation butwith an enriched solution in the micro-scale.

1.1 Part’s outline

In this second part of the present dissertation a multiscale hom*ogenization modelfor composite structures is addressed. The proposed formulation can simulate thebehavior of three-dimensional structures in non-linear range.

Following this objective, in Chapter 2 a review of the state of the art of themore relevant hom*ogenization theories is shown. The asymptotic hom*ogenizationtheory introduces the concept of two or more scale lengths, therefore it lays thefoundations of what is today known as multiscale hom*ogenization using RVEconcept. At the end of the chapter a review of the most important multiscaleapproaches found in the literature is made.

Chapter 3 presents the formulation of the two-scale hom*ogenization devel-oped in this study. In the beginning, the first-order hom*ogenization approachis described and then, using concepts of the second-order approach, a enhanced-first-order hom*ogenization is proposed. The BVPs in both scales and their com-putational implementation through of an efficient three-dimensional FEM is alsoshown.

In Chapter 4 the results obtained with the first-order hom*ogenization pro-posed are validated with other microscopic formulations. From the comparisonmade, it can be observed that the multiscale formulation obtains a good char-acterization of the composite without any special consideration because the mi-crostructure is modeled using an RVE. The computational run times and memoryused are also compared showing the advantages and drawbacks of the proposedmethod.

Chapter 5 presents a novel strategy for the non-linear simulation of compositestructures using multiscale hom*ogenization approaches. This strategy is capableof reducing a 98% the computational time required in a non-linear analysis. Theformulation is implemented in the FE code PLCd showing mesh independent atboth scales as the develop approach conserves the energy through the scales.

Finally, in Chapter 6 the conclusions of the formulations presented in thispart of the dissertation are addressed in detail.

Chapter 2

State of the art

The modeling of heterogeneous composite materials have a high degree of com-plexity because their constitutive behavior is strongly dependent on microstruc-tural effects. The development of formulations based on multiple scales, capableof predicting the response of a heterogeneous material phenomenologically ac-cording to the information derived from a study at the micro-mechanical, is anatural choice to address the problem.

Numerous efforts have been made to mathematically model composite mate-rials and structures with hom*ogenization methods by using suitable multiscaletechniques with relatively good approximation to the real global response of thecomposite. The most significant multiscale techniques, based in different theo-retical principles, are:

2.1 The effective medium approximation

The method proposed originally by Eshelby [27] consists to find the currentstresses in an elastic solid when a region of it, normally called inclusion (seeFigure 2.1), suffers a change of shape and size which, if the surrounding ma-terial was absent, is represented by a uniform hom*ogeneous deformation. Bymeans of the Eshelby solution a number of very important boundary value prob-lems can be solved, like: i) the solution for an ellipsoidal inclusion embeddedwithin an elastically mismatched matrix, ii) the solution for an ellipsoidal cav-ity in an elastic solid and iii) the solutions for circular and elliptical cracks inan elastic solid. Moreover, several theories which estimate the elastic proper-ties of composite materials use the Eshelby solution. Further developments byHashin [43], obtain expressions for the elastic moduli and their values bounds ofmany-phase heterogeneous materials using an approximate method based on thevariational theorems of the elasticity theory and on a concentric-spheres model.

67

68 Chapter 2. State of the art

Besides, Mori and Tanaka [94] developed a method of calculating the averageinternal stress in the matrix of a material containing inclusions with transforma-tion strain. The obtained actual stress in the matrix is the average stress plusthe locally fluctuating stress, the average of which vanishes in the matrix. Thedeveloped method also shows that the average stress in the matrix is uniformthroughout the material and independent of the position of the domain wherethe average treatment is carried out.

Figure 2.1: Ellipsoidal region of the Eshelby inclusion [1].

2.2 The self-consistent method

It may be considered as an extension of the effective medium approximation.The method is proposed by Hill [47] and it uses similar concepts to the Hershey-Kroner theory of crystalline aggregates. The macroscopic elastic moduli of two-phase composites taking into account the inhom*ogeneity of stress and strainfields are estimated. It is required phases with the character of matrix and ofeffectively ellipsoidal inclusions, but they can have any concentrations in thecomposite. The model of three phases or generalized self-consistent model givenby Christensen and Lo [12] is a more elaborate version. The method embeds thespherical (or cylindrical fibers) in a cap (spherical or cylindrical) which representsthe elastic properties of the matrix. Then, this set is embedded in an infinitemedium with the effective properties of the composite material to be determined.Finally, the effective elastic shear modulus of the material is obtained integratingthe differential equations governing the behavior of the three-phase boundaryconditions and of the applied loads.

A more recent work by Mercier and Molinari [87], which is based on the in-teraction law (postulated by Molinari [92] and validated by Mercier et al. [86]

2.3. Bounding methods 69

on the Eshelby problem) proposed two self-consistent schemes for perfectly disor-dered materials. The first one is valid for any non-linear behavior, and the secondscheme is used to aggregates with phases having the same strain rate sensitivity.Both schemes predict accurately the overall response of the composite materialand they are able to capture the strain and stress histories of the componentstoo.

2.3 Bounding methods

The bounding methods provide the lower and upper limits to the total stiffnessof the system or of the composites. In general, these methods obtain a simplerexpressions for the effective elastic properties through the minimum potentialand complementary energies. The most relevant of those are:

2.3.1 The classical bounds of Voigt and Reuss

The Voigt approach determines the elastic moduli by averaging stresses, expressedin terms of strains. The method assumes strain uniformity throughout the com-posite material. Therefore, the average strain of each phase is equal to the appliedstrain in the composite, which is similar to the rule of mixture assumption. Hence,considering that each component is linear-elastic, the following relationship canbe obtained

C =

n∑i=1

Ci · φi, (2.1)

where C is the effective elastic tensor of the composite, Ci and φi are the elasticconstitutive tensor and the volume fraction of the i− th phase, respectively.

On the other hand, Reuss proposed to determine the elastic moduli by aver-aging strains, expressed in terms of stresses but assuming stress uniformity andthen all components have the same stress. Therefore, the following estimationfor the effective elastic tensor of the composite is obtained as

C =

[n∑i=1

φiCi

]−1

. (2.2)

The Voigt and Reuss methods give an upper and lower bounds for the elasticmoduli of a composite with an arbitrary random micro geometry. The Voigtapproach gives the upper bound of the elastic moduli for the compound mate-rial, while the Reuss approach gives the lower bound. These bounding methodsdepend only of the phase volume fractions, and of course of the elastic consti-tutive tensors of the components, but do not require any further information orassumption in respect of the microstructure. The expressions obtained for thebounding effective moduli are valid even for anisotropic component materials

70 Chapter 2. State of the art

2.3.2 The variational bounding method

The authors Hashin and Shtrikman [44] use the variational formulation to ob-tained the upper and lower bounds for the effective elastic moduli of quasi-isotropic and quasi-hom*ogeneous multiphase materials of arbitrary phase geom-etry. The obtained bounds are close enough to provide a good estimate to theeffective moduli when the ratios between the different phase moduli are not toolarge. The method obtains analytical expressions for the elastic constants ofa heterogeneous material with random isotropic distribution of phases. Later,Walpole [135, 136] generalized the bounding method of Hashin and Shtrikmanfor materials with several phases, which may be arbitrarily anisotropic.

A variational method for bounding the effective properties of nonlinear com-posite materials with isotropic components with full variational principle statuswas proposed by Ponte Castaneda [112]. The author proposed two dual versionsof the new variational principle and it is demonstrated their equivalence to eachother and to the classical variational principles. The approaches are used to de-termine the bounds and to estimate the effective energy functions of nonlinearcomposites in the context of the deformation theory of plasticity. For completelyanisotropic composites simpler forms of the classical bounds of Voigt and Reussare recovered from the new variational principles. Also, for isotropic, particle-reinforced composites, as well as for transversely isotropic, fiber-reinforced com-posites simpler forms for nonlinear Hashin-Shtrikman bounds are obtained.

Lahellec and Suquet [64] proposed a new method for determining the overallbehavior of composite materials which can be composed by nonlinear inelasticcomponents. The evolution equations describing the constitutive behavior ofthe components can be reduced to the minimization of an incremental energyfunction by using an implicit time-discretization scheme. The alternative mini-mization problem is rigorously equivalent to a nonlinear thermoelastic problemwith a transformation strain which is a nonuniform field. Comparisons with full-field simulations show that the present model is good as long as the variationalprocedure is accurate in the purely dissipative setting, when elastic deformationsare neglected. If this is the case, the model accounts in a very satisfactory man-ner for the coupling between reversible and irreversible effects and is therefore anaccurate model for treating nonlinear viscoelastic and elasto-viscoplastic materi-als. Lahellec and Suquet [65] in a second part, of the described work, proposed aproper modification of the second-order procedure of Ponte Castaneda and leadsto replacing, at each time-step, the actual nonlinear viscoelastic composite bya linear viscoelastic one. The linearized problem is even further simplified byusing an effective internal variable in each individual component. The resultingpredictions are in good agreement with exact results and improve the predictionsof the previous work. The analytical models presented in the previous sectionsare reasonably able to predict equivalent material properties for relatively simplegeometries and low volume fraction of the second component inclusions. But

2.4. The asymptotic hom*ogenization theory 71

they are, in general, incapable to obtain the evolution of stresses and strainsin the microstructure. Moreover, the actual heterogeneous materials cannot betreated with these models because normally these composites have an arbitrarymicrostructural morphology.

2.4 The asymptotic hom*ogenization theory

This theory has proven to be a powerful technique for the analysis of structuralarrangements of two or more scale lengths. Through the use of asymptotic expan-sions of the displacement and stress fields, and appropriate variational principles,the hom*ogenization methods can provide not only the effective (hom*ogenized)material parameters, but also distributions of stresses and strains at the two lev-els. Bensoussan et al. [6] proposed an asymptotic expansion of the solution interms of a parameter ε, which is the ratio of the period of the structure to atypical length in the domain. The link from the microscopic to the macroscopicdescription of the behavior of the system is given by solving the problem in twoscales defined by the spatial variables x and y, where x is a macroscopic quan-tity and y = x/ε is a microscopic quantity; y is associated with the small lengthscale of the inclusions or heterogeneities. The asymptotic problem is formulatedin mathematical terms as a family of partial differential operators, dependingon the small parameter ε. The operators may be time independent or time de-pendent, steady or of evolution type, linear or nonlinear, etc. The coefficientsof the operators are periodic functions in all or in some variables with periodsproportional to ε. Since ε is assumed to be small, it is has a family of operatorswith rapidly oscillation coefficients. The standard classical formulation of thistheory is found in the work of Sanchez-Palencia [119, 120]. The two-scale processintroduced in the partial differential equations of the problem produces equationsin x, in y and in both variables. Normally, the equations in y are solvable if themicroscopic structure is periodic, and this leads to deduction of the macroscopicequations for the global behavior in x. It is emphasized that the hom*ogenized co-efficients depend on the local structure of the medium. Fish et al. [28] presenteda generalization of the mathematical hom*ogenization method based on double-scale asymptotic expansion to account for damage effects in heterogeneous media.A closed-form expression relating local fields to the overall strain and damage isderived. Non-local damage theory is developed by introducing the concept ofnon-local phase fields (stress, strain, free energy density, damage release rate,etc.) in a manner analogous to that currently practiced in concrete. Numericalresults of the model were found to be in good agreement with experimental data.

72 Chapter 2. State of the art

2.5 hom*ogenization using the RVE concept

The multiscale hom*ogenization based on the use of a unit cell or RVE has emergedas one of the most promising methods to compute the response of composite struc-tures. The unit cell is defined as a microscopic subregion that is representativeof the entire microstructure in an average sense. The RVE is employed to obtainthe effective properties for the hom*ogenized material because it is assumed thatit must contain a sufficient number of heterogeneities.

A thorough examination of the several representative volume element defi-nitions found in the literature can be consulted in the review made by Gitman[35]. Moreover, Ostoja-Starzewki [106] points out that the RVE is very clearlydefined in two situations only: i) unit cell in a periodic microstructure, as shownFigure 2.2a and ii) volume containing a very large (mathematically infinite) set ofsub-scale elements, possessing statistically hom*ogeneous and ergodic propertiesas shown Figure 2.2b.

Figure 2.2: Representative volume element models: (a) unit cell approach; (b) statisticaland ergodic approach [1].

On the other hand, Suquet [125] gives the basic principles of hom*ogenizationto obtain the constitutive equations for hom*ogenized properties of a heteroge-neous material. The process is resumed in the following three-step scheme andin Figure 2.3.

• Definition of a representative volume element. The size of this should belarge enough to contain a sufficient number of micro heterogeneities of whichthe constitutive behavior on these individual constituents is assumed to beknown.

• Microscopic boundary conditions are obtained based on the macroscopicinput variables (e.g. strain tensor), taking into account the geometry, con-stitutive laws, etc.

2.5. hom*ogenization using the RVE concept 73

• Macroscopic output variables are obtained based on the solution of themicroscopic behavior of the RVE. Macroscopic properties of the equivalenthom*ogeneous medium are evaluated.

Figure 2.3: Representative hom*ogenization scheme [1].

Terada et al. [128] conducted a study for multiscale hom*ogenization of theconvergence of the overall material properties when the unit cell size is increased.For general heterogeneous media that reveal irregular material distribution in amicro scale, the study showed that the periodic boundary condition provides themost reasonable estimates among the class of possible boundary conditions forstatistical hom*ogeneous media. The work demonstrated that it is not strictlyrequired the periodicity of RVE geometry in evaluating the effective properties.Moreover, at non-linear range the mechanical behaviors are more sensitive to thesize of RVE than those of linear case. The authors concluded that with moreaccurate geometric models of the RVE, and with larger RVE regions analyzed,the analysis provides a better understanding of the actual phenomena in themicroscopic region, as well as a better characterization of the overall mechanicalmaterial performance.

2.5.1 Multiscale computational hom*ogenization

Renard and Marmonier [115] were the first to use a finite element discretization tomodel heterogeneous materials with a multiscale approach. The method consistsin solving two finite element problems, one for each scale. In the micro scale thegeometry of the RVE is meshed and hom*ogenization rules are used to link thiswith the macro scale problem.

74 Chapter 2. State of the art

Guedes and Kikuchi [38] studied the mechanical behavior of linear elastic 2Dand 3D composite materials through the hom*ogenization method. This studywas one of the first that used hom*ogenization in three dimensions. The methodwas implemented via a finite element technique to analyze a composite with aperiodic microstructure. The implementation enabled the calculation of hom*oge-nized material mechanical properties, these characterized the overall behavior ofthe composite, as well as the analysis of the local performance of the material, asit enables the computation of local stress and strain distribution within the mi-crostructure of the composite. An adaptive finite element method was introducedin order to improve the accuracy of the numerical results. An error measure issuggested for the hom*ogenized material constants, based on the a priori errorestimations and the numerical implementation.

Non-linearity and FE2 hom*ogenization

Swan [126] presented a computational stress and strain controlled hom*ogenizationmethods for inelastic periodic composites within the framework of displacementfinite element. The implementation of the stress controlled method employs apenalty formulation to insure that the displacement solution satisfies the linear-periodic decomposition. The methods were assessed on a complete set of ho-mogenization computations for an elasto-plastic composite, the strain-controlledmethod was easier to implement and was demonstrated that it has more compu-tationally efficient than the others.

Later, Smit et al. [124] presented a hom*ogenization method for large de-formations and viscoelastic material behavior on microscopic and macroscopiclevel. The hom*ogenization method was implemented in a multi-level two dimen-sional finite element program with meshes on macroscopic level (mesh of entirestructure) and microscopic level (meshes of RVEs). The local macroscopic stressis obtained by applying the local macroscopic deformation on a unique RVEthrough imposing appropriate boundary conditions and averaging the resultingstress field. To each macroscopic integration point a unique discretized RVEis assigned that provides the local macroscopic stress tensor and the tangentialstiffness matrix. A separate iterative finite element procedures on the RVE isused in each iteration cycle of each macroscopic increment. The following Figure2.4 shows the proposed scheme by the authors. The computational cost was themain disadvantage of this implementation. The increase in computational timewith respect to a single-scale analysis on the same macroscopic mesh was of 160times for the structure analyzed.

Michel et al. [89] presented a review of several problems which are specificof composites with periodic microstructure composed of linear or non-linear con-stituents. The study obtains the estimation of phenomenological macroscopicconstitutive models through the analysis of a microscopic representative volumeelement proposing the concept of “macroscopic degrees of freedom”. A general

2.5. hom*ogenization using the RVE concept 75

Figure 2.4: Schematic representation of a multiscale finite element program [124].

framework permitting either a strain or stress control was proposed, and theimplementation of different types of boundary conditions was presented.

Haj-Ali and Muliana [40] proposed a three dimensional micromechanical mod-eling approach for the non-linear viscoelastic behavior of pultruded composites.A sub-laminate model is used to provide for a nonlinear equivalent continuum ofa layered medium. The system is idealized using a weighted-average response oftwo simplified micromodels with fiber and matrix. The proposed micromechani-cal framework was able to generate the effective anisotropic non-linear viscoelas-tic response as a direct outcome of the micromechanical hom*ogenization subcell.The same authors [41] extended the method to an integrated micromechanicaland structural framework for the non-linear viscoelastic analysis of laminatedcomposite materials and structures. The micromechanical model is numericallyimplemented within a shell-based non-linear finite element by imposing a planestress constraint on its 3D formulation. The micromechanical model provides theeffective non-linear constitutive behavior for each Gauss point. The formulationwas validated with several experimental creep tests available in the literature. Fi-nally, the work presents examples for non-linear viscoelastic structures, like a lam-inated panel and a composite ring. In a latter work, Haj-Ali et al. [39] extend theabove described method for the analysis of thick-section fiber reinforced plastic(FRP) composite materials and structures. The proposed modeling framework isapplied to a pultruded composite system. Non-linear 3D micromechanical modelsrepresenting the different composite layers are used to generate through-thicknesscomposite’s effective responses. The nested non-linear micromechanical modelsare implemented at each integration point in the finite element structural analy-sis. The results obtained demonstrated good prediction capabilities for effectiveproperties and for multi-axial non-linear behavior of pultruded composites.

Matsui et al. [80] made a feasibility study and introduced a parallel algorithmto achieve the computational efficiency. The focus was to the inhom*ogeneous

76 Chapter 2. State of the art

deformation of the overall structure, which may imply the loss of periodicityassumed in the initial state. The study presented an efficient algorithm for thedeconcentration of computational loads by using a PC-cluster system. A simplenumerical example for three dimensional heterogeneous structure was made. Theauthors concluded that the two-scale analysis is still expensive, even if a PC-cluster system for parallel computations is available at non-linear range.

Ladeveze et al. [63, 62] proposed a mixed, multilevel domain decompositionmethod or more accurately, as a “structure decomposition” method. The multi-scale computational strategy consists of describing the structure as an assemblyof simple components: substructures and interfaces. Each of these entities hasits own variables and equations. The distinction between the micro and macrolevels is made only at the interfaces, where forces and displacements are split intomacro contributions and micro complements. A substructure is subjected to theaction of its environment (the neighboring interfaces) defined by the force anda velocity distribution on its boundary. An interface between two substructurestransfers both the velocity and the force distributions.

Oller et al. [100] extended the work presented by Zalamea [143], with atwo-scale numerical hom*ogenization method that assumes the periodicity of theinternal structure of the material to address the problem of the steep gradientin the macroscopic field. To prevent the steep gradient in the macroscopic fieldvariables, which is produced by local boundary effects, the authors proposed alocal refinement of the finite element mesh. The objective was to maintain theperiodic condition on the boundaries of the cells near to the perturbation.

De Souza Neto and Feijoo [22] discussed some equivalence relationships forlarge strain multiscale solid constitutive models based on the volume averagingof the microscopic stress and deformation gradient fields over a representativevolume element. The work established that the volume averaging of the firstPiola-Kirchhoff stress over the reference configuration of an RVE is mechanicallyequivalent to the spatial averaging of Cauchy stress over the deformed RVE con-figuration. Whenever such conditions are met, multiscale constitutive modelsresulting from either the reference or spatial stress averaging are identical.

High-order multiscale hom*ogenization

The multiscale methods mentioned in the above sections are commonly denom-inated as first-order hom*ogenization approaches because in the mathematicalformulation considers only the first gradient of the macroscopic displacementfield.

Kouznetsova and Geers proposed what is called second-order hom*ogeniza-tion [59, 60], which is an extension of the first-order theory. In this case, themacroscopic deformation gradient tensor and its Lagrangian gradient is used tosolve the boundary value problem at the microstructural scale. The second-orderapproach allows solve problems in the presence of localization phenomena with-

2.5. hom*ogenization using the RVE concept 77

out loss of precision in the solution because the Lagrangian tensor is taken intoaccount. The main drawbacks of this method are its computational cost andcomplex implementation.

Geers, Coenen and Kouznetsova [31, 13] proposed a computational hom*og-enization technique for thin-structured sheets based on the hom*ogenization forfirst and second-order continua. The three dimensional heterogeneous sheet isrepresented by a hom*ogenized shell continuum for which the constitutive re-sponse is obtained from the nested analysis of a microstructural representativevolume element, incorporating the full thickness of the sheet and an in-planerepresentative cell of the macroscopic structure. At an in-plane integration pointof the macroscopic shell, the generalized strains (the membrane deformation andthe curvature) are used to formulate the boundary conditions for the micro RVEproblem. All microstructural constituents are modeled at the RVE as an ordinary3D continuum. Upon proper averaging of the RVE response, the macroscopicgeneralized stress and the moment resultants are obtained.

Chapter 3

Multiscale hom*ogenizationformulations

3.1 Introduction

3.2 General considerations

The no lineal transformation between the reference configuration of the body Ωand the current configuration of the same body Ωc is defined as: φ : Ω→ Ωc | x =φ (X), where x ∈ Ωc and X ∈ Ω are respectively the current and the referencepositions of the material point. Therefore, the linear mapping for an infinitesimalmaterial line element is

dx = F · dX, (3.1)

where the deformation gradient tensor is defined by

F =∂φ

∂X= ∇x. (3.2)

Here, the gradient operator ∇ is taken respect to the reference configuration X.Nevertheless, if now a finite material line within a finite volume is considered,

the expression given by (3.1) does not apply any more. However, a Taylor seriesexpansion (centered at Xo) can be used to obtain an expression for the finitematerial line 4x in the current configuration as

4x = F (Xo) · 4X +1

2G (Xo) : 4X 4X +O

(4X3

o

), (3.3)

where the gradient of the deformation gradient tensor is defined as

G =∂

∂X

(∂φ

∂X

)= ∇F. (3.4)

79

80 Chapter 3. Multiscale hom*ogenization formulations

Figure 3.1: Macrostructure and microstructure around of the point Xo.

It can be shown that the third-order tensor G has the symmetry propertyGijk = Gikj .

3.3 First-order hom*ogenization approach

Let us consider a solid domain (or body Ω) with a periodic or quasi-periodicmicrostructure that can be represented by a Represent Volume Element. In thisbody, it is possible to establish two scale levels, a macro scale (or structuralscale) for the macrostructure, and the other one micro scale (or sub scale) for themicrostructure. The microstructural scale is defined using a RVE which charac-terizes the microstructure of the material. Let us also consider an infinitesimalmaterial point Xo in the reference configuration of the structure, and the RVEaround this considered point as Figure 3.1 is showing.

The called principle of separation of scales [32] establishes that: the mi-crostructural length scale lµ is assumed to be much smaller than the macrostruc-tural characteristic length l, which is the length over the macroscopic space. Inother words, the principle says that the existing periodical microscopic dimensionaround of the macrostructural point (Xo) must be smaller than the characteristicmacrostructural dimension. If this principle is satisfied, the current configurationor deformed position of a material point in the RVE xµ ∈ Ωcµ can be approximatedas

xµ (Xo,Xµ) ∼= xoµ + F (Xo) · 4Xµ + w (Xµ) , (3.5)

3.3. First-order hom*ogenization approach 81

Figure 3.2: Reference and current configuration of the RVE.

where 4Xµ = Xµ − Xoµ, and Xµ ∈ Ωµ is the reference configuration or non-

deformed position of the material point in the RVE and Xoµ and xoµ are the origin

of the reference and the current coordinate system on the RVE, respectively (seeFigure 3.2). The extra term w is a microstructural displacement fluctuation field.

To simplify the symbolic manipulation of the formulation is convenient to setthe coordinate system’s origin as

Xoµ = 0 and xoµ = 0. (3.6)

Later, it will be proved that with these values, the rigid body motion of the RVEis avoided. Considering these restrictions, the expression given by (3.5) can berewritten as

xµ (Xo,Xµ) ∼= F (Xo) ·Xµ + w (Xµ) . (3.7)

3.3.1 Displacement field on the RVE

The displacement field uµ at the RVE is defined by

uµ = xµ −Xµ, (3.8)

and taking into account (3.7) in the previous equation,

uµ (Xo,Xµ) ∼= [F (Xo)− I] ·Xµ + w (Xµ) , (3.9)

where I is the second-order unit tensor.

82 Chapter 3. Multiscale hom*ogenization formulations

3.3.2 Kinematically admissible displacement fields andboundary conditions in the RVE

The displacement fields in the RVE that are kinematically admissible are obtainedas a result of the coupling between the macrostructure and the microstructure.This linkage is based on the average theorems and they have been initially pro-posed for infinitesimal deformations by Hill [46]. Later, Hill [48] and Nemat-Nasser [96] extended these to finite deformations.

The first of the averaging relations postulated that the volume average of themicrostructural deformation gradient tensor Fµ over the RVE must be equal tothe macroscopic deformation gradient tensor F. In the considered point Xo thisis

F (Xo) =1

Vµ

∫Ωµ

Fµ (Xo,Xµ) dV, (3.10)

where Vµ is the volume of the RVE in the reference configuration.Considering (3.7) it is possible to obtain the deformation gradient tensor in

the microstructural scale as

Fµ (Xo,Xµ) = ∇xµ (Xo,Xµ) ∼= F (Xo) +∇w (Xµ) , (3.11)

and using this relation, the right hand size of (3.10) is

1

Vµ

∫Ωµ

Fµ (Xo,Xµ) dV =1

Vµ

∫Ωµ

∇xµ (Xo,Xµ) dV,

= F (Xo) +1

Vµ

∫Ωµ

∇w (Xµ) dV.

(3.12)

Equation (3.12) can be rewritten as

F (Xo) =1

Vµ

∫Ωµ

Fµ (Xo,Xµ) dV − 1

Vµ

∫Ωµ

∇w (Xµ) dV, (3.13)

or

F (Xo) =1

Vµ

∫Ωµ

∇xµ (Xo,Xµ) dV − 1

Vµ

∫Ωµ

∇w (Xµ) dV. (3.14)

Finally, applying the divergence theorem, in the right hand size of (3.14), thiscan be also rewritten in term of surface integral as

F (Xo) =1

Vµ

∫∂Ωµ

xµ (Xo,Xµ) N dA− 1

Vµ

∫∂Ωµ

w (Xµ) N dA, (3.15)

where ∂Ωµ is the RVE boundary domain in the reference configuration, and Ndenotes the outward unit normal on ∂Ωµ.

3.3. First-order hom*ogenization approach 83

Figure 3.3: Normal vectors to the surfaces in the reference configuration of a CubicRVE.

Clearly, to satisfy the first average theorem, the integrals that depend of thedisplacement fluctuation in both (3.14) and (3.15) must vanish. Therefore,

∫Ωµ

∇w (Xµ) dV = 0 (3.16)

and

∫∂Ωµ

w (Xµ) N dA = 0. (3.17)

Noting Figure 3.2 and considering that the reference geometry configurationof the RVE is originally a cube, as the figure is showing, the integral restrictionon the RVE boundary can be splitted in the different surfaces of the ∂Ωµ domain.Besides, taking the reference coordinate system that is shown in Figure 3.3, theoutward unit normal of the cubic faces satisfy: N−X = −N+

X , N−Y = −N+Y and

N−Z = −N+Z . Here, the subscript makes reference to the axis which is perpen-

dicular to the considered face and the superscript defines the position of the faceon the axis. Therefore, considering this geometry, the expression given by (3.17)

84 Chapter 3. Multiscale hom*ogenization formulations

may be rewritten as(∫N+X

w dAyz −∫N−X

w dAyz

) N+

X

+

(∫N+Y

w dAxz −∫N−Y

w dAxz

) N+

Y

+

(∫N+Z

w dAxy −∫N−Z

w dAxy

) N+

Z = 0.

(3.18)

Equation (3.18) shows that the boundary restriction on the displacement fluctu-ation field can be splitted on the different surface pairs (X, Y and Z) of the RVEboundary.

Previous equations from (3.16) to (3.18) can be used to obtain the differentdisplacement fluctuation fields kinematically admissible in the microstructurallevel. Several models have been defined that assume different fluctuation fields,they are mentioned in the following.

(i) Taylor model (or zero fluctuations): The expression given by (3.16) isverified when

w, sufficiently regular | w (Xµ) = 0, ∀ Xµ ∈ Ωµ. (3.19)

This model gives hom*ogeneous deformation in the microstructural scale level(see (3.24)).

(ii) Linear boundary displacements (or zero boundary fluctuations): The ex-pression given by (3.17) is verified when

w, sufficiently regular | w (Xµ) = 0, ∀ Xµ ∈ ∂Ωµ. (3.20)

The deformation of the RVE boundary domain for this class are fully pre-scribed.

(iii) Periodic boundary fluctuations:

The key kinematical constraint for this class is that the displacement fluctu-ation must be periodic on the different faces of the RVE. That is, for each pairX+µ ,X

−µ

of boundary points the expression given by (3.18) is verified when

w, sufficiently regular | w(X+µ

)= w

(X−µ), ∀ pairs

X+µ ,X

−µ

∈ ∂Ωµ. (3.21)

(iv) Minimal constraint (or uniform boundary traction):

In this constraint the nontrivial solution of (3.17) is obtained.

3.3. First-order hom*ogenization approach 85

3.3.3 Microscopic and macroscopic strain tensor

Considering a infinitesimal deformation framework the strain tensor in the mi-crostructural level can be obtained as

Eµ (Xo,Xµ) = 12

(Fµ (Xo,Xµ) + FTµ (Xo,Xµ)

)− I

= 12

(F (Xo) + FT (Xo)

)− I

+ 12

(∇w (Xµ) + (∇w (Xµ))

T),

(3.22)

and, if (3.10) is satisfied it can be proved that taking the volume average ofthe microscopic strain tensor over the RVE domain the following relationship isobtained,

1

Vµ

∫Ωµ

Eµ (Xo,Xµ) dV =1

2

(F (Xo) + FT (Xo)

)− I = E (Xo) . (3.23)

Here, E (Xo) is the macroscopic strain tensor. Therefore, it is possible to rewrite(3.22) as

Eµ (Xo,Xµ) = E (Xo) + Ewµ (Xµ) , (3.24)

where Ewµ = 1

2

(∇w + (∇w)

T)

= ∇sw is the contribution of the displacement

fluctuation field to the microscopic strain tensor and∇s is the symmetric gradientoperator. Because (3.10) is verified the volume average of Ew

µ over the RVEdomain is equal zero.

3.3.4 Hill-Mandel principle and RVE equilibrium

The Hill-Mandel energy condition [47, 74], also referred to as the macro-hom*ogeneity condition, states that the virtual work of the point Xo consideredmust be equal to the volume average of the virtual work in the RVE to anykinematically admissible displacement field, this principle can be formulated as

S : δE (Xo) =1

Vµ

∫Ωµ

Sµ : δEµ dV, (3.25)

where S and Sµ are the macroscopic and microscopic stress tensor, respectively.Using (3.24), the principle is rewritten as

S : δE (Xo) =1

Vµ

∫Ωµ

Sµ dV : δE (Xo) +1

Vµ

∫Ωµ

Sµ : δEwµ (Xµ) dV. (3.26)

Taking the macroscopic stress tensor S as the volume average of the microstruc-tural stress tensor Sµ in the RVE domain, which is similar to the first average

86 Chapter 3. Multiscale hom*ogenization formulations

relation (see (3.10))

S (Xo,Xµ) ≡ 1

Vµ

∫Ωµ

Sµ (Xo,Xµ) dV, (3.27)

equation (3.26) will be satisfied if∫Ωµ

Sµ : δEwµ (Xµ) dV =

∫Ωµ

Sµ : ∇sδw dV = 0, (3.28)

Therefore, the RVE’s variational equilibrium equation is∫Ωµ

Sµ : ∇sδw dV = 0, (3.29)

which must be satisfied for any kinematically admissible displacement fluctuationfield w (see Section 3.3.2).

It is possible to observe that because of the symmetry of the stress tensor Sµit can be proved that Sµ : (∇a) = Sµ : (∇a)

T, where a is a first order tensor,

the (3.28) also can be rewritten as∫Ωµ

Sµ : δEwµ (Xµ) dV =

∫Ωµ

Sµ : ∇δw (Xµ) dV = 0. (3.30)

3.3.5 Microscopic and macroscopic stress tensor

The hom*ogenized stress tensor in the macroscopic level is given by (3.27). Themicroscopic stress tensor can be obtained as

Sµ (Xo,Xµ) = Cµ (Xµ) : Eµ (Xo,Xµ) ,

= Cµ (Xµ) : E (Xo) + Cµ (Xµ) : Ewµ (Xµ) ,

(3.31)

where Cµ is the material constitutive tensor in the RVE. Then, the macro stresstensor is

S (Xo,Xµ) = C : E (Xo) +1

Vµ

∫Ωµ

Cµ : Ewµ (Xµ) dV, (3.32)

where,

C ≡ 1

Vµ

∫Ωµ

Cµ dV (3.33)

is a constitutive tensor which can be considered a microstructural material prop-erty.

Equation (3.32) shows that the stress tensor S depends of the macroscopicstrain tensor E and also, of the microscopic strain tensor Ew

µ , which is obtained

3.4. Enhanced-first-order hom*ogenization approach 87

with the displacement fluctuation field of the RVE. Moreover, the microscopicposition vector Xµ does not appear explicitly in the microstructural strain tensorexpression (see (3.24)). Consequently, this variable does not appear in the mi-crostructural stress tensor either. Therefore, the periodic microstructure aroundthe macro point Xo does not have to be modeled with its exact dimensions. A nondimensional RVE with the internal distribution and volume fractions of the sim-ple materials is enough to obtain the microscopic strain and stress fields. Thisis one of the principal advantages of this first-order hom*ogenization approachcompared to other multiscale high-order approaches.

On the other hand, it can be observed that the kinematically admissibledisplacement fluctuation option used to satisfy the boundary condition in theRVE problem affects the final macroscopic stress tensor obtained, as occurs inthe Taylor model case. This means that if there is a null displacement fluctuationfield in the total RVE domain, the stress tensor S obtained only depend of thestrain tensor E and the constitutive tensor C. In other words, the Taylor modelcondition returns the classical mixing theory results.

3.4 Enhanced-first-order hom*ogenization ap-proach

A new enhanced first order hom*ogenization is proposed in the following, in orderto enrich the displacement field of the micro model with second order informationavailable in the macro model. The deformed position of a material point in theRVE given by (3.5), used in the first-order approach, can be improved if thesecond-order term of (3.3) is included. Then, it is possible to propose a newapproximation of the current configuration of the RVE as

xµ (Xo,Xµ) ∼= xcµ + F (Xo) · 4Xµ +1

2G (Xo) : 4Xµ 4Xµ + w (Xµ) , (3.34)

and setting the coordinate system’s origin as defined in (3.6), the proposed de-formed position of the RVE is

xµ (Xo,Xµ) ∼= F (Xo) ·Xµ +1

2G (Xo) : Xµ Xµ + w (Xµ) . (3.35)

Therefore, the proposed displacement field uµ on the RVE (see (3.8)) can beobtained now as

uµ (Xo,Xµ) ∼= [F (Xo)− I] ·Xµ +1

2G (Xo) : Xµ Xµ + w (Xµ) . (3.36)

Noting that an extra term appears in the proposed microscopic displacementfield by including the gradient of the deformation gradient tensor G in (3.34).

88 Chapter 3. Multiscale hom*ogenization formulations

This extra second-order term is a new linking term between the macroscopicand microscopic scales. The proposed displacement field in the RVE is enhancedbecause it reaches more information from the macro scale.

3.4.1 Kinematically admissible displacement fields andboundary conditions in the RVE

The first of the average postulates (see (3.10)) is used again to obtain the admis-sible displacement fields. The microscopic deformation gradient considering theexpression given by (3.35) is

Fµ (Xo,Xµ) = ∇xµ (Xo,Xµ) ∼= F (Xo) + G (Xo) ·Xµ +∇w (Xµ) . (3.37)

And, the volume average of this deformation gradient Fµ over the RVE is

1

Vµ

∫Ωµ

Fµ (Xo,Xµ) dV =1

Vµ

∫Ωµ

∇xµ (Xo,Xµ) dV,

= F (Xo) + G (Xo) ·1

Vµ

∫Ωµ

Xµ dV

+1

Vµ

∫Ωµ

∇w (Xµ) dV.

(3.38)

It can be proved that if the RVE geometry in the reference configuration isoriginally a cube, as shown in Figure 3.2, and the position of the origin of thecoordinate system is defined at the center of the RVE, then the first moment ofvolume of the RVE is ∫

Ωµ

Xµ dV = 0. (3.39)

Therefore, (3.38) can be rewritten as

F (Xo) =1

Vµ

∫Ωµ

Fµ (Xo,Xµ) dV − 1

Vµ

∫Ωµ

∇w (Xµ) dV, (3.40)

or

F (Xo) =1

Vµ

∫Ωµ

∇xµ (Xo,Xµ) dV − 1

Vµ

∫Ωµ

∇w (Xµ) dV. (3.41)

Equation (3.41) can be rewritten in terms of surface integrals applying the diver-gence theorem as

F (Xo) =1

Vµ

∫∂Ωµ

xµ (Xo,Xµ) N dA− 1

Vµ

∫∂Ωµ

w (Xµ) N dA. (3.42)

Equation (3.42) is exactly the same than the one obtained for the first-orderapproach (see (3.15)) then, the restrictions on the displacement fluctuation field

3.4. Enhanced-first-order hom*ogenization approach 89

are the same as those shown in (3.16)-(3.18), and the different displacementfluctuation fields kinematically admissible presented in Section 3.3.2 are still validfor this enhanced-first-order approach.

Extra kinematic restrictions and boundary conditions because of G

The next step consists in obtaining the kinematic restrictions result of includingthe new term G in the microscopic displacement field uµ. In other words, someextension of the first average theorem needs to be proposed in term of the gradientof the deformation gradient. In the following, a natural extension for the firstaverage theorem is presented. The main drawback of this proposal is that arrivesto restrictions on the derivative displacement fluctuation field and therefore, ahigh-order problem on the RVE must be considered. To avoid this situation, andcontinue using the classical first-order boundary value problem on the RVE, thealternative extension proposed by Kouznetsova [59] is also presented.

Natural extension of the first average theorem The first natural possi-bility for this extension could be

G (Xo) =1

Vµ

∫Ωµ

Gµ (Xo,Xµ) dV. (3.43)

Note that (3.43) is similar to (3.10) but in this case, the volume average of themicrostructural gradient of the deformation gradient tensor Gµ over the RVEmust be equal to the macroscopic gradient of the deformation gradient G in theconsidered point Xo.

Considering (3.37) and (3.4) the gradient of the deformation gradient in themicrostructural scale is

Gµ (Xo,Xµ) = ∇ (∇xµ (Xo,Xµ)) ∼= G (Xo) +∇ (∇w (Xµ)) . (3.44)

Using (3.44) and taking the volume average over the RVE it is possible to obtain

G (Xo) =1

Vµ

∫Ωµ

Gµ (Xo,Xµ) dV − 1

Vµ

∫Ωµ

∇ (∇w (Xµ)) dV, (3.45)

or

G (Xo) =1

Vµ

∫Ωµ

∇ (∇xµ (Xo,Xµ)) dV − 1

Vµ

∫Ωµ

∇ (∇w (Xµ)) dV. (3.46)

And, applying the divergence theorem in the last expression

G (Xo) =1

Vµ

∫∂Ωµ

∇xµ (Xo,Xµ) N dA− 1

Vµ

∫∂Ωµ

∇w (Xµ) N dA. (3.47)

90 Chapter 3. Multiscale hom*ogenization formulations

Figure 3.4: Normal vectors to the lines in the YZ surface of the Cubic RVE.

Similarly as in the first-order approach, to satisfy the proposed extension of thefirst average theorem, the integrals that depend of the displacement fluctuationin (3.45) and (3.47) must vanish, then∫

Ωµ

∇ (∇w (Xµ)) dV = 0, (3.48)

and ∫∂Ωµ

∇w (Xµ) N dA = 0. (3.49)

The last expression represents a new extra integral restriction on the derivativedisplacement fluctuation field. Taking the same consideration than before re-garding the geometry of the RVE (see Figure 3.3), the boundary integration in(3.49) can be splitted in(∫

N+X

∇w dAyz −∫N−X

∇w dAyz

) N+

X

+

(∫N+Y

∇w dAxz −∫N−Y

∇w dAxz

) N+

Y

+

(∫N+Z

∇w dAxy −∫N−Z

∇w dAxy

) N+

Z = 0.

(3.50)

Some components of the integrals can also be rewritten in terms of line-boundary integrals applying the divergence theorem. For example, if the firstleft integral in the first term in (3.50) is taken, the line-boundary of this surface

3.4. Enhanced-first-order hom*ogenization approach 91

integral can be separated in four different lines, two perpendiculars to Y axis, andthe other two perpendiculars to Z axis, as it is shown in Figure 3.4. Because ofthe RVE geometry considered, these lines boundary have the property of N−X|Y =

−N+X|Y and N−X|Z = −N+

X|Z . Then, with this information the considered integral

can be rewritten as∫N+X

∇w dAyz=

∫N+X

∇Xw dAyz

+

(∫N+X|Y

w dLz −∫N−X|Y

w dLz

) N+

X|Y

+

(∫N+X|Z

w dLy −∫N−X|Z

w dLy

) N+

X|Z ,

(3.51)

where ∇X represents the derivative with respect to the X axis, this term cannotbe reduced to a line-integral using the divergence theorem. It can be seen thatwhen the Periodic boundary fluctuations condition is the kinematically admissibleoption used for the displacement fluctuation field on the RVE, the two rightterms on (3.51) are satisfied directly. The points on the opposing lines are apair boundary points that have the same displacement fluctuation because of thekinematic condition imposed. Applying this same procedure to the rest of theterms of expression (3.50), this equation can be rewritten as(∫

N+X

∇Xw dAyz −∫N−X

∇Xw dAyz

) N+

X

+

(∫N+Y

∇Y w dAxz −∫N−Y

∇Y w dAxz

) N+

Y

+

(∫N+Z

∇Zw dAxy −∫N−Z

∇Zw dAxy

) N+

Z = 0.

(3.52)

The previous expression represents an extra restriction on the displacement fluc-tuation field that makes it kinematically admissible in the RVE. A possible setof boundary conditions that satisfies this restriction is∫

N+X

∇Xw dAyz =

∫N−X

∇Xw dAyz,∫N+Y

∇Y w dAxz =

∫N−Y

∇Y w dAxz,∫N+Z

∇Zw dAxy =

∫N−Z

∇Zw dAxy.

(3.53)

92 Chapter 3. Multiscale hom*ogenization formulations

Equation (3.52) is analogous to (3.18) but it is written in terms of derived dis-placement fluctuation field, in this case, on the normal direction of the pairsurfaces (see Figure 3.3). Therefore, to satisfy any kinematic restriction, as forexample (3.53), obtained from (3.52) a high-order problem on the microscopicscale must be considered because the restriction of the displacement fluctuationfield is written on its derivate.

Alternative extension of the first average theorem An alternative to theproposed extension of the averaging theorem given by (3.43) should be found tokeep a classical boundary value problem on the microstructural RVE problem.With this aim Kouznetsova [59] proposed another alternative extension of thefirst average theorem. The proposed condition imposes that the second momentof area of the deformed RVE, given in terms of the microscopic displacements,must be equal to the second moment of area of the RVE expressed in terms ofmacroscopic deformation variables [54]. Considering the above, the expressiongiven by (3.37) is multiplied by Xµ and integrated over the RVE volume to obtain

∫Ωµ

∇0xµ (Xo,Xµ) Xµ dV = F (Xo) ∫

Ωµ

Xµ dV

+ G (Xo) ·∫

Ωµ

Xµ Xµ dV

+

∫Ωµ

∇w (Xµ) Xµ dV .

(3.54)

Knowing that the first moment of volume of the undeformed RVE is zero (see(3.39)), and defining the second moment of volume of the undeformed RVE asJ =

∫Ωµ

Xµ Xµ dV . Equation (3.54) can be rewritten as

G (Xo) · J =

∫Ωµ

∇xµ (Xo,Xµ) Xµ dV −∫

Ωµ

∇w (Xµ) Xµ dV, (3.55)

replacing the following relationships

∇xµ (Xo,Xµ) Xµ = ∇ (xµ (Xo,Xµ) Xµ)− xµ (Xo,Xµ) I, (3.56)

and

∇w (Xµ) Xµ = ∇ (w (Xµ) Xµ)−w (Xµ) I, (3.57)

3.4. Enhanced-first-order hom*ogenization approach 93

it is obtained

G (Xo) · J =

∫Ωµ

∇ (xµ (Xo,Xµ) Xµ) dV

−∫

Ωµ

∇ (w (Xµ) Xµ) dV

−∫

Ωµ

xµ (Xo,Xµ) dV I +

∫Ωµ

w (Xµ) dV I.

(3.58)

Using (3.35) it can be shown that∫Ωµ

xµ (Xo,Xµ) dV I =1

2G (Xo) : J I +

∫Ωµ

w (Xµ) dV I, (3.59)

which is used to obtain the final version of the sought expression

G (Xo) · J + 12G (Xo) : J I =

∫Ωµ

∇ (xµ (Xo,Xµ) Xµ) dV

−∫

Ωµ

∇ (w (Xµ) Xµ) dV ,

(3.60)

Applying the divergence theorem on the right hand size of the equation, it canbe rewritten in term of surface integral as

G (Xo) · J + 12G (Xo) : J I =

∫∂Ωµ

xµ (Xo,Xµ) Xµ N dA

−∫∂Ωµ

w (Xµ) Xµ N dA.

(3.61)

It is possible to make a parallelism between (3.15) and (3.61). The additionalcondition regarding the second moment of area of the deformed RVE given by(3.54) requires that the influence of the displacement fluctuation field shouldvanish, then ∫

∂Ωµ

w (Xµ) Xµ N dA = 0 (3.62)

Equation (3.62) is a boundary restriction for the displacement fluctuation field,then it is not necessary a high-order boundary value problem, at microscopicscale, to satisfy the new boundary conditions deduced from it. Considering againthe cubic geometry in the reference configuration defined previously for the RVE(see Figure 3.3), the restriction given by (3.62) can be splitted in the different

94 Chapter 3. Multiscale hom*ogenization formulations

surfaces of the domain as(∫N+X

w Xµ dAyz −∫N−X

w Xµ dAyz

) N+

X

+

(∫N+Y

w Xµ dAxz −∫N−Y

w Xµ dAxz

) N+

Y

+

(∫N+Z

w Xµ dAxy −∫N−Z

w Xµ dAxy

) N+

Z = 0.

(3.63)

The last expression is used in Section 3.5.2 to obtain the boundary valueproblem on the RVE for the enhanced-first-order approach. In the case of Periodicboundary fluctuations condition, it can be proved that the expression (3.63) isautomatically satisfied if∫

N−X

w dAyz = 0 ,

∫N−Y

w dAxz = 0 and

∫N−Z

w dAxy = 0. (3.64)

Therefore, the extra boundary condition required in this case is that the integralof the periodic displacement fluctuations on the RVE surfaces must be zero.

3.4.2 Microscopic and macroscopic strain tensor

In this enhanced-first-order hom*ogenization approach, the strain tensor of themicrostructure for an infinitesimal deformation approach can be written as

Eµ (Xo,Xµ) = 12

(Fµ (Xo,Xµ) + FTµ (Xo,Xµ)

)− I

= 12

(F (Xo) + FT (Xo)

)− I

+ 12

(G (Xo) ·Xµ + (G (Xo) ·Xµ)

T)

+ 12

(∇w (Xµ) + (∇w (Xµ))

T).

(3.65)

Knowing that (3.10) is satisfied and using equation (3.39), the resulting expres-sion of the volume average of the microscopic strain tensor over the RVE domainis the same than (3.23), which was obtained previously in Section 3.3.3. There-fore, the microscopic strain tensor can be rewritten as

Eµ (Xo,Xµ) = E (Xo) + EGµ (Xo,Xµ) + Ew

µ (Xµ) , (3.66)

where EGµ = 1

2

(G ·Xµ + (G ·Xµ)

T)

is a new term in the microscopic strain

tensor, resulting from including the second-order term G in the formulation.Using the expression given by (3.39), it can be proved that the volume averageof this new term EG

µ over the RVE domain is equal to zero.

3.4. Enhanced-first-order hom*ogenization approach 95

Figure 3.5: Macro volume ΩM around point Xo and its micro structure.

3.4.3 Hill-Mandel principle and RVE equilibrium

When the second-order of the Taylor series expansion given by (3.3) is usedto improve the approximation of the deformed position of a material point inthe RVE (see (3.34)), it is assumed that exists a macroscopic finite volume ΩMaround the considered point Xo, as it is shown in Figure 3.5. This finite volumemust be smaller than the characteristic macroscopic dimension. Therefore, theHill-Mandel principle should be applied now not only taking into account thevirtual work of the point Xo, but considering the volume average of the virtualwork in the macro volume ΩM . This can be stated as

1

VM

∫ΩM

S : δE dV =1

Vµ

∫Ωµ

Sµ : δEµ dV (3.67)

The macroscopic deformed position of a material point in ΩM around thepoint Xo can be approximated with a second-order approach using (3.3) as

4x ∼= F (Xo) · 4X +1

2G (Xo) : 4X 4X, (3.68)

and the approximated macroscopic deformation gradient is

F ∼= F (Xo) + G (Xo) · 4X. (3.69)

The macroscopic strain tensor in the ΩM domain for infinitesimal deformationapproach can be then approximated as

E ∼= 1

2

(F (Xo) + FT (Xo)

)− I +

1

2

(G (Xo) · 4X + (G (Xo) · 4X)

T)

(3.70)

orE ∼= E (Xo) + EG (Xo,X) , (3.71)

96 Chapter 3. Multiscale hom*ogenization formulations

where EG = 12

(G · 4X + (G · 4X)

T)

.

Taking into account (3.66) and (3.71), the expression given by (3.67) can berewritten as

1

VM

∫ΩM

S dV : δE +1

VM

∫ΩM

S : δEG dV =1

Vµ

∫Ωµ

Sµ dV : δE

+1

Vµ

∫Ωµ

Sµ : δEGµ dV +

1

Vµ

∫Ωµ

Sµ : δEwµ dV

(3.72)

and because of the symmetry of the stress tensor, it can be proved that S :(∇a) = S : (∇a)

Tand S : (G.a) = S : (G.a)

T, where a is a first order tensor.

Then, (3.72) is finally

1

VM

∫ΩM

S dV : δE +1

VM

∫ΩM

S 4X dV... δG =

1

Vµ

∫Ωµ

Sµ dV : δE

+1

Vµ

∫Ωµ

Sµ Xµ dV... δG +

1

Vµ

∫Ωµ

Sµ : ∇sδw dV .

(3.73)Following the same procedure used in Section 3.3.4 to satisfy the Hill-Mandelprinciple, it is necessary to define the following tensors:

S ≡ 1

VM

∫ΩM

S dV ≡ 1

Vµ

∫Ωµ

Sµ dV, (3.74)

where S is the hom*ogenized stress tenor, which is obtained as the volume averageof the stress tensor around the point Xo, and

Q ≡ 1

VM

∫ΩM

S 4X dV ≡ 1

Vµ

∫Ωµ

Sµ Xµ dV. (3.75)

where Q is the hom*ogenized second-order stress tensor in the point Xo, which isa the three-order tensor. Finally, the RVE’s variational equilibrium equation is∫

Ωµ

Sµ : ∇sδw dV = 0, (3.76)

that must be satisfied for any kinematically admissible displacement fluctuationfield w shown in Section 3.4.1.

3.4.4 hom*ogenized stress and second-order stress tensor

The microscopic stress tensor can be obtained as

Sµ = Cµ : E (Xo) + Cµ : EGµ (Xo,Xµ) + Cµ : Ew

µ (Xµ) , (3.77)

3.4. Enhanced-first-order hom*ogenization approach 97

then, the hom*ogenized stress tensor at the macroscopic scale given by (3.74) is

S =1

Vµ

∫Ωµ

Cµ dV : E (Xo) +1

Vµ

∫Ωµ

Cµ : EGµ (Xo,Xµ) dV

+1

Vµ

∫Ωµ

Cµ : Ewµ (Xµ) dV

(3.78)

or

S = C : E (Xo) + B... G (Xo) +

1

Vµ

∫Ωµ

Cµ : Ewµ (Xµ) dV, (3.79)

where

B ≡ 1

Vµ

∫Ωµ

Cµ Xµ dV. (3.80)

The tensor B can be also considered a microscopic material property. This con-stitutive tensor relates gradient of the deformation gradient tensor G, to thehom*ogenized stress tensor S, and generates a coupling effect. This tensor B isanalogous to the called bending-extension coupling matrix used in plates or shellstheories [4].

Equation (3.79) shows that the hom*ogenized stress tensor S in the point Xo

depends of the macroscopic tensor E and G, of the microscopic displacementfluctuation field w and also, of the vector position Xµ of the RVE. Consideringnow a particular case where the simple materials within the RVE are symmetri-cally located respect to the coordinate system’s origin, which has been placed onthe RVE geometric center (see Section 3.4.1). It can be proved that taking thissymmetric distribution of the simple materials the value obtains for the constitu-tive tensor B is zero. Therefore, the hom*ogenized stress tensor for this case canbe rewritten as

S = C : E (Xo) +1

Vµ

∫Ωµ

Cµ : Ewµ (Xµ) dV. (3.81)

The macroscopic second-order term G and of the vector position Xµ do not affectthe hom*ogenized stress tensor. The expression given by (3.81) is the same thanthe one obtained for first-order hom*ogenization given by (3.32).

On the other hand, the hom*ogenized second-order stress tensor can be ob-tained using (3.75) and the microscopic stress tensor given by (3.77) as

Q =1

Vµ

∫Ωµ

Cµ Xµ dV : E (Xo)

+1

Vµ

∫Ωµ

(Cµ Xµ) : EGµ (Xo,Xµ) dV

+1

Vµ

∫Ωµ

Cµ : Ewµ (Xµ) Xµ dV ,

(3.82)

98 Chapter 3. Multiscale hom*ogenization formulations

or

Q = B : E (Xo) + D... G (Xo) +

1

Vµ

∫Ωµ

Cµ : Ewµ (Xµ) Xµ dV, (3.83)

where

D =1

Vµ

∫Ωµ

(Cµ Xµ) Xµ dV. (3.84)

The tensor D is also considered a microscopic material property, which is ob-tained with the RVE model, as it is done with tensors C and B. Taking intoaccount the symmetric distribution materials inside the RVE, the expression forthe hom*ogenized second-order stress can be written as

Q = D... G (Xo) +

1

Vµ

∫Ωµ

Cµ : Ewµ (Xµ) Xµ dV. (3.85)

Equation (3.85) shows that the second-order stress tensor Q depends on themacroscopic tensor G and on the microscopic displacement fluctuation field w.But it also depends on the vector position Xµ of the material point in the RVE.In addition, the tensor D does not vanish because of the symmetric materialsdistribution.

3.4.5 Final remarks on the enhanced-first-order hom*oge-nization approach

In the enhanced-first-order approach is lost the benefit shown by the first-orderapproach regarding the possibility of using a non dimensional RVE. The micro-scopic strain (3.66) and stress (3.77) tensor now have an explicit dependence withthe material vector position Xµ in the RVE. This dependence has been observedalso in the hom*ogenized second-order stress tensor (3.85). While the hom*ogenizedstress tensor (3.81) is not dependent of Xµ when a symmetric internal distribu-tion of the materials is taken. Besides, to satisfy (3.72), which is obtained fromthe Hill-Mandel condition extended to the enhanced-first-order approach, it isnecessary to impose that Ωµ ≡ ΩM . Thus the RVE’s dimension used to char-acterize the microstructure should be equal than the size of the finite volumearound the considered point Xo. Then, the enhanced-first-order hom*ogenizationprocedure can be understand as a domain decomposition where the sub-domainshave a periodic microstructure.

However, the enhanced-first-order hom*ogenization is better than the first-order hom*ogenization from a microscopic point of view. Although the hom*oge-nized stress tensor for both theories is the same, the microscopic displacementfield, the microscopic strain and, the stress tensors are not equal. The enhanced-first-order approach gets a better approximation of the microscopic behavior be-cause it accounts for the information provided by the macroscopic second-order

3.5. Macroscopic and microscopic formulation 99

term G. Therefore, in a non-linear analysis, the initiation and the evolution ofthe non-linear performance of the microstructure will be better characterized ifthe enhanced-first-order formulation is used.

On the other hand, a detailed analysis of the formulation shows that thefirst-order hom*ogenization theory is contained in the enhanced-first-order one.Therefore, when the principle of separation of scales is verified, the results ob-tained using the enhanced formulation will be the same than the ones obtainedusing the first-order hom*ogenization. In other words, if the periodic microstruc-tural length scale lµ is much smaller than the structure characteristic length l (seeFigure 3.1), the contribution of the extra term considered in (3.35) to improvethe first-order approach is negligible.

The variational equilibrium equation obtained in the microstructure for bothhom*ogenization formulations is the same (3.29) and (3.76). However, in theenhanced-first-order hom*ogenization formulation extra boundary conditions mustbe satisfied. High-order boundary conditions (3.53) are obtained if the naturalextension of the first average theorem is proposed (3.43). In consequence, thiskind of extra boundary conditions require a high-order microscopic equilibriumequation. To avoid this situation an alternative extension of the average the-orem given by (3.60) is used to conserve a first-order microscopic variationalproblem. New extra boundary conditions using (3.60) for the case of Periodicboundary fluctuations are obtained and shown in the expression (3.64). Finally,for the enhanced-first-order approach, a macroscopic second-order stress tensoris obtained from the RVE solution. This high-order stress tensor should be con-sidered in the macroscopic scale theory.

3.5 Macroscopic and microscopic formulation

In the following, the boundary value problem (BVP) in the macroscopic and inthe microscopic scales are presented.

3.5.1 Macroscopic BVP

A BVP is considered for the macrostructural scale of a domain Ω with a pe-riodic internal microstructure. The kinematics of the problem is related to adisplacement field on the macroscopic scale, which provides the displacement ofeach material point of the domain Ω. From a continuum mechanics approach themacroscopic BVP is

∂Sij∂Xj

+ fi = 0 in Ω,

ui = ui in ∂Ωu,

SijNj = ti in ∂Ωt,

(3.86)

100 Chapter 3. Multiscale hom*ogenization formulations

where Sij is the macroscopic stress tensor, and fi is the internal body force as-sociated to the mass forces of the material. The boundary of Ω (∂Ω) is defineddisjointedly by the surfaces ∂Ωu where the macroscopic displacement is known ui(Dirichlet’s condition) and ∂Ωt where the macroscopic surface load ti are known(Neumann’s condition) with ∂Ωu ∪ ∂Ωt = ∂Ω and ∂Ωu ∩ ∂Ωt = ∅. Finally, Nj

are the components of an outward vector normal to the surface ∂Ωt.The resolution of the BVP given by (3.86) consists on the determination of

the macroscopic displacement field corresponding to the solution u ∈ VΩ, whereVΩ is the set of continuous and sufficiently regular functions with zero-valued in∂Ωu. The partial differential equation in the macroscopic BVP above presentedcan be rewritten in a weak form (or variational form) as∫

Ω

∂Sij∂Xj

vi dV +

∫Ω

fivi dV = 0 ∀ v ∈ VΩ, (3.87)

where vi are the called test functions. Equation (3.87) can be rewritten, applyingthe divergence theorem, as∫

Ω

Sij∂vi∂Xj

dV =

∫Ω

fivi dV +

∫∂Ω

tivi dA ∀ v ∈ VΩ. (3.88)

Considering infinitesimal deformations, the macroscopic strain and stress ten-sor are

Eij =1

2

(Fij + FTij

)− Iij =

1

2

(∂ui∂Xj

+∂uj∂Xi

)in Ω,

Sij =1

Vµ

∫Ωµ

Sµ dV in Ω.

(3.89)

3.5.2 Microscopic BVP

The obtained variational equilibrium statement (or the virtual work equation) inthe microstructure (see (3.29) and (3.76)) can be written as∫

Ωµ

Sµ : ∇sw dV = 0 ∀ w ∈ VΩµ . (3.90)

Considering again an infinitesimal deformation, the microscopic strain tensor is

Eµ =1

2

(Fµ + FTµ

)− I = ∇suµ in Ωµ, (3.91)

where∇suµ is the symmetric gradient of the microscopic displacement field in theRVE and VΩµ is the set of continuous and sufficiently regular kinematically ad-missible RVE displacement fields. Further, it is assumed that in the microstruc-ture the constitutive behavior is described by conventional internal dissipative

3.5. Macroscopic and microscopic formulation 101

constitutive theories. Therefore, the microscopic stress tensor is obtained by in-tegrating the constitutive equations, knowing a set of internal variables α, forthe given strain tensor history. Then, it is

Sµ = Sµ(Eµ,α) = Sµ(∇suµ,α). (3.92)

With the above at hand, the resolution of the microstructure problem consistson the determination of the microscopic displacement field uµ ∈ VΩµ of thevariational problem for a given macroscopic deformation gradient tensor F, andits gradient G in enhanced-first-order approach case. Therefore, to complete theBVP in the microscopic scale it is necessary to define the boundary conditionused to obtain kinematically admissible displacement fields from the solution of(3.90).

First-order approach

In Section 3.3.2, the different displacement fluctuation fields, kinematicallyadmissible in an hom*ogenization analysis, were obtained. Using (3.9) is possibleto obtain the boundary condition for each case in terms of the displacement fieldas

(i) Taylor model (or zero fluctuations):

uµ = (F− I) ·Xµ, in ∀ Xµ ∈ Ωµ.

In this model is not necessary to solve the BVP defined by (3.90).

(ii) Linear boundary displacements (or zero boundary fluctuations):

uµ = (F− I) ·Xµ, in ∀ Xµ ∈ ∂Ωµ.

In this case a classical BVP with only Dirichlet conditions is obtained.

(iii) Periodic boundary fluctuations:

uµ(X+µ )− uµ(X−µ ) = D1(F− I) ·N+

X , in ∀ pairsX+µ ,X

−µ

∈ ∂Ωµ|NX

.

uµ(X+µ )− uµ(X−µ ) = D2(F− I) ·N+

Y , in ∀ pairsX+µ ,X

−µ

∈ ∂Ωµ|NY

.

uµ(X+µ )− uµ(X−µ ) = D3(F− I) ·N+

Z , in ∀ pairsX+µ ,X

−µ

∈ ∂Ωµ|NZ

.

In this case, the boundary condition is a constraint boundary condition.

102 Chapter 3. Multiscale hom*ogenization formulations

(iv) Minimal constraint (or uniform boundary traction):

∫N+X

uµdAyz −∫N−X

uµdAyz = D1(F− I) ·N+X , in ∀ Xµ ∈ ∂Ωµ|NX

.∫N+Y

uµdAxz −∫N−Y

uµdAxz = D2(F− I) ·N+Y , in ∀ Xµ ∈ ∂Ωµ|NY

.∫N+Z

uµdAxy −∫N−Z

uµdAxy = D3(F− I) ·N+Z , in ∀ Xµ ∈ ∂Ωµ|NZ

.

For this case, the boundary condition is an integral constraint boundary con-dition.

Enhanced-first-order approach

In Section 3.4.1, the different displacement fluctuation fields kinematicallyadmissible were obtained for the enhanced-first-order approach. Besides, withthis approach exists an extra restriction on the displacement fluctuation fieldbecause the term G is introduced in the microscopic displacement field. Using(3.36) it is possible to obtain the boundary condition for each case in terms ofthe displacement field as

(i) Taylor model (or zero fluctuations):

uµ = (F− I) ·Xµ +1

2G : Xµ Xµ, in ∀ Xµ ∈ Ωµ.

In this model is not necessary to solve the BVP defined by (3.90).

(ii) Linear boundary displacements (or zero boundary fluctuations):

uµ = (F− I) ·Xµ +1

2G : Xµ Xµ, in ∀ Xµ ∈ ∂Ωµ.

A classical BVP with only Dirichlet conditions is obtained because the extraboundary restriction (3.62) is automatically verified.

3.5. Macroscopic and microscopic formulation 103

(iii) Periodic boundary fluctuations:

uµ(X+µ )− uµ(X−µ ) = D1(F− I) ·N+

X +(D1)

2

2N+X ·G ·N+

X

+ D1G : N+X X−µ , in ∀ pairs

X+µ ,X

−µ

∈ ∂Ωµ|NX

.

uµ(X+µ )− uµ(X−µ ) = D2(F− I) ·N+

Y +(D2)

2

2N+Y ·G ·N+

Y

+ D2G : N+Y X−µ , in ∀ pairs

X+µ ,X

−µ

∈ ∂Ωµ|NY

.

uµ(X+µ )− uµ(X−µ ) = D3(F− I) ·N+

Z +(D3)

2

2N+Z ·G ·N+

Z

+ D3G : N+Z X−µ , in ∀ pairs

X+µ ,X

−µ

∈ ∂Ωµ|NZ

.

To satisfy the extra boundary restriction given by (3.64) extra boundary con-ditions are required. These are:

∫N−X

uµdAyz = −1

2D1D2D3(F− I) ·N+

X +1

8(D1)2D2D3G : N+

X N+X

+1

24(D2)3D3G : N+

Y N+Y

+1

24D2(D3)3G : N+

Z N+Z , in ∀ Xµ ∈ ∂Ωµ|N−X

.∫N−Y

uµdAxz = −1

2D1D2D3(F− I) ·N+

Y +1

8D1(D2)2D3G : N+

Y N+Y

+1

24(D1)3D3G : N+

X N+X

+1

24D1(D3)3G : N+

Z N+Z , in ∀ Xµ ∈ ∂Ωµ|N−Y

.∫N−Z

uµdAxy = −1

2D1D2D3(F− I) ·N+

Z +1

8D1D2(D3)2G : N+

Z N+Z

+1

24(D1)3D2G : N+

X N+X

+1

24D1(D2)3G : N+

Y N+Y , in ∀ Xµ ∈ ∂Ωµ|N−Z

.

In this case, one boundary condition is a constraint boundary condition andthe extra boundary condition is an integral constraint boundary condition.

3.5.3 Consequence of the boundary condition considered

The RVE has a finite dimension, which is opposed to the theoretically infinitemicrostructure considered, which create the intrinsic problem of the non-physical

104 Chapter 3. Multiscale hom*ogenization formulations

RVE edges. As a result, the election of the boundary condition in the RVEproblem is essential to characterize the real behavior of the microstructure. Theboundary conditions shown in the above sections are obtained from the restrictionon the microscopic displacement fields by the first average postulate (3.10) andthey are used to impose the driving macroscopic deformation gradient tensorF (or strain tensor) on the RVE. Besides, these boundary restrictions shouldincorporate the presence of the surrounding material of the RVE without (orminimizing) the introduction of spurious effects.

Effects on the characterization of the material properties

It has been shown in Section 3.3.5 that the boundary condition used in the RVEproblem affects the macroscopic stress tensor obtained and therefore it also affectsthe hom*ogenized constitutive tensor. In example, the expression (3.32) showsthat using the Taylor model condition, which in fact is not a strictly boundarycondition, the result provides an upper bound of the estimated hom*ogenizedmicroscopic stiffness, which is exactly the same that is predicted by the classicalmixing theory.

On the other hand, the Minimal constraint provides a lower bound of theestimated effective microstructural stiffness. This boundary condition imposesthe macroscopic strain tensor on the RVE in the weakest sense. It has beenshown that the resulting boundary distribution of the microscopic stress tensorin the RVE is uniform and equal to the macroscopic stress tensor in this boundaryrestriction [90, 21].

The Linear boundary displacements condition is a too restrictive constraintand it overestimates the hom*ogenized microscopic stiffness [14]. A conventionalBVP with full Dirichlet’s condition is obtained for the microstructure when thiscondition is addressed.

It has been shown in the literature that in general the Periodic boundaryfluctuations provide a better apparent stiffness estimate for both periodic as wellas random microstructures [128, 133, 90, 55, 56, 91, 110]. This condition makesthat the RVE self adjoint by point to point (pairs of points) coupling of bound-ary displacements, thereby it naturally incorporates the mechanical response ofthe surrounding material. Moreover, an anti-periodic condition of the boundaryforces is automatically fulfilled in the microscopic problem because the boundarypoints of the RVE are really considered as internal points of the structure.

Effects on the non-linear response

Besides obtaining effective properties of the material at the macrocopic scale,RVE models have been also used to study damage and strain localization effects[33, 72, 57]. There is a significant interaction between both phenomena, whenthe damage occur within the RVE, small zones with high strains coalesce to form

3.6. Finite element implementation 105

a strain localization band. If the macroscopic loading increases continuously, themicroscopic strain localizes on this band, accelerating the damage developmentand reducing the load capacity and finally resulting in strain softening in themacroscopic scale.

The boundary conditions imposed in the microscopic BVP need to addressthe beginning and the development of the localization band up to the point ofmacroscopic localization. Too restrictive boundary constraints, such as the Lin-ear boundary displacements case will suppress the development of the microscopiclocalization band, and searching an alternative localization pattern, it will over-estimate the predicted macroscopic strain softening load.

The Periodic boundary fluctuations condition allows the onset of a localizationpath compatible with the periodicity constraints, such as horizontal, vertical,multiple parallel inclined bands and etc.

In order to do not restrict the development of the localization band within theRVE the Minimal constraint conditions have been used as an alternative to theothers showing that the localization paths can cross the RVE in arbitrary manner[88, 51]. However, besides the underestimation of the effective microstructuralstiffness, this boundary condition is also sensitive to spurious localization in weakzones near the RVE boundary, resulting in an unrealistically early onset of macro-scopic failure [14].

Base on the different performance described, in this work the Periodic bound-ary fluctuations condition will be used in the computational implementation ofthe microscopic BVP for both hom*ogenization approaches considered, first-orderand enhanced-first-order.

3.6 Finite element implementation

The numerical implementation of the developed hom*ogenization approaches ismade through the FEM. This method splits the total domain of the problemin finite N-subdomains, known as Finite Element (FE) where the unknown dis-placement field is approximated using known displacements of some points (callednodal points) and shape functions.

When the FEM is used to solve the weak formulation of a BVP the totalintegration domain of the problem is separated in N-subdomains which are theFE domains. The value of the definite integral in the FE domain is obtainedusing some numerical integration procedure. In general, the numerical integrationmethods approximate the integral value as the weighted sum of the evaluatedintegrand in a finite set of points called integration points. These integrationpoints, and their weights, depend on the accuracy required and on the integrationmethod. In the following, the selected method will be the Gaussian quadraturerule which uses n-points (xi) called Gauss points and n-weights (wi) to obtainthe numerical approximate value of the integral.

106 Chapter 3. Multiscale hom*ogenization formulations

To solve the BVP at the macroscopic scale for linear and non-linear range aFEM implementation is used together with a Newton-Raphson iterative scheme.Therefore, it will be necessary to know the macroscopic stress tensor on the in-tegration points of the FEs in the macro domain. To achieve the stress tensorin the FE domain it is required a constitutive model to describe the compos-ite behavior. The multiscale hom*ogenization approaches above described can bechosen as the constitutive models. Consequently, the macroscopic stress tensor isobtained through the analysis of the microscopic scale. The macroscopic defor-mation gradient tensor (F), for the first-order hom*ogenization, and its gradienttensor (G), for the enhanced-first-order hom*ogenization, are used to solve themicroscopic BVP. The microscopic solution is used to obtain the hom*ogenizedstress tensor required to solve the macroscopic BVP through the FEM.

The macrostructural point Xo considered in the formulation represents phys-ically the Gauss points on the FE. Therefore, the macroscopic values of thedeformation gradient tensor F (Xo) and the gradient of the deformation gradienttensor G (Xo), required by the microscopic BVP, are obtained in these Gausspoints when the macroscopic BVP is solved.

3.6.1 Microscopic numerical implementation

As has been done at the macroscopic scale, the BVP at the microscopic scale issolved using the FEM. Therefore, the unknown displacement field in the BVPis reduced to a finite degrees of freedom, which are the nodal displacements atthe nodal points of the FEs. At this scale the microscopic displacement fieldobtained from the solution of the RVE must satisfy the boundary conditionsdefined previously, in Section 3.5.2.

If the Taylor model condition is considered, there is no BVP to solve becausethe displacement field on the total microscopic domain is constrained by the con-dition. For the case of Linear boundary displacements condition, a conventionalBVP is obtained with a Dirichlet’s condition on the whole boundary domain.Finally, the cases of Periodic boundary fluctuations and Minimal constraint con-ditions defined a BVP with constraint boundary conditions.

The restrictions of degrees of freedom on the boundary domain can be ac-counted through several methods such as elimination of redundant unknowns,penalty methods and Lagrange multipliers [89]. The last two methods have thedisadvantage that they lead to ill-conditioned stiffness matrix or to an increaseof the number of degrees of freedom of the problem.

To avoid these two disadvantages, here it is proposed solving the RVE by anelimination of redundant unknowns [36]. The constraint condition for the Peri-odic boundary fluctuations is shown in Section 3.5.2, where it is possible to observethe redundant boundary unknowns of the BVP. In these boundary constraint ex-pressions it is possible identify master unknowns (the unknowns to solve) and

3.6. Finite element implementation 107

slave unknowns. In Appendix B.1 the master-slave kinematic relationships arederived for the Periodic boundary fluctuations condition.

Linear implementation

Following a conventional notation, the FE approximation of the variational prob-lem (3.90), for a given discretization h, consists in the determination of the un-known vector uµ ∈ Vh

Ωµ of the microscopic global nodal displacement such as∫Ωhµ

BT : Sµ(B · uµ,α) dV · w = 0 ∀ w ∈ VhΩµ , (3.93)

where Ωhµ denotes the discretized domain of the RVE, B is the global strain-displacement matrix (see Appendices B.3 and B.4), w is the global vector ofnodal displacement fluctuation, α is a set of internal variables and Vh

Ωµ is theset of finite-dimensional nodal displacement vectors associated with the FE dis-cretization h of the RVE domain Ωµ. If a linear case is considered, the microscopicstress tensor is

Sµ = Cµ : Eµ = Cµ : B · uµ, (3.94)

therefore, (3.93) can be written as[∫Ωhµ

BT : Cµ : B dV · uµ]· w = 0 ∀ w ∈ Vh

Ωµ , (3.95)

or[K · uµ] · w = 0 ∀ w ∈ Vh

Ωµ , (3.96)

where,

K ≡∫

Ωhµ

BT : Cµ : B dV. (3.97)

Here, K is the global stiffness matrix.Equation (3.96) does not include the boundary constraint on the unknown

nodal displacement field because of the Periodic boundary fluctuations conditionas it was explained in the preceding section. If this conditions are applied usingan elimination of redundant unknowns method, the following system of equationsis obtained

Kr · ur = RHS, (3.98)

where, Kr is the reduced global stiffness matrix, ur the reduced global nodaldisplacements vector and RHS is the Right-Hand Side global vector which is ob-tained as a result of the applied boundary restriction. Section B.2.1 of AppendixB.2 shows how the elimination of redundant unknowns is addressed to obtain theexpression given by (3.98).

108 Chapter 3. Multiscale hom*ogenization formulations

Non-linear implementation

The solution of the FE approximation of the microscopic variational problemgiven by (3.90) in the non-linear case is addressed by a Newton-Raphson iterativescheme. Therefore, the microscopic displacement field for a typical iteration (k)in the micro-scale is obtained according to the following update expression

u(k)µ = u(k−1)

µ + d(k)µ , (3.99)

where d(k) ∈ VΩµ is the unknown iterative displacement field. Considering thatthe microscopic stress tensor can be obtained as

S(k)µ = S(k−1)

µ (E(k−1)µ ,α(k−1)) + δS(k), (3.100)

and taking the following approximation

δS(k) ∼= T(k−1)µ : ∇sd(k)

µ , (3.101)

with

T(k−1)µ ≡ dSµ

dEµ

∣∣∣∣u(k−1), α(k−1)

(3.102)

denoting the microscopic tangent constitutive tensor. With the above at hand,and for the given discretization h, the FE approximation of the microscopic prob-lem consists in solving the unknown iterative (k) of the global nodal displacementvector d(k) ∈ Vh

Ωµ of the following equation[∫Ωhµ

BT : S(k−1)µ (B · u(k−1)

µ ,α(k−1)) dV

+

∫Ωhµ

BT : T(k−1)µ : B dV · d(k)

µ

]· w = 0 ∀ w ∈ Vh

Ωµ ,

(3.103)

or[F(k−1) + K(k−1) · dkµ] · w = 0 ∀ w ∈ Vh

Ωµ , (3.104)

where,

F(k−1) ≡∫

Ωhµ

BT : S(k−1)µ (B · u(k−1)

µ ,α(k−1)) dV, (3.105)

and

K(k−1) ≡∫

Ωhµ

BT : T(k−1)µ : B dV (3.106)

is the global tangent stiffness matrix of the RVE.Like in the linear case, (3.104) does not include the boundary constraint over

the RVE boundary because of the Periodic boundary fluctuations condition. If

3.7. Final remarks 109

this condition is applied using an elimination of redundant unknowns method, theexpression given by (3.104) can be rewritten as (see Section B.2.2 of AppendixB.2)

K(k−1)r · d(k)

r = RHS(k−1), (3.107)

where, K(k−1)r is the reduced global tangent stiffness matrix, d

(k)r the reduced

iteration (k) of the global nodal displacements vector and RHS(k−1) is the globalRHS vector of the previous iteration.

3.7 Final remarks

Based on the characteristics of the formulations developed, as well as on theimplementation of these formulations when using the finite element method, inthe following are included some final remarks regarding the implications of usingthe first-order or the enhanced-first-order models.

Let us assume, for the sake of simplicity, that the FE mesh of the macroscopicmodel has a single integration point. In this case, the macroscopic finite volumeΩM around the point Xo considered in the formulation is related with the FEdomain as Ωe = ΩM , where Ωe is the FE domain. Taking into account the con-siderations made in Section 3.4.5 it can be concluded that Ωe = Ωµ, which meansthat for the integration point of the FE, the RVE domain must be geometricallyequal to the FE domain. Consequently, for an enhanced-first-order approach theRVE dimension is related with the discretization mesh used in the macroscopicBVP.

The macroscopic BVP presented in Section 3.5.1 does not take into accountthe hom*ogenized second-order stress tensor Q obtained in the enhanced-first-order hom*ogenization. Besides, if the RVE materials are symmetrically dis-tributed in it, the estimated hom*ogenized stress tensor obtained is the samewith both hom*ogenization approaches. Therefore, the hom*ogenization methodconsidered does not improve the macroscopic results obtained for the BVP. Toimprove the macroscopic results, a high-order FE or enhanced FE mesh mustbe considered. To account for the hom*ogenized second-order stress tensor Qin the macroscopic scale a second-order macroscopic formulation must be used[59, 60, 31].

3.7.1 Linear FE in the macroscopic mesh

In linear finite element the interpolation functions are first-order polynomialsand consequently, the displacement field in the domain of the FE is a first-orderfunction. The strain tensor is obtained by differentiating the displacement field,and then the strain tensor in the FE will be a constant function. Therefore, a

110 Chapter 3. Multiscale hom*ogenization formulations

fine FE mesh on the macroscopic BVP should be used to obtain an accurateapproximation of the strains and stresses.

When linear finite elements and a first-order hom*ogenization are used to solvethe macroscopic problem, the RVE is just a representative sub-domain of the pe-riodic microstructure that does not have any significance on real microscopicdimension as has been shown in Section 3.3.5. The constant value of the macro-scopic gradient tensor F in the integration point of the macroscopic FE is usedto define the BVP in the RVE. From the solution of the microscopic problem themacroscopic stress tensor is obtained for the considered integration point.

In this case, the solution of the microscopic BVP with the enhanced-first-order hom*ogenization is an inefficient procedure because the value of G in theintegration point of the macroscopic linear element has partial or even zero in-formation.

3.7.2 High-order FE in the macroscopic mesh

To improve the FEM approach high-order elements can be used. Quadratic finiteelements use second-order polynomials as interpolation functions to approximatethe displacement field within the FE’s domain. The deformation gradient tensorF of this element is a first-order function, while gradient of the deformation gra-dient tensor G, which is obtained deriving twice the displacement, is a constantfunction on the FE domain.

The enhanced-first-order hom*ogenization approach needs at least quadraticelements in the macroscopic mesh in order for the functions F and G not to havezero value in the FE domain. Quadratic elements need more than one Gausspoint to obtain the best integration approximation. As mentioned before, theRVE dimension must be related with the FE dimension in this enriched first-order hom*ogenization. For quadratic elements, the RVE must represent the sub-domain within the FE associated to the Gauss point. Because the deformationgradient F is not a constant function on the Gauss point domain, the RVE shouldnot be modeled with a different periodic sub-domain of the microstructure.

In this case, the microscopic results depend of the dimension of the RVE.Therefore, to obtain the best possible approximation of the strain and stressfields using the enhanced-first-order approach, and high-order finite elements inthe macroscopic mesh, the RVE geometry should represent the real volume ofthe surrounding domain in the Gauss point. If not, the analysis will have anassociated error due to the size mismatch.

3.8. Numerical example 111

3.8 Numerical example

The objective of this section is to show the advantages and drawbacks of theenhanced-first-order computational hom*ogenization with respect to the first-order approach through numerical examples.

Two numerical example have been analyzed with the same macroscopic ge-ometry, the first one uses a hom*ogeneous material and the numerical solutionobtained for both procedures are compared with the existing analytical solution.The second numerical simulation uses a matrix with a long fiber reinforcement.

3.8.1 Geometry, support scheme and mesh information

Macroscopic Beam model

The chosen macroscopic structure is a three-dimensional fixed support beam thatis subjected to a fixed displacement (dZ) at the free end. Figure 3.6 shows thedimensions and the support scheme on the geometry of the beam.

Figure 3.6: Dimensions of the geometry and support scheme of the structure simulated.

To study the numerical stability and convergence of the problem four meshsizes are simulated. Linear element and quadratic element are used in the differentmeshes of the numerical model for the first-order hom*ogenization case while onlyquadratic FEs are used for the enhanced-first-order approach. The linear FEis an hexahedron of 8 nodes and 8 Gauss points and the quadratic FE is anhexahedron with 20 nodes and 27 Gauss points.

112 Chapter 3. Multiscale hom*ogenization formulations

MeshLinear elements Quadratic elements

Elements Nodes Gauss Elements Nodes GaussMacro1 8x1x2 54 128 8x1x2 165 432Macro2 16x2x4 255 1024 16x2x4 869 3456Macro3 32x4x8 1485 8192 32x4x8 5433 27648Macro4 64x8x16 9945 65536 64x8x16 37937 221184

Table 3.1: Number of elements (in X, Y and Z directions), nodes and Gauss points ofthe meshes used in the beam structure.

Table 3.1 shows the more relevant information about the macroscopic meshesused for the numerical simulation. These meshes are graphically show in Figure3.7.

Mesh Macro1 Mesh Macro2

Mesh Macro3 Mesh Macro4

Figure 3.7: Different mesh sizes used in the macroscopic numerical model.

Microscopic RVE model

The geometry design of the RVE depends of the numerical simulation case. Forthe case of a hom*ogeneous material, the RVE is a simple cube with length L,this is shown in Figure 3.8a. In the second simulation case, the material definedis a composite with a 40% of cylindrical long fiber volume. The geometry of theRVE that represents this periodical microstructure is shown in the Figure 3.8b.

3.8. Numerical example 113

(a) RVE for hom*ogeneous material. (b) RVE for composite material.

Figure 3.8: RVE models for the two different numerical simulation cases.

It has been shown in previous section that the dimension of the RVE is an im-portant parameter for the enhanced-first-order hom*ogenization approach. More-over, this dimension L is directly related with the volume around the Gauss pointof the FE in the macroscopic mesh. Therefore, the value of the length L dependsof the dimension of the FE used in the numerical macro-model and of the num-ber of Gauss points of the FE. In Table 3.2 the number of elements in the beamheight, Z direction, and the value that takes the length L in the RVE for thedifferent macroscopic meshes used are shown. The value of the length L has beencalculated considering quadratic element on the macroscopic mesh.

Data Macro1 Macro2 Macro3 Macro4Num. elem. 2 4 8 16Length L [mm] 1.3525 0.6762 0.3381 0.1691

Table 3.2: Number of elements in Z direction of the beam and length L of the RVE forthe different macroscopic mesh sizes.

The RVE has been analyzed with just one FE model. Figure 3.9a shows themesh used in the RVE for the hom*ogeneous material case, which has 1000 FE,and Figure 3.9b shows the mesh of the RVE for the composite material that has1936 FE.

114 Chapter 3. Multiscale hom*ogenization formulations

(a) Mesh for hom*ogeneous material. (b) Mesh for composite material.

Figure 3.9: Mesh used on the RVE models for the different numerical simulations.

3.8.2 Results and analysis

Checkpoints and variables compared in the simulation

The reaction force in Z direction on the fixed support is a variable used for thecomparison. To compare not only this macroscopic variable another checkpointhas been designed. The macroscopic stress value is compared in the macroscopicGauss point closest to point A and the microscopic stress obtained in the RVE isalso compared for this same Gauss point. The microscopic stress value used forthe comparison is the one obtained in the Gauss point closest to point A withinthe RVE. The geometric point A is shown in Figure 3.6 of the beam simulated.The longitudinal stress values (SXX) and the shear stress values (SXZ) will becompared on this point.

hom*ogeneous material simulation

When the material used in the beam is a hom*ogeneous material it is possible toobtain the analytical solution for the support scheme shown in the Figure 3.6.The reaction force in Z direction is given by

RZ =

[l3

3EIyy+

6l

5GA

]−1

dZ , (3.108)

where E and G is the Young’s modulus and shear modulus of the material,respectively, while Iyy, A and l is the second moment of area, the cross sectionarea and the longitudinal length of the beam, respectively. Therefore, consideringan isotropic material with null Poisson’s ratio and taking a fixed displacementof dZ = 1mm, it is possible to address the values shown in Table 3.3 from theanalytical solution. The SXX and SXZ values shown in table correspond to theanalytical values obtained in point A.

3.8. Numerical example 115

Data E [MPa] G [MPa] RZ [N] SXX [MPa] SXZ [MPa]Values 26560 13280 600 100 0

Table 3.3: Material properties, reaction force and longitudinal and shear stresses inpoint A of the analytical solution

The numerical results obtained for the different approaches and meshes arepresented in a simplified form using tables and graphs. On the tables, the relativeerror or absolute error obtained when comparing the result with the analyticalsolution is also shown.

Table 3.4 shows the reaction force in Z direction obtained with the numericalsimulations. In this table, the results obtained with Linear Elements (LE) inthe macro-model and the First-Order (FO) hom*ogenization approach are shownin the first two columns. The following two columns show the results obtainedwith Quadratic Elements (QE) and the FO approach. And, in the last twocolumns, are included the results obtained with QE for the macro model, andthe Enhanced-First-Order (EFO) approach.

RZ [N] LE&FO % erel QE&FO % erel QE&EFO % erelMacro1 679.09 13.18 600.43 0.07 600.43 0.07Macro2 620.03 3.34 600.12 0.02 600.12 0.02Macro3 605.09 0.85 600.09 0.01 600.09 0.01Macro4 601.34 0.22 600.08 0.01 600.08 0.01

Table 3.4: Reaction force and relative error for the different approaches and meshes.

It is possible to observe that the results do not change when a Enhanced-first-order approach is used. This is because the EFO do not improve the macroscopicsolution, the macroscopic stress field obtained is the same than the one obtainedwith a FO approach, and therefore the reaction forces are also the same (see Table3.5). Another interesting conclusion obtained from the results is that an increasein the order of the macro FE model represents a meaningful improvement. Themesh Macro1 with QE obtains best results than the mesh Macro4 with LE, whichis surprising because Macho4 has 16 FE in the beam height. Figure 3.10 showsthe curves of reaction force vs number of finite elements in the beam height. Thiscurve shows clearly the result previously addressed.

Table 3.5 shows the macroscopic longitudinal stress obtained from the nu-merical simulations and the relative error of these numerical results using theanalytical result SXX = 100 [MPa] as reference. The stress values shown in thetable correspond to the ones obtained for the Gauss point closest to point A onthe beam meshes. It has to be noted that for large meshes, the relative erroris slightly increased by the distance between the Gauss point and the point A

116 Chapter 3. Multiscale hom*ogenization formulations

600

610

620

630

640

650

660

670

680

2 4 6 8 10 12 14 16

Rea

ctio

n fo

rce

[N]

Number of elements on the beam height

Lin. Ele. & First-orderQua. Ele. & First-order

Qua. Ele. & Enhanced-first-order

Figure 3.10: Reaction force vs number of elements in Z direction for the different ap-proaches.

considered.

SXX [MPa] LE&FO % erel QE&FO % erel QE&EFO % erelMacro1 69.43 30.57 86.08 13.92 86.08 13.92Macro2 84.02 15.98 93.00 7.00 93.00 7.00Macro3 91.82 8.18 96.50 3.50 96.50 3.50Macro4 95.86 4.14 98.25 1.75 98.25 1.75

Table 3.5: Macroscopic longitudinal stress and relative error for the values obtained inthe Gauss point closest to point A.

The improvements of the EFO approach can be seen when comparing themicrostructure results provided by the RVEs used in the macroscopic Gauss pointclosest to point A. The longitudinal stress and shear stress present in the followingtables and figures are the microstructure stress values of the Gauss point in theRVE closest to point A. Table 3.6 shows the value of the longitudinal stressobtained within the RVE for the Gauss point closest to point A. This table showsthat the results provided by the EFO model are always closer to the analyticalones, as the model is capable of capturing the bending effects on the material.

Figure 3.11a shows the stress obtained as a function of the number of elementsin the beam height. The general behavior of the stress when the number ofelements increase is similar. When a EFO approach is used the estimation of thestress is good even for large elements. The relative error for 2 elements case isaround 1% which represents a very good estimation.

Figure 3.11b shows the longitudinal stress as a function of the approach used

3.8. Numerical example 117

SXX [MPa] LE&FO % erel QE&FO % erel QE&EFO % erelMacro1 69.43 30.57 86.08 13.92 98.66 1.34Macro2 84.02 15.98 93.00 7.00 99.59 0.41Macro3 91.82 8.18 96.50 3.50 99.87 0.14Macro4 95.86 4.14 98.25 1.75 99.96 0.04

Table 3.6: Longitudinal stress in the RVE (Gauss point closest to point A) and relativeerror for the different approaches and meshes.

in the numerical simulation. The improvement of the stress obtained when theapproach changes can be approximated as a linear function. If the number ofelements in the height increase the slope of the function decreases. Therefore,the benefit of change the approach is more significant for meshes with low numberof elements.

118 Chapter 3. Multiscale hom*ogenization formulations

65

70

75

80

85

90

95

100

105

2 4 6 8 10 12 14 16

Lon

gitu

dina

l str

ess

(~ p

oint

A)

[MP

a]

Number of elements on the beam height

Lin. Ele. & First-orderQua. Ele. & First-order

Qua. Ele. & Enhanced-first-order

(a) SXX vs number of elements on the beam height.

65

70

75

80

85

90

95

100

105

LE&FO QE&FO QE&EFO

Lon

gitu

dina

l str

ess

(~ p

oint

A)

[MP

a]

2 E.B.T.4 E.B.T.8 E.B.T.

16 E.B.T.

(b) SXX vs approach used in the simulation.

Figure 3.11: Longitudinal stress obtained close to point A for the different meshes andapproaches used.

3.8. Numerical example 119

Macroscopic SXX [MPa] RVE SXX [MPa]L

E&

FO

QE

&F

OQ

E&

EF

O

Figure 3.12: Macroscopic and Microscopic (RVE closest to point A) SXX field for themesh Macro3.

As an example of the macroscopic and microscopic longitudinal stress fieldobtained for the different approaches the Figure 3.12 shows SXX for the mesh caseMacro3. For the FO approach this figure shows an uniform stress distribution inthe RVE, this is because the formulation only uses the macroscopic deformationgradient to solve the RVE. This occurs independently of the macro elementsuses, LE or QE. The macroscopic improvement observed in the QE model isbecause the solution of the macroscopic problem is better when this kind ofelement is used. For the same macroscopic solution, if an EFO approach isused, the approximation of the microscopic stress field improves. The RVE stressfield shows a not uniform distribution (see QE&EFO case) because the EFOformulation can considered second-order effects in the microstructure.

The results obtained for the shear stress in the RVE for the Gauss point

120 Chapter 3. Multiscale hom*ogenization formulations

closest to point A are shown in Table 3.7. This table also shows the absoluteerror, here it is not used the relative error because for this variable the analyticalvalue obtained in the geometric point A is SXZ = 0 [MPa]. Figure 3.13a showsthe shear stress obtained for the different approaches as a function of the numberof elements in the height of the beam. From the table and the figure it is possibleto observe that the main improvement in the shear stress results it is presentedwhen the FE is changed. This phenomenon can be observed more clearly inFigure 3.13b which shows the shear stress as a function of the approach used.However, the use of the EFO approach improves the shear stress obtained for allmeshes considered. The reason for this improvements is, as has been pointed outwith the SXX value, the capacity that the EFO formulation gives to the RVEmodel to account for the second-order effects existing in the macro model.

SXZ [MPa] LE&FO eabs QE&FO eabs QE&EFO eabsMacro1 -22.14 22.14 -3.90 3.90 -1.88 1.88Macro2 -12.26 12.26 -1.68 1.68 -0.48 0.48Macro3 -6.43 6.43 -0.77 0.77 -0.13 0.13Macro4 -3.29 3.29 -0.37 0.37 -0.04 0.04

Table 3.7: Microscopic shear stress (Gauss Point closest to point A) and absolute errorfor the different approaches and meshes.

3.8. Numerical example 121

-25

-20

-15

-10

-5

2 4 6 8 10 12 14 16

She

ar s

tres

s (~

poi

nt A

) [M

Pa]

Number of elements on the beam height

Lin. Ele. & First-orderQua. Ele. & First-order

Qua. Ele. & Enhanced-first-order

(a) SXZ vs number of elements on the beam height.

-25

-20

-15

-10

-5

LE&FO QE&FO QE&EFO

Seh

ear

stre

ss (

~ po

int A

) [M

Pa]

2 E.B.T.4 E.B.T.8 E.B.T.

16 E.B.T.

(b) SXZ vs approach used in the simulation.

Figure 3.13: Shear stress obtained close to point A for the different meshes and ap-proaches used.

122 Chapter 3. Multiscale hom*ogenization formulations

Composite material

In the following numerical simulations, the material used is a composite withlong fibers. The RVE used to simulate the internal structure of this compositeis shown in Figure 3.8b, while the FE mesh used is shown in Figure 3.9b. Thematerial used for the matrix is an elastic isotropic material (resin epoxy HSCEpikote 4652) with a Young’s modulus of Em = 4.52 [GPa] and a Poisson’s ratioof vm = 0.36. The long fiber material considered is a carbon fiber (Grafil TR30S3K carbon fiber) with a Ef = 235 [GPa] and vf = 0.21. The materials propertieshave been taken from the work of Perez et al. [109].

Table 3.8 shows the Z direction reaction force obtained for the different ap-proaches and meshes used in the numerical simulation. Figure 3.14 shows thesesame reaction forces plotted against the number of elements on the beam height.This figure shows that the global performance provided in the different modelsfor the hom*ogeneous material is also provided for the composite material. Thisis: changing the hom*ogenization approach, FO or EFO, does not change the re-action force obtained and the use of QE represents a meaningful improvement ofthe results obtained.

RZ [N] LE&FO QE&FO QE&EFOMacro1 1629.76 1537.74 1537.96Macro2 1559.39 1530.45 1530.46Macro3 1537.43 1529.36 1529.34Macro4 1531.30 1529.10 1529.14

Table 3.8: Reaction force for the different approaches and meshes.

For the case of composite materials it is necessary make a difference betweenthe results obtained for the hom*ogenized composite ( macroscopic result), andthe results obtained for the composite components in the RVE solution.

In Table 3.9 are shown the longitudinal and shear stresses for the compos-ite obtained for the different approaches and meshes used on the beam. Themacroscopic stresses values shown in the table correspond to the ones obtainedin the Gauss point closest to point A. The composite stresses obtained for theQE&FO and QE&EFO cases are the same, as has been seen in Table 3.5. Figure3.15 shows the longitudinal stress and Figure 3.16 the shear stress obtained forthe composite. The curves show the stress obtained for the different approachesvs the number of element on the beam height. The composite stresses valuesobtained for QE&FO and QE&EFO are the same for all mesh analyzed becausethe EFO approach enriches the microscopic stress field but does not improve themacroscopic hom*ogenized stress.

3.8. Numerical example 123

1520

1540

1560

1580

1600

1620

1640

2 4 6 8 10 12 14 16

Rea

ctio

n fo

rce

[N]

Number of elements on the beam height

Lin. Ele. & First-orderQua. Ele. & First-order

Qua. Ele. & Enhanced-first-order

Figure 3.14: Reaction force vs number of elements in Z direction for the different ap-proaches.

DataComposite SXX [MPa] Composite SXZ [MPa]

LE&FO QE&FO QE&EFO LE&FO QE&FO QE&EFOMacro1 185.54 221.36 221.39 -25.19 -9.21 -9.21Macro2 218.02 238.29 238.29 -15.02 -4.17 -4.17Macro3 235.96 247.13 247.13 -8.17 -1.98 -1.98Macro4 245.71 251.59 251.59 -4.27 -0.96 -0.96

Table 3.9: Macroscopic longitudinal stress and shear stress for the different approachesand meshes.

124 Chapter 3. Multiscale hom*ogenization formulations

180

190

200

210

220

230

240

250

260

2 4 6 8 10 12 14 16

Com

posi

te lo

ng. s

tres

s (~

poi

nt A

) [M

Pa]

Number of elements on the beam height

Lin. Ele. & First-orderQua. Ele. & First-order

Qua. Ele. & Enhanced-first-order

Figure 3.15: Composite SXX obtained close to point A for the different meshes andapproaches used.

-25

-20

-15

-10

-5

2 4 6 8 10 12 14 16

Com

posi

te s

hear

str

ess

(~ p

oint

A)

[MP

a]

Number of elements on the beam height

Lin. Ele. & First-orderQua. Ele. & First-order

Qua. Ele. & Enhanced-first-order

Figure 3.16: Composite SXZ obtained close to point A for the different meshes andapproaches used.

3.8. Numerical example 125

The improvements on the microscopic results when an EFO approach is usedcan be seen in Figure 3.17. The microscopic stress fields shown in the figurescorrespond to the RVE of the macroscopic Gauss point closest to point A forthe bean mesh Macro3. Figures on the left side present the longitudinal stressdistribution obtained in the RVE for the fiber component while the right sideshows the results obtained for the matrix component. The longitudinal fiberstress distribution for the FO approach is almost uniform for both kind of FEconsidered. While in the case of EFO approach the fiber stress distributionin the RVE is more realistic considering the bending macroscopic state. Theclassical linear distribution in the longitudinal stress expected for a bending loadis achieved around the average value obtained for the FO approach. The stressdistribution for the matrix present a similar behavior to the fiber component. Itcan be observed that the maximum stress for the fiber and matrix within theRVE are obtained in the Gauss points closest to point A, which is an expectedresult.

To quantify the improvement on the microscopic solution due to the EFO ap-proach, Table 3.10 shows the maximum values of the longitudinal stress for thefiber and matrix components within the RVE. These stress values are graphicallyrepresented in Figure 3.18a for the fiber material and in Figure 3.18b for the ma-trix component. From the figures is clearly seen that the response of componentschange when the approach is changed.

DataFiber SXX [MPa] Matrix SXX [MPa]

LE&FO QE&FO QE&EFO LE&FO QE&FO QE&EFOMacro1 454.56 543.31 616.13 11.11 11.18 14.10Macro2 534.71 584.90 622.41 11.69 12.01 13.67Macro3 578.97 606.60 625.56 12.23 12.46 13.25Macro4 603.02 617.54 627.07 12.53 12.68 12.98

Table 3.10: Longitudinal stress of the components in the RVE for the different ap-proaches and meshes.

126 Chapter 3. Multiscale hom*ogenization formulations

Fiber SXX [MPa] Matrix SXX [MPa]

LE

&F

OQ

E&

FO

QE

&E

FO

Figure 3.17: Fiber and matrix longitudinal stress field in the RVE of the Gauss pointclosest to point A for the mesh Macro3.

3.8. Numerical example 127

440

460

480

500

520

540

560

580

600

620

640

2 4 6 8 10 12 14 16

Fib

er lo

ng. s

tres

s [M

Pa]

Number of elements on the beam height

Lin. Ele. & First-orderQua. Ele. & First-order

Qua. Ele. & Enhanced-first-order

(a) Fiber SXX obtained in the RVE vs number of elements on the beam height.

10

11

12

13

14

15

2 4 6 8 10 12 14 16

Mat

rix

long

. str

ess

[MP

a]

Number of elements on the beam height

Lin. Ele. & First-orderQua. Ele. & First-order

Qua. Ele. & Enhanced-first-order

(b) Matrix SXX obtained in the RVE vs number of elements on the beam height.

Figure 3.18: Longitudinal stress of the fiber and matrix components in the RVE closeto point A for the different meshes and approaches used.

Chapter 4

Numerical comparison withother formulations

In the following chapter a comparison of the results provided by the hom*oge-nization approach presented in the previous chapter with other formulations isshown. The other two formulations used in the comparison are: a Micro modeland the Serial-Parallel theory.

4.1 Introduction

The objective of these simulations is look into the strengths and weaknesses of themultiscale hom*ogenization framework developed in this study. And, to knownthe computational cost required by the proposed formulation in comparison withother approaches used to analyze composite materials. The other formulationsused for the comparison have been selected because they obtain the global be-havior of the composite from the analysis of its microstructure. In the following,a brief description of each one of these numerical approaches is presented.

Micro models

In these models, the constituent materials forming the composite are modeledexplicitly. Therefore, the response of the composite arises naturally. Each singlematerial is modeled with its own constitutive law. These models are very powerfulbecause they do not need to take any hypothesis on the microstructural behavior.However, as it will be shown, their biggest limitation is their computational costand in most cases their use is not practical.

129

130 Chapter 4. Numerical comparison

Serial-parallel mixing theory

The Serial-Parallel (SP) mixing theory could be defined as a phenomenologicalhom*ogenization because it is based on the classical mixing theory. This formu-lation has been proposed by Rastellini et al. [114] and it assumes as principalhypothesis that the components of the composite behave as parallel materials inthe fibers alignment direction and as serial materials in the orthogonal direction.The theory makes the composite behavior dependent on the constitutive laws ofthe component materials and of their morphological distribution inside the com-posite. The proposed composite constitutive model is based on the appropriatemanagement of the constitutive models of component phases within a contin-uum framework by making use of suitable “closure equations” that characterizethe composite micro-mechanics. For more details regarding the formulation, seeSection 2.2.6 in Part I.

The potential of the SP approach is that predicts accurately the response ofcomposites in the linear and non linear range (i.e. delamination failure) as hasbeen proved in several papers [75, 76, 77, 78, 79, 108, 109]. For this reason, thistheory has been selected for the comparison.

hom*ogenization approach selected

The first-order hom*ogenization approach is used in the simulation because theaim of the comparison is to know the advantages and drawbacks of the generalhom*ogenization framework developed. Furthermore, the simulation is not lookingfor precision results on the microstructure, and it is conducted in the linear range.

4.2 Geometry and numerical models

A clamped beam with a vertical load at mid-span is the structure used to comparethe theories described previously: micro model, serial-parallel mixing theory andfirst-order hom*ogenization approach. Figure 4.1 shows the beam’s geometry,supports and loads. A macro numerical model will be used to simulate thebehavior of the structure with the different theories. In the case of the first-orderhom*ogenization approach another micro numerical model is necessary. The micronumerical model will have the internal structure of the composite.

4.2.1 Macro and micro numerical models

The macro FE model used is the half of the beam because the symmetry of thestructure (see Figure 4.1). Figure 4.2 shows the macro numerical model with oneof the meshes used. The finite element used is a first order hexahedron element.In order to obtain the real behavior of the structure with the FE model it isnecessary to impose symmetric boundary conditions. The symmetry plane, the

4.2. Geometry and numerical models 131

Figure 4.1: Geometric of the beam studied.

right face of Figure 4.2, normal to X-axis, and the X displacement is set to zeroin this face. To simulate the fixed support, the nodes’ movements in the left crosssection are also restricted. The applied load is a Z direction fixed displacementof -0.1 mm in the rigth cross section nodes (symmetry plane).

As said before, to simulate the microstructure in the first-order hom*ogeniza-tion a micro numerical model is required. The RVE’s geometry chosen is a cubewith unit length sides. The finite element used is the same than the macro nu-merical model. The different RVE models that will be used are shown in theFigures. 4.3 and 4.6.

Figure 4.2: Beam numerical model

4.2.2 Simple materials and composite description

The simple materials in all studied cases are isotropic elastic materials. Thecomposite material is a laminate. And, the laminate consists of several layers ofmaterial 1, called lamina 1 henceforth, and a combination of layers of material 2(lamina 2) or material 3 (lamina 3). The volumetric participation of lamina 1 is

132 Chapter 4. Numerical comparison

always a 50%.

Table 4.1 contains the mechanical properties of all the materials considered inthe composite. In this table, E is the Young’s modulus, G is the Shear modulusand ν is the Poisson’s ratio. The “Color ref.” is the color used to represent thematerial in the RVEs, as it is shown in Figs. 4.3 and 4.6. The lamina 3 has thesame properties as lamina 2, with the only difference of the shear modulus, whichis reduced by 10. This is done to emulate the effect of a degraded lamina 2.

Simple mat. Color ref. E (GPa) G (GPa) νLamina 1 Black 210 80.76 0.3Lamina 2 Grey 3.5 1.46 0.2Lamina 3 White 3.5 0.146 0.2

Table 4.1: Mechanical properties of the simple materials

4.3 Comparison for several material configura-tions

In this section, several examples are presented to compare the behavior of thedifferent theories. The result used to compare them is the reaction force, in Zdirection, at the fixed support obtained for a fixed Z displacement applied at thesymmetry plane (See Figure 4.1) .

4.3.1 Undamaged case

The undamaged case is the first one used to compare all theories. In this casethe laminate contains 50% of lamina 1 and 50% of lamina 2, which properties aredefined in Table 4.1.

The model using the SP mixing theory defines the composite material assum-ing that the parallel behavior is obtained in X and Y direction, while the rest ofdirections have a serial behavior. The hom*ogenization formulation uses a RVEmade with 8 elements that also contains 50% of lamina 1 and 50% of lamina 2.The RVE is shown in Figure 4.3. Finally, the micro-model is defined discretizingthe different layers in which the laminate is divided.

A convergence analysis of SP and hom*ogenization theories has been made.The quantity of finite elements in the macro-model FE mesh has increased untilthe difference, between two consecutive results, is negligible. Figure 4.4 presentsthe results obtained for the different meshes analyzed. The micro-model usedto compare the results obtained with the different theories has been made with196608 hexahedron elements, which results in 648999 degrees of freedom.

4.3. Comparison for several material configurations 133

Figure 4.3: RVE used for the undamaged case.

The reaction force obtained with the SP theory is 905.9 N, with the first-order hom*ogenization is 908.3 N and with the micro-model is 919.0 N. It can beconcluded that the three theories provide almost the same result, as the differencebetween the reaction force value is lower than a 1% which is a really good result.Besides, all theories allow knowing not only the global performance of the beamanalyzed, but also the specific response of each lamina to the loads applied.

Figure 4.4: Convergence analysis results.

4.3.2 Global damage case

The objective for this example is to compare the responses obtained when one ofthe laminate materials suffers some sort of degradation. To analyze this problem,five different simulations have been performed in which the mechanical propertiesof one of the laminates is reduced. More specifically, the degradation is applied

134 Chapter 4. Numerical comparison

on the shear strength of lamina 2, which is reduced progressively until reachingthe value of lamina 3 (see Table 4.1). Therefore, the new laminate consists 50% oflamina 1 and 50% of a new lamina that can be completely undamaged (propertiesof lamina 2) or with properties corresponding to 12.5%, 25%, 50% and 100% ofdamage (this last case, corresponds to lamina 3). The specific mechanical valuesconsidered are shown in Table 4.2.

Property 12.5% 25% 50% 100%G (GPa) 1.295 1.131 0.803 0.146E (GPa) 3.5 3.5 3.5 3.5

Table 4.2: Mechanical properties of the degradated Lamina 2

The composite is simulated with the SP mixing theory and with the first-order hom*ogenization, using the same material characterization and RVE thatwere used in the undamaged case. Figure 4.5 shows the results obtained forthe conducted simulations. This figure shows that, as it is expected, the resultsobtained for both theories are again exactly the same. The results obtained alsoshow that as the shear stiffness of one of the layers is reduced, the global stiffnessof the beam decreases. This effect can be understood as a delamination failure,as has been previously shown by Martinez et al. [76, 78]. Results also show thatthe reduction of global stiffness of the beam is not linear with the reduction ofthe shear strength of one of the laminas, being larger as the layer stiffness getssmaller.

Figure 4.5: Reaction force obtained in the global damage case.

4.3. Comparison for several material configurations 135

4.3.3 Local damage case

In this example, the objective is to compare the responses obtained when materialdamage takes place, not in all layers, but just in some of them. The compositeconsidered has always 50% of lamina 1, and a 50% of lamina 2 (undamaged anddamaged). It is assumed, like in the previous case, that a totally damaged lamina2 is numerally represented by the lamina 3. The comparison is made for thecases in which there are 0%, 12.5%, 25%, 50% and 100% of layers damaged. Thesimulation corresponding to 0% and 100% damaged have been already conductedin two previous simulations. The simulations corresponding to intermediate caseshave been studied with the three methods being compared in this chapter: first-order hom*ogenization, SP and a micro-model.

For SP theory, the composite is obtained combining two different laminateswith the SP formulation. One laminate has 50% of lamina 1 and 50% of lamina2 and the other laminate has 50% of lamina 1 and 50% of lamina 3. The volumefraction of these laminates in the composite depend in the amount of layersassumed to be damaged. For the hom*ogenization approach, the amount of layersdamaged is represented with the RVE. Figure 4.6 shows the RVEs considered toaccount for 50%, 25% and 12.5% of damaged layers, respectively. In this figure,the darker elements correspond to lamina 1, the light-grey elements correspond tolamina 2 (undamaged) and the white elements correspond to lamina 3 (damaged).Finally, the micro-model has been simulated discretizing each one of the laminaof the beam.

a) b) c)

Figure 4.6: RVEs containing 50%, 25% and 12.5% of damaged layers.

Figure 4.7 presents the results obtained with different simulations performed.This figure shows that for the extreme cases, this is for 0% or 100% of lamina3, the results obtained with different theories are almost equal. However, theresults obtained when there are some layers damaged do not have the sameagreement, especially when comparing the results obtained with the SP theorywith the ones obtained with the hom*ogenization method or the micro-model.While the decrease in the resultant reaction force with the SP theory is equal tothe one obtained in the case of considering global damage (see Figure 4.5), thisdecrease is substantially larger when considering the hom*ogenization method or

136 Chapter 4. Numerical comparison

a micro-model. These two theories provide nearly the same results.

Figure 4.7: Reaction force obtained in the local damage case.

The explanation for the difference in the response obtained for the differentmodels is obtained from the models themselves. The serial-parallel theory obtainsthe response of the composite assuming certain iso-stress and iso-strain boundaryconditions that regularizes the response of the material if it is defined with severallaminates. Therefore, the response of the structure and the result obtained issimilar to the one obtained when this damage was present in the whole structure.On the other hand, with the hom*ogenization and the micro-model theories, thedamaged layer is discretized specifically and it is possible for the simulation tocapture the dislocation that takes place, as it is shown in Figure 4.8 for the threecases considered. This dislocation is the responsible for the drop of the stiffnessand the fast decrement on the value of the reaction obtained.

a) b) c)

Figure 4.8: RVEs with 50%, 25% and 12.5% of damaged layers under load.

4.4. Run times and memory used 137

4.3.4 Local damage case in a localized region of the beam

At the light of previous results one may think that the SP theory is not capable ofrepresenting delamination processes. In this example it is shown that under somecirc*mstances this simulation is possible. Here, the beam has been simulated withtwo different laminates. The central band contains 50% of lamina 1 and 50% oflamina 3 (damaged); while the rest of the beam is simulated with 50% of lamina1 and 50% of lamina 2. Figure 4.9 shows the FE mesh of the beam macro-model.This example is simulated with the SP and the hom*ogenization theory. Thehom*ogenization theory uses the RVE shown in Figure 4.3.

Figure 4.9: FE mesh of the macro-model of the beam with two laminates.

In this case, the reaction force obtained is exactly the same for both, theSP and the hom*ogenization models: 663.9 N and 666.4 N, respectively. Thisexample shows that the SP theory is capable of providing the same results as thehom*ogenization theory when the response of the RVE fulfills the parameters inwhich is based the SP theory: iso-strain and iso-stress behavior. On the otherhand, if the RVE does not fulfill this behavior (i.e. there is a dislocation in it),the SP theory is not capable of predicting accurately the material response, as itwas shown in previous example.

4.4 Run times and memory used

One of the main drawbacks that has a hom*ogenization approach nowadays is itscomputational cost. Therefore, in order to know the performance of this formu-lation it is necessary not only to compare the results obtained with it, but alsoto compare the computational cost in terms of time and memory requirements.

To do these comparisons, in the case of the computational hom*ogenization,two different strategies have been considered. In the first one, the mechanical

138 Chapter 4. Numerical comparison

properties of the composite (stiffness matrix) are calculated at the beginning ofthe analysis, and these properties are used afterwards during the complete ofthe simulation. This case is named H-OneRVE. The other case corresponds toanalyze the RVE each time that it is necessary to know the stress provided bythe RVE for a given strain value. This case is named H-AllRVE. If the problem islinear, the results obtained in both cases are the same. However, in a non linearcase, it is necessary to simulate the problem with an H-AllRVE strategy in orderto capture properly the non linear response of the material.

Item Micro Model H-OneRVE H-AllRVE SP TheoryReal Time [Min:Seg] 6:46 0:01 2:27 0:02CPU Time [Min:Seg] 8:44 0:03 9:31 0:17Memory [Mbytes] 2690,00 7,45 7,45 15,82Reaction Force [N] 236,09 224,69 224,69 576,68

Table 4.3: Times and memory used to 50% located damage case.

Table 4.3 shows the computational times and memory required to conduct thesimulations with a localized damage of a 50%. The real time and cpu time arediscriminated because part of the FE code used is in parallel. The results showthat the CPU time in the Micro model and H-AllRVE are comparable. But, theCPU time of the H-OneRVE and SP theory are significantly better. Therefore,in terms of computational time, it is feasible to conduct a simulation with ahom*ogenization approach, as well as with the SP theory. However, this simulationmust be kept in the linear range. If the simulation is non linear, the H-AllRVEstrategy must be used, which makes the SP theory the only feasible option interms of computational time unless some non-linear strategy is developed tominimize the number of times in which the RVE has to be solved.

The main difference between the micro-model and the hom*ogenized modelis found in the memory requirements. While the computational time of the H-AllRVE and the micro-model are equivalent, the amount of memory required bythis last one is substantially larger (360 times larger). This difference is found be-cause the memory used is proportional to the FE mesh size of the numerical modeland, while the micro-model has to solve a problem with a very small discretiza-tion, the hom*ogenization framework only requires memory for the macro-problemand the RVE that is being solved. This difference makes unbearable solving largeproblems with micro-models and makes feasible using hom*ogenization methods,even if the problem is in non linear range.

Chapter 5

Non-linear extensionproposed for multiscalemethods

Most of the work on FE2 multiscale procedures are done on analyzing the nu-merical performance of RVE [137, 113] or on connecting different scales [32]. Ingeneral, in this kind of hom*ogenization methods the elastic properties of the mi-crostructure are obtained solving the microstructural problem at the beginning ofthe structure problem. The main drawback of hom*ogenization methods is theircomputational cost for a non-linear analysis because it is required solving theRVE in every integration point at the macrostructural problem, and for everytime step, in order to know the non-linear limit and then the behavior of themicrostructure in non-linear range. Non-linear performance has also the problemthat the dissipated energy of both scales is not always related [5].

In order to improve the computational cost of the multiscale hom*ogenizationsome strategies use model-order reduction techniques [142, 93, 45]. These meth-ods use the Proper Orthogonal Decomposition (POD) to obtain the reduced setof empirical shape functions. Besides, [45] proved that the common approachof replacing the non-affine term by an interpolant constructed taking only PODmodes arrives to ill-posed formulations. An enriched approximation space withthe span of the gradient of the empirical shape functions is proposed to avoidthis ill-posedness. However, these kind of procedures do not solve the completestructure.

Because of this, in the following to overcome this major drawback, this chapterpresents a new procedure to reduce computational cost of multiscale simulations.The chapter looks also into the problem of localization and energy dissipationacross the scales, as the proposed method must be consistent [1]. In the following

139

140 Chapter 5. Non-linear extension

the formulation and algorithm schemes of the proposed procedure are described.

5.1 Introduction

The main advantage of the FE2 method compared to a micro model is the re-duced computer memory requirements. To solve the same problem, the amountof memory required by the classical FE micro model method is substantiallylarger than FE2 procedure [107]. This difference is found because the memoryused is proportional to the FE mesh size and, while the FE micro model has tosolve a problem with a very small discretization, the FE2 procedure only requiresmemory for the macrostructural problem and the RVE that is being solved. How-ever, if the material reaches non-linear behavior, the computational cost of FE2

method becomes as large as the one required by the micro model case, as theRVE has to be solved for each integration point when a real structure is solved.Because of this, a new Non-Linear Strategy (NLS) is proposed in this work.

Continuum mechanics establishes the limit between linear and non-linear per-formance of materials using a comparison criterion that compares a given combi-nation of stresses with a threshold value (Von-Mises, Mohr-Coulomb, etc.). Thisapproach cannot be used in an hom*ogenization double scale solution directly asdifferent strain-stress states may lead to different failure modes of the composite.

A possible solution is to analyze the RVE at every time step. However, as hasbeen already stated, this is extremely expensive and very ineffective, because thestructural non-linear behavior often occurs in a small part of its domain.

The non-linear strategy developed consists in the definition of a comparisoncriterion, based on the macrostructural deformation of the Gauss point, but takesinto account that the RVE failure takes place in a small part of the domain. Inthe following section it is explained in detail. It has to be noted that the pro-cedure proposed has been developed for a first-order hom*ogenization approach.Further study will be required if it wants to be extended to the enhanced-first-order hom*ogenization approach, as in this case it should account not only for themacrostructural Gauss point deformation, but also for its gradient.

5.2 General concepts of the proposed approach

Here is proposed to develop a comparison function that looks a maximum level ofan elastic energy density that can be applied to the RVE before its failure. This isdone with the definition of an activation function for each single integration pointat the structural scale. It is important to remark that the proposed approachdoes not use a model reduction strategy, instead it is solving the actual structure,but only when it is strictly necessary.

The NLS is composed of two different procedures, a non-linear activation

5.2. General concepts of the proposed approach 141

function and a smart first step calculation. In the following are defined both ofthem.

5.2.1 Non-linear activation function

The definition of a Non-Linear Activation Function (NLAF) is based on the factthat any given material begins its non-linear performance when a single particleof the material reaches its stress threshold. The objective of the NLAF is toknow whether any material point of the RVE has reached its non-linear limitusing hom*ogenized variables.

To do so, it defines a function f that relates the elastic energy density (Ψe)of an integration point of the RVE with the maximum elastic energy (Ψe

Limit)that can be applied to this material point, before reaching the non-linear range.Therefore, f is defined as

f =Ψe

ΨeLimit

, 0 < f < 1. (5.1)

In other words, f provides a value of how far is a material point in the microstruc-ture to reach the non-linear state.

In order to know how far is the whole RVE to reach the non-linear perfor-mance, it is necessary to use the information obtained for all the integrationpoints of the RVE and transform it into a single representative number. This isdone with the assumption stated before that the failure of the macrostructure willstart when the first integration point of the microstructure fails. Therefore, thef parameter of the RVE corresponds to the maximum f value of all integrationpoints of the RVE. Then

maxf1, ..., fn = f , (5.2)

where the overline at the variables refers to the structure scale or hom*ogenizedvariables. Finally, the limit elastic energy density at the macrostructure scale isobtained with the following expression

ΨeLimit =

Ψe1

f, (5.3)

where Ψe1 is the elastic energy density for the strain state used to calculate f .

The process described can be schematized as it is shown in Figure 5.1.The NLAF is defined as

Ψe − ΨeLimit ≤ 0, (5.4)

where Ψe is the elastic energy density of the macro structural integration point,which is calculated in each load step of the simulation.

142 Chapter 5. Non-linear extension

ΨeLimit =

Ψe1

f

f =Ψe

ΨeLimit

RVE

ε1

Figure 5.1: Non-linear activation function scheme.

The NLAF previously defined is only valid for the strain state used to calculateΨeLimit (see (5.3)). If the strain state varies, it may also change the non-linear

failure mode and, therefore, the limit elastic energy density calculated may be nolonger valid. Therefore, (5.4) is valid while the strain state in the material remainsproportional to the one used to obtain Ψe

Limit. To quantify this proportionalitythe next expression is proposed

ε1 : εi‖ ε1 ‖‖ εi ‖

= 1 ⇐⇒ Comparable States, (5.5)

where the subscript i refers to the current i-nth deformation state and ‖ . ‖ isthe norm’s mathematical symbol. In case this proportionality is lost, it will berequired to calculate again the new limit elastic energy density of the RVE. Thisis summarized in the following flow diagram (see Figure 5.2).

It can be easily seen that with the proposed procedure the RVE must be solvedfor each macro integration point on the first time step, in order to calculate theelastic energy density limit using (5.3). Afterwards, it only will be necessaryto solve the RVE again if the strain state of the integration point becomes nonproportional to the calculated originally or if the NLAF is not satisfied, whichmeans that the RVE becomes non-linear. Therefore, if only few elements of thestructure reach the non-linear state, only these elements will have to be solvedin the non-linear analysis.

5.2.2 Smart first step

As said before, at the beginning of the analysis it is required to solve the RVE forevery single integration point of the macrostructure to obtain its Ψe

Limit. Thiscalculation process can be extremely expensive in terms of computational cost.

In order to reduce this computational cost, it is proposed a Smart First Step(SFS) strategy. This strategy consist in solving the RVE only if the deformation

5.2. General concepts of the proposed approach 143

Ψei − Ψe

Limit ≤ 0

σep = C : εiΨe

i = 12 σep : εi

Solve Micro

no

ΨeLimit = Ψe

i

σi ; CT

ivi

εimpivi−1

Exit

εref.:εi

‖εref.‖‖εi‖ ≈ 1yes

Solve Micro

no

ΨeLimit =

Ψei

f

f

σi = σe.p.

ivi = ivi−1

mp = mat. propertiesiv = internal variablesep = elastic predictori = umpteenth step

yes

Figure 5.2: Non-linear strategy algorithm scheme.

applied to it is different to all other deformation states considered previously.Therefore before calculating the Ψe

Limit of the RVE, the SFS procedure comparesthe deformation between the current and the all previous integration points al-ready calculated (see (5.5)). If the SFS finds one comparable strain state, thecurrent RVE takes the values of the RVE already solved. If none of the previousmicrostructures solved have a comparable state, the actual RVE is calculated.Figure 5.3 shows the scheme of the described algorithm.

It will be shown, in the validation examples, that this procedure reducessignificantly the computational cost of the first step load in the simulation.

5.2.3 Numerical hom*ogenized tangent constitutive tensor

The proposed NLS has been implemented in PLCd [111], a parallel finite elementcode that works with 3D solid geometries (see Appendix B.5). In the code, aNewton-Raphson scheme is adopted to solve the non-linear problem. To facilitatethe convergence of the whole problem, the tangent constitutive tensor at theintegration point is necessary to obtain the global tangent stiffness matrix.

144 Chapter 5. Non-linear extension

csk = εk:εi

‖εk‖‖εi‖ csk>cscs = cskkp = kyes

nok=k+1

εi ; Ψei

k=1 ; cs=0

cs ≈ 1

cs ; kp

Solve Micro

no

ΨeLimit,i =

Ψei

f

f

Ψetk = Ψe

Limit,i

εtk = εi

tk = tk+1

Exit

yes

ΨeLimit,i = Ψe

kp

loop from k=1 to k=tk

cs = comp. statestk = total k pointskp = k pointeri = th gauss point

Figure 5.3: Smart first step algorithm scheme.

A perturbation method is used to obtain a numerical approximation of thehom*ogenized tangent constitutive tensor of the RVE in the integration point. Themethod implemented is analogous to the one proposed by Martinez et al. in [77](see Section B.5.2 in Appendix B.5). Being the only difference that in currentprocedure the perturbations must be applied on the RVE instead of applyingthem to a constitutive equation. The n small perturbations (δεj) are appliedto the hom*ogenized or structural strain vector. The RVE is solved n times andas result gives the n stress vector δjσ. Therefore, the j columns of the tangentconstitutive tensor for the RVE can be obtained as

Ctj ≡

δjσ

δεj. (5.6)

The calculation of the tangent stiffness tensor is necessary to obtain a good

5.3. Energy dissipation in a multiscale analysis 145

convergence of the problem but it is computationally expensive. This showsagain the necessity to reduce the number of times in which this calculation isperformed, and proves the necessity of having a non-linear strategy to conductthe simulation.

5.3 Energy dissipation in a multiscale analysis

The solution of material non-linear problems with a numerical double scale ho-mogenization procedure not only should be affordable computationally, but theresults obtained in the non-linear process must be also correct. Therefore, theprocedure must dissipate the same energy in both scales. In order to conservethe dissipated energy through the scales, the following methodology is proposed.

5.3.1 Fracture energy

Fracture mechanics presents the fracture energy per unit of area, Gf , as a prop-erty of the material. This energy can be calculated as

Gf =Wf

Af, (5.7)

where, Wf is the energy dissipated by the fracture at the end of a quasi-staticprocess, and Af is the total fractured area. This fracture energy is the linkbetween the fracture mechanics and the constitutive model based on classicalsolid mechanics. The constitutive model must satisfy:

The good representation of behavior of a set of points inside of a finite domain.

The same energy dissipated by the total volume as the one dissipated by thesolid in the real fracture process.

Considering a simple tensile test, the constitutive model must verify the fol-lowing condition of dissipation

Wf = GfAf︸ ︷︷ ︸Fracture Mechanics

≡ Πd =

∫v

gfdV︸ ︷︷ ︸Solid Mechanics

(5.8)

where gf is the maximum specific energy dissipated by the constitutive model.Equation (5.8) states that the energy delivered to the tensile test must be equalto the energy dissipated by the constitutive model. In solid mechanics, the dissi-pation phenomena is located in a volume that can be represented as Vd = AfLf ,where Lf is a fracture length. For FE method the localization phenomena inone strip of finite elements is sought, therefore Lf is commonly approximated by

146 Chapter 5. Non-linear extension

some reference length of the finite element. This length is a parameter that ac-counts for the amount of energy dissipated by the fractured material. Replacingthe volume Vd in (5.8) the following expression is obtained

Wf = GfAf = gfAfLf . (5.9)

From (5.9) the relation between the material parameter Gf and the specific en-ergy dissipated gf is found

gf =GfLf

. (5.10)

5.3.2 Localization at the microstructural scale

In multiscale procedure, the specific energy at the macro structural scale is ob-tained as

gf =1

Vm

∫Ωm

gmf dVm, (5.11)

where the index m is used to reference the microstructural scale variables.

Figure 5.4: Representative volume of the subscale.

Taking the same consideration than macroscale solid behavior, now the dis-sipation phenomena is located at microstructural level (see Figure 5.4). In suchcase, we have the following dissipated equation

gf =1

Amf lgmf A

mf lf , (5.12)

where, Amf is the RVE cross section area, l is the length in normal direction ofAmf and lf is the fracture length at the microstructure (RVE). With (5.12) ispossible to obtain the specific energy gmf dissipated at the microstructural scalelevel as

gmf =gf l

lf=

Gf l

Lf lf. (5.13)

5.3. Energy dissipation in a multiscale analysis 147

Equation (5.13) shows the relationship between the gmf and Gf which ensuresto dissipate the same energy by the solid mechanics, using a multiscale method,than the one obtains with a tensile test. The validity of this relation is proved inthe following example.

5.3.3 Validation example

A simple tensile numerical test over a material sample is simulated. The objectiveof this example is to analyze the objectivity of the response obtained using theproposed FE2 method. The same test using a classical one scale FE methodis also solved for comparison purposes. The geometry, the supports and thedisplacements scheme of the simulated structure is presented in Figure 5.5. Theapplied fixed displacement is represented by the arrows in the figure.

Figure 5.5: Structure simulated in the tensile test.

Material

The simple material used in the tensile test takes the properties shown in Table5.1. The constitutive model chosen is an explicit scalar damage model withexponential softening [123, 97]. For this particular case, where the stress state isuniform and there is only one simple material, and in order to help the localizationof the softening problem, the elastic limit is increased in some elements (drawnwith gray color in Figure 5.6) up to a value of σLimit = 102MPa.

Properties E [GPa] ν [−] σLimit [MPa] Gf[kJ/m2

]Values 100 0 100 20

Table 5.1: Simple material properties used in the tensile test.

FE meshes

The finite element used to solve the problem is a first order hexahedron element.The example is solved for different combinations of finite element meshes. Figure5.6 shows the different mesh sizes used in the simulation. The left side of Figure5.6 shows the three different meshes used for the structural scale. The meshMacro1 has 10 finite elements, the mesh Macro2 has 84 elements and finally the

148 Chapter 5. Non-linear extension

mesh Macro3 has 656 elements. On the other hand, the right side of Figure 5.6shows the two different meshes used for the microstructural scale. The meshMicro1 has 125 finite elements and the mesh Micro2 has 729 elements.

Figure 5.6: Different meshes used in the tensile test.

Results

The results obtained with the different mesh combinations are shown graphicallyin the Figure 5.7. As can be observed from the figure, the results are equal forall combinations, and for both methods.

For the case considered, it is possible to validate the numerical results withanalytical calculations, knowing the area of the specimen, the Young’s modulusand the maximum tensile stress that can be applied, the maximum load anddisplacement in the beam is

Fmax = σLimit.A = 10kPa,dFmax = L.σLimit

E = 0.1mm.

It is also possible to calculate analytically the dissipated energy at the end of thetest as

Wf = Gf .A = 2J.

5.3. Energy dissipation in a multiscale analysis 149

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5

Tra

ctio

n F

orce

[kN

]

Longitudinal Displacement [mm]

Macro1Ma1-Mi1Ma1-Mi2

Macro2Ma2-Mi1Ma2-Mi2

Macro3Ma3-Mi1Ma3-Mi2

Figure 5.7: Traction force vs displacement curves obtained in the tensile test.

If this energy is calculated from the numerical models, the following table isobtained:

Energy [J] Macro1 Macro2 Macro3One scale 1.728 1.737 1.748Micro1 1.752 1.741 1.777Micro2 1.713 1.761 1.812

Table 5.2: Dissipated energies obtained in the tensile test.

The difference between the estimated value and the ones show in Table 5.2 isbecause at the numerical analysis the simulation has been stopped at 0.5mm. Ithas to be noted also that the dissipation obtained with all mesh configurations ispractically the same. Which proves the consistency of the formulation proposed.

The localization of non-linear phenomena in one strip of finite elements at thestructural scale is shown in the Figure 5.8, for the analysis made with Macro2mesh. This figure shows that damage is concentrated in the central zone of thematerial sample, and therefore, the displacement too. In the proposed multiscalemethod the localization phenomena must be observed also at the microstructuralscale. As an example, Figure 5.9 shows the microstructural displacement anddamage obtained at the end of one tensile numerical test. In the figure can beobserved that both results are localized in one strip of finite elements in the RVEmeshes.

150 Chapter 5. Non-linear extension

Displacement Damage

Figure 5.8: Macrostructural results obtained at the end of the tensile test.

Micro 1 Micro 2

Dis

pla

cem

ent

Dam

age

Figure 5.9: Microstructural results obtained at the end of the tensile test.

5.4. Numerical examples of non-linear analyses 151

5.4 Numerical examples of non-linear analyses

5.4.1 Tensile test of a plate with a hole.

The objective of this example is to show the performance of the NLS developed,as well as to analyze the failure of the structure localizes in a strip of elements.The test is a tensile test made on a plate with a hole in its center. Due tothe symmetry of the geometry and of the load applied, only a quart of the realstructure is simulated. Figure 5.10 shows the modeled geometry, the supports andthe displacements scheme in the numerical model. The applied fixed displacementis represented by the arrows in Figure 5.10.

Figure 5.10: Simulated structure of the plate with a hole.

Material

Table 5.3 shows the properties of the simple material used. The constitutivemodel of the material is the same (explicit scalar damage) that has been used inthe previous validation example 5.3.3.

Properties E [GPa] ν [−] σLimit [MPa] Gf[kJ/m2

]Values 100 0.15 100 10

Table 5.3: Simple material properties used in the plate with a hole.

FE meshes

To analyze the response’s objectivity in the test, two finite element meshes havebeen used for the macroscopic model. Figure 5.11 shows the mesh sizes employed.

152 Chapter 5. Non-linear extension

Mesh1 has 360 finite elements while Mesh2 is more dense and has 2880 elements.The microstructural model and finite element meshes are the same than the onesused in previous validation example 5.3.3.

Mesh1 Mesh2

Figure 5.11: Different meshes used in the plate with a hole.

Results

Figure 5.12 shows the traction force vs displacement curves obtained for thedifferent mesh combinations. This figure shows that the results are almost equal.Therefore, the result obtained with the proposal method is mesh independent.The curves show than the maximum force does not pass of 80 KPa and it isobtained for an applied displacement of 0.08mm.

10

20

30

40

50

60

70

80

90

0 0.02 0.04 0.06 0.08 0.1 0.12

For

ce [

kN]

Displacement [mm]

Mesh1-Mi1Mesh1-Mi2Mesh2-Mi1Mesh2-Mi2

Figure 5.12: Forve vs Displacement for the plate with a hole.

The dissipated energy for the different mesh configurations used in this ex-

5.4. Numerical examples of non-linear analyses 153

ample is shown in Table 5.4. From the Table 5.4 can be observed than the worstdifference between two results is less than 2%.

Energy [J] Mesh1 Mesh2Micro1 3.152 3.135Micro2 3.192 3.169

Table 5.4: Dissipated energy in the plate with a hole.

To fully understand the behavior of the structure under the applied loadseveral figures for different load state are presented. Figure 5.13 and Figure5.14 show the results obtained for Mesh1 and Mesh2, respectively. In the figures,strain and stress in Y direction and scalar damage are presented for four differentfixed displacement steps.

The figures show how at the beginning of the test (label a) d=0.05mm), themaximum strain and stress are located at the inner border of the hole. Then,the non-linear process starts there and, as a consequence, the damage increasesin that zone. Due to constitutive model used, when the damage increases in thematerial the stress decreases. As the applied displacement continues increasing,the structure transfers the load to non damaged zones. Therefore, the zone withmaximum stress moves from the inner border to the central part and the strainand damage move on as a constitutive response. At the end of the test (label d)d=0.11mm in the figures), the maximum stress is located in the right externalborder of the plate. It is important to mention that during the test, the stressin the structure never takes values over the limit imposed (see Table 5.3) as canbe observed in Figures 5.13 and 5.14. Finally, the figures show how the model iscapable of localizing all damage in a single strip of finite elements.

154 Chapter 5. Non-linear extension

Y strain direction Y stress direction Damage

a)d=

0.05

mm

b)

d=

0.07

mm

c)d=

0.0

9m

md)

d=

0.1

1mm

Figure 5.13: Results obtained in the plate with a hole to Mesh1.

5.4. Numerical examples of non-linear analyses 155

Y strain direction Y stress direction Damage

a)

d=

0.0

5mm

b)

d=

0.07

mm

c)d=

0.0

9m

md)

d=

0.1

1mm

Figure 5.14: Results obtained in the plate with a hole to Mesh2.

156 Chapter 5. Non-linear extension

Computational times

To show the advantage of using the NLS proposed, in comparison with a fullclassical FE2, the calculation times are presented herein. A full FE2 solves themicrostructural problem for every integration point of the structure, and forevery time step. This procedure does not distinguish between linear range andnon-linear range.

Tables 5.5 and 5.6 present the calculation times required to solve the shown ex-ample in the same desktop computer, an Intel R© CoreTM i7-2600 CPU @ 3.40GHzwith 8GB of RAM. The tables show the times used by the FE2 and by theFE2 with the NLS incorporated. The speed ratio column has the relation timesbetween both methodologies.

Table 5.5 has the total real times necessary to complete the numerical test,up to d = 0.115mm, for all mesh configurations. The speed ratio variable showsthat the advantage of using the developed strategies increases specially when thesize of the macrostructure’s mesh increases. This is an expected result because,in larger meshes, the proportion between linear and non-linear elements becomesalso larger.

Model FE2 FE2+NLS Speed ratioMesh1-Micro1 1:21:53 0:28:19 2.89Mesh1-Micro2 8:41:19 3:10:44 2.73Mesh2-Micro1 11:19:49 2:29:28 4.55Mesh2-Micro2 76:40:33 18:39:33 4.11

Table 5.5: Computation times requested to solve the plate with a hole [hs:min:seg].

On the other hand, it is important to mention that when a RVE becomesnon-linear, its computational cost is more expensive than when it is linear. Thisis because, besides the possible iteration required by the RVE to obtain thecorrect non-linear solution, the estimation of the tangent constitutive tensor byperturbation method requires to solve the RVE six more times (see Subsection5.2.3).

Consequently, when the number of non-linear elements in a problem increase,the efficiency of the proposed method decreases. For the analyzed example,if the simulation is stopped at the maximum admissible force in the structure(around d=0.08mm in Figure 5.12) which probably the most interesting valuefor an engineer, the speed ratio would be better. To prove this, let’s considerthe Mesh2-Micro2 simulation. In this case, when the maximum load is applied(d=0.08mm) there are only 392 elements in non-linear range, instead of 576elements for d=0.115mm, and therefore at this load step the speed ratio is of 7instead of 4.11.

This simulation is also used to validate the effect of the Smart First Step

5.4. Numerical examples of non-linear analyses 157

procedure. To do so, Table 5.6 shows the computational times consumed for thefirst step in each one of the simulations conducted. The times shown prove thatusing SFS strategy improves highly the computational efficiency also for smallmesh sizes, as speed ratio variable shows. The table also shows that the numberof RVE solved by the SFS is independent of the mesh used in the microstructuralproblem.

Modelwithout SFS with SFS Speed

ratioTime RVE solved Time RVE solvedMesh1-Micro1 0:17.9 2880 0:01.0 151 17.9Mesh1-Micro2 1:48.3 2880 0:06.5 151 16.7Mesh2-Micro1 2:12.0 23040 0:02.6 303 50.8Mesh2-Micro2 14:05.9 23040 0:12.5 303 67.7

Table 5.6: First step computation times in the plate with a hole [min:seg].

5.4.2 Industrial Component

In order to validate the efficiency obtained with the NLS when it is applied to thesolution of a real structure, in the following is included the non-linear simulationof an structural component. In this case, the structure selected for analysis is theindustrial component shown in Figure 5.15. The geometry of this engine stiffenerhas been proposed in the framework of M-RECT Project. The stiffener is linkedon one side to the gearbox, and on the other side to the engine. This componenthas the objective of improving the connection between engine and gearbox, aswell as changing the dynamic properties of the overall structure.

Figure 5.15: Engine stiffener part.

158 Chapter 5. Non-linear extension

Materials

The material that will be used for the stiffener, different from the one usedin M-RECT, is a laminated thermoplastic composite. Therefore, the materialproperties vary through the laminate thickness and respect to the laminate’sreference direction. The composite is made with three orthotropic sheets (seeFigure 5.15). The two external sheets (drawn in blue) have a thickness of 1.5mmeach one and, the core sheet (drawn in gray) has a thickness of 5.5mm.

The external laminae is composed by carbon fibers in an epoxy matrix. Theperiodic microstructure of the external sheets can be represented by the RVEshown in the Figure 5.16. The laminate has a 40% of cylindrical long fibervolume.

a) Geometry of the microstructure (RVE)

b) Mesh used in the RVE

Figure 5.16: Geometry and mesh of the RVE used in the external sheets.

The matrix is an isotropic material, simulated with an explicit scalar damageconstitutive model with exponential softening (resin epoxy HSC Epikote 4652).The long fiber is modeled with an elastic constitutive model (Grafil TR30S 3Kcarbon fiber). The properties of these simple materials are shown in Table 5.7[109].

Finally, the FE mesh employed to analyze the RVE is shown in Figure 5.16.The mesh uses 1464 first order hexahedra finite elements.

On the other hand, the core sheet of the engine stiffener is a TenCate com-mercial product, Cetex TC1200 PEEK 5HS LAMINATE. The properties of thismaterial have been obtained from TenCate website [127] and are shown in Table

5.4. Numerical examples of non-linear analyses 159

Material E [GPa] ν [−] σLimit [MPa] Gf[J/m2

]Epoxy matrix 4.52 0.36 68 780Carbon fiber 235 0.21 4410 -

Table 5.7: Simple material properties from Perez et al. (2013).

5.8. For the simulation, the core material is modeled using an elastic constitutivemodel.

Propertie [GPa] E(0o) E(90o) G(In plane)TenCate lamina 56.1 55.6 4.5

Table 5.8: TenCate Lamina properties.

Mesh and boundary conditions

Figure 5.17 label a) shows the mesh used to simulate the engine stiffener. Themesh has 355.302 first order tetrahedra finite elements. The external laminaerequires 108.041 elements while the core lamina has 247.261 elements.

The nodes that will be restricted and the laminate’s reference direction areshown in Figure 5.17 label b). The nodes with green color are over the face incontact with the gearbox. These nodes have a zero movement restriction in alldirections. On the other hand, the nodes drawn in yellow are on the face incontact with the engine. In this case, the restriction on these nodes is a fixeddisplacement in X direction. The laminate’s reference direction is the long fiberlongitudinal direction in the external laminae.

Results

The analysis conducted on the first step to evaluate the elastic energy densityavailable in each integration point allows defining a “possible damage” map of thestructure, as it is shown in Figure 5.18 label a), where f is presented. The bluezones in the external sheets have an f near to zero, which means that in thesezones the relation between the elastic energy density required by the load, Ψe,and the available elastic energy density in the material under that load, Ψe

Limit,is very small. On the other hand, regions with f values near to one can be definedas critical zones, as in these Ψe is very close to Ψe

Limit. Therefore, these zonesare the regions where non-linear process has more possibilities to start. In thisexample, the SFS has required analyzing 6.514 RVEs to determine the thresholdfunctions of the whole structure. This quantity represents only the 6% of theelements on the external laminae.

160 Chapter 5. Non-linear extension

a) Mesh used in the engine stiffener

b) Nodes restricted and laminate’s reference direction

Figure 5.17: Mesh and boundary conditions used in the engine stiffener.

The numerical simulation has been stopped for a fixed X displacement of1.36mm. The hom*ogenized stress at the end of the analysis in the laminate’sreference direction is shown in Figure 5.18 label b). From the figure it is observedthat the maximum absolute stress is a compressive stress and it is located nearto the face in contact with the gearbox. The maximum tensile stress is locatedin the same region but in the opposite external sheet.

Figure 5.18 label c) shows the scalar hom*ogenized damage at the end of thetest. The damaged area shown in figure has a relation with the previous resultspresented. It is on the maximum compressive stress zone (see label b)) and it isa critical zone in Figure 5.18 label a).

To understand the internal structure behavior in the damaged zone it is nec-essary to observe the mechanical performance of the most damaged RVE. Incurrent simulation the RVE selected is the one with the maximum hom*ogenizeddamage, in the engine stiffener mesh (see Figure 5.18 label c)). Figure 5.19 shows

5.4. Numerical examples of non-linear analyses 161

a) Parameter f at the first step

b) hom*ogenized stress on the reference direction

c) Scalar hom*ogenized damage at external sheet

Figure 5.18: Results obtained in the engine stiffener.

the results obtained for this RVE. This shows the stress in the RVE’s local X axisat the beginning (label a) at first step) and at the end (label b) at last step) ofthe test. The shear stress in XY and YZ direction is also shown in the figure forthe first step of the analysis. Finally, the matrix scalar damage variable in theRVE is shown for the last step.

From Figure 5.19 can be observed that X compression stress is the dominantstate in the RVE but, its failure is produced by shear in the matrix material. Inorder to account for the high strength threshold of the carbon fiber (see Table5.7), and to reduce computational cost, this material has been simulated withan elastic constitutive model. Figure 5.19 shows that this assumption is correct,as the maximum fiber stress reached in the analysis (label b) at last step) is faraway of its strength threshold as it was expected.

162 Chapter 5. Non-linear extension

The Figure 5.19 shows that the external sheet has interlaminar delaminationin the damaged zone (see Figure 5.18 label c)). It is important to stand out thatalthough the damage in the matrix is located in a small zone its global effect ismeaningful. Figure 5.18 label c) shows that some elements have lost about 75%of its original load capacity.

a) at first step b) at last step

Xst

ress

dir

ecti

on

a) XY direction b) YZ direction

Fir

stst

epsh

ear

stre

ss

a) Whole RVE b) Only matrix

Last

step

dam

age

Figure 5.19: Results obtained in the RVE with maximum hom*ogenized damage.

This example has shown that it is possible to solve real problems with a non-linear hom*ogenization scheme. However, to see the advantages of the proposedNLS procedure, it is necessary to analyze the computational times required by

5.4. Numerical examples of non-linear analyses 163

Methods FE2 FE2+NLS Speed ratioTime [hs:min] 782:46 11:36 67.4

Table 5.9: Computation times requested by the simulation.

the simulation, these are shown in Table 5.9. The FE2 computational time hasbeen evaluated based on the time required to solve one RVE and the number ofsteps and iterations required by the simulation. Table 5.9 shows that a FE2 hasa computational cost that makes unfeasible these sort of simulations. In currentcase, the simulation requires more than 32 days and 14 hours to be completed.The proposed method has a really good computational time, less than 12 hours,and it is capable of speeding up the process at a speed ratio of 67.4. This impliesa reduction of more than a 98% of the computational time required to conductthe calculation. In other words, it makes feasible a numerical analysis that waspreviously unfeasible.

Chapter 6

Numerical hom*ogenization.Concluding remarks

The multiscale hom*ogenization framework proposed in Part II of the disserta-tion, which is described in detail in Chapter 3, has been proved a competitivealternative to model three-dimensional composite structures. An extension ofthe first-order hom*ogenization approach developed has been proposed to con-sider high-order effects in the microstructure. The called Enhanced-first-orderhom*ogenization presented in Section 3.4 preserves a classical first-order BVP atthe microstructure scale but also at the macrostructure scale. Therefore, withthis new approach bending effects can be seen in the microscopic scale withoutincrease the complexity of the macroscopic formulation and its computationalimplementation. However, there is no improvements in the macrostructure solu-tion considering the one obtained with the first-order hom*ogenization approach,as has been proven in the numerical example present in Section 3.8. The elim-ination of redundant unknowns implemented, for both cases of hom*ogenizationapproaches developed, to solve the microscopic BVP through a cubic RVE hasproven to be an efficient option to satisfy the boundary conditions.

For linear analysis, the comparison presented in the Chapter 4 proves thatthe multiscale hom*ogenization has many advantages over other theories, such asmicro models or the SP theory, as it is capable of capturing complex responses ofthe material (such as dislocations) with an affordable computational cost. Thehom*ogenization approach can represent accurately effects such as a local damagedlamina because the internal structure of the composite is physically modeled inthe RVE. The SP theory cannot account automatically for such effects, unlessthey are present in the whole finite element, as has been shown in the examplespresented in Sections 4.3.2 and 4.3.4. However, the main advantage of the SPtheory is that it is capable of conducting non-linear analysis without increasing

165

166 Chapter 6. Concluding remarks

substantially the computational cost of the simulation.The comparison of the computational cost presented in Section 4.4 by the

different formulations has shown that, in terms of computational time, the costof the first-order hom*ogenization for elastic simulations that solve the RVE onlyat the beginning to characterize the material, and the SP theory is comparable.Besides, this computational cost is substantially lower than the one requiredby a micro model or for non-linear simulations conducted with a classical FE2

hom*ogenization framework. However, the main difference between the full FE2

hom*ogenization approach and the micro models is found in the memory requiredby the simulation, being the cost of this last one 360 times larger.

A proposed extension to non-linear range of the FE2 computational hom*og-enization is proposed and presented in Chapter 5. The developed formulationuses a non-linear activation function defined in the structural scale that is ob-tained by solving the microstructure scale. The activation function predicts ifa material point (or integration point) in the structure is in linear or non-linearrange. Therefore, the approach proposed only analyzes the non-linear integra-tion points by solving the microscopic BVP using the RVE. Section 5.2 shows theformulation developed to obtain the non-linear strategy proposed in this study.Besides, a smart first step had to be also developed to obtain in an efficient waythe activation function.

The purpose of the non-linear procedure developed is to solve non-linear prob-lems, and the first requirement to meet is conserve the dissipated energy throughthe scales. Section 5.3 describes in detail how the fracture length concept appliedto one scale continuum mechanics is extended to multiscale hom*ogenization ap-proach. It is shown that the non-linear formulation presented is energy consistentand mesh independence at the macroscopic and microscopic scale.

The main objective of the proposed non-linear strategy is to reduce signifi-cantly the computational times requested by a multi-scale approach. The platewith a hole example presented in Section 5.4.1 shows how the computationaltimes are reduced around four times. Besides, the mesh independence and en-ergy consistency of the proposed methodology is proved again. The example alsoshows how the localization phenomena in the structural scale, in this case theplate with a hole, appears naturally from the microstructure scale. Finally, anengine stiffener has been solved in Section 5.4.2 to prove the large computationaladvantage of the proposed procedure when a real industrial component is simu-lated. The computational time is less than 12 hours comparing to 32 days andalmost 15 hours required by a full FE2 approach. In addition, the method pre-dicted the failure zone naturally and the mode failure of the composite’s internalstructure.

Conclusions

In this final chapter the main contributions of this study are presented in termsof achievements and in terms of concluding remarks. Future improvements andfuture lines of work based on the expertise gained and on the main difficultiesfound while developing this research are suggested at the end.

Achievements

The main aim of the present work was to develop a comprehensive constitu-tive formulation for the analysis of three-dimensional composite structures inlinear and non-linear range. In this context, a phenomenological hom*ogenizationmethod for composites using carbon nanotubes as reinforcement was developedin the first stage of this study. Then, with the objective to reach a broader inter-nal structures composite a multiscale hom*ogenization procedure was proposed.A first-order hom*ogenization approach and an extension of this, to account high-order effects in the microstructure, called enhanced-first-order hom*ogenizationwere developed. An efficient and consistent methodology was developed to ad-dress non-linear FE2 hom*ogenization analysis for realistic composite structures.

In Part I, the composite constitutive model proposed, based on the mixingtheory, for reinforced composites with carbon nanotubes has been presented.The developed formulation relates the reinforcement and the matrix componentsthrough an interface material. Therefore, it is possible to consider non-linearphenomenons by using non-linear constitutive laws to characterize the interfacematerial.

It has been shown that the elastic properties estimated with the proposedformulation are in good agreement with experimental data obtained from theliterature. The validation of the composite non-linear response, provided by thepresented constitutive model, has been performed using the experimental data oftwo different composites made with MWCNTs randomly distributed.

The developed formulation has been used to predict and compare the mechan-ical properties of a straight beam subjected to four-point bending with differentmaterial configurations. In addition, a non-linear analysis has also been made

167

168 Conclusions

using the same structure and composites showing the capacity of the proposedmethodology to obtain the complete response of the composite structures behav-ior.

A visco-elastic analysis of the beam structure used previously has been con-ducted for a sinusoidal load. For this simulation, a visco-elastic constitutive modelwas defined in the matrix and interface components. The results of the analysisproved the improvement in the capacity of energy dissipation for the compositewith MWCNTs. Finally, the effect of the nanotubes angle in the mechanicalproperties of the composite obtained was studied. It has been proven that thenanotubes distribution has an significant influence in the composite achieved.

In Part II, the proposed FE2 hom*ogenization framework and its extension tonon-linear range have been presented. A first-order hom*ogenization approach andan enhanced-first-order hom*ogenization, which is able to consider second-ordereffects of the macrostructure in the microstructure, have been considered. Theenhanced-first-order hom*ogenization approach has proven to be able to obtain amore realistic microstructural solution as compared with the first-order approachwhen a bending macrostructural state is considered.

The developed formulation has shown to be a competitive alternative tomodel three-dimensional composite structures. For linear analysis, the presentedmethodology has proved to have many advantages over other theories, as it iscapable of capturing complex responses of the material with an affordable com-putational cost. However, for non-linear analysis of structures the computationaltime of the computational hom*ogenization approach is extremely expensive, al-though the procedure shows a significant reduction on the memory requirements,when compared with a micro model.

To outcome these problems, a novel extension for FE2 hom*ogenization ap-proach to non-linear range is developed. The proposed formulation uses a non-linear activation function in the macrostructural scale and its task is predictif a material point on the structure is in linear or non-linear range. Becauseof this, the developed strategy only solves the microstructural scale, through aBVP in the RVE, for the macroscopic non-linear integration points found. Inaddition, a smart first step had to be also developed to obtain in an efficientway this activation function. The fracture length concept has been extended tomultiscale hom*ogenization framework so it has allowed that the developed ap-proach is energy consistent and mesh independence at the macrostructural andmicrostructural scale.

The example of the plate with a hole shows how the localization phenomenonon the structural appears naturally from the microstructural scale. The pre-sented examples also show the significantly reduction of the computational timesof the developed non-linear strategy. For the engine stiffener example, the com-putational time is less than 12 hours comparing to 32 days and almost 15 hoursrequired by a classical FE2 approach. In addition, in the examples the method

Conclusions 169

predicted the failure zone naturally and the mode failure of the internal structure.

Concluding remarks

As result of this work a multiscale composite constitutive model to analyze com-plex materials has been achieved. With the models developed is possible toconduct linear and non-linear simulations of structural components. Multiscaleanalysis can be phenomenological or computational and depending of the needsit may be decided for one option or another, or even a mix of them.

In the case of the phenomenological model, the constitutive formulation de-veloped has been focused in reinforced composite using CNTs, and is based onthe classical mixing theory. The presented model has proven that is possible toconsider CNTs reinforcements in composites through numerical simulations witha low computational cost. In addition, this phenomenological hom*ogenizationapproach can be considered as a constitutive equation manager as the theoryof mixtures is. Therefore, the developed methodology could be easily extendedto other composites which using reinforcements with similar behavior such asnanofibers or short fibers.

With the developed model the non-linear performance of the composite isprovided by each constitutive model and the load transfer capacity of the interfaceregion is also affected if the interface is damaged. The formulation has includedthis effect affecting the parallel length with the damage level of the interface.Therefore, if the interface is fully damaged the contribution of the CNTs to thecomposite is null.

However, as every phenomenological method, the developed constitutivemodel requires some parameters to be calibrated such as the geometric param-eters and mechanical properties of the interface component. These, togetherwith the intrinsic problems of the mixing theory, are the main drawbacks of theproposed procedure.

In the case of the computational hom*ogenization, the presented FE2 frame-work has proven to be able to simulate structural components in non-linear rangewith an affordable computational cost. It has been also shown that when themacroscopic second-order term is considered in the formulation more realisticmicrostructural solutions are achieved.

The decision of following a first-order hom*ogenization approach and then im-prove it to obtain a enhanced-first-order formulation over using a full second-ordercomputational approximation was based on the fact that the computational im-plementation of the high-order formulation reached at the macrostructural levelgenerally involve elements that use larger number of nodes, degrees of freedomand boundary conditions.

The solution of the microstructural problem has been addressed using a Peri-odic boundary fluctuations condition. It has been shown that the other boundary

170 Conclusions

conditions overestimate or underestimate the characterization of the microstruc-ture. However, the periodic boundary conditions have proven to obtain a goodestimate of the hom*ogenized microstructural stiffness. In non-linear range, thiscondition also allows an strain localization band within the RVE without spuriouseffects.

The softening problem and the strain localization band in the FE2 approachwere addressed in both scales through the fracture length concept. The conven-tional method used in FEM to approximate the fracture length by some referencelength of the finite element was extended to FE2 hom*ogenization. Therefore,the specific energy at the microstructural scale is related to the fracture energythrough both reference lengths now. The developed procedure has shown to be agood and simple computationally way to conserve the energy through the scalesand ensures the objectivity of the FE2 response.

The hom*ogenization approaches developed have been complemented with aNon Linear Strategy, developed to decide whether the RVE has to be solvedor not in a non-linear problem. The strategy has been applied to the solutionof a real engineering structure proving that with it is possible to reduce thecomputational cost of a non-linear simulation by a 98%, when compared with aFE2 hom*ogenization procedure.

Finally, as concluding remark, it can be said that the different formulationsand numerical procedures presented in this work, together with the simulationsconducted to prove their validity, have shown that it is possible to accomplishmultiscale analyses of engineering composites taking into account their non-linearbehavior. This is an important achievement that contributes to make of thismodels, in a near future, the new simulation standard for composite materials.

Future work

In the following is described some of the further work that can be derived fromcurrent research:

- In general, the carbon nanotubes tend to agglomerate, also they have the un-dulation and misalignment problems during the manufacture of the composite.The effect of these issues on the final properties and response of the compositesome time is meaningful. Therefore, the introduction of some corrector param-eters to control these phenomenons would be a significant improvement for thephenomenological hom*ogenization proposed.

- General extension of the phenomenological hom*ogenization developed in Part Ito consider any kind of short reinforcement and thereby extend its application toshort fiber reinforced composites and to the new generation of concrete materialsreinforced with steel short fiber.

Conclusions 171

- Use the developed hom*ogenization framework, specially the non-linear strategytogether to the enhanced-first-order hom*ogenization approach, to solve engi-neering problems in which the non-linear behavior of complex composites playan important role.

- Implementation of new constitutive laws for the simulation of simple materialssuch as plasticity in the microstructural scale. Therefore, other failure phenom-ena may be studied in the macrostructural scale.

- Extension of the multiscale hom*ogenization implemented to different elementtheories for the simulation of the structure or macroscopic scale, such as shelland plates formulations.

- In many composites can be observed two or more reinforcement levels. Forexample, a composite of long fiber reinforced matrix where the matrix alsois reinforced with nanotubes. It is possible to think in a multiscale/multi-method approach to analyze this kind of materials. The FE2 hom*ogenizationwill simulate an RVE of long fibers embedded in a matrix and, this last material,will be simulated with the phenomenological hom*ogenization proposed.

- Improve the efficiency of the parallelization process in order to get a fully opti-mized approach. In the current parallel implementation the calculation processon the macrostructural scale is subdivided in an efficient dynamic way usingthe OpenMP philosophy. The calculation process on the microstructural scaleis addressed in a serial process. A new parallel implementation using a MessagePassing Interface (MPI) method in the macrostructure and then a OpenMPmethod in the microstructure will reduce the computational times of the simu-lations.

Appendices

173

Appendix A

Constitutive models

The formulation developed is Section 4.2 require that all composite componentsmust fulfill the expression given by (4.1). Therefore, it is possible to use anyconstitutive law to describe the mechanical performance of the different compo-nents. In this appendix are defined the models that are used in this dissertationto characterize the different composite component defined.

A.1 Elastic constitutive model

The specific Helmholtz free energy for this material can be written as

Ψ (ε, θ) = Ψe (ε) + Ψt (θ) =1

2ε : C : ε+ Ψt (θ) . (A.1)

And the local form of the Clausius-Duhem inequality given by (A.7) can beexpressed in this case as

σ : ε− ηθ −[∂Ψe

∂ε: ε+

∂Ψt

∂θθ

]− 1

θq.∂θ

∂x≥ 0 (A.2)

or (σ − ∂Ψe

∂ε

): ε−

(η +

∂Ψt

∂θ

)θ − 1

θq.∂θ

∂x≥ 0. (A.3)

Therefore, to ensure compliance with the second thermodynamic law

σ.=∂Ψe

∂εη.= −∂Ψt

∂θ. (A.4)

175

176 Appendix A. Constitutive models

A.2 Elasto-plastic constitutive model

For this material case, the specific Helmholtz free energy, considering uncoupledelasticity is

Ψ (εe, p, θ) = Ψe (εe) + Ψp (p) + Ψt (θ) =1

2εe : C : εe + Ψp (p) + Ψt (θ) , (A.5)

where Ψe is the specific elastic free energy, Ψp is the specific plastic free energy, Ψt

is the specific temperature free energy, p is a internal variable tensor associatedwith plastic behavior. The total deformation of the material tensor is split intoits elastic, εe and plastic, εp parts. This is

ε = εe + εp (A.6)

The local form of the Clausius-Duhem inequality for this material can beexpressed as

Ξ = σ : ε− ηθ − Ψ− 1

θq.∂θ

∂x≥ 0, (A.7)

and using the above expression it can rewritten as

σ : (εe + εp)− ηθ −[∂Ψe

∂εe: εe +

∂Ψp

∂p.p+

∂Ψt

∂θθ

]− 1

θq.∂θ

∂x≥ 0 (A.8)

or (σ − ∂Ψe

∂εe

): εe −

(η +

∂Ψt

∂θ

)θ + σ : εp − ∂Ψp

∂p.p− 1

θq.∂θ

∂x≥ 0. (A.9)

being σ the stress tensor, η the entropy, and q the vector field of heat flow. Toensure compliance with the second thermodynamic law it must be defined

σ.=∂Ψe

∂εeη.= −∂Ψt

∂θP

.= −∂Ψp

∂p(A.10)

where P is the thermodynamic tensor associated with the internal variable tensorp. Finally, the mechanical dissipation for a material point is

Ξm = Ξp = σ : εp + P.p ≥ 0. (A.11)

A.3 Elasto-damage constitutive model

In this case, the expression of the Helmholtz free energy is

Ψ (ε, d, θ) = Ψe (ε, d) + Ψt (θ) = (1− d) Ψeo (ε) + Ψt (θ)

= (1− d)1

2ε : C : ε+ Ψt (θ),

(A.12)

A.4. Visco-elastic constitutive model 177

where d is a internal variable associated with the damage. The local form of theClausius-Duhem inequality given by (A.7) for this material can be expressed as

σ : ε− ηθ −[∂Ψe

∂ε: ε+

∂Ψe

∂dd+

∂Ψt

∂θθ

]− 1

θq.∂θ

∂x≥ 0 (A.13)

or (σ − ∂Ψe

∂ε

): ε−

(η +

∂Ψt

∂θ

)θ − ∂Ψe

∂dd− 1

θq.∂θ

∂x≥ 0. (A.14)

To ensure compliance with the second thermodynamic law it must be defined

σ.=∂Ψe

∂εη.= −∂Ψt

∂θD

.= −∂Ψe

∂d, (A.15)

being D the thermodynamic scalar associated with the internal scalar variable d.And, the mechanical dissipation for a material point is

Ξm = Ξd = D.d ≥ 0. (A.16)

A.4 Visco-elastic constitutive model

The present visco-elastic constitutive model is a generalized Maxwell model [99],which is an alternative general form to summarize the Kelvin and the Maxwellsimplified model in a single formulation. Therefore, this model tends to the basicKelvin model when C1 →∞ or it transforms into the basic Maxwell model whenC∞ → 0 (see Figure A.1). Then, this formulation is useful and suitable for therepresentation of different types of viscous behavior in solids.

Figure A.1: Scheme of the generalized visco-elastic Maxwell model [99].

178 Appendix A. Constitutive models

The stress state at any time can be expressed asσ∞

(t) = C∞ε(t)

σi(t) = C1(ε(t)− εi(t)) = ξ1εi(t).

(A.17)

Considering the equilibrium condition the following relation can be written

σ(t) = σi(t) + σ∞

(t) = C1(ε(t)− εi(t)) + C∞ε(t) = ξ1εi(t) + C∞ε(t), (A.18)

starting that C0 = C∞ + C1 and operation algebraically in (A.18), the stressequation is obtained as

σ(t) = C0ε(t) + C1εi(t). (A.19)

Taking into account the second expression in (A.17), the differential equation forthe inelastic strain is obtained as

C1ε(t) = C1εi(t) + ξ1ε

i(t) ⇒ ε(t)

r1=εi(t)

r1+ εi(t) (A.20)

where, the time delay is defined by r1 = ξ1/C1. Applying a strain ε(t) since atime t ≥ τ0, the solution for εi(t) obtained from previous differential equation is

εi(t) = 0 ∀ t < τ0

εi(t) =

∫ t

−∞

1

r1e−(t−s)/r1ε(s) ds ∀ t ≥ τ0,

(A.21)

and using this solution into (A.19), the following expression for the stress isobtained

σ(t) = 0 ∀ t < τ0

σ(t) = C0ε(t)−C1

r1

∫ t

−∞

e−(t−s)/r1ε(s) ds ∀ t ≥ τ0.(A.22)

Defining now the uniaxial relaxation function G(t), as the inverse of the uniaxialcreep function

G(t) = [J(t)]−1

= C∞ + C1e−t/r1 , (A.23)

and, taking into account the inversion of the relaxation function for this particularmodel, the uniaxial creep function is

J(t) =1

C∞

[1− C1

C0e−(C∞r1C0

)t]. (A.24)

The integration of the (A.22) can be made by parts, and therefore the stress canbe written as the following compact form

σ(t) = 0 ∀ t < τ0

σ(t) =

∫ t

−∞

G(t− s)dε(s)ds

ds ∀ t ≥ τ0.(A.25)

A.4. Visco-elastic constitutive model 179

Numerical solution of the integral

To obtain the solution of the visco-elastic problem the convolution integral in(A.25) must be solved. Considering that the function J(t) or G(t) comply withthe semi-group property, and therefore the functions inside the integral meet thecondition

f(a+ b) = f(a)f(b), (A.26)

the convolution integral can be avoided by carrying out the following time inte-gral,

I(t) =

∫ t

−∞

f(t− s)g(s) ds = I(t−∆t) +

∫ t

t−∆t

f(t− s)g(s) ds. (A.27)

The integral of the previous time I(t − ∆t) is used in each time step and theintegration is carried out only at the current time integral ∆t. The exponen-tial function found in the model, based on spring-damping analogies, lead toexponential relaxation function that comply with the semi-group property.

Multiaxial extension of the visco-elastic model

The multiaxial extension for the expression of the stress given by (A.22) is writtenat t+ ∆t through the following approximation

σij(t+ ∆t) = C0ijkl

[εkl(t+ ∆t)− C1

C0r1

∫ t+∆t

−∞

e−(t+∆t−s)/r1εkl(s) ds

], (A.28)

where C0ijkl is the multiaxial extension of C0 and it can be also written as

C0ijkl = C

∞

ijkl + C1ijkl, the C

∞

ijkl is the classic elastic constitutive tensor for thematerial without viscous effect. The integral solution can be carried out withouta convolution, it is then rewritten as

σij(t+ ∆t) = C0ijklεkl(t+ ∆t)− C0

ijkl

C1

C0r1

∫ t

−∞

e−(t−s)/r1εkl(s) ds

−C0ijkl

C1

C0r1

∫ t+∆t

t

e−(t+∆t−s)/r1εkl(s) ds,

(A.29)

integrating the third term of the right hand side using the trapezoidal rule andreordering the expressions obtained, the stress can be written as

σij(t+ ∆t) = C0ijklεkl(t+ ∆t)

[1− C1

C0r1

∆t

2

]+ σij(t)e

−(∆t)/r1

−C0ijklεkl(t)e

−(∆t)/r1

[1 +

C1

C0r1

∆t

2

].

(A.30)

Appendix B

Computationalimplementation

B.1 Microscopic Kinematic relationships

Taking a structured FE mesh on the boundary of the RVE shown in Figure B.1, itis possible identify easily master and slave nodes. The right part of figure showsthe chosen master nodes (named with a letter) and the slave nodes (named witha letter and number). The quantity of periodic nodes depends on the FE mesh.Table B.1 shows master nodes and the slave nodes located on the edges andsurfaces of the RVE shown in Figure B.1. Moreover, Table B.2 shows the eightperiodic vertices nodes in the RVE (see left part of Figure B.1). In these verticesnodes, also it is possible to identify a master node and seven slave nodes. Inthe following, the vertex node “1” will be the master vertex node and the othersseven vertices nodes (“2” ,“3” , ... and “8”) are the slaves nodes.

Master nodes a b c d e fSlave nodes a1, a2, a3 b1, b2, b3 c1, c2, c3 d1 e1 f1

Table B.1: Master and slave periodic nodes denomination.

Nodes 1 2 3 4 5 6 7 8

Table B.2: Periodic vertices nodes in the RVE.

Considering a generic master node “a” on the axis X and its slave nodes“a1”, “a2” and “a3” the position vector difference between their in the reference

181

182 Appendix B. Computational implementation

Figure B.1: Master and slaves nodes in a general hexagonal RVE.

configuration is

Xa1 −Xa =

Xµ

D2/2−D3/2

−

Xµ

−D2/2−D3/2

=

0D2

= D2N+Y

Xa2−Xa =

Xµ

D2/2D3/2

−

Xµ

−D2/2−D3/2

=

0D2

D3

= D2N+Y +D3N

+Z

Xa3−Xa =

Xµ

−D2/2D3/2

−

Xµ

−D2/2−D3/2

=

00D3

= D3N+Z

(B.1)

Following the same procedure is possible to obtain the position vector differ-ence for the other master and slave nodes as

Xb1 −Xb = D1N+X , Xb2 −Xb = D1N

+X +D3N

+Z , Xb3 −Xb = D3N

+Z ,

Xc1 −Xc = D1N+X , Xc2 −Xc = D1N

+X +D2N

+Y , Xc3 −Xc = D2N

+Y ,

Xd1 −Xd = D3N+Z , Xe1 −Xe = D1N

+X , Xf1 −Xf = D2N

+Y .

And, for the vertices nodes are

X2 −X1 = D1N+X , X3 −X1 = D1N

+X +D2N

+Y ,

X4 −X1 = D2N+Y , X5 −X1 = D3N

+Z ,

X6 −X1 = D1N+X +D3N

+Z , X7 −X1 = D1N

+X +D2N

+Y +D3N

+Z ,

X8 −X1 = D2N+Y +D3N

+Z .

B.1. Microscopic Kinematic relationships 183

where D1, D2 and D3 are the X, Y and Z direction size of the RVE.

B.1.1 Master-slave kinematic relationships

In the following sections the kinematic master-slave relationships will be obtainedfor the first-order and enhanced-first-order hom*ogenization approaches.

First-order approach

Taking into account that Periodic boundary fluctuations condition are consideredin the microscopic BVP and the proposed displacement field given by (3.9) ispossible to write the displacement of the slave node “a1” as a function of thedisplacement of the master node “a” as

ua1= ua + (F− I) · (Xa1

−Xa) , (B.2)

and using the above expression

ua1= ua +D2 (F− I) ·N+

Y . (B.3)

To simplify the final expressions is defined: sm1 = D1 (F− I) · N+X , sm2 =

D2 (F− I) ·N+Y and sm3 = D3 (F− I) ·N+

Z . Therefore, it can be shown that

ua1= ua + sm2, ua2

= ua + sm2 + sm3, ua3= ua + sm3,

ub1 = ub + sm1, ub2 = ub + sm1 + sm3, ub3 = ub + sm3,

uc1 = uc + sm1, uc2 = uc + sm1 + sm2, uc3 = uc + sm2,

ud1= ud + sm3, ue1 = ue + sm1, uf1

= uf + sm2.

(B.4)

And, for the vertices nodes

u2 = u1 + sm1, u3 = u1 + sm1 + sm2,

u4 = u1 + sm2, u5 = u1 + sm3,

u6 = u1 + sm1 + sm3, u7 = u1 + sm1 + sm2 + sm3,

u8 = u1 + sm2 + sm3

(B.5)

Enhanced first-order approach

Considering again that Periodic boundary fluctuations condition are consideredin the microscopic BVP and the proposed displacement field given by (3.36), itis possible to rewrite the displacement of the slave node “a1” as a function of thedisplacement of the master node “a” for the enriched approach as

ua1 = ua + (F− I) · (Xa1 −Xa) +1

2(Xa1

·G ·Xa1−Xa ·G ·Xa) (B.6)

184 Appendix B. Computational implementation

using (B.1), last expression can be rewritten as

ua1= ua +D2 (F− I) ·N+

Y +(D2)

2

2N+Y ·G ·N+

Y +D2Xa ·G ·N+Y . (B.7)

To simplify the final expressions is defined

smG1 = (D1)2

2 N+X ·G·N

+X , smG

2 = (D2)2

2 N+Y ·G·N

+Y , smG

3 = (D3)2

2 N+Z ·G·N

+Z ,

smG12 = D1D2N

+X ·G·N

+Y , smG

13 = D1D3N+X ·G·N

+Z , smG

23 = D2D3N+Y ·G·N

+Z ,

SMG1 = D1N

+X ·G, SMG

2 = D2N+Y ·G, SMG

3 = D3N+Z ·G.

Therefore, it can be shown that the slaves nodes are

ua1= ua + sm2 + smG

2 + SMG2 ·Xa,

ua2 = ua + sm2 + sm3 + smG2 + smG

3 + smG23 + (SMG

2 + SMG3 ) ·Xa,

ua3= ua + sm3 + smG

3 + SMG3 ·Xa,

ub1 = ub + sm1 + smG1 + SMG

1 ·Xb,

ub2 = ub + sm1 + sm3 + smG1 + smG

3 + smG13 + (SMG

1 + SMG3 ) ·Xb,

ub3 = ub + sm3 + smG3 + SMG

3 ·Xb,

uc1 = uc + sm1 + smG1 + SMG

1 ·Xc,

uc2 = uc + sm1 + sm2 + smG1 + smG

2 + smG12 + (SMG

1 + SMG2 ) ·Xc,

uc3 = uc + sm2 + smG2 + SMG

2 ·Xc,

ud1 = ud + sm3 + smG3 + SMG

3 ·Xd,

ue1 = ue + sm1 + smG1 + SMG

1 ·Xe,

uf1= uf + sm2 + smG

2 + SMG2 ·Xf .

(B.8)

B.1. Microscopic Kinematic relationships 185

And, taking into account that the position vector of the master vertex node “1”is: X1 = −D1

2 N+X − D2

2 N+Y − D3

2 N+Z , the slaves vertices nodes are

u2 = u1 + sm1 − smG12

2 − smG13

2 ,

u3 = u1 + sm1 + sm2 − smG13

2 − smG23

2 ,

u4 = u1 + sm2 − smG12

2 − smG23

2 ,

u5 = u1 + sm3 − smG13

2 − smG23

2 ,

u6 = u1 + sm1 + sm3 − smG12

2 − smG23

2 ,

u7 = u1 + sm1 + sm2 + sm3,

u8 = u1 + sm2 + smG3 − smG

12

2 − smG13

2 .

(B.9)

Figure B.2: Master and slaves nodes on the negative faces of the RVE.

For the enhanced-first-order approach an extra boundary restrictions must besatisfied. In the Periodic boundary fluctuations considered these extra boundaryconditions are integral boundary constraints on each negative face of the RVE.The extra boundary constraints described in Section 3.5.2 can be rewritten as

Ayz · uyz = Hyz, in ∂Ωhµ|N−X

,

Axz · uxz = Hxz, in ∂Ωhµ|N−Y

,

Axy · uxy = Hyz, in ∂Ωhµ|N−Z

.

(B.10)

186 Appendix B. Computational implementation

where,

Hyz = −1

2D1D2D3(F− I) ·N+

X +1

8(D1)2D2D3G : N+

X N+X

+1

24(D2)3D3G : N+

Y N+Y +

1

24D2(D3)3G : N+

Z N+Z ,

Hxz = −1

2D1D2D3(F− I) ·N+

Y +1

8D1(D2)2D3G : N+

Y N+Y

+1

24(D1)3D3G : N+

X N+X +

1

24D1(D3)3G : N+

Z N+Z ,

Hxy = −1

2D1D2D3(F− I) ·N+

Z +1

8D1D2(D3)2G : N+

Z N+Z

+1

24(D1)3D2G : N+

X N+X +

1

24D1(D2)3G : N+

Y N+Y

and,

Ayz =

∫N−X

Nyz dAyz, Axz =

∫N−Y

Nxz dAxz, Axy =

∫N−Z

Nxy dAxy.

Here, Nyz, Nxz and Nxy are the shape functions on the negative face Y Z, XZand XY of the RVE, respectively. And, from Figure B.2 it is possible to writethe displacement vectors of the nodes on the different negative faces as

uyz = u1|u4|u5|u8|ub|ub3 |uc|uc3 |ue,uxz = u1|u2|u5|u6|ua|ua3

|uc|uc1 |uf,uxy = u1|u2|u3|u4|ua|ua1 |ub|ub1 |ud.

In the previous displacement vectors of the nodes on the negative faces of theRVE it is possible identify masters and slaves nodes. Therefore, using (B.8) and(B.9) the boundary constraints (B.10) above obtained can be written in terms ofmaster nodes only as

Amyz · umyz = Hm

yz, in ∂Ωhµ|N−X

,

Amxz · umxz = Hm

xz, in ∂Ωhµ|N−Y

,

Amxy · umxy = Hm

yz, in ∂Ωhµ|N−Z

.

(B.11)

where,umyz = u1|ub|uc|ue,umxz = u1|ua|uc|uf,umxy = u1|ua|ub|ud.

B.1. Microscopic Kinematic relationships 187

and, as an example, the term of the matrix Amyz for the ub of the master nodes

on the negative face Y Z is

Amyz|b = Ayz|b + Am

yz|b3 ,

and the contribution to Hmyz for the ub3 of the slave nodes on the negative face

Y Z isHmyz|b = Am

yz|b3 · (sm3 + smG3 + SMG

3 ·Xb).

The master nodes on the different negative faces of the RVE must verify(B.11). Therefore, with the aim to find redundant unknowns, it is possible toidentify another slave extra node by each negative face which can be obtained asa function of the other master nodes. Then,

us1yz = −[As1yz]−1 ·A(m−1)

yz · u(m−1)yz + Hm

yz, in ∂Ωhµ|N−X

,

us2xz = −[As2xz]−1 ·A(m−1)

xz · u(m−1)xz + Hm

xz, in ∂Ωhµ|N−Y

,

us3xy = −[As3xy]−1 ·A(m−1)

xy · u(m−1)xy + Hm

yz, in ∂Ωhµ|N−Z

.

(B.12)

188 Appendix B. Computational implementation

B.2 Elimination of the slave degrees of freedom

In the present section the reduction of the degrees of freedom on the equationsystem is shown for the different cases.

B.2.1 Linear implementation

Introducing the concept of master nodes and slave nodes into the equation system,it can be rewritten as

Kii Kim Kis

Kmi Kmm Kms

Ksi Ksm Kss

uiumus

·

wiwmws

= 0 ∀ w, (B.13)

where the subscripts i, m, and s refer to the degrees of freedom of the internalnodes, master nodes and slave nodes, respectively. If the contribution of eachtype of degree of freedom is separated

Kii

Kmi

Ksi

ui+

Kim

Kmm

Ksm

um+

Kis

Kms

Kss

us ·

wiwmws

= 0 ∀ w.

(B.14)

Considering that Periodic boundary fluctuations condition are taken in themicroscopic problem, it is possible to write in matrix form the master-slave kine-matic relationship presented in the Section B.1.1 of the Appendix B.1 as

us = [Ssm] um+ 4d . (B.15)

Introducing (B.15) in (B.14), the following expression is obtained Kii

Kmi

Ksi

ui+

Kim

Kmm

Ksm

um+

Kis

Kms

Kss

[Ssm] um

+

Kis

Kms

Kss

4d ·

wiwmws

= 0 ∀ w.

(B.16)

B.2. Elimination of the slave degrees of freedom 189

and, it is possible to arrange the above equation as Kii

Kmi

Ksi

ui+

Kim +KisSsmKmm +KmsSsmKsm +KssSsm

um+

Kis

Kms

Kss

4d ·

wiwmws

= 0 ∀ w.

(B.17)

Thanks to the boundary conditions considered for the problem, it can beshown that the relationships of the displacement fluctuation field between theboundary master and slave nodes, written in a matrix form are

ws = [Ssm] wm , (B.18)

therefore, it is possible to reduce the system of equations shown in (B.17) as[Kii

Kmi + STsmKsi

]ui +

+

[Kim +KisSsm

Kmm +KmsSsm + STsmKsm + STsmKssSsm

]um+

+

[Kis

Kms + STsmKss

]4d

·

wi

wm

= 0 ∀ w.

(B.19)

Defining the reduced stiffness matrix, the reduced displacements vector, andthe Right-Hand Side (RHS) vector as

[Kr] =

[Kii Kim+KisSsm

Kmi+STsmKsi Kmm+KmsSsm+STsmKsm+STsmKssSsm

], (B.20)

ur =

uium

and (B.21)

RHS = −[

Kis

Kms + STsmKss

]4d

, (B.22)

therefore, the starting problem has been reduced to

[Kr] ur − RHS · wr = 0 ∀ w. (B.23)

And finally, the reduced equation system to be solved is

Kr · ur = RHS. (B.24)

190 Appendix B. Computational implementation

Equation (B.24) shows that the degrees of freedom of the slave nodes are notincluded in the reduced displacements vector. The reduced equation system hasless degrees of freedom than the original one and it also satisfies automaticallythe boundary conditions considered.

B.2.2 Non-linear implementation

Introducing the concept of master nodes and slave nodes into the equation system,it can be rewritten as

Fi

Fm

Fs

(k−1)

+

Kii Kim Kis

Kmi Kmm Kms

Ksi Ksm Kss

(k−1)

di

dm

ds

(k)

·

wi

wm

ws

= 0 ∀ w,

(B.25)

where the subscripts i, m, and s refer to the degrees of freedom of the inter-nal nodes, master nodes and slave nodes, respectively. Considering the updateformula of the Newton-Raphson method for the master degrees of freedom as

um(k) = um(k−1) + dm(k). (B.26)

It is possible to write (B.15) in the current k − th micro problem iteration as

us(k) = [Ssm]um(k) + 4d(t), (B.27)

where the superscript t is associated to the current iteration of the macro problem.Then, using (B.26) the above expression can be written as

us(k) = [Ssm]um(k−1) + 4d(t) + [Ssm]dm(k),

= us(k−1) + ds(k),(B.28)

and the following relationships is found

ds(k) = [Ssm]dm(k). (B.29)

B.2. Elimination of the slave degrees of freedom 191

Introducing (B.29) in (B.25)

Fi

Fm

Fs

(k−1)

+

Kii

Kmi

Ksi

(k−1)

di(k)

+

Kim +KisSsm

Kmm +KmsSsm

Ksm +KssSsm

(k−1)

dm(k)

·

wi

wm

ws

= 0 ∀ w.

(B.30)

Considering (B.18) in the above expression, it is possible to reduce the system ofequations shown in (B.25) as

Fi

Fm + STsmFs

(k−1)

+

[Kii Kim+KisSsm

Kmi+STsmKsi Kmm+KmsSsm+STsmKsm+STsmKssSsm

](k−1)di

dm

(k)

·

wi

wm

= 0 ∀ w,

(B.31)or in its reduced form using the definitions of the previous section

−RHS(k−1) + [Kr](k−1)dr(k)

· wr = 0 ∀ w. (B.32)

Therefore, the reduced equation system to be solved is

K(k−1)r · d(k)

r = RHS(k−1). (B.33)

Equation (B.33) shows that the degrees of freedom of the slave nodes are notincluded in the displacements vector, even for the non-linear case, and the reducedequation system also satisfies automatically the boundary conditions.

192 Appendix B. Computational implementation

B.3 Derivatives of the shape functions

The finite element method uses the shape functions to approximate the contin-uous displacement field within the finite element domain. In general, the shapefunctions are defined in an iso-parametric domain, then

N = N (ξ) . (B.34)

The derivatives of the shape functions are computed in the iso-parametric domainas

Bξ =∂N

∂ξ; Hξ =

∂

∂ξ

(∂N

∂ξ

)=∂Bξ

∂ξ. (B.35)

Considering that the finite element domain is approximated as

X = N (ξ) · X, (B.36)

where X are the coordinate values of the finite element nodes in the elementdomain. Then, the Jacobian matrix which transforms the iso-parametric domainto the finite element domain, is defined as

J =∂X

∂ξ=∂N

∂ξ· X = Bξ · X (B.37)

and

J−1 =∂ξ

∂X=[Bξ · X

]−1(B.38)

The shape functions and their derivatives have to be calculated in the finiteelement domain

N (X) ;∂N

∂X;

∂

∂X

(∂N

∂X

). (B.39)

Therefore, the values on the finite element domain are obtained as following:

Value of the shape functions

N (X) = N (ξ) (B.40)

First derivative of the shape functions

B =∂N

∂X=∂N

∂ξ· ∂ξ∂X

= Bξ · J−1 (B.41)

B.3. Derivatives of the shape functions 193

Second derivative of the shape functions

H =∂

∂X

(∂N

∂X

)=

∂

∂X

(Bξ · J−1

)=

∂

∂ξ

(Bξ · J−1

)· J−1, (B.42)

and considering the above expression

H = Hξ · J−1 · J−1 + Bξ ·∂J−1

∂ξ· J−1. (B.43)

Taking intro account thatJ · J−1 = I (B.44)

and

∂J

∂ξ· J−1 + J · ∂J−1

∂ξ= 0 ⇐⇒ ∂J−1

∂ξ= −J−1 · ∂J

∂ξ· J−1. (B.45)

It is possible to obtain the unknown term as

∂J−1

∂ξ= −J−1 · X · ∂Bξ

∂ξ· J−1 = −J−1 · X ·Hξ · J−1. (B.46)

Therefore,H =

(Hξ −Bξ · J−1 · X ·Hξ

)· J−1 · J−1. (B.47)

194 Appendix B. Computational implementation

B.4 Derivation of displacement field, deforma-tion gradient and gradient of the deforma-tion gradient tensor

Displacement field

Taking into account the consideration made in Section B.3, the displacement fieldin the finite element domain can be approximated as

u = N(ξ) · u, (B.48)

where u are the nodal displacement values of the finite element.

Deformation gradient tensor

If the current configuration position of the structure is approximated as

x = u + X, (B.49)

where the deformation gradient tensor is obtained

F = ∇0x =∂u

∂X+ I (B.50)

then,F = B · u + I. (B.51)

Gradient of the deformation gradient tensor

G = ∇0F =∂

∂X(B · u) (B.52)

finally,G = H · u. (B.53)

B.5. FEM implementation in PLCd code 195

B.5 FEM implementation in PLCd code

The numerical implementation of the hom*ogenization method is carried outthrough of a coupled solution scheme in two scales by means of PLCd program.The PLCd code is an implicit FEM code developed in FORTRAN by Prof. Oller’sgroup at CIMNE [111]. The program solves solid mechanics problems which caneither be linear or non-linear and take into account small or large deformationhypothesis. It has been developed to treat a large variety of composite materialsthrough the use of different composite theories and can support materials withgeneral anisotropy.

The basic problem in a general non-linear analysis is to find the state equilib-rium of a body in function of the applied loads, by using an incremental solutionapproach. In the case of a static or quasi-static analysis, in which the time effectdoes not affect the equilibrium equations, the time factor is only a convenientvariable which denote different intensities of applied load. In a classical FEMimplementation the non-linear response is effectively carried out using a step-by-step incremental solution, in which the total applied load is divided into severalnumber of load steps. Then, in each load step an iterative process must be per-formed until the solution for the equilibrium equation is achieved. The iterativeprocess is conducted with a Newton-Raphson procedure. Figure B.3 shows aflow diagram of the incremental solution and the iterative method implementedin PLCd code at macroscopic scale.

196 Appendix B. Computational implementation

Initial material and geometricproperties + boundary conditions

Calculate elastic stiffness:K = BT : C : B

Fext(n) = Fext

(n−1) +4Fext(n)

u(n) = u(n−1) + K−1 · 4Fext(n)

u(k=1) = u(n) , Fext = Fext(n)

F(k) = B · u(k) + I , G(k) = H · u(k)

Obtain S(k), Ctan

(k) through

CONSTITUTIVE MODEL

Calculate internal force:Fint

(k) = BT : S(k)

Calculate tangent stiffness:T(k) = BT : Ctan

(k) : B

4F =∣∣∣Fext − Fint

(k)

∣∣∣ ≤ tol

END

u (k

)=

u (k

−1)

+T

−1

(k−

1)·4

F

u (n

−1)

=u (

k)

,K

=T

(k)

Load loop

N-R loop

n = 1, nload

k = 1, kmax

k = k + 1n = n + 1

n = nload

No

Yes

Figure B.3: PLCd Flow diagram of the FEM implementation at the macro-scale.

B.5. FEM implementation in PLCd code 197

B.5.1 PLCd FE2 implementation

In the PLCd program, the macro-scale problem is solved by a classical FE imple-mentation and the micro-scale problem is also solved using a FE implementation.The general framework of the solution consists in solving for each Gauss point ofeach FE at the macro-scale, another FE problem defined by the RVE in order toaddress the response at the micro-scale that will give the hom*ogenized solutionof the Gauss point at the macro-scale.

However, considering that all materials in the micro-scale have an elastic be-havior and assuming a perfect contact between the component materials, theelastic constitutive tensor for the composite which relates the macroscopic ho-mogenized variables remains constant. Therefore, the expression given by (3.32)in the first-order hom*ogenization and (3.81) in the enhanced-first-order hom*og-enization can be expressed in a similar way as in the classical expression forhom*ogeneous materials as

S = C : E, (B.54)

where C is a tensor formed by the elastic constants of the hom*ogenized composite,called hom*ogenized elastic constitutive tensor and S and E are the macroscopicor hom*ogenized stress and strain tensors.

Rewriting (B.54) using a Voigt notation Si = CijEj , where i, j = 1 · · · 6 it ispossible to observe that one way to obtain this constitutive tensor of the materialis by applying a macroscopic strain tensor on the RVE in order to compute themicroscopic stress tensor Sµ. Then, the macroscopic stress tensor in the RVEis calculated for each macroscopic strain tensor applied according to expressiongiven by (3.27) or (3.74).

Considering a three-dimensional problem, it is necessary six independentstrains cases to obtain the hom*ogenized constitutive tensor, each one accountingfor a different principal direction. The macroscopic strain tensor for each direc-tion is designed taking a unit value on the direction analyzed and zero for theothers. For example, in the X direction case is

Ej = [1, 0, 0, 0, 0, 0]⇒ Si =1

Vµ

∫Ωµ

Sµ dV ⇒ Ci1 = Si[E1]−1 = Si. (B.55)

Figure B.4 shows a flow diagram with the implementation in PLCd code tocalculate the hom*ogenized constitutive tensor.

198 Appendix B. Computational implementation

Initial material and RVEproperties + boundary condition

Calculate elastic stiffness:k = bTµ : Cµ : bµ

Using Voigt notation:Si = CijEj |i;j=1,··· ,6

S(k)i = 0 , E

(k)j = 0 , E

(k)k = 1

Calculate reduced system:

k(k)r , rhs(k) ⇒ u(k)

r = [k(k)r ]−1 · rhs(k)

u(k)s = Ssm · u(k)

m +4d(k)

u(k)µ = u(k)

i |u(k)m |u(k)

s

F(k)µ = bµ · u(k)

µ + I

Obtain S(k)µ through

CONSTITUTIVE MODEL

Calculate macroscale stress:

S(k)i = 1

Vµ

ˆ

Ωµ

S(k)µ dV

Calculate k-column as: Cik = S(k)i

hom*ogenized constitutive tensor: C

k=

k+

1

k-loopk = 1, ..., 6

k = 6

Figure B.4: Flow diagram of the hom*ogenized elastic constitutive tensor calculation inPLCd code.

B.5. FEM implementation in PLCd code 199

Non-linear FE2 implementation

When a non-linear analysis is addressed by a FE2 numerical hom*ogenization thenon-linear response of the hom*ogenized composite is obtained through the anal-ysis of the RVE. Therefore, the macroscopic stress tensor is obtained throughthe solution of one RVE for each Gauss point at the macro-scale FE mesh. InFigure B.5 is possible to observe a flow diagram of the FE2 numerical imple-mentation developed for the FEM code used. The micro-scale problem is solvedusing another FE problem. Therefore, when the non-linear behavior begins ona component material in the RVE it is necessary an iterative process to addressthe solution. Figure B.6 shows the flow diagram of the FE implementation atthe micro-scale. The flow diagrams shown that for each non-linear macro-scalestep there are one macro-scale iterative process and several micro-scale iterativeprocesses for each non-linear RVE.

200 Appendix B. Computational implementation

Initial material and geometricproperties + boundary conditions

Calculate elastic stiffness:K = BT : C : B

Fext(n) = Fext

(n−1) +4Fext(n)

u(n) = u(n−1) + K−1 · 4Fext(n)

u(k=1) = u(n) , Fext = Fext(n)

F(k) = B · u(k) + I , G(k) = H · u(k)

Obtain S(k), Ctan

(k) through

the solution of the RVEs

Calculate internal force:Fint

(k) = BT : S(k)

Calculate tangent stiffness:T(k) = BT : Ctan

(k) : B

4F =∣∣∣Fext − Fint

(k)

∣∣∣ ≤ tol

END

u (k

)=

u (k

−1)

+T

−1

(k−

1)·4

F

u (n

−1)

=u (

k)

,K

=T

(k)

Initial RVE properties+ boundary conditions

Calculate: C

Load loop

N-R loop

n = 1, nload

k = 1, kmax

k = k + 1n = n + 1

n = nload

No

Yes

Figure B.5: PLCd flow diagram of the FE2 implementation.

B.5. FEM implementation in PLCd code 201

Macro input F(k) and G(k)+ RVE properties

Calculate elastic stiffness:k = bTµ : Cµ : bµ

Calculate reduced system:

kr , rhs ⇒ u(1)r = k−1

r · rhs

u(j)s = Ssm · u(j)

m +4du(j)µ = u(j)

i |u(j)m |u(j)

s

F(j)µ = bµ · u(j)

µ + I

Obtain S(j)µ , Ctan

µ(j) through

CONSTITUTIVE MODEL

Calculate internal force:Fintµ(j) = bTµ : S(j)

µ

Calculate tangent stiffness:t(j) = bTµ : Ctan

µ(j) : bµ

Calculate reduced system: t(j)r , rhs(j)

4rhs =∣∣rhs− rhs(j)

∣∣ ≤ tol

Calculate macro output: S(k) , Ctan

(k) END

u(j)

r=

u(j−

1)r

+[t

(j−

1)r

]−1·4

rhs

N-R loopj = 1, jmax

j = j + 1No

Yes

Figure B.6: PLCd flow diagram of the FE2 implementation at the micro-scale.

202 Appendix B. Computational implementation

B.5.2 Numerical tangent constitutive tensor

The perturbation method used to obtain the hom*ogenized tangent tensor of theRVE is described in this section. It follows the same procedure defined in [77].The tangent constitutive tensor (Ctan) is defined as

S = Ctan

: E. (B.56)

The matrix description given by (B.56) can be written for orthotropic materialsas

S1

...

Sn

=

Ctan

11 · · · Ctan

1n

.... . .

...

Ctan

n1 · · · Ctan

nn

E1

...

En

. (B.57)

The stress vector rate can be obtained as the sum of n stress vectors, which arethe product of the j component of the strain vector rate and the j column of thetangent tensor. Then

S ≡n∑j=1

δjS =

n∑j=1

Ctan

j .δEj , (B.58)

where

Ctan

j =[

Ctan

1j Ctan

2j · · · Ctan

nj

]T. (B.59)

Equation (B.58) can be used to obtain the j column of the tangent constitutivetensor as

Ctan

j =jS

Ej

≡ δjS

δEj(B.60)

The perturbation method consists in defining n small variations, or perturbations,of the strain vector δEj , to obtain n stress vectors δjS to obtain the numericalapproach given by (B.60) of the tangent constitutive tensor. Figure B.7 is showinga flow diagram of the implementation in the PLCd of the perturbation methoddescribed to obtain the macroscopic tangent constitutive tensor.

B.5. FEM implementation in PLCd code 203

Macro input E, S and RVE properties

Calculate elastic stiffness: k=bTµ :Cµ:bµ

Using Voigt notation: Si=Ctan

ij Ej |i;j=1,··· ,6

S(k)i = S , E

(k)j = E , E

(k)k = E

(k)k + δE

Calculate reduced system:

k(k)r , rhs(k) ⇒ u(k)

r(1) = [k(k)r ]−1 · rhs(k)

u(k)s(p) = Ssm · u(k)

m(p) +4d(k)

u(k)µ(p) = u(k)

i(p)|u(k)m(p)|u

(k)s(p)

Obtain S(k)µ(p), Ctan

µ(p) through

CONSTITUTIVE MODEL

Calculate internal force and tangentstiffness: Fintµ(p)=bTµ :S(k)

µ(p) ; t(p)=bTµ :Ctan

µ(p):bµ

Calculate reduced system: t(p)r , rhs(k)

(p)

4rhs(k)=∣∣rhs(k)−rhs(k)

(p)

∣∣≤tol

Calculate delta macroscale stress:

δS(k)i = S− 1

Vµ

´

Ωµ S(k)µ dV

Calculate k-column as: Ctan

ik = δS(k)i /δE Ctan

k=k

+1

k-loop

N-R loop

k = 1, ..., 6

p = 1, pmax

p = p + 1

u(k)

r(p

)=u(k

)r

(p−

1)+

[t(p

−1)

r]−

1·4

rhs(k

)

Yes

k = 6

No

Figure B.7: Flow diagram of the perturbation method implemented at PLCd to obtainthe macroscopic tangent constitutive tensor.

204 Appendix B. Computational implementation

B.5.3 PLCd parallelization and memory improvements

Parallelization tasks

The objective of parallelizing the code is to reduce the computing time in order toimprove the overall performance of the FE2 hom*ogenization method developed.The parallelization of the FEM code has been done using OpenMP (http://www.OpenMP.org).

OpenMP is a shared memory application program interface that can be im-plemented on a broad range of architectures. It consists in a set of compiler di-rectives, run-time library routines and environment variables that can be addedto sequential FORTRAN code in order to specify how the work is to be accessedand shared among threads that will execute on different processors or cores andto order accesses to the shared data as needed.

According to Amdahl’s law, to obtain a maximum speedup or scalability,in a parallel program, is important parallelize the maximum percentage of thecode. Therefore, the parallelization of a computational code is an unfinished task.Because of this, during the study many sequential loops in the PLCd have beenparallelized to improve the scalability, such as convergence element loop, writingdatabase no-converged to converged element loop, assemble sparse stiffness matrixelement loop and assemble residual forces vector element loop.

On the other hand, in a non-linear analysis, the tasks balance between thecores is a significant variable to maintain the speedup. Then, an efficient par-allelization strategy in the constitutive element loop (see Figure B.8) has beenaddressed with the aim to improve the tasks balance in the threads during anon-linear analysis. The “OMP SET SCHEDULE(type,chunk)” instruction isused to configure the parallel strategy. In the default static type strategy thenumber of elements is divided into the number of threads and they are assignedto the threads at the beginning of the loop. Therefore, when a thread receivesmany elements with a non-linear behavior the tasks balance is lost. This situa-tion is more critical in a FE2 implementation because the composite constitutivemodel is another micro FE problem. To conserve a good tasks balance a dynamictype strategy with a chunk of 1 is set. The dynamic strategy assigns the chunkquantity of elements to the threads at the beginning and then, when a threadhas finished with that task, a new chunk number of elements is assigned to it. Inother works, the tasks assigned to each thread is dynamic.

Memory manager

The advantage of the reduced memory requirement of the multiscale approachin linear range is lost when the method is extended to non-linear case. The non-linear behavior of the component materials in the RVE depend of the internalvariables and strain tensor history. Therefore, the internal RVE database for eachGauss point of the macro-scale mesh should be saved in the memory. To minimize

B.5. FEM implementation in PLCd code 205

the memory requirements for non-linear analyses a management procedure toassign dynamic memory is implemented in PLCd code. While all componentmaterials in the RVE of the macro Gauss point are in the elastic range the codedoes not reserve memory for the internal RVE database. However, if some microGauss points of the RVE of a macro Gauss point begin with a non-linear behavioronly for these Gauss points is reserved space in memory. With this memorystrategy, the memory requirement for non-linear analyses through a multiscalehom*ogenization is reduced to the minimum required.

206 Appendix B. Computational implementation

Initial material and geometricproperties + boundary conditions

Calculate elastic stiffness tensor:K = BT : C : B

u = K−1 · Fext

F(k) = B · u(k) + I , G(k) = H · u(k)

Obtain S(k), Ctan

(k) through

CONSTITUTIVE MODEL

Calculate internal force:Fint

(k) = BT : S(k)

Update matrix:T(k) = BT : Ctan

(k) : B

Assemble Fint(k) and T(k)

for whole structure

4F =∣∣∣Fext − Fint

(k)

∣∣∣ ≤ toler

END

u (k

)=

u (k

−1)

+T

−1

(k−

1)·4

F

Element loop

Gausspoint loop

k = k + 1No

Yes

Figure B.8: PLCd flow diagram of the parallel element loop implementation at themacro-scale.

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