Representative Volume Elements and Unit Cells: Concepts, Theory, Applications and Implementation

Numerical methods to estimate material properties usually involve analysis of a representative volume element (RVE) or unit cell (UC). The representative volume element (RVE) or unit cell (UC) is the smallest volume over which a measurement can be made that will yield a value representative of the whole. RVEs and UCs are widely used in the characterisation of materials with multiscale architectures such as composites. However, finite element (FE) software packages such as Abaqus and Comsol MultiPhysics do not offer the capability for RVE and UC modelling directly on their own. To apply them to analyse RVEs and UCs, the generation of the FE models for them, the imposition of boundary conditions, and the extraction of directly relevant results are essentially the responsibility of the user. These have tended to be incorrectly implemented by users! For the first time, this book will provide a comprehensive account on correct modelling of RVEs and UCs, which will eliminate any uncertainties and ambiguities.

The book offers a complete and thorough review on the subject of RVEs and UCs, establishing a framework on a rigorous mathematical and mechanical basis to ensure that basic concepts, such as symmetry and free body diagrams, are applied correctly and consistently. It also demonstrates to readers that rigorous applications of mathematics and mechanics are meant to make things clear, consistent, thorough and, most of all, simple and easy to follow, rather than the opposite as many perceive. As a result, the book shows that the appropriate use of RVEs and UCs can deliver an effective and reliable means of material characterisation. It not only provides a much needed comprehensive account on material characterisation but, more importantly, explains how such characterisation can be conducted in a consistent and systematic manner. It also includes a ready-to-use open source code for UCs that can be downloaded from a companion site for potential users to utilise, adapt and expand as they wish.

• The theories presented in this book will give users more confidence when applying RVE and UC models to analyse materials of complex architectures with accuracy and efficiency

• Systematic explanations of RVE and UC theories have been included, as well as their applications in composites

• It illustrates in detail how to set up UC models and provides an open source code to implement via Abaqus

Preface xi

Part One: Basics

1. Introduction d background, objectives and basic

concepts 3

1.1 The concept of length scales and typical length scales in physics and

engineering 3

1.2 Multiscale modelling 4

1.3 Representative volume element and unit cell 5

1.4 Background of this monograph 6

1.5 Objectives of this monograph 6

1.6 The structure of this monograph 8

References 10

2. Symmetry, symmetry transformations and symmetry conditions 11

2.1 Introduction 11

2.2 Geometric transformations and the concept of symmetry 12

2.3 Symmetry of physical fields 16

2.4 Continuity and free body diagrams 24

2.5 Symmetry conditions 28

2.6 Concluding remarks 41

References 42

3. Material categorisation and material characterisation 43

3.1 Background 43

3.2 Material categorisation 45

3.3 Material characterisation 60

3.4 Concluding remarks 64

References 64

4. Representative volume elements and unit cells 67

4.1 Introduction 67

4.2 RVEs 68

4.3 UCs 71

4.4 Concluding remarks 76

References 77

5. Common erroneous treatments and their conceptual sources

of errors 79

5.1 Realistic or hypothetic background 79

5.2 The construction of RVEs and their boundary 82

5.3 The construction of UCs 84

5.4 Post-processing 96

5.5 Implementation issues 98

5.6 Verification and the lack of ‘sanity checks’ 101

5.7 Concluding remarks 102

References 103

Part Two: Consistent formulation of unit cells and

representative volume elements

6. Formulation of unit cells 107

6.1 Introduction 107

6.2 Relative displacement field and rigid body rotations 108

6.3 Relative displacement boundary conditions for unit cells 114

6.4 Typical unit cells and their boundary conditions in terms of relative

displacements 115

6.5 Requirements on meshing 176

6.6 Key degrees of freedom and average strains 177

6.7 Average stresses and effective material properties 179

6.8 Thermal expansion coefficients 182

6.9 “Sanity checks” as basic verifications 183

6.10 Concluding remarks 185

References 187

7. Periodic traction boundary conditions and the key degrees of

freedom for unit cells 189

7.1 Introduction 189

7.2 Boundaries and boundary conditions for unit cells resulting from

translational symmetries 193

7.3 Total potential energy and variational principle for unit cells under

prescribed average strains 197

7.4 Periodic traction boundary conditions as the natural boundary

conditions for unit cells 198

7.5 The nature of the reactions at the prescribed key degrees of freedom 202

7.6 Prescribed concentrated ‘forces’ at the key degrees of freedom 209

7.7 Examples 211

7.8 Conclusions 219

References 220

8. Further symmetries within a UC 223

8.1 Introduction 223

8.2 Further reflectional symmetries to existing translational symmetries 225

8.3 Further rotational symmetries to existing translational symmetries 251

8.4 Examples of mixed reflectional and rotational symmetries 291

8.5 Centrally reflectional symmetry 303

8.6 Guidance to the sequence of exploiting existing symmetries 314

8.7 Concluding statement 315

References 317

9. RVE for media with randomly distributed inclusions 319

9.1 Introduction 319

9.2 Displacement boundary conditions and traction boundary conditions

for an RVE 320

9.3 Decay length for boundary effects 323

9.4 Generation of random patterns 327

9.5 Strain and stress fields in the RVE and the sub-domain 330

9.6 Post-processing for average stresses, strains and effective properties 336

9.7 Conclusions 345

References 345

10. The diffusion problem 347

10.1 Introduction 347

10.2 Governing equation 347

10.3 Relative concentration field 351

10.4 An example of a cuboidal unit cell 353

10.5 RVEs 355

10.6 Post-processing for average concentration gradients and diffusion fluxes 356

10.7 Conclusions 360

References 360

11. Boundaries of applicability of representative volume elements

and unit cells 361

11.1 Introduction 361

11.2 Predictions of elastic properties and strengths 361

11.3 Representative volume elements 363

11.4 Unit cells 366

11.5 Conclusions 366

References 367

Part Three: Further developments

12. Applications to textile composites 371

12.1 Introduction 371

12.2 Use of symmetries when defining an effective UC 381

12.3 Unit cells for two-dimensional textile composites 383

12.4 Unit cells for three-dimensional textile composites 398

12.5 Conclusions 414

References 415

13. Application of unit cells to problems of finite deformation 417

13.1 Introduction 417

13.2 Unit cell modelling at finite deformations 419

13.3 The uncertainties associated with material definition 433

13.4 Concluding remarks 436

References 437

14. Automated implementation: UnitCells© composites

characterisation code 439

14.1 Introduction 439

14.2 Abaqus/CAE modelling practicality 441

14.3 Verification and validation 450

14.4 Concluding remarks 456

References 456

Index 459