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

**Contents **

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

**Preface**

It was in early 1995 when I was attending an international conference where I was intrigued by a presentation dealing with a sophisticated material behaviour of a unidirectional composite where micromechanical finite element analysis was conducted on a unit cell. My attention was not caught by the high level of sophistication of material behaviour as I can hardly remember any of it by now. It was the peculiar shape of the unit cell, which was a trapezium as the highlighted part of the hexagon on the front cover of this book but with the inclined side curved. As no justification was provided either in the presentation or the paper included in the conference proceeding, it had to be logged in my mind as a mystery. The same day, whilst queuing up for lunch, the presenter happened to queue right in front of me. I seized the opportunity to enquire why the side of his unit cell was curved. ‘It has to be in order to keep the angles at both sides of it at right angles.’ He answered but apparently not enough to clear the position. I followed by querying the significance of those right angles. He replied mysteriously, ‘If not, there would be stress singularities there!’ He disclosed all he knew but I was instantly convinced on the spot beyond any doubt that something was wrong with the boundary conditions.

I took the problem back and started a little search of the literature. To my surprise, I realised the appalling state-of-the-art as described in details in the Chapter 5 of this monograph. I was both attracted to the problem and in the meantime had a strong sense of duty to put it right. It took me over two years to come up with my first paper on the subject of unit cells and it took almost the same duration to get it eventually published in the Proceeding of the Royal Society London A in 1999. I thought that it was done and dusted. Soon, I realised that the account could be improved by a more systematic approach. As a result, two more papers were published, one on twodimensional and one on three-dimensional unit cells, respectively. Again, I thought that was it!

Before long, I became frustrated again by the fact that when unit cells were employed as published in the literature they were often presented without providing the boundary conditions as if the unit cell was just a geometric shape of some kind. I then devised a number of cases to illustrate the fact that unit cells of rather different appearances could share the same boundary conditions and hence represent the same material to be characterised, whilst unit cells sharing the identical appearance but subject to different boundary conditions could represent materials of completely different characteristics.

The story kept evolving along the line. Whilst reflecting on the subject in a chapter for the Comprehensive Composite Materials II, I realised I had published nearly thirty papers in reputable academic journals, some with collaborators, on the very subject of unit cells and representative volume elements, in addition to contributions to various conferences. The chapter did not have enough room to accommodate the contents available, and I concluded that it was the time to wrap it up into a single coherent account to benefit future users as monographs are meant for, but, for me personally, it would truly be a time when I could put the matter behind me, hopefully.

Over a good number of years after the major publications of mine on unit cells, I kept receiving requests for help through emails. Apparently, simply sending copies of those papers proved to be not enough as people tended to get back to me with more questions. Well-posed formulation delivers appropriate boundary conditions for the unit cells. If one is to use a single word to describe such boundary conditions, it has to be ‘tedious’. Without being too negative, the ‘tedious’ boundary conditions are also systematic, which is a very important characteristics. Eventually, I generated a set of templates which seemed to have worked well in helping the followers of my papers.

Around 2010, a postdoctoral researcher of mine, Dr Laurent Jeanmeure, suggested that the unit cells I generated as demonstrated through the templates could be automated by writing a piece of code in Python to drive Abaqus/CAE as a secondary development of the FEM code. Within the duration of his project, he managed to demonstrate the feasibility of the approach. A substantial development did not start until I met Dr Qing Pan a year or so later, who was a postgraduate student then at Nanjing University of Aeronautics and Astronautics, China. He was undoubtedly a genius in programming and voluntarily helped to have many of the formulated unit cells coded using Python. The code was then named UnitCells_, a form of which is to be made available on a special website Elsevier will provide. Qing was subsequently recruited onto a PhD course at the University of Nottingham under my supervision. Eventually, we managed to code on that platform most unit cells of practical significance, which took care of the tedious part of the implementation of unit cells. UnitCells_ has demonstrated itself as an useful and reliable platform for systematic multiscale characterisation of composite materials, from unidirectional fibre reinforced ones to laminates made of them and from particulate reinforcements to textile preform reinforcements. The provision of the truthful records of the development along this line is to serve as my sincere acknowledgement to these two close collaborators of mine on the subject of unit cells.

To highlight my list of acknowledgements, the anonymous presenter as I referred to at the beginning of the preface and his co-authors should hold a special place, even though my publications tended to be negative on their particular choice of the unit cell. Without their induction to the field, I might have never found myself ploughing in this field, let alone harvest any crop from it.

I would also like to express my gratitude to my collaborators/co-authors for various relevant publications. Some of them brought their problems to me as challenges, without which many of the topics I have studied on the subject of unit cells and representative volume elements would not even have been contemplated. Acknowledgements are due to a current PhD student under my supervision, Mr Mingming Xu, whose work on parameterisation of 3D woven composites was quoted in one of the subsections of Chapter 12.

I would also like to thank my co-author Dr Elena Sitnikova for her enthusiasm, commitment to the ethos and willingness to sacrifice her offwork time to make the publication of this monograph possible.