This book pursues optimal design from the perspective of mechanical properties and resistance to failure caused by cracks and fatigue. The book abandons the scale separation hypothesis and takes up phase-field modeling, which is at the cutting edge of research and is of high industrial and practical relevance. Part 1 starts by testing the limits of the homogenization-based approach when the size of the representative volume element is non-negligible compared to the structure. The book then introduces a non-local homogenization scheme to take into account the strain gradient effects. Using a phase field method, Part 2 offers three significant contributions concerning optimal placement of the inclusion phases. Respectively, these contributions take into account fractures in quasi-brittle materials, interface cracks and periodic composites. The topology optimization proposed has significantly increased the fracture resistance of the composites studied.

Daicong Da is a research associate in the Mechanical Engineering department at the University of Wisconsin-Madison, USA. His research interests cover structural topology optimization and its applications.

Introduction ix

Part 1. Multiscale Topology Optimization in the Context of Non-separated Scales 1

Chapter 1. Size Effect Analysis in Topology Optimization for Periodic Structures Using the Classical Homogenization 3

1.1. The classical homogenization method 4

1.1.1. Localization problem 4

1.1.2. Definition and computation of the effective material properties 7

1.1.3. Numerical implementation for the local problem with PER 9

1.2. Topology optimization model and procedure 10

1.2.1. Optimization model and sensitivity number 10

1.2.2. Finite element meshes and relocalization scheme 12

1.2.3. Optimization procedure 14

1.3. Numerical examples 16

1.3.1. Doubly clamped elastic domain 17

1.3.2. L-shaped structure 19

1.3.3. MBB beam 24

1.4. Concluding remarks 25

Chapter 2. Multiscale Topology Optimization of Periodic Structures Taking into Account Strain Gradient 29

2.1. Non-local filter-based homogenization for non-separated scales 30

2.1.1. Definition of local and mesoscopic fields through the filter 30

2.1.2. Microscopic unit cell calculations 33

2.1.3. Mesoscopic structure calculations 39

2.2. Topology optimization procedure 41

2.2.1. Model definition and sensitivity numbers 41

2.2.2. Overall optimization procedure 42

2.3. Validation of the non-local homogenization approach 43

2.4. Numerical examples 45

2.4.1. Cantilever beam with a concentrated load 46

2.4.2. Four-point bending lattice structure 52

2.5. Concluding remarks 55

Chapter 3. Topology Optimization of Meso-structures with Fixed Periodic Microstructures 57

3.1. Optimization model and procedure 58

3.2. Numerical examples 61

3.2.1. A double-clamped beam 61

3.2.2. A cantilever beam 64

3.3. Concluding remarks 66

Part 2. Topology Optimization for Maximizing the Fracture Resistance 67

Chapter 4. Topology Optimization for Optimal Fracture Resistance of Quasi-brittle Composites 69

4.1. Phase field modeling of crack propagation 71

4.1.1. Phase field approximation of cracks 71

4.1.2. Thermodynamics of the phase field crack evolution 72

4.1.3. Weak forms of displacement and phase field problems 75

4.1.4. Finite element discretization 76

4.2. Topology optimization model for fracture resistance 78

4.2.1. Model definitions 78

4.2.2. Sensitivity analysis 80

4.2.3. Extended BESO method 85

4.3. Numerical examples 87

4.3.1. Design of a 2D reinforced plate with one pre-existing crack notch 88

4.3.2. Design of a 2D reinforced plate with two pre-existing crack notches 93

4.3.3. Design of a 2D reinforced plate with multiple pre-existing cracks 96

4.3.4. Design of a 3D reinforced plate with a single pre-existing crack notch surface 98

4.4. Concluding remarks 101

Chapter 5. Topology Optimization for Optimal Fracture Resistance Taking into Account Interfacial Damage 103

5.1. Phase field modeling of bulk crack and cohesive interfaces 104

5.1.1. Regularized representation of a discontinuous field 104

5.1.2. Energy functional 106

5.1.3. Displacement and phase field problems 108

5.1.4. Finite element discretization and numerical implementation 111

5.2. Topology optimization method 114

5.2.1. Model definitions 114

5.2.2. Sensitivity analysis 116

5.3. Numerical examples 119

5.3.1. Design of a plate with one initial crack under traction 120

5.3.2. Design of a plate without initial cracks for traction loads 123

5.3.3. Design of a square plate without initial cracks in tensile loading 125

5.3.4. Design of a plate with a single initial crack under three-point bending 128

5.3.5. Design of a plate containing multiple inclusions 130

5.4. Concluding remarks 133

Chapter 6. Topology Optimization for Maximizing the Fracture Resistance of Periodic Composites 135

6.1. Topology optimization model 136

6.2. Numerical examples 138

6.2.1. Design of a periodic composite under three-point bending 138

6.2.2. Design of a periodic composite under non-symmetric three-point bending 146

6.3. Concluding remarks 151

Conclusion 153

References 157

Index 173