Computational Homogenization of Heterogeneous Materials with Finite Elements

ISBN-13: 9783030183851 / Angielski / Miękka / 2020 / 223 str.

Julien Yvonnet

Opis

This monograph provides a concise overview of the main theoretical and numerical tools to solve homogenization problems in solids with finite elements. Starting from simple cases (linear thermal case) the problems are progressively complexified to finish with nonlinear problems. The book is not an overview of current research in that field, but a course book, and summarizes established knowledge in this area such that students or researchers who would like to start working on this subject will acquire the basics without any preliminary knowledge about homogenization. More specifically, the book is written with the objective of practical implementation of the methodologies in simple programs such as Matlab. The presentation is kept at a level where no deep mathematics are required.

Spis treści

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Why computational homogenization? . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Brief historical and recent advances . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Industrial applications and use in commercial softwares . . . . . . . . . . 3

1.4 Position of the present monograph as compared to available other

books on that topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Overview and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Review of classical FEM formulations and discretizations . . . . . . . . . . . 5

2.1 Steady-state thermal problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Strong form of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Weak forms of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 2D FEM discretization with linear elements . . . . . . . . . . . . . . 7

2.1.4 Assembly of the elementary systems . . . . . . . . . . . . . . . . . . . . 12

2.1.5 Prescribing Dirichlet conditions . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Strong form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 2D discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.4 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Conduction properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 The notion of RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Localization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Averaged quantities and Hill-Mandel lemma . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Averaging theorem: temperature gradient . . . . . . . . . . . . . . . . 31

3.3.2 Averaging theorem: heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.3 Hill-Mandel lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Computation of the effective conductivity tensor . . . . . . . . . . . . . . . . 33

3.4.1 The superposition principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.2 Definition of the effective conductivity tensor . . . . . . . . . . . . 34

v

vi Contents

3.5 Periodic boundary conditions for the thermal problem: numerical

implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.6 Numerical calculation of effective conductivity with 2D FEM . . . . . 39

3.6.1 Transverse effective conductivity . . . . . . . . . . . . . . . . . . . . . . . 39

3.6.2 Computation of the out-of plane properties using a 2D RVE 41

3.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Elasticity and thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1 Localization problem for elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Averaged quantities and Hill-Mandel lemma . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Averaging theorem: strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.2 Averaging theorem : stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Definition of the effective elastic tensor . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.1 Strain approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2 Stress approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Computations of the effective properties with FEM . . . . . . . . . . . . . . 55

4.4.1 2D case: transverse effective properties . . . . . . . . . . . . . . . . . . 55

4.4.2 Computation of out-of-plane elastic properties using a 2D

RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.3 Full 3D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.5 Periodic boundary conditions for 2D elastic problem: practical

implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.6 Extension to thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7 Second-order linear homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8 Filter-based homogenization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

9 Nonlinear Computational Homogenization . . . . . . . . . . . . . . . . . . . . . . . . 81

References