Representative Volume Elements and Unit Cells: Concepts, Theory, Applications and Implementation (Woodhead Publishing Series in Composites Science and Engineering) 0081026382, 9780081026380

Table of contents :
Cover
Representative Volume Elements and Unit Cells: Concepts, Theory, Applications and Implementation
Copyright
Dedication
Preface
Part One: Basics
1 . Introduction — background, objectives and basic concepts
1.1 The concept of length scales and typical length scales in physics and engineering
1.2 Multiscale modelling
1.3 Representative volume element and unit cell
1.4 Background of this monograph
1.5 Objectives of this monograph
1.6 The structure of this monograph
References
2 . Symmetry, symmetry transformations and symmetry conditions
2.1 Introduction
2.2 Geometric transformations and the concept of symmetry
2.2.1 Reflectional transformation and reflectional symmetry
2.2.2 Rotational transformation and rotational symmetry
2.2.3 Translational transformation and translational symmetry
2.2.4 Symmetry as a mathematical study
2.3 Symmetry of physical fields
2.4 Continuity and free body diagrams
2.5 Symmetry conditions
2.5.1 Reflectional symmetry
2.5.2 180° rotational symmetry
2.5.3 Translational symmetry—one-dimensional scenario as an introduction
2.5.4 Translational symmetry conditions in three-dimensional scenarios
2.6 Concluding remarks
References
3 . Material categorisation and material characterisation
3.1 Background
3.2 Material categorisation
3.2.1 Homogeneity
3.2.2 Anisotropy
3.2.2.1 Reflectional symmetry
3.2.2.2 Rotational symmetry
3.3 Material characterisation
3.4 Concluding remarks
References
4 . Representative volume elements and unit cells
4.1 Introduction
4.2 RVEs
4.2.1 Representativeness
4.2.2 Zone affected by boundary effects and the concept of decay length
4.3 UCs
4.3.1 Regularity
4.3.2 The role of translational symmetries
4.3.3 Identification of cells based on the available translational symmetries
4.3.4 Mapping from the unit cell to any other cell and the relationship between paired pieces of the boundary of the unit cell
4.4 Concluding remarks
References
5 . Common erroneous treatments and their conceptual sources of errors
5.1 Realistic or hypothetic background
5.2 The construction of RVEs and their boundary
5.3 The construction of UCs
5.3.1 Problems associated with the abuse of reflectional symmetries
5.3.2 Rotational symmetries
5.3.3 Translational symmetries
5.3.4 Redundant boundary conditions
5.3.5 Incomplete use of available symmetries present in the microstructure
5.3.6 A unit cell as an assembly of multiple cells
5.3.7 Essential and natural boundary conditions
5.4 Post-processing
5.5 Implementation issues
5.6 Verification and the lack of ‘sanity checks’
5.7 Concluding remarks
References
Part Two: Consistent formulation of unit cells and representative volume elements
6 . Formulation of unit cells
6.1 Introduction
6.2 Relative displacement field and rigid body rotations
6.3 Relative displacement boundary conditions for unit cells
6.4 Typical unit cells and their boundary conditions in terms of relative displacements
6.4.1 2D unit cells
6.4.1.1 An introduction to 2D idealisation
6.4.1.2 2D unit cells with translational symmetries along coordinate axes
6.4.1.3 2D unit cell with translational symmetries along two non-orthogonal directions
6.4.1.4 2D unit cells in presence of more than two translational symmetries
6.4.2 3D unit cells
6.4.2.1 Introduction
6.4.2.2 3D unit cell with translational symmetries along three non-coplanar axes
6.4.2.3 3D unit cell for SC packing
6.4.2.4 3D unit cell for FCC packing
6.4.2.5 3D unit cell for body centred cubic packing (BCC)
6.4.2.6 3D unit cell for close packed hexagonal packing (CPH)
6.4.2.7 A unit cell for laminated composites
6.4.2.8 A unit cell from Cn rotational symmetry (Li et al., 2014)
6.5 Requirements on meshing
6.6 Key degrees of freedom and average strains
6.7 Average stresses and effective material properties
6.8 Thermal expansion coefficients
6.9 “Sanity checks” as basic verifications
6.10 Concluding remarks
References
7 . Periodic traction boundary conditions and the key degrees of freedom for unit cells
7.1 Introduction
7.2 Boundaries and boundary conditions for unit cells resulting from translational symmetries
7.3 Total potential energy and variational principle for unit cells under prescribed average strains
7.4 Periodic traction boundary conditions as the natural boundary conditions for unit cells
7.5 The nature of the reactions at the prescribed key degrees of freedom
7.6 Prescribed concentrated ‘forces’ at the key degrees of freedom
7.7 Examples
7.7.1 A 2D square unit cell
7.7.1.1 Prescribed average strains
7.7.1.2 Prescribed average stresses
7.7.2 A 2D hexagonal unit cell
7.7.3 A 3D rhombic dodecahedron unit cell for FCC packing
7.8 Conclusions
References
8 . Further symmetries within a UC
8.1 Introduction
8.2 Further reflectional symmetries to existing translational symmetries
8.2.1 One reflectional symmetry
8.2.1.1 Boundary conditions under a symmetric loading (any of σx0, σy0, σz0 and τyz0 or their combination)
8.2.1.2 Boundary conditions under an antisymmetric loading (any of τxz0 and τxy0 or their combination)
8.2.1.3 Unification of formulation of the boundary conditions for single reflectional symmetry
8.2.2 Two reflectional symmetries
8.2.2.1 Boundary conditions under σx0, σy0  and σz0
8.2.2.2 Boundary conditions under τyz0
8.2.2.3 Boundary conditions under τxz0
8.2.2.4 Boundary conditions under τxy0
8.2.2.5 Unification of formulation of the boundary conditions for two reflectional symmetries
8.2.3 Three reflectional symmetries
8.2.3.1 Boundary conditions under σx0, σy0  and σz0
8.2.3.2 Boundary conditions under τyz0
8.2.3.3 Boundary conditions under τxz0
8.2.3.4 Boundary conditions under τxy0
8.2.4 Various examples of application
8.2.4.1 Application to the 2D UC for square packed UD composites
8.2.4.2 Application to the UC (UC) for the simple cubic packing (SC)
8.3 Further rotational symmetries to existing translational symmetries
8.3.1 One rotational symmetry
8.3.1.1 Boundary conditions under a symmetric loading (any of σx0, σy0, σz0 and τxy0 or their combination)
8.3.1.2 Boundary conditions under an antisymmetric loading (any of τyz0 and τxz0 or their combination)
8.3.2 Two rotational symmetries
8.3.2.1 Boundary conditions under σx0, σy0  and σz0
8.3.2.2 Boundary conditions under τyz0
8.3.2.3 Boundary conditions under τxz0
8.3.2.4 Boundary conditions under τxy0
8.3.3 Application to 3D 4-axial braided composites where more symmetries are present
8.3.3.1 Boundary conditions under σx0, σy0, σz0 or any combination of them (symmetric)
8.3.3.2 Boundary conditions under τyz0 (antisymmetric)
8.3.3.3 Boundary conditions under τxz0 (symmetric)
8.3.3.4 Boundary conditions under τxy0 (antisymmetric)
8.4 Examples of mixed reflectional and rotational symmetries
8.4.1 Hexagonal packing
8.4.1.1 Boundary conditions under σx0, σy0  and σz0
8.4.1.2 Boundary conditions under τyz0
8.4.1.3 Boundary conditions under τxz0
8.4.1.4 Boundary conditions under τxy0
8.4.2 Plain weave
8.4.2.1 Boundary conditions under σx0, σy0, σz0 or any combination of them
8.4.2.2 Boundary conditions under τyz0
8.4.2.3 Boundary conditions under τxz0
8.4.2.4 Boundary conditions under τxy0
8.5 Centrally reflectional symmetry
8.6 Guidance to the sequence of exploiting existing symmetries
8.7 Concluding statement
References
9 . RVE for media with randomly distributed inclusions
9.1 Introduction
9.2 Displacement boundary conditions and traction boundary conditions for an RVE
9.3 Decay length for boundary effects
9.4 Generation of random patterns
9.5 Strain and stress fields in the RVE and the sub-domain
9.6 Post-processing for average stresses, strains and effective properties
9.7 Conclusions
References
10 . The diffusion problem
10.1 Introduction
10.2 Governing equation
10.3 Relative concentration field
10.4 An example of a cuboidal unit cell
10.5 RVEs
10.6 Post-processing for average concentration gradients and diffusion fluxes
10.7 Conclusions
References
11 . Boundaries of applicability of representative volume elements and unit cells
11.1 Introduction
11.2 Predictions of elastic properties and strengths
11.3 Representative volume elements
11.4 Unit cells
11.5 Conclusions
References
Part Three: Further developments
12 . Applications to textile composites
12.1 Introduction
12.1.1 Background
12.1.2 Composites made of woven preforms
12.1.3 Composites made of braided preforms
12.2 Use of symmetries when defining an effective UC
12.3 Unit cells for two-dimensional textile composites
12.3.1 Idealisations in the thickness direction
12.3.2 Plain weave
12.3.3 Twill weave
12.3.4 Satin weaves
12.3.5 2D 2-axial braid
12.3.6 2D 3-axial braids
12.4 Unit cells for three-dimensional textile composites
12.4.1 3D weaves
12.4.2 3D braids
12.5 Conclusions
References
13 . Application of unit cells to problems of finite deformation
13.1 Introduction
13.2 Unit cell modelling at finite deformations
13.2.1 Boundary conditions
13.2.2 Stress averaging
13.2.3 An assertion
13.2.4 Verification through FE modelling using Abaqus
13.2.4.1 Applying nodal displacements at Kdofs
13.2.4.2 Applying concentrated forces at Kdofs
13.2.5 Procedure for post-processing
13.2.6 Rotations
13.3 The uncertainties associated with material definition
13.4 Concluding remarks
References
14 . Automated implementation: UnitCells© composites characterisation code
14.1 Introduction
14.2 Abaqus/CAE modelling practicality
14.2.1 Selection of the shape of the unit cell
14.2.2 The dimensions of the unit cell and the unit system
14.2.3 Meshing to satisfy geometric periodicity
14.2.4 Element selection and mesh density
14.2.5 Imposition of relative displacement boundary conditions
14.2.6 Definition of constituent materials
14.2.7 Load case generation
14.2.8 Flowchart of the UnitCells© code
14.2.9 Available types of unit cells and possible multiscale modelling
14.3 Verification and validation
14.4 Concluding remarks
References
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
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W
Y
Back Cover

Citation preview

REPRESENTATIVE VOLUME ELEMENTS AND UNIT CELLS Concepts, Theory, Applications and Implementation

SHUGUANG LI University of Nottingham Nottingham, United Kingdom

ELENA SITNIKOVA University of Nottingham Nottingham, United Kingdom

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright Ó 2020 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-102638-0 For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisition Editor: Gwen Jones Editorial Project Manager: Isabella C. Silva Production Project Manager: Debasish Ghosh Cover Designer: Mark Rogers Typeset by TNQ Technologies

To Harry SL To Genevieve ES as special individuals of the generation that has yet the entire canvas to paint their future on

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

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Automated implementation: UnitCellsÓ composites characterisation codePreface

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 UnitCellsÓ, 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. UnitCellsÓ has demonstrated itself as an useful and reliable platform for systematic multiscale characterisation of composite materials, from unidirectional fibre reinforced ones

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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. Shuguang Li July 2019

CHAPTER 1

Introduction d background, objectives and basic concepts 1.1 The concept of length scales and typical length scales in physics and engineering A length scale is a range of lengths as a valid measure involved in a physical or engineering problem. Beyond the range, the physical or engineering problem will no longer hold in its formulation and validity, usually because of the restrictions introduced in defining the problem. Many modern materials involve different length scales in different aspects of the behaviour of the material. Traditionally, the length scale in engineering is macroscopic, in which engineering artifacts are designed, manufactured and deployed. It has therefore been taken for granted in conventional studies, such as theoretical mechanics, solid, fluid and continuum mechanics, mechanisms and structures, etc. Modern science and technology have allowed the human vision to be extended to substantially more refined scales. In the meantime, materials can now be manipulated at a mesoscopic (typically around millimetre), microscopic (from a few to tens of microns) and, nowadays, nanoscopic (from tens to hundreds of nanometers) scales. For practical applications of fibre reinforced composites, micro-, meso- and macroscopic scales are common places. The discussion in this monograph will be confined to these scales. Even so, it crosses three typical length scales. Without being overly restrictive in the narrative in the text, length scales will be referred to in a relative relationship as a lower length scale and an upper length scale. If the microscopic scale is considered as the lower one, the upper one could be mesoscopic as well as macroscopic, whilst mesoscopic scale is the upper one for microscopic but lower one for macroscopic scale. Mesoscopic scale is particularly relevant to the modelling the behaviour of plies in conventional laminates and fibre tows in textile composites. Sometimes in this monograph, length scales might be directly referred to as macroscopic, mesoscopic and microscopic if it is more relevant to the topic under discussion. This is usually associated with specific applications, for instance, unidirectionally fibre reinforced (UD) composites, where fibre diameters are conventionally restricted to a narrow range in microns Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00001-3

© 2020 Elsevier Ltd. All rights reserved.

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practically. It will therefore be helpful to use the relevant length scale to make the context easier to associate with reality, especially for those not very familiar with the terminologies of multiscale modelling.

1.2 Multiscale modelling Multiscale modelling refers to a theoretical approach in which analysis of the material is conducted at one length scale but the outcomes of the analysis deliver useful information about the material at another length scale. The purpose of it is usually to characterise a material by analysing it at a lower length scale, taking full account of the architectures in the construction of the material and the properties of different constituents quantifiable at the lower length scale. As the outcome of material characterisation exercise, average properties of the material at the upper length scale are obtained, which are sometimes also referred to as effective or macroscopic properties. Throughout this monograph, all three terms will be used interchangeably. The same applies to the physical fields, such as stresses and strains. To facilitate multiscale modelling, some assumptions are usually employed in a theoretical approach. An essential assumption is that there exist a finite volume of the material whose behaviour represents that of the bulk of the material. The basis of the assumption is the uniformity or homogeneity of the material at one length scale. Usually, it will be considered as the upper length scale. Up to this point, there is nothing new as this is the basis for the characterisation of the material conventionally. For instance, this is the consideration behind physical characterisation testing, where test coupons are meant to represent the material. Multiscale modelling differs from conventional practices by minimising the representative volume to such an extent that its dimensions fall into a length scale lower. As a result, the representative volume can be perceived to be infinitesimally small and can be effectively considered as a material point at the upper length scale. On the other hand, the representative volume is of a finite domain at the lower length scale in which the physical problem can be re-established after taking account of the intricacies within this representative volume. The objective of introducing a representative volume is to allow the formulation of a properly posed mathematical problem d a boundary value problem d that can be solved one way or another. Given the limited availability and applicability of analytical approaches, numerical solutions become the norm nowadays, typically, using the finite element method (FEM). Upon solving the problem, the desirable physical fields, such as

Introduction d background, objectives and basic concepts

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stress, strain, displacement, temperature, etc. become known. Their average values are expected to be corresponding values respectively at the material point at the upper length scale, from which the constitutive relationships of the material at the upper length scale can be obtained and the material is thus characterised.

1.3 Representative volume element and unit cell A key step in multiscale modelling in general is the appropriate definition of the representative volume. Once it has been specified as a certain mathematical domain of definitive dimensions, it is usually called a representative volume element (RVE). The key requirement for an RVE is its representativeness, i.e. its behaviour in an average sense should be a truthful representation of that of the material at the upper length scale. The applicability of the RVE relies on the homogeneity of the material and the uniformity of the physical fields defining the constitutive relationship of the material at the upper length scale, either physically or statistically. Obviously, an RVE representative enough for one aspect of material behaviour may not be so for another aspect of the behaviour of the same material. A special type of RVEs is called unit cells (UCs), which are formulated based on the regularity in the architecture of the material at the lower length scale. The regularity can be a direct geometric representation or some kind of idealisation based on statistical features at the lower length scale. In the case of statistical-based measure, one has to be aware of the implications coming with the measure taken in order to achieve the desired idealisation, as some of them could compromise the representativeness of the UC in some respects, as will be emphasised appropriately in this monograph where the issues emerge. The physical homogeneity at upper length scale and the geometric regularity at the lower length scale can be undermined to different extents due to the manufacturing practicality. Appropriate allowances should be kept on the accuracy of the results predicted from the multiscale modelling. One might use the statistical homogeneity and idealised regularity as a reference position if one wishes to quantify the effects of the deviation from the physical homogeneity at the upper length scale or the geometric regularity at the lower length scale by incorporating the actual deviation into the model. However, this should not be turned into a routine practice for every exception encountered in reality as one might miss the benefits of multiscale modelling completely.

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1.4 Background of this monograph Given the establishment of multiscale modelling as a methodology in modern physical sciences and engineering, the use of RVEs and UCs has become a popular means of analysis as can be found in vast number of papers at conferences and in academic journals. A fair number of PhD theses around the world contain results obtained from RVEs or UCs one way or another. However, the presentations of most of such contents in most of these publications tend to share some commonality: being somewhat incomplete. A scientific measure of publications is that the results can be reproduced by others following the same procedure. An anomaly with RVEs and UCs, in particular, is that the results cannot be reproduced because the procedures provided are often short of a key piece of information, namely, the boundary conditions. Even if they are provided in some of the publications, they are mostly presented without much justification, if any. As has been mentioned earlier, the objective of RVEs or UCs is to deliver a boundary value problem mathematically. Any solution to a boundary value problem will become rather absurd if the boundary conditions have not been specified. As readers will find later in this monograph, boundary conditions for UC are not formidable to obtain but they are by no means trivial, either. It should not be too difficult for readers to conclude why they have often been absent. This sets the background to many years of dedicated work by the authors who believe that it is time to bring perfect clarity to this subject, which is the ultimate aim of this monograph.

1.5 Objectives of this monograph Multiscale modelling using RVEs and UCs has been widely employed as an effective tool for material characterisation, but it is meaningful only if it has been carried out systematically and consistently in a reliable and efficient manner. Any inconsistency could cast doubt on the applicability of multiscale modelling. The main objective of this monograph is to ensure the crucial consistency and to offer a comprehensive procedure by providing a full account of relevant concepts and principles. The subject can thus be established as a coherent study to fill the gaps and resolve misperceptions and confusion over it, whilst presenting readers and potential users a crystal clear account as the subject deserves. On the other hand, the RVEs and UCs have their own boundaries of applicability which may not always be defined clearly in the literature and hence

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have not always been duly respected. As a result, they can easily be subjected to abuses. Light will also be shed to this aspect as a part of the monograph. Much of the efforts in this monograph will be placed on the formulation of the RVEs and UCs to facilitate multiscale modelling. Some of the considerations are generic and hence applicable to each of the RVEs or UCs constructed, whilst others may be specific and apply to special aspects of the problem. The numerical solver will be confined to FEM by default and hence the emphasis will not be on how to solve complicated governing equations. Rather, it is to prepare the problem in such a way that FEM can be employed correctly and effectively. In particular, one will find that the most demanding aspect is the formulation of the boundary conditions so that they ensure a truthful representation of the physical or geometric characteristics of the material architecture. Given the objective as set above, the demand on the readers and, in particular, the users will not be the mathematical sophistication. However, one must be prepared for two challenges. Firstly, some of the mathematical concepts, such as symmetry, appear to be fairly common sense but there are more contents in them. Whilst the common sense part can be easily found in textbooks and hardly needs much explanation, the rest have hardly been covered and taught appropriately in any subject. As a result, when symmetry is encountered, e.g. in the analysis of symmetric structures, it is usually the simplest scenarios that are dealt with, e.g. reflectional symmetry. For the other types of symmetry, the readers are usually left to improvise as if it is meant to be loose or a self-explanatory subject. A significant effort will be made in this monograph to fill in this gap by producing a long-missing comprehensive account. For this part, readers will find familiar topics presented in great details but every now and then subtleties emerge, accounts on which in the literature are always very confusing. One such example is the use of translational symmetries which lays the basis for the so-called periodic boundary conditions. Careful user must have noticed that such periodic boundary conditions are in fact not always periodic, reproducing a modern version of an ancient Chinese paradox: “White horses are not horses” (Gongsun Long, 320e250BC) (Lucas, 2012). A subject cannot be consistent and clear enough if one still has to face paradoxes like this. It will be shown that confusions of this kind are due to the missing links in understanding, where misperception overshadows the simple truth, which turns out to be simple only after it has been revealed and connected coherently with existing knowledge framework.

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The second challenge will be that the derivations and elaborations can often be tedious with contents being seemingly repetitive, yet presented again and again from place to place. On the other hand, waiving them would easily make the presentation hard to comprehend and hence difficult to reproduce if one wishes to follow the procedure. The authors have therefore chosen to sacrifice conciseness for clarity. The actual implementation of the procedures formulated is in fact even more tedious. Fortunately, a vast amount of repetitive tasks can be taken by computers, with the availability of modern computing power and versatile software. The responsibility of the users reduces to issuing correct commands for computers to execute, in particular, by prescribing correct boundary conditions, as has been demonstrated in the code the authors and their co-workers developed over the recent years as published on a designate website provided by Elsevier. Multiscale nature is one of characteristics of composites. For material design and characterisation, it is often desirable to derive effective properties of a composite at an upper length scale from those of its constituents at a lower scale. The methodology established in present monograph, as also demonstrated through the code mentioned above, offers a powerful means to facilitate this process after laying it on a firm basis of a rigorously formulated theoretical framework.

1.6 The structure of this monograph The ethos of this monograph stems from a chapter the authors contributed to Comprehensive Composite Materials II (Li and Sitnikova, 2018) but as a much substantialised, broadened and detailed account on the same subject. In order to deliver the objectives as set, this monograph will be presented in three parts, consisting of fourteen chapters altogether. The basics, including concepts and principles, are covered in Part I after this introductory chapter, some of which, though basic, may not have been found anywhere else as a comprehensive account, e.g. the concept of symmetry in Chapter 2, especially their association with physical fields, the concept of material categorisation and the role of rotational symmetries in material categorisation in Chapter 3, and the logical relationship between RVEs and UCs in Chapter 4. As a highlight of this part, it finishes with an account in Chapter 5 on the common mistakes in the applications of RVEs and UCs as a result of oversights or abuses of the basic concepts and principles. Part II is a substantial part of this monograph addressing the formulation of RVEs and UCs, in particular, the boundary conditions, including both

Introduction d background, objectives and basic concepts

9

the general considerations as well as those specific for particular problems, such as applications of the concepts and principles laid down in Part I of the monograph. Out of the diversity of various applications, readers are expected to reach convergence back to the basic concepts and principles. In other words, whilst RVEs and UCs may differ one from another, leading to different shapes and expressions of boundary conditions, constantly exacerbated by the lack of uniqueness in approaches and choices, the underlying considerations never change. This is the key message conveyed by Chapter 6. There are a few convenient treatments, or the lack of the need of any treatment, that emerge once the UCs have been systematically formulated, for instance, the post-processing of the results from an FE analysis of a UC. They could be perceived as fortunate coincidences. However, they can be mathematically proven. Chapter 7 offers such an account as the profound and rigorous theoretical basis of UCs, which provides solid mathematical basis behind the establishment of UCs. For the purpose of defining UCs, translational symmetries alone would be sufficient and this is the proper way of constructing UCs. However, the presence of other symmetries, reflectional and rotational, whilst useful, could be the source of confusions. Their proper use has been elaborated reasonably comprehensively in Chapter 8 to dispel any doubt and uncertainty. The formulation of RVEs specifically is covered in Chapter 9, where the key concept is their representativeness. Prescriptions of boundary conditions as well as the post-processing of the results all have to reflect this key consideration. In terms of post-processing, the distinction between intuition and mathematical rigour has been elaborated as another example of proper theoretical exercise. The established practices from the concept to applications are generic and hence apply to other physical disciplines. As an example, Chapter 10 crosses the boundaries of disciplines and an account has been generated for the subject of diffusion problems which underlies a number of rather different physical processes, from heat conduction to fluid flow through porous materials. Part II finishes with an account in Chapter 11 on the applicability of RVEs and UCs, warning the users that the boundaries of applicability should not be crossed without grave considerations and reasonable justifications. Further applications, both in breadth and depth, are explored in Part III over a couple of carefully selected topics in which the authors have firsthand experience. Whilst addressing the popular applications of textile composites in order to disentangle confusing considerations as present on the topic, very much owing to the abundance of the available symmetries in the architecture of textile preforms, it has also offered a good opportunity

10

Representative Volume Elements and Unit Cells

in Chapter 12 for the readers to recapitulate the basic concepts and principles as laid down earlier in previous parts of the monograph. Another very tempting topic on finite deformation is presented in Chapter 13. Applications beyond the regime of small deformation are fairly common in engineering nowadays, whether as deliberate design or blind extrapolation. Most of modern FE codes are equipped with relevant facilities. However, applications of RVEs and UCs in the finite deformation regime cannot be taken light-heartedly. This has been fully demonstrated in this chapter. In many ways, the chapter is more revealing than conclusive. A number of crucial issues lying under the surface do not seem to be clear at all, and typically remain unnoticed by the users. In particular, whilst there has been the lack of established and consistent theory on the changes of the material’s principal axes for anisotropic materials as a result of finite deformation, FE codes, such as Abaqus as demonstrated, tend to make a rather arbitrary assumption on users’ behalf and yet keeps the details hidden from the users. To be alerted by such practices is an understatement in this respect. The part as well as the whole monograph complete with a demonstration in Chapter 14 of a piece of software UnitCells© developed by the authors and their co-workers as a secondary development of Abaqus, where the formulations as presented through the monograph, that are based on clear concepts and principles, have been consistently and successfully implemented. After all, for engineers as the authors would label themselves, theories are established for applications and implementations. The whole monograph does just that, no more and no less, at least it is intended to do so and it is for the readers to tell whether this target was hit. The authors hope that the monograph will be of value to researchers in material sciences and many sections of engineering. It is deemed to be unsuitable as a textbook, but some topics, in particular, the account on symmetries, the concept of material categorisation and significance of correct boundary conditions in boundary value problems, should offer informative supplements to students of all levels, as they should have been taught that ideally, but never practically due to the lack of availability of established information.

References Li, S., Sitnikova, E., 2018. 1.18 An excursion into representative volume elements and unit cells. In: Beaumont, P.W.R., Zweben, C.H. (Eds.), Comprehensive Composite Materials II. Elsevier, Oxford. Lucas, T., 2012. Why white horses are not horses and other Chinese puzzles. Logique et Analyse 55, 185e203.

CHAPTER 2

Symmetry, symmetry transformations and symmetry conditions 2.1 Introduction Geometric patterns often have special feature called symmetry. Symmetry as a subject in mathematics often switches too quickly to the abstract study of group theory. It therefore distances itself from engineering applications. On the other hand, for engineering students, the study of symmetry never makes a systematic subject to be taught. It is usually introduced in an intuitive manner as a part of structural mechanics or finite element analysis for some specific applications. As a terminology, symmetry has been widely used. However, users of the terminology commonly show a state of confusion when a slightly more scientific description is required. For instance, no one would hesitate to spot the mirror, or reflectional, symmetry about the central axis in the shape as shown in Fig. 2.1(a). On the other hand, when a shape as shown in Fig. 2.1(b) is concerned, a range of different terms may emerge, such as skewed symmetry or antisymmetry, without giving much clarity to the concept but only implying that it is being different from reflectional symmetry.

(a)

(b)

Fig. 2.1 Symmetric shapes: (a) reflectional and (b) rotational symmetry. Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00002-5

© 2020 Elsevier Ltd. All rights reserved.

11

j

12

Representative Volume Elements and Unit Cells

A proper mathematical description of type of symmetry for the shape as shown in Fig. 2.1(b) is 180 rotational symmetry about its center. The point made through this example is that clear definitions are necessary in order to understand the nature of different types of symmetry, and they will become crucial if one intends to take advantage of them in serious applications. Otherwise, confusion is inevitable.

2.2 Geometric transformations and the concept of symmetry The reflection and rotation as illustrated through Fig. 2.1 are conveniently associated with respective geometric transformations. In fact, there is yet another very important transformation, which is called translation. A simplistic illustration of each of the three types of transformations is shown in Fig. 2.2. They will be presented systematically in detail as follows.

2.2.1 Reflectional transformation and reflectional symmetry The geometric transformation mapping a point from one side of an axis to the opposite side of the axis at an equal distance from the axis is called a reflectional transformation made to that point, as demonstrated in Fig. 2.2(a). When a reflectional transformation is applied to every point in a shape, it is a transformation made to the shape concerned, e.g. the dark gray triangle is transformed to the light gray one in Fig. 2.2(a). In general, the axis defining a reflectional transformation does not have to be a coordinate axis, but it has been chosen so in Fig. 2.2(a) for the ease of presentation. Generalising the above description from a two-dimensional scenario to three-dimensional problems, a reflection will be about a plane. Similar to the axis defining a reflectional transformation in two-dimensional case, the plane defining a reflectional transformation does not have to be a coordinate plane, or be parallel to a coordinate plane. In practical applications, it

(a)

y

(-x,y)

(b)

y

(x,y) x

(c) (x,y)

(-x,-y)

x

y x

(x+Δ, y)

Fig. 2.2 Transformations: (a) reflectional; (b) rotational; and (c) translational.

Symmetry, symmetry transformations and symmetry conditions

13

often appears to be inclined, whilst coincidence with a coordinate plane being a special case. A reflectional transformation is usually denoted Sp, where the subscript p refers to plane of reflection, p-plane. Sometimes, when the plane of reflection is not specified, notation S is simply used. A reflectional symmetry can then be defined properly through the transformation introduced above: if an object or geometric pattern keeps its shape unchanged after a reflectional transformation, it is reflectionally symmetric. It is easy to see that the shape in Fig. 2.1(a) possesses reflectional symmetry according to the definition given.

2.2.2 Rotational transformation and rotational symmetry A rotational transformation in a general three-dimensional case maps a point to another point in the same plane perpendicular to the axis of rotation the same distance away from of the axis of rotation. Fig. 2.2(b) offers a twodimensional illustration with an angle of rotation of 180 , where the axis of rotation degenerates to a point in the plane perpendicular to the axis. As far as such a rotational transformation is concerned, the angle of rotation can be of an arbitrary magnitude. However, for its applications to the symmetry problems, it will have to be equal to 360 /n with n being a positive integer. In general, the axis of a rotation does not have to be a coordinate axis but it has been chosen to coincide with the z-axis (pointing out of the page) in Fig. 2.2(b) for the ease of presentation. For a rotational transformation with the angle of rotation of 360 /n, the rotational transformation is usually denoted as Can if the rotation is about axis a by an angle of 360 /n, or Cn without specifying the axis of rotation. The most commonly encountered angle of rotation is 180 , i.e. C2, as shown in Fig. 2.2(b). If an object keeps its shape unchanged after a rotational transformation, it is called rotationally symmetric. Depending on the number n which can take different values, rotational symmetry can have a variety, e.g. Ca2 , Ca3 , etc., unlike the reflectional symmetry which is unique about a given plane or axis. The reflectional symmetry is undoubtedly the most familiar type to the readers to such an extent that sometimes it is perceived as the only type of symmetry or the most important type of symmetry. This perception is echoed by the fact that in the commercial FE codes, e.g. Abaqus (2016), it is the only type of symmetry available. Yet, the rotational symmetry as introduced here is just as important as its reflectional counterpart, in

14

Representative Volume Elements and Unit Cells

particular, in the analysis of composites (Li and Reid, 1992). For instance, reflectional symmetries can be absent from many micro/meso/macro structures where rotational symmetries are often available. In particular, laminates of arbitrary layup usually do not show any reflectional symmetry. However, they can be 180 rotationally symmetric about the axis perpendicular to the plane of the laminate (Li and Reid, 1992).

2.2.3 Translational transformation and translational symmetry Translational symmetry is often overlooked. However, in many ways, it is probably the most important type of symmetry in applications to material characterization and the formulation of unit cells as is the subject of the present book (Li, 1999, 2001, 2008; Li and Wongsto, 2004). Without it, materials cannot be logically homogenized and unit cells cannot be appropriately established, as will be discussed in Chapter 5. Whilst reflectional and rotational symmetries could be present in a shape of a finite extent, a translational symmetry is present only in patterns of an infinite extent. It is precisely the reason why the translational symmetry plays a unique role in the formulation of unit cells and cannot be replaced by other types of symmetry, since no other type of symmetry can reduce an infinite extent a material occupies to a unit cell of finite extent. A pattern of translational symmetries is shown in Fig. 2.3 which should be considered as a patch extracted from an infinite extent. To define the concept of translational symmetry, one can start with the definition of the respective transformation, as was done in the previous

y

η

Arrows for directions of translations

ξ x

Fig. 2.3 A pattern of translational symmetry, assuming infinite extent with only a finite patch shown.

Symmetry, symmetry transformations and symmetry conditions

15

subsections. Translational transformation which maps a point to another along a given direction x by a distance of D is illustrated in Fig. 2.2(c). It is denoted as TxD , or TD without specifying the direction of translation. A pattern (of infinite extent) which keeps its configuration unchanged after a translational transformation, is translationally symmetric. Such objects are also described as periodic, with the periodicity being in the direction of translational symmetry and the respective translation as the period. Like periodicity, if there is a translational symmetry of TD, translational transformations TnD, where n is an integer, are all symmetric. For practical considerations, one is interested in the one with the smallest magnitude of the symmetric translation in that direction. A pattern can have multiple translational symmetries in different directions and each can have a different magnitude of translation. The directions do not have to coincide with coordinate axes and they do not necessarily have to be perpendicular to each other, either. For instance, the twodimensional pattern in Fig. 2.3 has translational symmetries along x-, y-, x- and h-directions, respectively, with different translations. In general, for an N-dimensional pattern, amongst all available translational symmetries, a maximum of N of them are independent. Specifically, amongst x, y, x and h in Fig. 2.3, only two are independent. Taking x and h as the independent ones, x is a combination of x and h with a positive and a negative translation, respectively, whilst y is a combination of x and h with a positive translation in each of them. The availability of redundant translational symmetries gives users more choices but may become confusing. However, since this comes with practical problems and is beyond users’ choice, one has to deal with them as they are. Appreciation of the available symmetries and understanding of the interrelationships amongst them is the key to come out of the confusion, if any. Practical applications of symmetries in the context of the present book are to reduce the size of the object to be analyzed. In this respect, the use of reflectional symmetry, S, halves the size and a rotational symmetry, Cn, reduces the size to 1/n of it. For an object of infinite extent, both a half and 1/n are still infinite in extent. The translational symmetry is the only type of symmetry that can reduce the size from an infinite extent to a finite one. For instance, TxD can bring the size down from infinite in the x-direction to a segment of length D in the x-direction. Despite this important special feature of translational symmetry, it has never been made a popular topic to be covered in any engineering subject.

16

Representative Volume Elements and Unit Cells

2.2.4 Symmetry as a mathematical study Apart from the types of symmetries as introduced above, there can be others. In terms of reflections, there can be a reflection about a plane as introduced above, about an axis (identical to 180 rotation), about a point, etc. However, a careful examination will reveal that they are combinations of reflection and rotation and hence are not independent. For instance, reflection about a point, or central reflection, is the combination of a reflection about a plane followed by a 180 rotation about the axis perpendicular to the plane of reflection, or verse versa, as the sequence makes no difference. Therefore, the three types as introduced, viz. reflection, rotation and translation, are three generic types of symmetries, although sometimes convenience can be found with some special combinations of them, e.g. the central reflection (Li and Zou, 2011). As a transformation, the one mapping a point to itself is called an identical transformation. Whilst it is the most trivial type, it is perhaps the most important one mathematically. If all symmetries as introduced above are put together as a set with each symmetry as an element of the set, mathematicians can form a group after introducing certain operations between the elements of the set. If one wishes to get to the bottom of intrinsic nature of symmetries, the so-called group theory as a mathematical study will be the way forward, which is often beyond the scope of engineering and certainly this book. Interested readers are referred to (Humphreys and Prest, 2004), for instance, where the contents are extremely abstract and sophisticated for a non-mathematician. Fortunately, for engineering applications, a clear definition of symmetries as provided in previous subsections is usually sufficient, at least for the purpose of this book.

2.3 Symmetry of physical fields As sketched in Fig. 2.2, geometric transformations can be expressed as mapping from original P at the coordinate (x, y, z) to its image P0 at (x0 , y0 , z0 ). For practical formulation of unit cells as will be presented in Chapter 6 systematically, they can be expressed mathematically as follows. Reflection about plane p parallel to yz-plane, which will be referred to as x-plane, defined by x ¼ x0, is written as: Sp :

Pðx  x0 ; y; zÞ / P 0 ð  ðx  x0 Þ; y; zÞ;

(2.1)

Symmetry, symmetry transformations and symmetry conditions

17

Rotational transformation about an axis parallel to the x-axis, defined by y ¼ y0 and z ¼ z0, becomes: Ca2 :

Pðx; y  y0 ; z  z0 Þ / P 0 ðx;  ðy  y0 Þ;  ðz  z0 ÞÞ;

(2.2)

Translation along axis x by Dx, Dy and Dz is defined as: TaD :

Pðx; y; zÞ / P 0 ðx þ Dx ; y þ Dy ; z þ Dz Þ:

(2.3)

In order to generalise the definition, one can similarly introduce reflections about other coordinate planes, and even planes other than coordinate planes. This is, however, of little practical use as far as the present book is concerned, since one can usually select the coordinate system in such a way that the reflection plane coincides with one of the coordinate planes. A special case when x0 ¼ 0 corresponds to the reflection about the x-plane. Equally, rotations can be readily expressed about other axes. A special case of x0 ¼ y0 ¼ 0 defines the rotation is about the z-axis. When any one of translational distances,Dx, Dy and Dz, vanishes, the translation is in a plane. If two of them vanish, the translation is along the coordinate axis associated with the non-vanishing value. Objects or patterns keeping their shapes under transformations as introduced above are of respective symmetries. Only the geometric considerations have been given so far. Physical problems are all defined in certain geometric domains. In the discussion that follows, the concept of geometric symmetry will be extended to physical fields. In defining the symmetry of a physical field, a default assumption is that the domain in which the physical field is defined is symmetric in geometry, reflectionally, rotationally and/or translationally, without which, the symmetry of a physical field is simply irrelevant. As was for the geometry, the concept of mapping of physical fields can be introduced. For a physical scalar field, f, the following transformations can be defined, corresponding to their geometric counterparts as defined in Eqs. (2.1)e(2.3), respectively. Sp :

Reflection : Rotation :

Ca2 :

Translation :

fðx  x0 ; y; zÞ / fð  ðx  x0 Þ; y; zÞ

(2.4)

fðx; y  y0 ; z  z0 Þ /fðx; ðy  y0 Þ; ðz  z0 ÞÞ (2.5)

TaD :

fðx; y; zÞ / fðx þ Dx ; y þ Dy ; z þ Dz Þ

(2.6)

18

Representative Volume Elements and Unit Cells

Consider the reflectional transformation as an example. In the case of a one-dimensional field, i.e. when 4 is a single variable function, the transformation can be illustrated graphically as shown in Fig. 2.4 where the domain for the definition of the problem is the interval [x1, x2] which is symmetric about point x0. For a one-dimensional domain, its geometric reflectional symmetry is implied if point x0 bisects interval [x1, x2]. The reflectional transformation maps the blue curve to the red one in a one-to-one manner. With respect to the transformations as specified above, a scalar physical field is reflectionally, rotationally and translationally symmetric if fð  ðx  x0 Þ; y; zÞ ¼ fðx  x0 ; y; zÞ;

(2.7)

fðx; ðy  y0 Þ; ðz  z0 ÞÞ ¼ fðx; y  y0 ; z  z0 Þ;

(2.8)

fðx þ Dx ; y þ Dy ; z þ Dz Þ ¼ fðx; y; zÞ;

(2.9)

respectively. It is reflectionally, rotationally and translationally antisymmetric if fð  ðx  x0 Þ; y; zÞ ¼ fðx  x0 ; y; zÞ;

(2.10)

fðx; ðy  y0 Þ; ðz  z0 ÞÞ ¼ fðx; y  y0 ; z  z0 Þ;

(2.11)

fðx þ Dx ; y þ Dy ; z þ Dz Þ ¼ fðx; y; zÞ

(2.12)

Examples of one-dimensional fields that are reflectionally symmetric and reflectionally antisymmetric about the axis x ¼ x0 are shown in Fig. 2.5 as blue and red curves, respectively. If an antisymmetric field is continuous

y (x-x0)

(-(x-x0))

x x1

x0

x2

Fig. 2.4 An example of the reflectional transformation of the one-dimensional field.

19

Symmetry, symmetry transformations and symmetry conditions

y Symmetric

ϕ (x) ϕ (x )

x1

x0

x x2

Antisymmetric Fig. 2.5 Example of symmetric and antisymmetric reflections of one-dimensional fields.

at x ¼ x0, it is necessary that the field takes zero value at x ¼ x0 because zero is the only number which keeps its value after a change of its sense. Symmetry and antisymmetry are rather familiar concept in mathematics when they are called even and odd functions if x0 ¼ 0. It can be seen that an appropriate use of the terminology of antisymmetry is to associate it with the nature of the physical fields and not with the geometry, since for any physical field to be symmetric or antisymmetric, a precondition is that the domain for the physical field is geometrically symmetric. Calling the shape in Fig. 2.1(b) antisymmetric could therefore be misleading. Physical fields defined over a symmetric domain are not always symmetric or antisymmetric. In fact, symmetric or antisymmetric fields are special cases of generally asymmetric fields. However, one can prove for linear systems that any asymmetric field can be presented as the sum of a symmetric and an antisymmetric components. This applies to linear systems only because the underlying principle is the superposition law which is valid only for linear systems. In the construction of RVEs or UCs, it can be found that displacement field lacks the desirable translational symmetry. However, given the nature of the particular form of displacement involved, the superposition principle will be employed then by splitting the asymmetric displacement field into a translationally symmetric relative displacement field and another piecewise constant displacement field corresponding to a piecewise-defined rigid body translation. This will be fully elaborated in Section 2.5.3 with practical applications being given in Chapter 6. Over a symmetric domain, the transformations of a vectorial physical field, j, are defined as follows.

20

Representative Volume Elements and Unit Cells

8 9 > = < jx > Reflection Sx : jy / > ; : > jz ðxx0 ;y;zÞ 9 8 > = < jx > ¼ jy > > ; : jz ððxx0 Þ;y;zÞ

8 j0 9 > < x= j0y > : ; j0z ðx0 x0 ;y0 ;z0 Þ (2.13)

8 9 8 j0 9 > > = < jx > < x= 2 j0y jy / Rotation Cz : > > > : ; : ; jz ðx;yy0 ;zz0 Þ j0z ðx0 ;y0 y0 ;z0 z0 Þ 9 8 > = < jx > ¼ jy > > ; : jz ðx;ðyy0 Þ;ðzz0 ÞÞ 8 9 8 j0 9 j > > > x < = < x= D j0y jy / Translation Tx : > > ; : > : ; jz ðx;y;zÞ j0z ðx0 ;y0 ;z0 Þ 8 9 > = < jx > ¼ jy > ; : > jz ðxþDx ;yþDy ;zþDz Þ

(2.14)

(2.15)

In the case of a two-dimensional vectorial field defined over a twodimensional domain, an illustration of the reflectional transformation about x ¼ x0 is given in Fig. 2.6, where arrows show the values of the field at the discrete points as picked for the purpose of illustration. y

y

ψ (x,y)

ψ′ (x′,y′) Ω′

Ω

x

x x0

x0

Fig. 2.6 An example of the reflectional transformation about axis x ¼ x0 of a twodimensional vectorial field defined over domain U.

21

Symmetry, symmetry transformations and symmetry conditions

A vectorial field, j, defined over a geometrically symmetric domain is called symmetric under a respective transformation if 8 9 9 8 > > < jx > = = < jx > Reflection : / jy ¼ jy (2.16) > > > : > ; ; : jz ððxx0 Þ;y;zÞ jz ðxx0 ;y;zÞ 9 8 j > > x = < jy ¼ Rotation : / > > ; : jz ðx;ðyy0 Þ;ðzz0 ÞÞ

8 9 > = < jx > jy > ; : > jz ðx;yy0 ;zz0 Þ (2.17)

8 9 8 9 > > < jx > = = < jx > Translation : / jy ¼ jy : > > : > ; ; : > jz ðxþDx ;yþDy ;zþDz Þ jz ðx;y;zÞ It is antisymmetric if 8 8 9 9 j j > > > > x x < < = = jy ¼ jy Reflection : / > > > : : > ; ; jz ððxx0 Þ;y;zÞ jz ðxx0 ;y;zÞ

(2.18)

(2.19)

9 8 8 9 > > = = < jx > < jx > jy ¼ jy Rotation: / > > > ; ; : : > jz ðx;ðyy0 Þ;ðzz0 ÞÞ jz ðx;yy0 ;zz0 Þ (2.20) 9 8 9 8 > > = = < jx > < jx > Translation : / jy ¼ jy : > > > ; ; : > : jz ðxþDx ;yþDy ;zþDz Þ jz ðx;y;zÞ

(2.21)

Examples of symmetric and antisymmetric reflection of a twodimensional vectorial field j(x,y) defined over a two-dimensional domain U are shown in Fig. 2.7(a) and (b), respectively. It can be seen that whilst the symmetric case is reasonably straightforward as it follows one’s intuition, the antisymmetric one is anything but intuitive. One will have to rely on its mathematical definition of antisymmetry in order to make correct use of it. Unfortunately, in the applications of unit cells, antisymmetric scenarios are

22

Representative Volume Elements and Unit Cells

(a)

y

(b)

ψ (x,y)

Ω

x0

x

y

ψ (x,y)

Ω

x

x0

Fig. 2.7 Example of (a) symmetric and (b) antisymmetric reflections about axis x ¼ x0 of a two-dimensional vectorial field defined over domain U.

routinely encountered under shear loading. Lack of adequate understanding of antisymmetry concept is likely to be the reason why the shear loading is often the aspect avoided in most accounts of unit cells modeling in the literature. In general, each component of a vectorial field is associated with a coordinate axis. If the direction of a coordinate axis reverses under a mapping transformation (reflection, rotation or translation), a symmetric transformation associated with the mapping reverses the sense of the corresponding component of the vectorial field, whilst if the direction of a coordinate axis is not affected by a mapping transformation, the associated component of the vectorial field keeps its sense. For example, under the reflectional transformation about the x-plane, the x component of the vector field in Fig. 2.7(a) reverses its sense as the x-axis does, whilst y component keeps its sense. An antisymmetric transformation does exactly the opposite, e.g. in Fig. 2.6(b) where the x component kept its sense but the y component has reversed its sense. The same considerations of symmetric and antisymmetric mapping can be made to tensorial fields. Since any component of a second rank tensor is associated with two axes, the interpretations of symmetry and antisymmetry will be more complicated. However, if one follows the considerations given to scalar and vectorial fields as 0th and the first rank tensors, the concept of symmetric and antisymmetric transformations can be generalized to a tensorial field of an arbitrary rank as follows. Each component of an rth rank tensorial field is associated with r coordinate axes allowing repetition as appropriate. Out of the r coordinate axes, assume that there are k (0  k  r) of them whose directions reverse under a mapping transformation, reflection, rotation or translation. A symmetric transformation keeps the sense

Symmetry, symmetry transformations and symmetry conditions

23

of the corresponding component of the tensorial field if k is an even number and, otherwise, it reverses its sense. An antisymmetric transformation reverses the sense of the corresponding component of the tensorial field if k is an even number whilst keeps its sense if k is odd. The above apparently applies to scalar and vectorial fields as special cases. Applying the definition of symmetric and antisymmetric mapping extended to tensorial fields, one can examine the strain as an example of second rank tensor under a mapping transformation 2 3 3 2 εxx εxy εxz εxx εxy εxz 6 7 7 6 (2.22) 4 εxy εyy εyz 5 / 4 εxy εyy εyz 5 : εxz εyz εzz P εxz εyz εzz P 0 Its diagonal components are always associated with the same axis twice and k is therefore even, either 0 or 2. A symmetric transformation will never change their sense whilst an antisymmetric one always will. A shear component is associated with two different axes and k can therefore be 0, 1 or 2 depending on the component and the transformation concerned. For instance, under the reflection about a plane perpendicular to x-axis, k ¼ 1 for εxy and εxz and k ¼ 0 for εyz, whilst under the rotation about an axis parallel to the z-axes, k ¼ 1 for εyz and εxz and k ¼ 2 for εxy. It can therefore be seen that shear is a lot more complicated and hence prone to confusion. An antisymmetric transformation reverses everything accordingly. Interested readers are encouraged to extend this to a fourth ranked tensor which features stiffness and compliance of a material. In general, a scalar physical field, or a component of a vectorial or tensorial physical field, is symmetric under a respective transformation, reflection, rotation or translation, if it keeps its magnitude and sense after a symmetry transformation. It is antisymmetric if it keeps its magnitude but reverses its sense after a symmetry transformation. Refer to Fig. 2.7 as appropriate to assist the interpretation of the statements made. The symmetry or antisymmetry of the components of a field is an important feature of the physical problem if it is present in the problem. Taking advantage of it, the problem can be simplified, often drastically. Usually, it involves interpreting the symmetry into some kind of conditions, in the case of unit cells, the boundary conditions for the unit cell. However, the symmetry alone as described is not sufficient and it will have to be topped up with another basic consideration, continuity, as will be addressed in the next section.

24

Representative Volume Elements and Unit Cells

2.4 Continuity and free body diagrams A basic feature of physical fields in most physical problems is their continuity. It is usually an essential requirement in the formulation of the physical problem. In a mechanical problem of deformation, for instance, continuity of displacement field is required by the deformation kinematics and hence it is a fundamental precondition for the finite element method, which is a displacement-based approach. The continuity should certainly not be compromised by the consideration of symmetries. In addition to the vectorial displacement field, a mechanical problem also involves tensorial fields of stress, s, and strain, ε. The continuity condition on stress or strain is a common place of confusion. Stress or strain field as a tensor does not have to be continuous, i.e. the six components of either of these tensors do not have to be fully continuous functions of coordinates. For stress field, the essential requirements on the continuity condition are defined by Newton’s third law, i.e. traction as a vector must be continuous in the direction of the normal to the plane where the traction is defined, as will be elaborated in this section. The stress components not exposed in the traction vector, i.e. two direct stresses and one shear stress in the plane parallel to the plane on which the traction is defined do not have to be continuous along the direction perpendicular to the plane. It should be pointed out that equilibrium condition is not the continuity condition for the stress field, at least as far as finite element applications are concerned. Equilibrium is associated with Newton’s first law. In an approach based on variational methods, such as finite elements, equilibrium is satisfied by energy minimization, and it is not a precondition like displacement continuity. In order to describe the continuity for various fields, it is essential that the tool of free body diagram (FBD) is not only used, but is also used correctly. Whilst FBD is basic in many subjects of physics and considered as the very basis of any subsequent development in these studies, it is not always used correctly by all means. The key source of confusion associated with the FBD is likely to be the understanding of Newton’s third law, in particular, the definition of action and reaction. Whilst everyone could recite their characteristics as being equal in magnitude, opposite in direction and acting along the same line, a centrepiece of the concept is often missing, i.e. that they act on different bodies. Without this consideration in place, action and reaction could easily be forces in equilibrium and hence could be canceled out, which leads to the fact that some forces failed to be represented.

25

Symmetry, symmetry transformations and symmetry conditions

FBD is an imaginary means of cutting a physical body apart. As long as the necessary continuity for all fields involved in the physical problem is maintained, the cut should not have violated any other considerations of the problem. However, by cutting the body apart, internal variables can be exposed and hence become visible. They can then be analyzed appropriately, e.g. in terms of equilibrium, symmetry, etc. Assume a body is cut into two parts as shown in Fig. 2.8(a) and the physical problem involves a vectorial field {u v w}T, where the third component, w, not shown in Fig. 2.8, points out of the page. The continuity means ðx; y; zÞ ¼ ðx0 ; y0 ; z0 Þ 8 9 8 09 > = > =

v ¼ v0 : > ; > ; : > : 0> w w

(2.23)

(2.24)

For the deformation problem, u, v and w are the displacements. Without such continuity, deformation would result in slits or overlaps at the cut in the body, which violates the deformation kinematics 2    3 vu 1 vu vv 1 vu vw þ þ 6 vx 2 vy vx 2 vz vx 7 6 7 2 3 6  εxx εyx εzx  7 6 1 vu vv  vv 1 vv vw 7 6 7 6ε 7 ε ε þ þ ¼ 6 7: 4 xy yy zy 5 6 2 vy vx vy 2 vz vy 7 6  7 εxz εyz εzz    6 7 1 vv vw vw 4 1 vu vw 5 þ þ 2 vz vx 2 vz vy vz (2.25) For Eq. (2.25) to hold, each of u, v and w is required to be continuous with respect to x, y and z due to the presence of partial derivatives with

(a)

Sy

u'

u ( x, y,z)

(b)

v'

v

( x', y',z' )

Sy' Sx

( x, y,z )

Sx'

( x', y',z')

Fig. 2.8 Free body diagrams: (a) continuity of displacements; (b) tractions on the surfaces exposed by a cut as actions and reactions.

26

Representative Volume Elements and Unit Cells

respect to all three coordinates in the matrix on the right hand side of Eq. (2.25). There are neither physical nor mathematical reasons for stresses to be continuous in the same way as the displacements, hence the following is not required in general 2 2 3 3 sx sxy sxz sx sxy sxz 6s 6 7 7 ¼ 4 sxy sy syz 5 : (2.26) 4 xy sy syz 5 sxz syz sz ðx;y;zÞ sxz syz sz ðx0 ;y0 ;z0 Þ To satisfy equilibrium equations vsx vsxy vszx þ þ þ fx ¼ 0; vx vy vz vsxy vsy vsyz þ þ þ fy ¼ 0; vx vy vz

(2.27)

vszx vsyz vsz þ þ þ fz ¼ 0; vx vy vz each stress component is only required to be continuous in certain directions. For instance, sx needs to be continuous only in the x-direction, as equilibrium only requires it to be differentiable with respect to x only. A practical example can be found in laminated structures where none of the in-plane stresses needs to be continuous in the z-direction. The mandatory continuity conditions associated with stresses are expressed in terms of traction 8 9 2 38 9 sx syx szx > > < Sx > < nx > = = 6s 7 fS g ¼ Sy ¼ ½sfng ¼ 4 xy sy szy 5 ny ; (2.28) > > : > : > ; ; sxz syz sz Sz nz where {n} is the outward normal to the surface exposed by a cut. On the surfaces of free bodies exposed by a cut, the tractions on both sides of the cut are action and reaction. Newton’s third law requires them to be equal in magnitude but opposite in direction 8 9 8 S0 9 S > > > x < = < x= Sy ¼  Sy0 : (2.29) > > : > ; : ; Sz Sz0

27

Symmetry, symmetry transformations and symmetry conditions

as shown in Fig. 2.8(b). Consider a body cut by a plane perpendicular to the x-axis. In this case, 8 9 8 9 > >

= < sx > = thus fSg ¼ sxy (2.30) fng ¼ 0 > > > : > ; : ; 0 sxz At x and xþ, i.e. on the two surfaces created in the free bodies, the outward normals are 8 9 8 9 1 1 > > > > < = < = 0 and fngjxþ ¼ (2.31) fngjx ¼ 0 > > > : > ; : ; 0 0 respectively. The tractions on them are therefore 9 9 8 8 > > = = < sx > < sx > and fSgjxþ ¼  sxy fSgjx ¼ sxy > > > > ; ; : : sxz sxz

(2.32)

respectively. Since fSgjx and fSgjxþ are action and reaction to each other, Newton’s third law requires that 9 9 8 8   > > = = < sx > < sx >   sxy  ¼ sxy  (2.33) fSgjx ¼ fSgjxþ i.e.  > > > > ; ; : : sxz sxz  þ x

x

It is clear that only the stress components shown above form the tractions as vectors on the exposed surfaces of the free bodies perpendicular to the xaxis. Their continuity as in Eq. (2.33) is required by Newton’s third law. Readers are reminded that the action and reaction act on different bodies and hence are not in equilibrium. It is obvious that the continuity associated with stress is not for the stress tensor as a whole, but only for those components exposed on the surface concerned to form the traction. Other stress components are irrelevant to the continuity consideration across the plane concerned, as Newton’s third law does not involve any of them. Eqs. (2.24) and (2.33) are the continuity conditions as obtained from the FBD. The argument can be applied to cutting planes perpendicular to y- and z-axes, respectively, as well as to planes orientated arbitrarily.

28

Representative Volume Elements and Unit Cells

2.5 Symmetry conditions Symmetry in the context of the present discussion is a geometric as well as a physical property. In order to apply it in physical disciplines, the concept has to be adapted to the physical fields under consideration, such as displacements, strains and stresses, so that the implications of symmetry can be elaborated to offer useful information to the physical study. The conditions on the physical fields implied by the symmetry will be called symmetry conditions. Physical fields usually have their senses. If a field is multiplied by 1, it is converted to its opposite sense. The involvement of the sense enriches the concept of symmetry, but also causes confusion as a by-product. Under a symmetry transformation in accordance with the geometric symmetry, a physical field could be symmetric, if the field keeps its sense under the symmetry transformation, or antisymmetric if the field changes to its opposite sense under the symmetry transformation. Apparently, whilst the word “symmetric” can be employed to describe geometries as well as physical fields, the word “antisymmetric” only applies to physical fields. If one is determined to abuse the terminology, it could be used to mean anything, which is of course not in the interest of the present monograph. The symmetry conditions will be derived for each of the three symmetries in this section. Without loss of generality, a mechanical problem of deformation will be considered, which involves a vector field of displacement, u, and a tensor field of stress, s. Whilst all of them play equally important roles in the physical problem, the symmetry conditions will be derived only for the displacement and stress because they provide necessary boundary conditions for the analysis of RVEs and UCs in the context of the present monograph.

2.5.1 Reflectional symmetry Assume the reflection is about a plane perpendicular to the x-axis located at x0. Split the domain by the symmetry plane in the sense of free body diagram as shown in Fig. 2.9(a). The continuity of the displacements and traction requires that 8 9 9 9 8 9 8 8     > > > >

= = =

= < sx > < sx >     v  ¼ v  and sxy  ¼ sxy  ; (2.34) > > > > > > : > ; ; ; : > ; : : w w sxz sxz þ þ   x0

x0

x0

x0

29

Symmetry, symmetry transformations and symmetry conditions

Fig. 2.9 (a) Free body diagram to show the continuity of displacements and stresses, and (b) symmetric and (c) antisymmetric reflectional symmetry of the displacements and stresses.

where out-of-plane components w and sxz are not shown but not difficult to envisage without being shown and they will be kept in the subsequent discussions as appropriate. If the physical fields under consideration are symmetric about the plane of reflection as shown in Fig. 2.9(b), the symmetry transformation requires 8 9 9 9 8 9 8 8     u u s s > > > > > > > > x x < = = = < = < <     v  ¼ v  and sxy  ¼ sxy  :    > > > > > > > > : ; ; ; : ; : : w w  sxz  þ sxz   þ x0

x0

x0

x0

(2.35)

30

Representative Volume Elements and Unit Cells

If the physical fields under consideration are antisymmetric, as in Fig. 2.9(c), the symmetry transformation requires 9 9 8 9 8 8 8 9     > > > > = = < u > = < sx > < sx >

=     and sxy  ¼ sxy  : v  ¼ v  > > > > > > > ; ; : ; : : : > ; w x w xþ sxz xþ sxz x 0

0

0

0

(2.36) In each case, symmetric or antisymmetric, one can eliminate those at ðÞjx by considering (2.34) and (2.35), or (2.34) and (2.36) as a set of simultaneous equations. For symmetric case, one obtains ujxþ ¼ 0

and

0

sxy jxþ ¼ sxz jxþ ¼ 0;

(2.37)

0

0

and for antisymmetric case, one obtains vjxþ ¼ wjxþ ¼ 0 0

0

and

sx jxþ ¼ 0;

(2.38)

0

which are conditions for displacements and stresses on the symmetry plane that have to be satisfied to meet the symmetric and antisymmetric requirements, respectively. The same argument as presented above also results in the following identities. In symmetric case vjxþ0 ¼ vjxþ0 ;

wjxþ0 ¼ wjxþ0 ;

and

sx jxþ0 ¼ sx jxþ0

(2.39)

and

sxy jxþ ¼ sxy jxþ :

(2.40)

and in antisymmetric case ujxþ ¼ ujxþ ; 0

0

sxz jxþ ¼ sxz jxþ 0

0

0

0

They can all be reduced to a 0 ¼ 0 type of identities and therefore they are always satisfied without imposing any extra requirement as boundary conditions. It should be noted that reflectional symmetry conditions come with constraints on displacements. In FE analysis, sometimes, it is necessary to introduce some constraints simply to rule out rigid body motions. The first equation in Eq. (2.37) has automatically eliminated rigid body translation in the x-direction and rigid body rotations about y- and z-axes. The displacement constraints in Eq. (2.38) prevent rigid body translations in the y- and z-directions and the rigid body rotation about the x-axis. It should also be pointed out that the vanishing conditions on the displacements as derived above are the consequence, and hence necessary conditions, of the symmetry concerned. Artificial assignment of non-vanishing values to these displacements, e.g. to falsify a translational symmetry, violates the reflectional symmetry considerations and should not be attempted.

Symmetry, symmetry transformations and symmetry conditions

31

2.5.2 180 rotational symmetry Like reflectional symmetries, 180 rotational symmetries can be either symmetric or antisymmetric depending on the loading conditions. The conditions implied by this symmetry can again be derived from two considerations, continuity and symmetry. Illustrated in Fig. 2.10(a) are continuity considerations as involved in the free body diagram. Under a symmetric transformation, the displacements and the traction on the surface are shown in Fig. 2.10(b) and (c), respectively, whilst the antisymmetric counterparts are shown in Fig. 2.10(d) and (e). In Fig. 2.10, the coordinate system is deliberately chosen to be different from that in Fig. 2.9. Materials having a reflectional symmetry about a plane perpendicular to the x-axis are categorized in exactly the same group as those having a rotational symmetry about the x-axis, as will be fully elaborated in Chapter 3. As far as the present discussion is concerned, rotational symmetries will be limited to 180 rotations, i.e. C2. The use of other angles in the context of polar coordinate applications can be found Chapter 6 as developed in (Li et al., 2014). Assume the 180 rotational symmetry is about the x-axis which is out of the plane shown in Fig. 2.10. The way to split the domain into two halves in the sense of free body diagrams is no longer unique. The partitioning surface does not even have to be a plane. The conditions for the surface are that it passes the axis of rotation and is rotationally symmetric about the same axis. For the sake of argument, it is chosen as the plane perpendicular to the y-axis as shown in Fig. 2.10, which certainly satisfies the conditions as stated above. The continuity of the displacement and stress fields requires 8 9 8 9   > >

=

=   v  ¼ v  ; (2.41) > > : > ; : > ; w y¼0þ ;z w y¼0 ;z 9 9 8 8   s s > > > > yx yx = = < <   sy  ¼ sy  : > > > > ; ; : : syz y¼0þ ;z syz y¼0 ;z

(2.42)

If the physical fields under consideration are symmetric, the symmetry transformation requires

32

Representative Volume Elements and Unit Cells

z

(a)

w|z

w'|z v'|z O' τ 'yz|-z σ 'y|-z

v|z

z

y

O

σy|-z τ yz|-z

-z

z

(b)

w|z v|z

z

v'|-z w'|-z O’

y

O

w|-z v|-z

-z

v'|z w'|z

z

(c)

τ 'yz|-z σ 'y| -z σ y|z O'

z y

O τ yz|z

τ 'yz|z σ 'y|z σy|-z

-z

τyz|-z

z

(d)

w'|-z

w|z

v'|-z O’

w'|z

v|z

z

y

O

w|-z

v'|z

v|-z

-z z

(e) σ 'y|-z τ 'yz|-z O'

σ 'y|z

τ 'yz|z

σy|z

z O τ yz|z

σy|-z

y

-z

τ yz|-z

Fig. 2.10 (a) Free body diagram to show the continuity of displacements and stresses, (b) symmetric rotational symmetry of the displacements; (c) symmetric rotational symmetry of the stresses; (d) antisymmetric rotational symmetry of the displacements and (e) antisymmetric rotational symmetry of the stresses.

Symmetry, symmetry transformations and symmetry conditions

8 9 8 9   u u > > > > < = < =   v  ¼ v   > > > > : ; : ; w y¼0þ ;z w y¼0 ;z

8 9 9 8   > > < syx > = = < syx >   sy  sy ¼ :   > > > > : ; ; : syz y¼0þ ;z syz y¼0 ;z

33

(2.43)

(2.44)

If the physical fields under consideration are antisymmetric, the symmetry transformation becomes 8 9 8 9   > >

= < u > =   v  ¼ v  ; (2.45)  > > > > : ; : ; w y¼0þ ;z w y¼0 ;z 9 9 8 8   > > = = < syx > < syx >   sy  ¼ sy  : > > > > ; ; : : syz y¼0þ ;z syz y¼0 ;z

(2.46)

After eliminating those on ðÞjy¼0 for symmetric case, the conditions obtained are as follows which should be imposed as boundary conditions. For the symmetric case, 8 9 8 9   u u > > > > > > > < > = < = v  ¼ v  and > > > >  > > > > : ; : ; w y¼0þ ;z w y¼0þ ;z 9 9 8 8 (2.47)   syx > syx > > >   > > > > = = < <  sy  sy ¼  > > > >  > > > > ; ; : : syz syz þ þ y¼0 ;z

y¼0 ;z

excluding the axis of rotation whilst the axis of the rotation is under the following special conditions vjy¼0;z¼0 ¼ wjy¼0;z¼0 ¼ 0; syz jy¼0;z¼0 ¼ syz jy¼0;z¼0:

syx jy¼0;z¼0 ¼ syx jy¼0;z¼0 and

(2.48)

34

Representative Volume Elements and Unit Cells

The remaining conditions at the axis of rotation are as given below, providing no constraints. ujy¼0;z¼0 ¼ ujy¼0;z¼0

and

sy jy¼0;z¼0 ¼ sy jy¼0;z¼0:

For the antisymmetric case, the respective expressions become 8 9 8 9   u> u > > >  > > > > < = < = ¼ and v  v  > > > >   > > > > : ; : ; w y¼0þ ;z w y¼0þ ;z 9 9 8 8   syx > syx > > >  > > > > = = < < sy  ¼ sy  > > > > > > > > ; ; : : syz y¼0þ ;z syz y¼0þ ;z

(2.49)

(2.50)

excluding the axis of rotation whilst the axis of the rotation is under the following conditions ujy¼0;z¼0 ¼ 0 and

sy jy¼0;z¼0 ¼ 0;

(2.51)

with the following trivial conditions offering no constraint at all vjy¼0;z¼0 ¼ vjy¼0;z¼0 ;

wjy¼0;z¼0 ¼ wjy¼0;z¼0 ;

syx jy¼0;z¼0 ¼ syx jy¼0;z¼0 and

syz jy¼0;z¼0 ¼ syz jy¼0;z¼0:

(2.52)

Equations (2.51) and (2.52) are conditions of displacement and stress components to be satisfied on the y-plane selected as the partition plane, for symmetric and antisymmetric cases, respectively. They will be the boundary conditions for the half domain to be analyzed on the y  0 side. Similar to the discussion in the previous subsection, the obtained rotational symmetry conditions for displacements eliminate some of the rigid body motions. The first equation in (2.48) does not only eliminate the two rigid body translations in the x-plane, it also constraints the x-axis from rotating about the y- and z-axis, respectively. The remaining rigid body motions after symmetric rotational transformation are translation in the x-direction and rotation about the x-axis. Under antisymmetric rotational symmetry transformation, Eq. (2.51) eliminates the x-direction rigid body translation and the condition on v in Eq. (2.50) eliminates the rigid body rotation about the x-axis, whilst the remaining four rigid body motions are still active and should be constrained separately before FE analysis can be conducted.

35

Symmetry, symmetry transformations and symmetry conditions

Similar boundary conditions can be obtained if one chooses a different partitioning surface, as the underlying considerations are the same. Equally, if the rotational symmetry is about axis other than the x-axis, derivation can be pursued by the readers in a similar fashion.

2.5.3 Translational symmetrydone-dimensional scenario as an introduction Since the translational symmetry is relatively the least familiar type of symmetry, it will be dealt with in steps, with the one-dimensional scenario as a first step into the problem. In one-dimensional space, a geometric object is a linear segment. It is always symmetric, both reflectionally and rotationally, about its center. However, for it to be translationally symmetric, it has to be infinite on both sides. A distinct difference between translation and other symmetries becomes obvious right at the start. As far as the geometry is concerned, the infinite object is translationally symmetric by any distance of translation. Introduce the x-axis along the direction of translation and assume that the physical field is translationally symmetric by a translation of Dx. The presence of translational symmetry for the geometry and the physical properties is a precondition to the present discussion. It is the users’ responsibility to ensure that the precondition is met. Obviously, not all physical fields in reality possess this type of symmetry. In fact, even with perfectly uniform stress distribution, where the translational symmetry for the stress field is obvious, the associated displacement field does not possess such translational symmetry. The displacement field will be the major focus of the following discussion. A segment of a characteristic length of Dx for the physical problem between x0 and x0þDx can be selected to produce the free body diagram. The displacement field, u, can be defined as a linear function of coordinate x. The so-called multiscale modeling is to establish certain relationship between the behaviors of the material at the upper and lower length scales. To illustrate this, this displacement field can be given in each of the length scales as follows. UðxÞ ¼ ε0 x þ u0 at the upper length scale

(2.53)

uðxÞ ¼ εðxÞx þ u0 at the lower length scale;

(2.54)

where u0 is the displacement at a reference point, the origin of the x-axis in the present case, i.e. u0 ¼ u(0), ε(x) the strain field at the lower length scale,

36

Representative Volume Elements and Unit Cells

which is a periodic function with a period of Dx, and ε0 the average of ε(x) and hence constant strain at the upper length scale. As shown in Fig. 2.11, the displacement field at both length scales is obviously not periodic and there is no translational symmetry for the displacement field, even though both stress and strain fields associated with it are periodic and hence translationally symmetric, and they are both uniform at the upper length scale. In Fig. 2.11, they are assumed to be uniform even at the lower length scale as a simplest example to help with revealing the lack of translational symmetry for displacement. The lack of translational symmetry in displacement fields has often been a source of confusion. In order to apply the translational symmetry to displacement properly, it is necessary to resort to the concept of relative displacement field instead. This concept will be employed in two different but closely related contexts. One is the difference between displacements at a point within the characteristic segment and that at the reference point within the same segment. It will be applied primarily to the lower length scale to establish the displacement field at this length scale. The other one is the difference between displacements at the corresponding points in different segments and it applies to both length scales. Select any point R at xR within segment [x0, x0 þ Dx] as a reference for the segment. Assume the point of interest is P at x within the segment. The displacement of P relative to that of R is given as uP  uR ¼ ux  uxR ;

(2.55)

where uR and uP are displacements at points R and P. The translational symmetry maps point R and P to every segment. In the nth segment, the relative displacement between the images of these two points, R0 and P0 , respectively, is uP 0  uR0 ¼ uxþnDx  uxR þnDx; x0

u(x0)

x0+Δx

u(x0+Δx)

(2.56) x0+nΔx

u(x +nΔx)

Fig. 2.11 Linear one-dimensional displacement field corresponding to a uniform strain field.

37

Symmetry, symmetry transformations and symmetry conditions

where uR0 and uP 0 are displacements at points R0 and P0 as the images of R and P under the translational symmetry transformation. A relative displacement field can thus be constructed segment by segment as Du ¼ uP  uR ¼ uP 0  uR0 : (2.57) It is identical in all segments and therefore there is a translational symmetry, or periodicity in the relative displacement field. Such a relative displacement field is defined in a piecewise manner. It is shown in Fig. 2.12(a) as red zigzag for the case of a uniform strain state where the displacement field itself is a continuous (linear) function. A more general case where the strain distribution is not uniform in the lower length scale as shown in Fig. 2.12(b). It is worth noting that the relative displacement field is not continuous. However, this should not be any cause for concern as long as the displacement field itself associated with it is continuous, as it is indeed the case. The displacement field can be recovered as uðx’Þ ¼ uP ’ ¼ uR’ þ Du;

(2.58)

where uR0 is a step function shown in Fig. 2.12 in brown. This function is also discontinuous; it corresponds to a piecewise-defined rigid body translation and hence remains constant in each segment. The superposition of uR0 and Du, as shown in blue, recovers the original displacement field as well as its continuity. It is now clear that periodicity is not present in the displacement field, in general. However, it is present in the relative displacement. With this being established, it can be appropriately dealt with to deliver the symmetry conditions under translation. To facilitate the derivation, without loss of generality, R is placed at x ¼ x0 and P at x ¼ x0þDx, i.e. the two ends of the segment taken as a free body, as shown in Fig. 2.13.

(a)

(b)

u(x)

x0

x 0 +Δ x

x0+nΔ x

x

u(x)

x0

x 0 +Δ x

x0+nΔ x

x

Fig. 2.12 Schematic illustration of the relative displacement field (zigzag) in contrast with the displacement field (plot shown in blue) corresponding to (a) uniform strain state; (b) non-uniform strain distribution in the lower scale.

38

Representative Volume Elements and Unit Cells

n = -1

n=0 x0−

x0+

n=1

x0− + Δx x0+ + Δx

Fig. 2.13 Neighboring segments with the one corresponding to n ¼ 0 as a free body.

The continuity requires that   þ       and u x u x0 ¼ u xþ 0 0 þ Dx ¼ u x0 þ Dx

(2.59)

whilst the translational symmetry gives     u x 0 þ Dx  u x0 ¼ DU ¼ ε0 Dx:

(2.60)

where U and ε0 are the displacement strain at the upper length  and  average þ scale, respectively. Eliminating u x þðn 1ÞDx , one obtains u x 0 0   þ   (2.61) u x0 þ Dx  u x0 ¼ ε0 Dx: This is the symmetry condition for displacement resulting from translation by Dx in a one-dimensional case. Expressed in this way, the relative displacement between the two ends of the free body segment is explicitly related to the average strain at the upper length scale. Similar, considering axial stress in the segment, and bearing in mind that stress is translationally symmetric, the continuity (action and reaction in terms of traction) requires    þ  F x (2.62) 0 þ Dx ¼ F x0 þ Dx Traction as stress times the outward normal. The outward normal   is defined   þ  for x0 þDx is in the positive direction of x whilst that for x0 þDx is in the negative direction. As a result, one obtains    þ  s x (2.63) 0 þ Dx ¼ s x0 þ Dx : The translational symmetry requires     þ s xþ (2.64) 0 ¼ s x0 þ Dx :  þ  Eliminating s x0 þDx , one obtains the symmetry condition in stress for the segment in a one-dimensional case as      s xþ (2.65) 0 ¼ s x0 þ Dx : It relates the stress at one end of the segment to that at the other end.

Symmetry, symmetry transformations and symmetry conditions

39

2.5.4 Translational symmetry conditions in threedimensional scenarios Generalising the discussion above from one-dimensional to threedimensional problems and assuming that the translational symmetry remains by Dx in the x-direction, applying the same argument as in the previous subsection, one has 8 9 8 9 8 9  > > >

=

= < DU > =  v  v  ¼ DV . (2.66) > > > > : > ; : > ; : ; w x þDx w DW xþ 0

0

Similarly, the traction symmetry condition is 9 9 8 8   > > = = < sx > < sx >   sxy  ¼ sxy  :  > > > > ; ; : : sxz x þDx sxz xþ 0

(2.67)

0

For practicality, only symmetric transformations of physical fields have been considered for translational symmetries. Antisymmetry is not a common phenomenon as far as translational symmetry is concerned and therefore will not be dealt with here. Multiscale material characterization is conventionally based on homogenization, which assumes that the materials occupies an infinite space at the lower length scale. The translational symmetry is the only type of symmetry available which can reduce an infinite extent to a finite domain for effective analysis, as has already been stated earlier in Section 2.2.3, and it therefore bears particular relevance to the subject of RVEs and UCs as is addressed in the present monograph. It is therefore essential that the translational symmetry is dealt with correctly and efficiently. Any attempt to bypass translational symmetry by using reflectional and/or rotational symmetry instead is mathematically flawed and therefore will have glitches embedded generically, as will be further elaborated in Chapter 5. A further necessary generalization is three-dimensional problems where the translational symmetry is along the direction of axis x which is inclined with respect to coordinate axes defined in the domain. The symmetry condition can be obtained as follows. Consider an arbitrary point P(x,y,z) and a reference point R in the representative segment along the direction of the translation. The relative displacement at P to R is also mapped to another segment in the same way and hence

40

Representative Volume Elements and Unit Cells

8 9 8 9 8 9 8 9     u u u > > > > > > > < = < = < =

=     v   v  ¼ v   v  :    > > > > > > > > : ; : ; : ; : ; w P w R w P0 w R0

(2.68)

Rearranging the above equation, one has 8 9 8 9 8 9 8 9     > > > >

=

=

=

=     v   v  ¼ v   v     > > > > > > > > : ; : ; : ; : ; w w w w 0 P P ðx0;y0;z0Þ ðx;y:zÞ 8 9 8 9   > >

=

=   ¼ v   v  (2.69) > > : > ; : > ; w w R0 R 8 9 8 9   > >

=

=   where v   v  is the relative displacement between points in  > > > > : ; : ; w w 0 R

R

different segments, each at a fixed position within the respective segment. Such relative displacement at lower length scale is apparently the same as that at the upper length scale, i.e. 8 9 8 9 8 9   > > >

=

= < DU > =   v   v  ¼ DV (2.70) > > > > : > ; : > ; : ; w w DW 0 R

R

where U, V and W are the upper length scale displacement components in the x-, y- and z-directions, respectively. The relative displacement at the upper length scale can be expressed in terms of the displacement gradient at the upper length scale as 2 3 vU vU vU 6 7 9 8 9 6 vx vy vz 78 6 7> Dx > DU > > = < = 6 vV vV vV 7< 6 7 (2.71) DV ¼ 6 7 Dy > > > vx vy vz 7> : ; : ; 6 6 7 DW 6 vW vW vW 7 Dz 4 5 vx vy vz where Dx, Dy and Dz are components of the translation vector associated with the translational symmetry concerned. Assuming that the translation in direction of x is by distance of d and it is associated with a unit base vector of n ¼ ð nx ny nz ÞT , the translation vector can be given as

Symmetry, symmetry transformations and symmetry conditions

41

8 9 8 9 Dx > > > < = = < nx > Dy ¼ nd ¼ ny d; (2.72) > > > : ; ; : > Dz nz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi where d ¼ ðDxÞ2 þ ðDyÞ2 þ ðDzÞ2 The relative displacement at the lower length scale between corresponding points related through the translational symmetry concerned can be related to the displacement gradient at the upper length scale as 2 3 vU vU vU 6 7 8 9 8 9 6 vx vy vz 78 9   6 7> nx > > > < =

=

= 6 vV vV vV 7   6 7 v   v  ¼6 7 ny d: > > > 6 vx vy vz 7> : > ; : > ; 7: nz ; w  ðx0;y0;z0Þ w  ðx;y:zÞ 6 6 vW vW vW 7 4 5 vx vy vz (2.73) The relative displacement between corresponding points on the opposite sides of the segment as given above delivers the displacement symmetry condition corresponding to the respective translational symmetry. Unlike displacement, stress field is periodic in presence of translational symmetry. As a result, the traction symmetry condition is obtained as 02 38 91 sx sxy sxz > =C < nx > B6 B4 sxy sy syz 7 C n 5 y @ A > ; : > sxz syz sz nz ðx0;y0;z0Þ 02 38 91 sx sxy sxz < nx = B6 7 C ¼B : (2.74) @4 sxy sy syz 5: ny ;A nz sxz syz sz ðx;y:zÞ

The two terms involved in the above relationship are tractions at P and respectively, as the corresponding parts on the opposite sides of the boundary.

P 0,

2.6 Concluding remarks A brief but sufficiently systematic account has been presented in this chapter on the geometric symmetries, reflectional, rotational and

42

Representative Volume Elements and Unit Cells

translational as three generic types. They have been extended to describe the special features of physical fields as often appear in practical problems. Systematic derivations of the symmetry conditions for the physical problem concerned under each of the three types of symmetries are presented, respectively. In the process of derivations, only two considerations were employed: the continuity as reflected in the use of a free body diagrams and the symmetry transformation on physical fields concerned, on the assumption that the domain in which the physical fields are defined is of the relevant geometrical symmetry, as well as the medium occupying the domain as the material under consideration. It has been made clear that the translational symmetry is the only symmetry type which can reduce the domain from infinite to finite. It will therefore have to be accepted as one of the essential elements in the study of UCs. Once this type of symmetry is appropriately formulated, the necessity of the concept of relative displacement field becomes apparent. In order to proceed with the study of UCs on a rigorous basis, it has to be formulated properly, which has been done in this chapter. On the contrary, both the reflectional and rotational symmetries can only reduce an infinite domain to semi-infinite, which is still infinite in extent. In this respect, reflectional and rotational symmetries have often been subjected to abuses, as will be discussed in Chapter 5.

References Abaqus Analysis User’s Guide, 2016 HTML Documentation, Dassault Systemes, Rhode Island, USA Humphreys, J.F., Prest, M.Y., 2004. Numbers, Groups and Codes. Cambridge University Press, Cambridge. Li, S., 1999. On the unit cell for micromechanical analysis of fibre-reinforced composites. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455, 815e838. Li, S., 2001. General unit cells for micromechanical analyses of unidirectional composites. Composites Part A: Applied Science and Manufacturing 32, 815e826. Li, S., 2008. Boundary conditions for unit cells from periodic microstructures and their implications. Composites Science and Technology 68, 1962e1974. Li, S., Kyaw, S., Jones, A., 2014. Boundary conditions resulting from cylindrical and longitudinal periodicities. Computers & Structures 133, 122e130. Li, S., Reid, S.R., 1992. On the symmetry conditions for laminated fibre-reinforced composite structures. International Journal of Solids and Structures 29, 2867e2880. Li, S., Wongsto, A., 2004. Unit cells for micromechanical analyses of particle-reinforced composites. Mechanics of Materials 36, 543e572. Li, S., Zou, Z., 2011. The use of central reflection in the formulation of unit cells for micromechanical FEA. Mechanics of Materials 43, 824e834.

CHAPTER 3

Material categorisation and material characterisation 3.1 Background Solid mechanics grew side by side with the metallurgy over the past few centuries. Since metallic materials were the major application of solid mechanics, in most textbooks, the homogeneity, isotropy and linearity of elastic characteristics are considered as the preconditions. Under such preconditions, the materials have been categorised and the category of these materials are as assumed, i.e. homogenous, isotropic and linearly elastic materials. For materials in this category, the material behaviour can be fully described by two material properties, viz. Young’s modulus, E, and Poisson’s ratio, n. The characterisation of these materials in terms of their elastic behaviour is simply the determination of these two properties. In most practical scenarios, the suppliers of materials concerned are responsible for providing them and there are standards available to follow in order to have these properties measured. With only one category of materials being of practical interest over a long period of time, it is therefore not surprising that the material categorisation has never been addressed. Industrial applications of composites upset the established preconditions to a large extent. However, since the early uses of composites mostly involved laminates based on unidirectional (UD) fibre-reinforced laminae, the complications associated with modelling the heterogeneity across the thickness of the laminate has been eased off with the establishment of laminate theories of various types, e.g. the classical laminate theory (Jones, 1998), the first order shear deformable laminate theory (Reissner, 1945; Mindlin, 1951), high order theories (Reddy, 1984) and the layer-wise theory (Mau, 1973). The behaviour of a laminated composite can be described as a two-dimensional problem, one way or another, with its properties obtained from its building blocks, i.e. UD laminae. The material properties required to describe the behaviour of UD composites turn out to be a reasonably straightforward extension of those for metals, although one might have to associate Young’s modulus and Poisson’s ratio with specific Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00003-7

© 2020 Elsevier Ltd. All rights reserved.

43

j

44

Representative Volume Elements and Unit Cells

directions, and the shear modulus becomes independent due to the anisotropy of the material. Standards (ASTM, 2014; British Standards Institution, 1997) for their measurements have been developed mostly by adapting their isotropic counterparts with shear (ASTM, 2012; 2013; British Standards Institution, 1998; Mohseni Shakib and Li, 2009) as an exception. For isotropic materials, there is no need to measure the shear modulus as it can be derived from E and n due to their isotropy. This explains the diversity in the methods of measuring shear properties for composites in comparison with measuring Young’s moduli and Poisson’s ratios. There has not been apparent need for material categorisation so far whilst composites have been applied to their contemporary level. An illustration of how the need for material categorisation was avoided can be given through an observation below. It is worth noting that most of users of composites were not aware of this need hence did not realise that they had skipped this step when developing products for the industrial scale applications. The material’s principal axes for UD composites are apparent and can be easily identified by anyone. They have always been taken naturally as the coordinate system in which material is to be characterised. UD composites therefore exhibit their orthotropy by default. However, in order to define a laminate as a structure, a unique coordinate system has to be employed, which does not always coincide with the material’s principal coordinate system, as a laminate usually comprises a significant number of off-axis plies. In such a structural coordinate system, an off-axis UD composite no longer appears as an orthotropic material anymore. An important step in the mechanics of composite materials at this point is to introduce coordinate transformation. With this, one avoids the need to characterise each offaxis ply as a generally anisotropic material. However, with a slightly more open mind, one would start to sense the restriction of the existing framework of material characterisation system if he/she cared to ask how accurately the orthotropy is kept by the material. Modern efforts seem to try to accommodate inaccuracy in fibre orientation under the consideration of misalignment, once again, to avoid a fundamental overhaul of the existing methods of material characterisation. The restriction the lack of material categorisation places on applying new materials in engineering design will be further revealed in the next observation. Due to the limited applicability of laminate theories in strength prediction, in aerospace industry, design allowables have conventionally been obtained experimentally for a range of potentially useful laminate layups. To fit to existing standards, only balanced layups, in which for any

Material categorisation and material characterisation

45

off-axis ply at a ply angle of q, there is another ply at -q, could be selected in order for the laminate to demonstrate an effectively orthotropic behaviour. Even so, given the unlimited combinations of layups, the process of obtaining the design allowables as an essential step is viewed as a significant commitment for any specific development. Whilst the acquisition of the design allowables is well-recognised as the process of material characterisation, the laminates have been limited only to balanced layups. A material categorisation exercise has been bypassed by having effectively orthotropic materials only. If materials are categorised into orthotropic and nonorthotropic, with the latter category being filtered out before it entered any serious consideration, the categorisation step has been essentially skipped as redundant. The consequence of this mentality leads to the following scenario. The modern development of science and technology gives rise to new materials, often with special micro/meso structures, e.g., auxetic materials, metamaterials and 3D textile composites, which promise a high potential in prospective applications. However, designers are often not able to take advantage of these materials, not because of the lack of need for them, but because the existing material characterisation system does not support materials of this category. The fundamental reason for this is the absence of the concept of material categorisation. Without materials being appropriately categorised, their characterisation becomes irrelevant. Before they can be characterised, their applications are simply beyond reach. By exposing the lack of material categorisation as a crucial factor preventing effective applications of new materials, the authors advocate the development and use of material categorisation as an essential procedure before characterisation is even contemplated. Such a procedure should be established systematically and, ideally, should be standardised to support the engineering acceptance of emerging new materials. Hopefully, the next section would serve as a first stab to the development along this line.

3.2 Material categorisation Any serious application of a material requires relevant properties of the material to be made known. The process of obtaining such properties is often referred to as material characterisation. Existing material characterisation procedures follow available standards. A common restriction among all existing standards is that they are applicable only to materials which exhibit effective orthortropy. As material characterisation is such an important

46

Representative Volume Elements and Unit Cells

Fig. 3.1 Different types of internal structure: (a) a twill weave and (b) a gyroid.

procedure, it would not be a perfectly professional attitude to jump to it blindly. To demonstrate this, consider a twill weave composite as an example, as shown in Fig. 3.1(a). If readers are challenged to characterise it, how should this be conducted? To be more specific, in order to cut specimens for testing, along which directions should they be cut? These directions should certainly be along the principal axes of the material. Given the orthogonal tow paths, one could intuitively choose the orthogonal axes in the plane of the fabric along the fibre tows as the principal axes of the material. However, apart from intuition, there is no justification whatsoever for such selection of the orientations of specimens. The same applies to materials of gyroidal internal structure as shown in Fig. 3.1(b). The development of modern materials keeps breaking the boundary of the conventional category of materials, which is the category of homogeneous, isotropic and linearly elastic materials. Therefore, the position of material characterisation without material categorisation needs a serious review. Categorisation is to put the material concerned into an appropriate category. It is therefore mostly qualitative, whilst following strict definitions of each category. The extent of heterogeneity, degree of anisotropy and severity of nonlinearity are always significant considerations in the material selection phase of any serious engineering design. They need to be categorised before subsequent quantitative material characterisation becomes meaningful. To date, a substantial volume of studies is available characterising material nonlinearity, such as hypo-elasticity, hyper-elasticity, plasticity, etc., in homogeneous and isotropic materials. Extension of these studies into the realm

Material categorisation and material characterisation

47

of heterogeneous and anisotropic materials is hard to contemplate before heterogeneity and anisotropy have been addressed appropriately. It will be attempted in this chapter under the assumption of linearly elastic material behaviour, as is the scope of this book.

3.2.1 Homogeneity In material categorisation, homogeneity is undoubtedly the most important descriptor to examine, although it is often taken for granted. Without it, the behaviour of a material will vary from point to point. There are materials, natural or engineered, such as bones and functionally graded materials, showing noticeable spatial variation of properties. With the latest developments of manufacturing technology for composites, such as 3D printing (Zhuo et al., 2017) and automatically steered tape lay (Gurdal and Olmedo, 1993), there has been a trend for laminates to deviate from their conventional construction as a layup of UD laminae. With curved fibre paths, effective properties of the material will vary with coordinates even along the same fibre tows. Material heterogeneity will be inevitable due to the anisotropy of the material and also possibly due to the varied fibre volume fraction. The characterisation of such materials in general becomes a rather specialised study and therefore will be beyond the scope of this book. Attention will be focused only on the class of materials which can be considered homogeneous at least at one length scale, whilst the material is expected to be used in engineering within this length scale or above. Heterogeneity will be present at lower length scales. One of the tasks of the so-called micromechanics is to homogenise the heterogeneity at a lower length scale so that the material can be treated as homogeneous for engineering applications at the upper length scale. A material can be considered as homogeneous at one length scale if it is of either a completely random or a perfectly regular architecture at a lower length scale. In order to homogenise it, one can find the concept of RVEs for materials of random structures and UCs for materials of regular structures at a lower length scale useful means. The underlying type of symmetry to facilitate this categorisation is translation. A material is effectively homogeneous if it possesses translational symmetries in three non-coplanar directions, either in a statistical sense for random architectures or a rigorous geometric sense for regular architectures, where the minimum distances of these translations determine the characteristic dimensions of the RVE or UC. Identifying such translational symmetry/symmetries is the first task of material categorisation. Without such translational symmetry, a material

48

Representative Volume Elements and Unit Cells

will have to be considered as generally heterogeneous, which will be very difficult to deal with as there is no industrial standards yet that could be applied to characterise it, let alone any consistent and meaningful applications of it. Given the above categorisation on homogeneity, a natural guidance for the subsequent material characterisation is that specimens employed should have sufficient number of RVEs or UCs within the gauge length in all directions, since the experimental results could be distorted if the number chosen is too low. However, there may not be a straightforward answer to the question as to how many of RVEs/UCs should be considered sufficient, as this may vary from material to material and from property to property. The users should be aware of this fact before deciding on the practical size for the specimens they are going to test and hence should devote some efforts into building up the experience on the material concerned. For this reason alone, existing standards should be followed with due diligence, in particular, when deciding on the dimensions of the specimen. As a good practice or guideline, a systematic procedure should be in the form of some kind of convergence study. Specimens of increasing dimensions should be tested until the point when further increase in dimensions would not make noticeable differences to the properties being measured. One can then take the minimum dimensions from the converged ones as the representative dimensions for the specimens. For materials of random architectures, the dimensions are physical sizes whilst for material of regular architectures they can be number of cells in each direction. In the latter case, incomplete cells should be avoided when possible, especially if the number of cells to be included is not particularly large.

3.2.2 Anisotropy Having established the homogeneity of a material, the next logical step is to categorise the material’s anisotropy. The elasticity of a generally anisotropic material is described by a full 66 stiffness or compliance matrix. The symmetry of these matrices is present due to the condition of the existence strain energy or complimentary strain energy density (Jones, 1998). Consider strain energy density, which, in general, is defined as follows 1 U ¼ Cij εi εj ; 2

(3.1)

where contracted notation has been used for strain and stiffness tensors. Stiffness matrix can be expressed via differentiation of strain energy density as

Material categorisation and material characterisation

v2 U v2 U ¼ Cij and ¼ Cji : vεi vεj vεj vεi

49

(3.2)

The existence of the partial derivatives requires them to be the same, i.e. v2 U v2 U ¼ hence Cij ¼ Cji : vεi vεj vεj vεi

(3.3)

This is also the continuity requirement for the partial derivatives involved. A meaningfully defined strain energy density function should possess such continuity in its partial derivatives as the minimum continuity requirement. The same applies to the compliance matrix if the complementary strain energy is considered instead. With the symmetry of the stiffness and compliance matrices, a most generally anisotropic material requires 21 independent elastic constants in order to characterise its elastic behaviour. Generally anisotropic materials are rarely applied in any practical problems since there are simply no established means to characterise them experimentally. Most engineering materials possess features that can be utilised in order to simplify the presentation of the material. A systematic simplification can be made if a principal axis and principal plane can be identified. In order to identify a principal axis or principal plane, the concept of symmetry can be used again following the development in Chapter 2, in particular, Section 2.3 regarding the symmetry and antisymmetry of tensorial physical fields. To categorise the anisotropy for the material’s elastic behaviour, references will be made to the strain tensor as the stimulus to a material as the system and the stress tensor as the response. The stimulus and response are so chosen in order to categorise the stiffness tensor C and they can be swapped around if one goes for the compliance tensor S instead. A plane is a principal plane if, when the material is subjected to a stimulus (e.g. strains) symmetric about this plane, the response (e.g. stresses) to the stimulus will be symmetric about the plane, and if the response to an antisymmetric stimulus is antisymmetric about the plane. The axis perpendicular to a principal plane is a principal axis. If there exist a principal axis or a principal plane in a material, it is called monoclinic, and it will have 13 independent elastic constants. If another principal axis perpendicular to the existing principal axis, or a principal plane perpendicular to the existing principal plane can be identified, the material will be categorised as orthotropic, which has 9 independent elastic constants.

50

Representative Volume Elements and Unit Cells

The tasks of categorisation of anisotropy is to identify if there exists any principal axis or plane in the material. A method for identifying the principal axis or planes is proposed here based on use of symmetries present in the material. In sections to follow, it will be proven that a plane of reflectional symmetry in the material is a principal plane and an axis of rotational symmetry is a principal axis. It is fairly straightforward to exhibit the effects of a reflectional symmetry about one plane on the material to be categorised and such materials are classified as monoclinic materials, although systematic derivation is not usually found in textbooks. One might argue that such derivation is straightforward and therefore can be waived. This is in fact true. However, without such straightforward derivation for a reflectional symmetry as a stepping stone, it would not be as simple to go through the derivation in order to show the effects of a rotational symmetry. As a result, rotational symmetry in material categorisation has never been even mentioned in the literature, let alone being established. Failure to categorising materials of the micro/ meso structures as shown in Fig. 3.1 is the proof, as will be elaborated in details in Section 3.2.2.2. For this reason, the effects of a reflectional symmetry are examined systematically first as follows. 3.2.2.1 Reflectional symmetry Consider the elasticity problem in which the material properties are fully described by its stiffness matrix C. The stress and strain components in their conventional sense are sketched on right hand side of Fig. 3.2, where they are shown only on the visible faces of the infinitesimal cubes. Consider a reflectional transformation about the x-plane, i.e. the plane perpendicular

z' x'

y'

V'zz,H'zz

W'xz,J'xz W'yz,J'yz

z Plane of reflection

W'xy,J'xy

V'yy,H'yy Image

V'xx,H'xx

Vzz,Hzz y x

Wyz,Jyz

Wxz,Jxz

Vxx,Hxx

Wxy, Jxy W'xz,J'xz

Wxy,Jxy

Vyy,Hyy Original

Fig. 3.2 Reflection of a stress state and the corresponding strain state about the xplane.

51

Material categorisation and material characterisation

to the x-axis. The reflected stress state and the corresponding strain state are shown on the left of the reflection plane. Under the same reflectional transformation, stiffness C is transformed to C0 , the stress state s to s0 and the corresponding strain ε state to ε0. Before and after the transformation, one has s ¼ Cε and s0 ¼ C0 ε0 : (3.4) Depending on the material it is applied to, the reflectional transformation may preserve its properties or upset them, as schematically shown in Fig. 3.3(a) and (b), respectively. The material is symmetric about the plane if its properties are preserved under the symmetry transformation. Identifying such symmetry is the first task of material categorisation. In absence of any symmetry, a material will have to be categorised as a generally anisotropic material. Assuming that a reflectional symmetry has been identified, the coordinate system can always be set up as in Fig. 3.2 with the reflection plane in Fig. 3.2 coincident with the plane of the reflectional symmetry, in which case the following equality holds C0 ¼ C.

(a)

(3.5)

y

y

C'

C

x

(b)

x

y

y

C

C'

x

x

Fig. 3.3 Schematic illustrations of materials under reflectional transformation about y-axis (or x-plane), imaging the material being a UD composite with the hatch pattern representing the fibres: (a) the material properties are preserved, and (b) the material properties are changed after the transformation.

52

Representative Volume Elements and Unit Cells

With the stress and strain states before and after the transformation being related as shown in Fig. 3.2, if they are presented referring to a common coordinate system as

8 0 9 sxx > > > > > > > 0 > > syy > > > > > > > > < s0 > =

8

9

ε0xx > 8 > 9 8 > > > > > εxx 9 s xx > > > > 0 > > > > > > εyy > > > > > > > > > > > > εyy > > > > syy > > > > > > > > > > > > > > > > > < 0 = = < < ε εzz = szz zz zz ¼ ¼ and ; > syz > > gyz > > > s0yz > g0yz > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > sxz > > > > gxz > > > > > 0 > 0 > > > > > > > > > sxz > gxz > > > ; ; : : > > > > > > > > g s xy xy : 0 ; : 0 ; sxy gxy

(3.6)

the following relationship holds:

8

9

s0xx > 9 > 8 > > > sxx > > > > 0 > > > > > s > > > > yy > > > > > > s yy > > > > > > > > > > < = 0 =

> > s > s yz > > > yz > > > > > > > > > > > > > > s > > > xz > 0 > > > s > ; > : xz > > > sxy : 0 > ; sxy 2 c11 c12 c13 c14 6c 6 21 c22 c23 c24 6 6 c31 c32 c33 c34 ¼6 6c 6 41 c42 c43 c44 6 4 c51 c52 c53 c54 c61

c62

c63

c64

¼ Cε0

c15 c25 c35 c45 c55 c65

9 38 c16 > εxx > > > > > > > ε > > c26 7 yy > > 7> > > 7> < c36 7 εzz = 7 . c46 7 > gyz > > 7> > > > 7> > > c56 5> > gxz > > > > : ; gxy c66

(3.7)

This can be rewritten as an equation with the minus signs in the stresses and strains being absorbed into the coefficient matrix as follows:

53

Material categorisation and material characterisation

8 9 2 sxx > c11 > > > > > 6 > > > syy > c21 > > > > 6 >

= 6 6 c31 zz ¼6 > syz > > 6 > 6 c41 > > > > 6 > > > sxz > > 4 c51 > > > : ; sxy c61 Given 2 c11 6c 6 21 6 6 c31 6 6c 6 41 6 4 c51

c12

c13

c14

c15

c22

c23

c24

c25

c32

c33

c34

c35

c42

c43

c44

c45

c52

c53

c54

c55

c62

c63

c64

c65

9 38 c16 > εxx > > > > > εyy > > > c26 7 > > 7> > > > 7> < c36 7 εzz = 7 : > gyz > c46 7 > 7> > > > 7> > gxz > > c56 5> > > > > : ; gxy c66

(3.8)

(3.4), one has c12 c22

c13 c23

c14 c24

c15 c25

c32

c33

c34

c35

c42

c43

c44

c45

c52

c53

c54

c55

3 c16 c26 7 7 7 c36 7 7 c46 7 7 7 c56 5

c61

c62 c63 c64 c65 c66 8 9 2 εxx > c11 c12 c13 > > > > > 6 > > εyy > > c21 c22 c23 > > > > 6 > < εzz > = 6 6 c31 c32 c33  h6 6 > gyz > c42 c43 > > 6 c41 > > > > 6 > > > 4 c51 c52 c53 gxz > > > > > : ; gxy c61 c62 c63

c14

c15

c24

c25

c34

c35

c44 c54

c45 c55

c64

c65

9 38 c16 > εxx > > > > > εyy > > > c26 7 > > 7> > > > > 7< c36 7 εzz = 7 ; gyz > c46 7 > > 7> > > > 7> > > gxz > c56 5> > > > > : ; g c66 xy (3.9)

which should hold at an arbitrary strain state ε. This yields the equality of the stiffness matrices as follows: 2

c11

6c 6 21 6 6 c31 6 6c 6 41 6 4 c51 c61

c12

c13

c14

c15

c22

c23

c24

c25

c32 c42

c33 c43

c34 c44

c35 c45

c52 c62

c53 c63

c54 c64

c55 c65

c16

3

2

c11

6 c26 7 7 6 c21 7 6 c36 7 6 c31 7¼6 6 c46 7 7 6 c41 7 6 c56 5 4 c51 c66 c61

c12

c13

c14

c15

c22

c23

c24

c25

c32 c42

c33 c43

c34 c44

c35 c45

c52 c62

c53 c63

c54 c64

c55 c65

c16

3

c26 7 7 7 c36 7 7. c46 7 7 7 c56 5 c66 (3.10)

54

Representative Volume Elements and Unit Cells

Obtaining (3.10) from (3.9) should not be considered as cancelling common factors which is not usually applicable to matrix or tensor operations, in general. From the equality in (3.10), all the components equal in magnitude and of opposite sense should be equal to zero; for instance, if c15 ¼ c15 , then c15 ¼ 0, because zero is the only number which can be considered as positive and negative at the same time. Therefore, c15 ¼ c25 ¼ c35 ¼ c45 ¼ c16 ¼ c26 ¼ c36 ¼ c46 ¼ 0: (3.11) Given the above, the stiffness matrix of the material, which is reflectionally symmetric about the x-plane, is reduced to 2 3 c11 c12 c13 c14 0 0 6c 0 7 6 21 c22 c23 c24 0 7 6 7 6 c31 c32 c33 c34 0 0 7 6 7 : (3.12) C¼6 0 7 6 c41 c42 c43 c44 0 7 6 7 4 0 0 0 0 c55 c56 5 0 0 0 0 c65 c66 If one resorts to the algebraic way of considering the effects of symmetry as described in Section 2.3 of Chapter 2 regarding the transformation of tensors, an alternative approach can be put forward. The components of the stiffness matrix in their contracted form can be expressed in terms of their tensorial counterparts as follows 2

c11 6c 6 21 6 6 c31 6 6c 6 41 6 4 c51 c61

c12 c22 c32 c42

c13 c23 c33 c43

c14 c24 c34 c44

c15 c25 c35 c45

c52 c62

c53 c63

c54 c64

c55 c65

3 2 c1111 c16 6 c c26 7 7 6 2211 7 6 c36 7 6 c3311 7¼6 6 c46 7 7 6 2c2311 7 6 c56 5 4 2c1311 c66 2c1211

c1122 c2222 c3322 2c2322

c1133 c2233 c3333 2c2333

2c1123 2c2223 2c3323 4c2323

2c1113 2c2213 2c3313 4c2313

2c1322 2c1222

2c1333 2c1233

4c1323 4c1223

4c1313 4c1213

3 2c1112 2c2212 7 7 7 2c3312 7 7 4c2312 7 7 7 4c1312 5 4c1212 (3.13)

Since the reflection is about the x-plane, only the x-axis reverses its direction under the reflectional transformation. The components inside the matrix on the right having odd number of ‘1’ in their subscripts change sense after the reflectional transformation, whilst the rest keep their sense. Thus, the reflectional symmetry leads to the following relationship for the those components cijkl ¼  cijkl .

(3.14)

Material categorisation and material characterisation

55

This argument brings the derivation straight to (3.10), which leads to exactly the same conclusion of vanishing components of the stiffness matrix as shown in (3.12). The appearance of the stiffness or compliance matrix as given in (3.12) is of a typical monoclinic characteristics, as signified by the vanishing coupling effects between direct stresses/strains and the two shear counterparts, s13 and s12 , exposed on the principal plane. The monoclinic characteristics are present only when the principal plane has been employed as one of the coordinate planes. Otherwise, the stiffness matrix would still be a full matrix, although one can prove that out of all components of the full matrix there are only 13 of them that are independent and the rest can be expressed in terms of the independent ones. If the coordinate system had been chosen such that the plane of reflectional symmetry is coincident with the y-plane, instead, the principal plane would be the y-plane and the vanishing components would be those coupling between direct stresses/strains and the two shear counterparts, s23 and s12 . The same argument applies to the case of the z-plane being the principal plane. The existence of a further reflectional symmetry about a plane perpendicular to the existing principal plane reduces the material to orthotropy. The presence of two perpendicular principal planes implies that the third plane perpendicular to the two principal planes is also a principal plane. The elastic behaviour of an orthotropic material can be fully described by 9 elastic constants. It should be noted that the existing industrial standards only apply to characterisation of materials with the degree of anisotropy no higher than orthotropy (ASTM, 2014; British Standards Institution,1997). Examples of abuses of the existing standards will be discussed in the next subsection in relation to the twill weave composites. Further categorisation along the line is possible. If a material is reflectionally symmetric about planes 45 apart intersecting at a common axis, it is squarely symmetric in the plane perpendicular to the axis. Its elastic behaviour can be fully described by 6 elastic constants. An example of such material can be a UD composite of squarely packed fibres over its transverse cross-section (Li, 1999, 2001), which is shown in Fig. 3.4(a). If a material is reflectionally symmetric about planes 22.5 or 120 apart intersecting at a common axis, it is axisymmetric in the plane perpendicular to the axis, and such material is conventionally referred to as transversely isotropic about the axis. Its elastic behaviour can be fully described by 5 elastic constants. An example of material with symmetry planes 120 apart can be a UD composite of hexagonally packed fibres over its transverse cross-section (Li, 1999, 2001), as shown in Fig. 3.4(b).

56

Representative Volume Elements and Unit Cells

(a)

(b)

Fig. 3.4 Example of (a) squarely symmetric and (b) transversely isotropic case.

If a material is squarely symmetric in two planes perpendicular to each other, it is cubically symmetric, and its elastic behaviour can be fully described by 3 elastic constants. A material transversely symmetric about two axes perpendicular to each other is isotropic, and its elastic behaviour can be fully described by 2 elastic constants. 3.2.2.2 Rotational symmetry The role of rotational symmetries in material categorisation has hardly been mentioned anywhere in the literature. Consider a 180 rotation about the x-axis as sketched in Fig. 3.5. Under the rotation, the original stress state and the corresponding strain state, shown on the right hand side of the figure, are transformed to those shown on the left hand side. If the material is symmetric under the rotation, (3.5) holds in this case as well. If the images of the stress and strain states in Fig. 3.5 are expressed in the original material coordinate system, one has 8 9 8 9 εxx > > > ε0xx > 8 0 9 8 > > 9 > > > > > > > > > > s sxx > > > > > xx > > > > > > > > > > > > > > > 0 > > > > > > > > ε ε yy > > > > > > > > 0 yy > > > > > > > > s > > > > s > > > > yy yy > > > > > > > > > > > > > > > > > > > > > > > > > > > > 0 > > > < = < s0 = > ε ε = = < < zz s zz zz zz ¼ ¼ and ; (3.15) 0 > > 0 > > > > > > s s > > > > g yz > > > > yz > > > > yz > > > > > > > gyz > > > > > > > > > > > > > > > > > > > > 0 > > > > > > > > > s s > > > > > > > xz > > 0 > xz > > > > > > g g > > > > > > > xz > xz > > > > > : 0 ; > ; > : > > > > > sxy > > sxy > > > > > > ; : g0xy ; : g xy

57

Material categorisation and material characterisation

y'

x'

V'zz,H'zz

W'xz,J'xz

y x

W'yz,J'yz

z'

Vzz, Hzz

z

W'xy,J'xy V'yy,H'yy

V'xx,H'xx

Wxy, Jxy

180q

Vxx, Hxx Vyy, Hyy

W'xz,J'xz

Image

Fig. 3.5 axis.

180

Wxz, Jxz Wxy, Jxy

Wyz, Jyz

Original

rotation of a stress state and the corresponding strain state about the x-

which is identical to (3.6). Following exactly the same procedure as adopted in dealing with the reflectional symmetry, an equality of the stiffness matrices before and after transformation is obtained as 2

c45 c55

3 2 c11 c16 6 7 c26 7 6 c21 7 6 c36 7 6 c31 7¼6 6 c46 7 7 6 c41 7 6 c56 5 4 c51

c65

c66

c61

c11 6c 6 21 6 6 c31 6 6c 6 41 6 4 c51

c12 c22

c13 c23

c14 c24

c15 c25

c32

c33

c34

c35

c42 c52

c43 c53

c44 c54

c61

c62

c63

c64

c12 c22

c13 c23

c14 c24

c15 c25

c32

c33

c34

c35

c42 c52

c43 c53

c44 c54

c45 c55

c62

c63

c64

c65

3 c16 c26 7 7 7 c36 7 7; c46 7 7 7 c56 5 c66 (3.16)

which is again identical to (3.13). Applying the same reasoning as was adopted previously regarding the components of equal magnitude and opposite sense, the final form of stiffness matrix becomes 2 3 c11 c12 c13 c14 0 0 6c 0 7 6 21 c22 c23 c24 0 7 6 7 6 c31 c32 c33 c34 0 0 7 6 7: (3.17) C ¼6 7 c c c c 0 0 41 42 43 44 6 7 6 7 4 0 0 0 0 c55 c56 5 0 0 0 0 c65 c66 Unlike the transformation in the reflectional symmetry where only the x-axis reverses its direction, in a rotational symmetry, both the y- and z-axes reverse their direction whilst the x-axis keeps its direction as shown in Fig. 3.5. Again, the same result can be obtained taking the algebraic route of the transformation of tensors as was described in Section 2.3 of Chapter 2. The

58

Representative Volume Elements and Unit Cells

stiffness matrix expressed in the tensorial form is identical (3.13), namely 2 3 2 c1111 c1122 c1133 c11 c12 c13 c14 c15 c16 6 6c 7 6 21 c22 c23 c24 c25 c26 7 6 c2211 c2222 c2233 6 7 6 6 c31 c32 c33 c34 c35 c36 7 6 c3311 c3322 c3333 6 7¼6 6c 7 6c c c c c c 41 42 43 44 45 46 6 7 6 2311 c2322 c2333 6 7 6 4 c51 c52 c53 c54 c55 c56 5 4 c1311 c1322 c1333 c61

c62

c63

c64

c65

c66

c1211

c1222

c1233

to that given by c1123

c1113

c2223

c2213

c3323

c3313

c2323

c2313

c1323

c1313

c1223

c1112

3

c2212 7 7 7 c3312 7 7: c2312 7 7 7 c1312 5

c1213 c1212 (3.18) Unlike in case of reflectional transformation, when the change in sense of certain components under symmetry transformation was associated with components having odd number of ‘1’ in the subscripts, in the present case the change of sense is due to the sums of the occurrences of ‘2’ and ‘3’ in the subscripts being odd, because both axes 2 and 3 reverse their directions under a rotational transformation, whilst all other components having even sums. This yields stiffness matrix expression as given by Eq. (3.17), which in turn is identical to (3.12). Thus the equivalence between a 180 rotational symmetry about an axis and a reflectional symmetry about a plane perpendicular to the axis of the rotational symmetry has been fully established as far as the categorisation of material is concerned. A reflectional symmetry introduces a principal plane and by definition, the axis perpendicular to a principal plane is a principal axis. A rotational symmetry defines a principal axis as its direct outcome. By definition, the plane perpendicular to the axis of rotational symmetry is a principal plane. The equivalence between these two types of symmetries in terms of material categorisation is, however, by no means to imply that these two symmetries are the same in all other respects. In fact, they are very different in many ways other than what their functions are in material categorisation. The equivalence established here between the two symmetries is particularly significant when dealing with materials where reflectional symmetry is absent, whilst the presence of rotational symmetry is obvious, such as twill woven and the gyroidal patterns as shown in Fig. 3.1. The role of rotational symmetries in material categorisation does not seem to have been well recognised in the literature to the best of the authors’ knowledge. The twill weave shown in Fig. 3.1(a) is 180 rotationally symmetric about a diagonal axis in the plane of the fabric which is marked in Fig. 3.6(a) as dash-dot line. It is also 180 rotationally symmetric about an axis perpendicular to the plane of the fabric as represented by a dot in

Material categorisation and material characterisation

59

Fig. 3.6 Textile preforms of various symmetries: (a) twill weave; (b) 3D 4-axial braid.

Fig. 3.6(a). It is therefore orthotropic but with its principal axes in the plane of the fabric being 45 off the direction of fibre tows. In general, the material is not orthotropic about axes along the directions of fibre tows. When one resorts to mechanical testing to characterise the material, the specimens must be cut in the principal directions. Testing specimens cut in the directions of fibre tows will not be supported by any existing standard. Another typical textile preform is 4-axial 3D braid, a unit cell of which is shown in Fig. 3.6(b). There is no reflectional symmetry in the structure. However, it is 180 rotationally symmetric about all three coordinate axes as shown in Fig. 3.6(b) and it is therefore orthotropic. Similarly, the gyroidal structure does not have any reflectional symmetry but one can easily identify one axis out of each of six faces of an appropriately chosen unit cell, about which the structure is 180 rotationally symmetric, as shown in Fig. 3.7(a). Given the equivalence as established above, if a material is 90 rotationally symmetric about an axis, or, in notations introduced in Section 2.2.2 of Chapter 2, is of C4 symmetry, it is squarely symmetric in the plane perpendicular to the axis. If a material is 45 or 120 rotationally symmetric about an axis, i.e. is of 8 C or C3 symmetry, it is transversely isotropic about the axis.

60

Representative Volume Elements and Unit Cells

Fig. 3.7 The perspectives of a gyroid: (a) C3 symmetry about the axis out of the page shown as a white blob when viewing the square boxed part as a unit cell, and (b) C3 symmetry about diagonal axis shown as the blob.

In a similar manner, one can define cubic symmetry and isotropy accordingly. The gyroidal shape as shown in Fig. 3.1(b) has one and a half periods in each of the directions. Losing an appropriate half of a period in each direction, a cell as marked by white square Fig. 3.7(a) is obtained, which is rotationally symmetric about three orthogonal axes, the one relevant to the perspective as shown in Fig. 3.7(a) has been indicated by a white blob representing the axis pointing out of the page. The material is therefore orthotropic. In addition, one can also observe that the structure has a 120 rotational symmetry about one of the diagonals connecting the far opposite corners of the cube as shown in Fig. 3.7(b) where the picture was taken in the direction along this diagonal axis. As a result, the material behaviours in the three orthogonal principal directions are identical, delivering the cubic symmetry for the material having a gyroidal micro/mesoscopic structure.

3.3 Material characterisation Once the material is categorised, material characterisation becomes relevant. Experimental techniques suitable for one category of materials do not necessarily apply to materials in another category. For instance, those suitable for isotropic materials are not expected to be directly applicable to orthotropic materials and, similarly, those applying perfectly well to

Material categorisation and material characterisation

61

orthotropic materials would be misleading when blindly employed for monoclinic materials. Same as categorisation, in order to characterise a material, one needs to specify an appropriate length scale at which the outcomes of the characterisation will be presented. This should be in line with the scale where homogeneity can be assumed, i.e. the upper length scale, as introduced in Section 1.1. The objective of micromechanical material characterisation is to obtain effective properties of the material at this scale from the properties of its constituents and the architecture of the material at a lower length scale, which will reduce the demand on conducting physical testing of the material. The most popular family of properties are the effective elastic constants, such as Young’s moduli, Poisson’s ratios and shear moduli. It is essential that any meaningful measurements should be conducted following the definitions of these properties. In fact, this is precisely what standards are supposed to offer if the properties are measured experimentally. Whilst following the requirements as specified in standards, users could easily miss the considerations behind standard, i.e. the definitions of the properties to be measured. However, this does not usually cause a problem provided that the experimental procedures follow the requirements as specified in the standards. When tests are conducted virtually, problems arise. No specimens have to be machined anymore and they no longer have to be in the shape as defined in the standards, such as a dumbbell shape for tensile tests. For instance, in a virtual test, a specimen can be a unit cell only. Users often become too preoccupied with trying to meet the requirements as set in standards, whilst turning a blind eye to the definitions of the properties to be measured, which in fact underline the standards. The authors have witnessed attempts of modelling the dumbbell specimens without giving any considerations as to why a dumbbell shape was introduced in experiments in the first place. Therefore, to avoid potential confusion when applying virtual tools for material characterisation, the definitions of elastic properties are reproduced below. Young’s moduli:

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Representative Volume Elements and Unit Cells

s1 ε1 s2 E2 ¼ ε2 s3 E3 ¼ ε3

E1 ¼

under the conditions

s2 ¼ s3 ¼ s23 ¼ s13 ¼ s12 ¼ DT ¼ 0

under the conditions

s1 ¼ s3 ¼ s23 ¼ s13 ¼ s12 ¼ DT ¼ 0

under the conditions

s1 ¼ s2 ¼ s23 ¼ s13 ¼ s12 ¼ DT ¼ 0 (3.19)

Poisson’s ratios: ε2 ε3 n12 ¼  and n13 ¼  ε1 ε1

under the conditions

s2 ¼ s3 ¼ s23 ¼ s13 ¼ s12 ¼ DT ¼ 0; ε3 n23 ¼  under the conditions s1 ¼ s3 ¼ s23 ¼ s13 ¼ s12 ¼ DT ¼ 0; ε2 (3.20) whilst their complementary counterparts can be obtained as follows E3 n23 ; E2 E3 ¼ n13 ; E1 E2 ¼ n12 : E1

n32 ¼ n31 n21

(3.21)

If one wishes to have them measured, they should be obtained as follows. ε1 under the conditions s1 ¼ s3 ¼ s23 ¼ s13 ¼ s12 ¼ DT ¼ 0; n21 ¼  ε2 ε2 ε1 n32 ¼  and n31 ¼  under the conditions ε3 ε3 s1 ¼ s2 ¼ s23 ¼ s13 ¼ s12 ¼ DT ¼ 0: Shear moduli:

(3.22)

Material categorisation and material characterisation

s23 g23 s13 ¼ g13 s12 ¼ g12

63

G23 ¼

under the conditions

s1 ¼ s2 ¼ s3 ¼ s13 ¼ s12 ¼ DT ¼ 0

G13

under the conditions

s1 ¼ s2 ¼ s3 ¼ s23 ¼ s12 ¼ DT ¼ 0

under the conditions

s1 ¼ s2 ¼ s3 ¼ s23 ¼ s13 ¼ DT ¼ 0

G12

(3.23)

The conditions presented above can be verbalised as follows: Young’s moduli and Poisson’s ratios should be measured under uniaxial stress states in appropriate directions, not under uniaxial strain state. To ensure a uniaxial stress state, the material should be free from other stresses. A uniaxial or pure shear stress state can be compromised by any undesirable constraint. For instance, if a specimen is constrained from sideway deformation whilst under axial stress, the stress state obtained will not be a uniaxial one. It is fair to say that the definitions of these properties have rarely caused confusion. It is the conditions the material is subjected to that are often casually imposed. This is obviously a blind spot in many practices. The authors have simply witnessed too many mistakes of this kind, and so often they were perceived as minor mistakes not worth correcting even after the users were made aware of them. Similarly, a shear modulus should be obtained under an effectively pure shear stress state. The simulation of this loading case tends to be more prone to mistakes. Amongst the publications on micromechanical analysis, nine out of ten of them tended to avoid involving shear, often without providing any reason for such omission. The same considerations can be given to all other physical disciplines, e.g. the diffusion problem. A key rule to bear in mind is that any effective property should be obtained according to its physical definition. Micromechanical analysis is to provide a means of virtual testing and it is the responsibility of the user to ensure that all required testing conditions are observed in the simulation as well as in the physical testing. In many ways, it should be a lot easier to observe these conditions in theoretical simulation than on the physical lab floor. Even so, there are probably as many pitfalls in virtual testing as in physical testing where mistakes can be made. Right attitude is often the key to success and nothing of this kind is meant to be easily achievable through casual tampering. As has been already stated previously, the material characterisation is meant to be an exercise for obtaining quantitative information about the material. Any avoidable error, no matter how large or small, should be

64

Representative Volume Elements and Unit Cells

eliminated whenever it is practical and is not at a significant cost. Any unavoidable error should come with an appropriate assessment in terms of its effects and should be duly reflected in the restrictions placed on the applicability of the employed material characterisation method. Bringing avoidable errors into any practical evaluation is not a so-called engineering approach; in fact, it is an abuse of engineering approaches. Professional engineering approaches are to keep unavoidable error under control. Following the definitions of material properties given here, micromechanical material characterisation will be pursued by analysing appropriately formulated RVEs or UCs which will be the subject of the Part II of this book. The analyses involved are to obtain material responses when subject to loading conditions as specified by the definition of the desired effective material properties to be evaluated. This approach does not mimic the requirements as specified in the standards, but reflects the spirit of the standards.

3.4 Concluding remarks The concept of material categorisation has been proposed formally alongside with material characterisation. They are equally important, with the former establishing the qualitative properties of the material in terms of its range of linearity, its length scale in which homogeneity can be assumed and its degree of anisotropy. Material characterisation is a quantitative process which should only be conducted once the material has been appropriately categorised as it is only then when one can be sure which standard is deemed to be suitable to follow. Multiscale modelling employed as a tool for material characterisation is a way of virtual testing. The key to fulfil a virtual test in order to measure a certain property of the material is to follow the definition of the property and this is more important than following the respective standard.

References ASTM Standard D3039/D3039M-14, 2014. Standard test method for tensile properties of polymer matrix composite materials. ASTM International, West Conshohocken, PA. ASTM Standard D3518/D3518M-13, 2013. Standard test method for in-plane shear response of polymer matrix composite materials by tensile test of a 45 laminate. ASTM International, West Conshohocken, PA. ASTM Standard D7078/D7078M-12, 2012. Standard test method for shear properties of composite materials by V-notched rail shear method. ASTM International, West Conshohocken, PA.

Material categorisation and material characterisation

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British Standards Institution, BS EN ISO 14129, 1998. Fibre-reinforced plastic composites d determination of the in-plane shear stress/shear strain response, including the in-plane shear modulus and strength, by the  45 tension test method. England: BSI, London. British Standards Institution, BS EN ISO 527-4, 1997. Plastics. Determination of tensile properties. test conditions for isotropic and orthotropic fibre-reinforced plastic composites. England:BSI, London. Gurdal, Z., Olmedo, R., 1993. In-plane response of laminates with spatially varying fiber orientations - variable stiffness concept. AIAA Journal 31, 751e758. Jones, R.M., 1998. Mechanics of Composite Materials. CRC Press, Boca Raton. Li, S., 1999. On the unit cell for micromechanical analysis of fibre-reinforced composites. In: Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455, pp. 815e838. Li, S., 2001. General unit cells for micromechanical analyses of unidirectional composites. Composites Part A: Applied Science and Manufacturing 32, 815e826. Mau, S.T., 1973. A refined laminated plate theory. ASME Journal of Applied Mechanics 40, 606e607. Mindlin, R.D., 1951. Influence of rotatory inertia and shear in flexural motion of isotropic elastic plates. ASME Journal of Applied Mechanics 18, 31e38. Mohseni Shakib, S.M., Li, S., 2009. Modified three rail shear fixture (ASTM D 4255/D 4255M) and an experimental study of nonlinear in-plane shear behaviour of FRC. Composites Science and Technology 69, 1854e1866. Reddy, J.N., 1984. A simple higher-order theory for laminated composite plates. ASME Journal of Applied Mechanics 51, 745e752. Reissner, E., 1945. The effect of transverse shear deformation on the bending of elastic plates. ASME Journal of Applied Mechanics 12. A68-77. Zhuo, P., Li, S., Ashcroft, I., Jones, A., Pu, J., 2017. 3D printing of continuous fibre reinforced thermoplastic composites. In: 21st International Conference on Composite Materials. Xi’an, p. 20, 25th August 2017.

CHAPTER 4

Representative volume elements and unit cells 4.1 Introduction When materials of micro/meso-structures at a lower length scale are analysed in order to determine their effective properties in an upper length scale, it is often necessary to resort to the concepts of representative volume elements (RVEs) or unit cells (UCs) at the lower length scale where analyses are conducted before the effective material properties in the upper length scale can be extracted. Such analyses are usually referred to as multiscale modelling. The use of RVEs and UCs has been an essential means to bridge between different length scales. The underlying justification for such multiscale modelling is the fact that the material at its upper length scale is homogeneous, which can be associated with one of the two scenarios at the lower length scale, the statistic uniformity or the regularity in the architecture at the lower length scale. They give rise to RVEs and UCs which have been found as useful tools to facilitate such multiscale modelling. The terminologies of RVEs and UCs are used interchangeably sometimes, resulting in a degree of confusion. It is therefore helpful for the subsequent discussion if they are logically defined as is the objective of this chapter. In the context of the present chapter, attention will be paid primarily to the geometric arrangements whilst the physical considerations will be delivered in a number of chapters to follow. It should be noted that the actually employed RVEs and UCs may vary from one physical field to another depending on the nature of the physical problem. What is presented in this chapter pertains to the basic considerations which are generic for most physical problems where these concepts are relevant. The readers are warned that the definitions introduced may not be identical to what they might have seen in the literature or what they might have intuitively in their mind. They are presented to follow the ethos of this monograph in the most logical way the authors could think of.

Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00004-9

© 2020 Elsevier Ltd. All rights reserved.

67

j

68

Representative Volume Elements and Unit Cells

4.2 RVEs 4.2.1 Representativeness RVEs are usually introduced at the lower length scale. Their definition rests on the keyword ’representativeness’. An RVE is representative if it is capable of reproducing properties as obtained from a volume of the material of an infinite size in the length scale concerned. This should be the criterion rather than any perception, no matter how commonly it has been referred to. The representativeness is usually defined in terms of the effective properties at the upper length scale of the material of interest. As long as the material is homogeneous at its upper length scale, an RVE can be introduced as an appropriate volume of the material at its lower length scale. In multiscale modelling, conventionally and also well-justified, any finite dimension at the upper length scale is considered as infinite at the lower length scale. The dimensions of an RVE must be finite at the lower length scale, which can then be considered as mathematical infinitesimal at the upper length scale. A volume of infinite dimension at the lower length scale will always be representative but not an ideal candidate as an RVE. An RVE is a sufficiently large volume of the material concerned in its lower length scale such that any larger volume will be equally representative. RVEs are commonly employed for problems involving random structure at their lower length scale, for instance, a unidirectionally (UD) fibre reinforced composite with microscopic image over its transverse cross-section as shown in Fig. 4.1. In this case, an RVE should be sufficiently large to be representative, whereas materials of regular structure at their lower length scale can be dealt with in a more deterministic way without relying on statistics as will be discussed in the next section. An appropriate RVE can always be obtained as long as the volume chosen is large enough. Apparently, for computational efficiency, one will be naturally interested in the minimum size of the RVE. Any volume of the material of a size smaller than that will no longer be representative. The minimum size of RVE may vary from material to material, from discipline to discipline, and sometimes from one effective property of interest to another. For instance, some properties, such as heat capacity, Young’s modulus of UD fibre reinforced composites in the fibre direction, etc., are dominated by the constituent volume fractions. For them, any RVE of the right volume fraction will serve the purpose. Other properties, especially those based on statistical homogeneity, such as Young’s modulus of UD composites

Representative volume elements and unit cells

69

Fig. 4.1 Random fibre distribution observed over the transverse cross section of a real unidirectional composite.

transverse to fibres, might need a substantial volume of the material in order to be representative. The representativeness of an RVE is judged based on the effective properties it characterises. It does not have to reproduce the appearance of any part of the material geometrically or the stress and strain distributions in any part. If in the lower length scale the structure of the material is irregular, no part can possibly be reproduced by any other part. When introducing RVEs for random structures at lower length scales, a common but fundamentally wrong practice is to falsify periodicity so that periodic boundary conditions can be imposed. Detailed critique will be presented in the next chapter.

4.2.2 Zone affected by boundary effects and the concept of decay length Hill (1963) proposed a criterion for the appropriate size of an RVE as a first attempt as follows. A chosen volume element can be loaded in two different ways, either through prescribed uniform displacements or by prescribing uniform traction over its boundary. They will lead to different results in terms of the predicted effective properties. As the size of the volume element

70

Representative Volume Elements and Unit Cells

increases, the difference reduces. The chosen volume element will be representative when the difference is sufficiently small. Whilst the criterion of Hill (1963) appears to be reasonable intuitively and has shown to offer a working approach, it could be overly demanding in terms of the actual size of the RVE. The difference between the two solutions in terms of the stress or strain fields obtained is only found in the neighbourhood of the boundary. Further into the material from the boundary, the difference diminishes, and the solution should be the same as if the analysis was conducted over a volume of the material of an infinite size. This can be justified on the basis of the Saint-Venant’s principle. Strictly speaking, the absolute difference between the two solutions in terms of the stress or strain fields obtained will never vanish. What vanishes is the difference between the numerical values of the concerned effective property, which are predicted relative to the volume of the RVE. As the volume increases, the ratio of the volume of the zone affected by boundary conditions to the total volume of the RVE reduces. The depth of the zone affected by boundary conditions, defined as a distance from the surface of the RVE to the point where the boundary effects diminish, is a characteristic measure. Although it varies from point to point along the surface of the RVE, it will be bounded and the upper bound will be called the decay length hereafter in this book. The decay length is a definitive value for a given material. The volume, Vs , of the zone affected by boundary conditions can be effectively viewed as a shell of the thickness of the decay length as the outermost part of an RVE. For example, consider a cubic RVE with the side length a, the decay length t and of volume V. As the size of the RVE increases, a increases whilst t remains constant. When t becomes negligible with respect to a, the ratio of the two volumes tends to vanish as   2  3 Vs a3  ða  tÞ3 3a2 t  3at 2 þ t 3 t t t ¼ ¼ ¼ 3 þ /0:  3 a a a V a3 a3 (4.1) Within the decay length, the stress and strain fields are bound to be different under prescribed uniform displacement and uniform traction. The fact that the predicted effective properties approach each other as the volume increases is simply the effect of dilution. This observation leads to an important conclusion. Since for a given material the decay length is a fixed value, by the distance of a decay length into the RVE, the stress and strain fields are no longer affected by the method of

r¼

Representative volume elements and unit cells

71

imposition of boundary conditions, i.e. whether it is done by applying a uniform displacement or a uniform traction. After stripping the shell of the thickness of the decay length off the RVE, the stress and strain fields in the core left behind should be the same as those obtained by analysing a volume of the material of an infinite size. A much more efficient and logical way of introducing RVEs therefore emerges. One does not have to increase the size of the RVE until the difference in predictions of properties is sufficiently diluted. As long as the core is of a representative constituent volume fraction, there will be no need to increase the size of the volume anymore, even if the difference as described by Hill is still significant. The average stresses and strains as obtained from the core should produce predictions of the effective properties of the material that will be as accurate as those produced by the stresses and strains obtained with a volume of the material of an infinite size. Detailed implementation of this approach will be presented in Chapter 9 of this book. To conclude the discussion on RVEs, judgement whether a volume element constitutes an RVE rests on its representativeness. Whilst it always comes down as a matter of the size of the volume element, it should never be size for the sake of size. There have been two different measures of the size. One is to use the size to dilute the errors introduced by inaccurate boundary conditions whilst the absolute errors remain unchanged, as was the case in (Hill, 1963). The other is to allow sufficient space to accommodate the zone affected by boundary effects whilst further into the RVE beyond this zone, the absolute errors are eliminated. As long as the part free from boundary effects is representative in its constituent volume fractions, it will be perfectly representative in other physical aspects. The second approach is apparently more favourable in all respects.

4.3 UCs 4.3.1 Regularity A unit cell (UC) is a portion of material at its lower length scale which reproduces all other parts of the material through appropriate symmetry transformations, so that the UC and its images fill up the space the material occupies in exactly the same way as the original material does without leaving any gap or causing any overlap. The existence of an appropriate UC implies strongly the presence of regularity in the architecture at the lower length scale of the material, which delivers the homogeneity of the material at its upper length scale. Homogeneity at the upper length scale can also

72

Representative Volume Elements and Unit Cells

result from the complete randomness of the architecture at the low length scale based on the statistic considerations. Because of the difference as stated, the use of UCs can be as a consequence of realistic modelling of the material architecture at its lower length scale, or an idealisation of the otherwise complete random structure at its lower length scale. An appropriate RVE as discussed in the previous section will have to be employed if the randomness in the lower length scale architecture has to be reflected in the model for the material at its upper length scale whose homogeneity is based on statistics. To fulfil the objective of filling up the space with a UC and its images under symmetry transformations without any gaps or overlaps alone, one can avoid the translational symmetry since it is a purely geometric exercise. Such geometric approach to the definition of the unit cells is perhaps the reason why wrong types of symmetry are often used in the literature on unit cell modelling, which eventually leads to fundamental mistakes as will be fully elaborated in the next chapter.

4.3.2 The role of translational symmetries When formulating UC, it is crucial to bear in mind that the space occupied by a UC is only meant to be a domain in which the physical problem concerned is defined. In order to formulate the problem, one has to deal with relevant physical fields defined in the domain, such as displacement, strain and stress. Some of the fields over the entire space, in particular, the displacement field, cannot be rationally reproduced from that in a UC without resorting to the translational symmetry. Note that in FEM, which is commonly employed for UC modelling, essential boundary conditions have to be expressed in terms of displacements, not strains or stresses. As a result, the boundary conditions for UCs cannot be formulated rationally without the use of the translational symmetry. As of present, the considerations of rationality are often neglected by casually employing reflectional and rotational symmetries alone in order to obtain the boundary conditions for UCs. Examples of such flawed approaches will be revealed and scrutinised in the next chapter. Assuming the material under consideration being homogenous in its upper length scale, it is always translationally symmetric. The use of translational symmetries does not carry any implication on the anisotropic characteristics of the material whilst offering a means for its homogenisation. A material is not necessarily reflectionally or rotationally symmetric in general unless it is of a special architecture at its lower length scale. All stress components are symmetric

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Representative volume elements and unit cells

under a translational symmetry transformation, but some stress components are antisymmetric under a reflectional or rotational symmetry transformation, adding extra complications to the modelling of the material for characterisation. As a rational approach, one should always exhaust the available translational symmetries first when defining a UC before exploring the benefits of additional reflectional and/or rotational symmetries if one is interested in the minimum size of the UC for a given material to be characterised. The subject of using additional reflectional and/or rotational symmetries to minimise the size of the unit cell will be elaborated in Chapter 8. The definition of a UC relies on correct interpretation of symmetries present in the structure. Given the possibility of redundant translational symmetries and the non-unique selection of the phase for a full period, even for the same pattern, unit cells of different appearances can be obtained. Sometimes, building blocks are naturally partitioned, such as fish scale pavement tiles in the two-dimensional case as shown in Fig. 4.2(a) which has been resorted to already in the previous chapter to introduce translational symmetries. Otherwise, as a generally applicable approach, the Voronoi diagram (Ahuja and Schachter, 1983) can be employed to tessellate the patterned structure as shown in Fig. 4.2(b), which brings a degree of uniqueness into the consideration (Li, 2001; Li and Wongsto, 2004). A Voronoi cell is generated in such a way that each side of it is a perpendicular bisector to the segment connecting the centres of the adjacent cells. Just as they represent the same physical problem, different shapes of the UCs as shown in Fig. 4.2 follow the same translational symmetries and therefore they should result in exactly the same outcomes in terms of stress distribution in the lower length scale and effective properties in the upper scale (Li, 1999, 2008). After the differences in the appearances of UCs have been trivialised, readers are reminded of one aspect of practicality. When the UCs are (a)

(b)

Arrows for directions of translations

Fig. 4.2 Unit cells of different shapes for the same pattern (a) natural partitioning and (b) Voronoi tessellation.

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analysed eventually using finite elements, different choices of the shapes could make significant differences to the generation and quality of the meshes. Some choices could leave awkward areas in the UC to mesh, e.g. those involving extremely sharp corners. It is therefore advisable that serious users of UCs ought to be ‘mesh-minded’ when deciding on the shape of the UC to be employed. Once the shape of a UC has been selected one way or another, the boundary of the UC has been determined. Establishing the correspondence between various parts of the boundary associated with the translational symmetries present in the structure in the lower length scale is a key step. This will the dealt with in the subsections to follow.

4.3.3 Identification of cells based on the available translational symmetries In a translationally symmetric pattern, after an appropriate tessellation, identical cells can be generated. Selecting any one of them as the unit cell (UC), such as the shaded one in Fig. 2.3, any other cell can be identified through its relationship to the UC in terms of the translational symmetry transformation between these two cells or mapping from the UC to the cell concerned. For the pattern as shown in Fig. 2.3, as argued in Section 2.2.3, there are four axes of translational symmetries and only two of them are independent. The spacing, or period, for each of the translations is different. For instance, the spacing along the x-axis is a and that along the y-axis is b as specified in Fig. 4.3, which is reproduced from Fig. 2.3 for the following discussion. If the x and the y axes are selected as the generic ones, the cell on the left hand y (1,1)

(-1,2) (0,1)

(-1,1)

η

(1,0) (0,0)

a b

(-1,0)

ξ

(1,-1)

(0,-1) (-1,-1)

(1,-2)

Fig. 4.3 Identification of cells with the help of independent translational symmetries present in the pattern.

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side shoulder of the shaded one can be obtained as a translation in the x-direction by -1 spacing and in the y-direction by 1 spacing. Employing these two translations, every cell in the pattern as shown in Fig. 4.3 can be identified by a pair of integers corresponding to the number of spaces of translation in the x- and y-directions, respectively, as shown in Fig. 4.3.

4.3.4 Mapping from the unit cell to any other cell and the relationship between paired pieces of the boundary of the unit cell A point (x, y) in the shaded cell is mapped to (x’, y’) in another cell through the following relationship  0   x x ¼ þ iap þ jbq (4.2) 0 y y where i and j are the number of spacings a and b, respectively, the cell translates in the x- and y-directions and p and q are the base vectors in the xand y-directions. For the given pattern, the base vectors are unit vectors of fixed directions. Through the mapping, the UC is capable of reproducing every other cell through two translations as identified by the two integers, i and j, as indicated. Partition the boundary of the UC into three pairs, coloured in red, yellow and green respectively. The red pair is related when i ¼ 1 and j ¼ 0 as     x x ¼ þ ap (4.3) y top right y bottom left the yellow pair is related when i ¼ 0 and j ¼ 1 as     x x ¼ þ bq y top y bottom and the green pair when i ¼ -1 and j ¼ 1 as     x x ¼  ap þ bq. y top right y bottom left

(4.4)

(4.5)

For these pairs of the pieces of the boundary, increments of coordinates can be obtained as       Dx x x ¼  ¼ ap (4.6) Dy y bottom left y top right

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Dx

 ¼

Dy and   Dx Dy

    x x  ¼ bq y top y bottom

    x x ¼  ¼ ap þ bq y bottom left y top right

(4.7)

(4.8)

respectively. After substituting them into the expressions of relative displacements as obtained in Eq. (2.71), the boundary conditions for a UC are obtained, which henceforth will be referred to as relative displacement boundary conditions for the UC. The pattern in Fig. 4.3 is for illustration purposes and it is twodimensional. It is straightforward to generalise it into three-dimensional, in which case there could be up to three independent translations in the three-dimensional space as will be shown through practical examples of unit cells in Chapter 6.

4.4 Concluding remarks A systematic account has been given on the concepts of UCs and RVEs, both of which are employed to represent the material for its characterisation. The difference between the two in terms of application is straightforward, namely, the former is intended for materials with regular structure at the lower length scale, whilst the latter is for materials of random structure. Difficulties that are typically encountered when formulating UCs and RVEs geometrically are briefly outlined. Specifically, formulation of UCs heavily relies on correct treatment of the symmetries, both in terms of geometry as well as of physical fields involved so that appropriate boundary conditions can be obtained from the symmetry conditions. Misuse of symmetries is the main cause of fundamental mistakes in formulating various aspects of UCs. For RVEs, one of the major modelling issues is how to ensure their representativeness. As has been explained, it can be resolved by making use of the measure of the decay length as a characteristic feature of the structure of the material. In the literature, another terminology, representative unit cells (RUCs), can sometimes be encountered. This was probably intended to reconcile between RVEs and UCs, but it does not help as it leads to more confusion. A logical reconciliation between RVEs and UCs is that a UC is always an RVE but not vice versa. In this sense, RUC is in fact tautology as ‘buttery butter’.

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In terms of applications, UCs and RVEs are designated for regular and random micro/meso structures, respectively. The first author wishes to confess that, in his publications in the past, he did not follow this rule of classification, e.g. in (Wongsto and Li, 2005; Li et al., 2009). The classification suggested above only became clear in his vision afterwards when more thought had been given to this issue in order to be logical as much as possible. With the concepts of UCs and RVEs being explained, their systematic formulation in terms of mathematical models, FE implementation and verification will be given in the chapters to follow. Given that the analysis of both UCs and RVEs is a boundary value problem mathematically, and FEM has been employed as the solver numerically, the formulation concerned will be primarily about the derivation for the boundary conditions for the respective boundary value problem.

References Ahuja, N., Schachter, B.J., 1983. Pattern Models. Wiley, New York. Hill, R., 1963. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357e372. Li, S., 1999. On the unit cell for micromechanical analysis of fibre-reinforced composites. In: Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455, pp. 815e838. Li, S., 2001. General unit cells for micromechanical analyses of unidirectional composites. Composites Part A: Applied Science and Manufacturing 32, 815e826. Li, S., 2008. Boundary conditions for unit cells from periodic microstructures and their implications. Composites Science and Technology 68, 1962e1974. Li, S., Singh, C.V., Talreja, R., 2009. A representative volume element based on translational symmetries for FE analysis of cracked laminates with two arrays of cracks. International Journal of Solids and Structures 46, 1793e1804. Li, S., Wongsto, A., 2004. Unit cells for micromechanical analyses of particle-reinforced composites. Mechanics of Materials 36, 543e572. Wongsto, A., Li, S., 2005. Micromechanical FE analysis of UD fibre-reinforced composites with fibres distributed at random over the transverse cross-section. Composites Part A: Applied Science and Manufacturing 36, 1246e1266.

CHAPTER 5

Common erroneous treatments and their conceptual sources of errors 5.1 Realistic or hypothetic background The way mathematics is taught to engineering students nowadays tends to leave them with a misperception that boundary conditions seem to occupy an insignificant position. This subject is loosely placed at an interdisciplinary spot of intersection between mathematics and physics, partial differential equations and ordinary differential equations, the boundary value problem and the initial value problem, and numerical solutions and analytical solutions. Conventional teaching of differential equations in mathematics starts from linear ordinary ones of constant coefficients for which solutions can be obtained as the sum of a particular solution and the complementary solution, where the latter can be obtained following a fixed rule, which involves integration constants to be determined through either initial or boundary conditions. As the determination of the integration constants can be essentially reduced to the solutions of a series simultaneous linear algebraic equations, which is usually a well-known topic by the time when the subject of differential equations is taught, most of the attention in the delivery of the subject therefore focuses naturally on the construction of the complementary and particular solutions. Whilst there is nothing wrong with the logic behind this, students are subconsciously left with an impression that boundary conditions are of secondary importance, if not unimportant. As the teaching moves into the subject partial differential equations, the scope usually narrows down to those conventionally referred to as the equations of mathematical physics. If analytical approaches are followed, which are unfortunately not a common practice for most of engineering studies nowadays, lecturers usually have to make a significant effort to emphasise the importance of boundary conditions, as the boundary conditions do not only determine the solutions, but often dictate the way how the equations are solved. The emphasis has been made to such an extent that the Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00005-0

© 2020 Elsevier Ltd. All rights reserved.

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j

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problem of partial differential equations is often simply called the boundary value problem, given the fact that the terminology of boundary value problems was originally introduced in ordinary differential equations as opposed to initial value problems. However, as the numerical solutions become more and more available, in particular, with the widespread use of the modern finite element analysis (FEA), the emphasis on the significance of boundary conditions has been side-lined again, although the finite element method (FEM) was meant to be a solver for boundary value problems in general. Conventional teaching of FEM places most of the efforts on the formulation of various finite elements, their stiffness matrices, in particular, and various solution techniques. If the teaching incorporates some underlying theoretical basis, one might be taught that the essential boundary conditions are associated with imposing displacement boundary values and natural boundary conditions are incorporated by applying loads on the boundary, distributed or concentrated. If the teaching is biased towards practical applications, boundary conditions are presented as rather procedural. In particular, loads on the boundary are no longer considered as boundary conditions. Users of FEM do not even need to distinguish between body forces and surface forces, except for their different dimensions, whilst in their analytical counterparts, body forces represent the right hand side of the governing equilibrium equation and hence are for the particular solution to account for, whereas the surface forces are associated with natural boundary conditions. The founding fathers of the FEM knew perfectly well the significance of the boundary conditions to the actual solutions of a physical problem. However, as far as the formulation of the FEM is concerned, the significance of boundary conditions has been understated, either explicitly or implicitly. Boundary conditions have perhaps been further trivialised by another fact in the FEM. An FEA cannot proceed without eliminating rigid body displacements. This gives rise to the fact that, as long as the rigid body displacements are constrained, a solution is guaranteed, assuming all other aspects of the analysis have been treated appropriately. In old days, obtaining a solution required so much expertise that solvers of boundary value problems were well capable of justifying the solutions they obtained, given their mathematical competence. With such competence becoming scarce and even redundant, as perceived by many, modern users of FEM are not left with much capability of assessing the correctness of their FEM solutions. This has been further exacerbated by the salesmen’s ‘black box’ claims that users do not need to understand anything except for knowing

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which button to push. Users have not been informed with sufficient emphasis that a solution obtained with incorrect boundary conditions, whilst remaining a valid solution, is the one to a problem physically different from what one is interested in, and hence irrelevant as far as his/her own practices are concerned. Modern users of FEM spend most of their efforts generating their FEM meshes, often complicated by ever increasingly demanding challenges, such as large deformation, inelasticity, contact conditions, etc., coupled with other physical problems. During the process, in order to facilitate trial runs, minimal boundary conditions are often imposed in order to keep the problem simple. This makes perfect sense as an intermediate stage of development, provided that the boundary conditions are revisited and put absolutely right afterwards once other problems get resolved. In reality, it is often not quite the case. Users often get too occupied by other complications to worry about boundary conditions. A practical consideration is that one could claim novelty in his/her work only by overcoming some of those sophisticated challenges. By the time one managed to obtain some seemingly promising results in resolving the complications involved, there is no longer incentive to revisit the matter of boundary conditions, the least juicy part of the exercise. In all fairness, boundary conditions are not the most exciting, and certainly not the most challenging, part of a typical FEM model nowadays, relative to other sophistications as cited above. Any user competent enough to deal with any of those sophistications should be capable enough of interpreting the physical constraints into appropriate boundary conditions to deliver a correct analysis of the problem. However, because of the persistent negligence of appropriate prescription of boundary conditions, an attitude became widely established that definition of boundary conditions is a business where scientific rigour becomes irrelevant and hence they could be attended without much dedication. For some, perhaps, the problem has become an exercise of hiding errors from being spotted by reviewers. A symptom of this attitude can be clearly identified in terms of unit cell applications where boundary conditions were not specified in most accounts, and even in those with boundary conditions provided, they were ‘proposed’, ‘assumed’, ‘approximated’, ‘suggested’, ‘taken to be’, without any justification or referencing, or sometimes, simply given. One way or another, the fact that the boundary conditions for unit cells can be in fact derived rigorously has been completely ignored. The ultimate consequence is that many serious analyses, including those published in top ranked academic journals, presented at professional conferences, and

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incorporated in technical reports, some of which might be in support of engineering artifacts the public perceives sound and safe, have glitches associated with boundary conditions. It is a fair statement to make before entering the discussion of various errors and their sources in the following sections that any solution, no matter how much sophistication has been incorporated, is unlikely to be trustworthy if the boundary conditions have not been prescribed correctly. This sets the background for the discussions to be presented in the present chapter and readers are free to argue how realistic or hypothetic the assumed background is. The fact is that most of the errors identified have close relationship with boundaries and boundary conditions.

5.2 The construction of RVEs and their boundary As was discussed in Chapter 4, a representative volume element (RVE) has to be representative by definition, i.e. it should preserve the characteristics of interest. In many applications where RVEs have to be employed, the randomness of the features involved is often one of the considerations underlying such characteristics. Otherwise, these features could have been idealized into a regular distribution which is much simpler to deal with. However, one can often find many accounts of RVEs of this type where this very characteristic is compromised. When forming RVEs for random structures, such as that in Fig. 5.1(a), at lower length scales, there is a practice of tampering the boundary of the RVE in order to imitate periodicity so that periodic boundary condition can be imposed. In this process, all features truncated by the boundary of the RVE are artificially duplicated on the opposite side, as shown in Fig. 5.1(b) by shaded circles, where darkly shaded ones result from multiple duplications as the original feature appears on two sides of the boundary. There are three obvious and fundamental issues associated with this practice. a) This interferes the constituent volume fractions, compromising its representativeness. b) There is not always space available to accommodate the truncated features. The truncated features can only be dealt with by a number of means: (i) selecting a specific patch where space happens to be available; (ii) allowing overlapping, as depicted in Fig. 5.1(b); (iii) artificially moving existing features around in order to free some space; and (iv) artificially deleting or avoiding overlapping features during the generation of the features, as illustrated in Fig. 5.1(c), giving rise to a ‘RVE’ as

Common erroneous treatments and their conceptual sources of errors

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Fig. 5.1 Illustration of falsified periodicity when generating RVEs: (a) RVE of a random structure; (b) tampered RVE; (c) the material represented by the RVE without overlapping; (d) RVE trimmed (e) the material represented by the RVE with overlapping; (f) material after removing/avoiding overlapping.

shown in Fig. 5.1(d). In each case, the randomness of the original structure will be compromised by the artificial interference. c) The falsified periodicity means periodicity in the structure at the length scale concerned, which can be made obvious if one lays up an array of such RVEs, as in Fig. 5.1(e) and (f). This spoils the randomness of the structure greatly as a systematic pattern is present in the structure. With the structure so systematically tampered, one should question if it is still worthwhile to model such an RVE with the features seemingly distributed at random but no longer representative in many ways. The point to make here is not that one is not allowed to make approximations. However, one should be aware and acknowledge at least that an approximation has been made. If it is meant to be a scientific investigation, some necessary justifications for the approximation need to be provided. In the case when no justifications is available, one can at least invite justifications from readers as his/her acknowledgement of the approximation made. Unfortunately, this does not seem to be the case in such practices. It is not uncommon for unjustified approximations made by someone to be reproduced by others without any further critical assessment. It leads to even more problematic scenario when the incorrect treatment is viewed as the norm merely because it is used by

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many and anything different from it is automatically considered as wrong, echoing a notorious quote “A lie repeated a thousand times becomes truth”. To conclude this section, it should be pointed out that appropriate approach to analysing untampered RVEs is available as will be addressed in Chapter 9 of this book. In particular, it will be demonstrated that periodic boundary conditions are neither necessary for the analysis of RVEs nor logical to support their representativeness.

5.3 The construction of UCs 5.3.1 Problems associated with the abuse of reflectional symmetries Reflectional symmetries have been perceived by many as the only type of symmetries, as they are the only type available in commercial FE codes. This is apparently wrong as was explained in Chapter 2. For micromechanical analysis, an essential assumption is that the domain to be dealt with is infinite in extent, due to the fact that the lower length scale features, e.g. the period of fibre distribution, are usually orders of magnitude smaller than the upper length scale dimensions in which the material is used. The objective of employing RVEs or UCs is to reduce this infinite domain into a finite extent without losing the representativeness of the characteristics of interest. A reflectional symmetry can reduce an infinite domain to semi-infinite, which sounds a significant reduction, but semiinfinite is still not finite in extent. To force a further reduction, the principles of reflectional symmetries have often been subjected to abuses, examples of which can be found in the literature. A typical approach is to use the symmetry twice with two ‘symmetry’ planes placed parallel to each other at a distance of a period. Any of these two planes is indeed a symmetry plane, given the infinite extent of the domain, but not both simultaneously. The first use of a reflectional symmetry has reduced the infinite domain to a semi-infinite one. Geometrically, the semi-infinite domain does no longer possess the same symmetry, as is schematically illustrated in Fig. 5.2. Mechanically, a reflectional symmetry imposes a condition that some displacement(s) vanish on the reflectional symmetry plane. If one attempts to prescribe the same reflectional symmetry conditions on two parallel planes simultaneously, this inevitably would result in a condition that the relative displacement(s) between these two planes have to vanish. This represents a specific type of deformation, such as plane strain, which can hardly

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Common erroneous treatments and their conceptual sources of errors

Infinite

Symmetry conditions

Infinite

Reflectional symmetry is no longer available

Fig. 5.2 Applicability of reflectional symmetry.

be relevant as the objective of a unit cell is to be representative and a plane strain problem can never be sufficiently representative for unit cell applications. In terms of effective properties, this would result in infinite effective Young’s or shear modulus in this direction, if one employs a unit cell formed this way. In fact, users of a unit cell so constructed would soon realise that their models were not quite right. They then have to artificially relax somehow the constraints imposed by the symmetry to allow for certain patterns of deformation prohibited by the symmetry considerations in place. Any relaxation therefore conflicts with the symmetry by definition and the consistency in the approach is thus compromised. The lack of consistency often precursors further confusion down the line. A typical example of such practice can be found in the so-called equivalent coordinate systems (Whitcomb et al., 2000; De Carvalho et al., 2011). Given what have been elaborated above, the so-called equivalent coordinate systems can never be properly equivalent as the origins of these coordinate systems displace by different amounts and in different directions. The displacements of the origins of such coordinate systems cannot be defined in relation to the reflectional symmetry. The root source of such errors could be traced back to the classical book on theory of elasticity by Timoshenko and Goodier (1969), where any plane perpendicular to the longitudinal axis was considered as plane of reflectional symmetry when the plane strain problem was introduced. This was ambiguous even there but fortunately did not cause issues in the definition of the plane strain problem because the displacement along the axes happened to

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Representative Volume Elements and Unit Cells

be zero as another requirement of the plane strain problem. However, this way of introducing the plane strain problem prohibits the consistent generalisation of the problem to the so-called generalized plane strain problem (Li and Lim, 2005) which has also been available in commercial FE codes, such as Abaqus (2016). If formulated properly, the plane strain problem should be a special case of the generalised plane strain problem. Another source of confusions associated with the use of reflectional symmetries is the fact that not every physical field is ‘symmetrically symmetric’ (Li and Reid, 1992), as have been explained in Subsection 2.5.1. If one describes a ‘symmetrically symmetric’ field as an even function, i.e. f ð xÞ ¼ f ðxÞ, then some fields are ‘antisymmetrically symmetric’, i.e. can be described as an odd function, f ð xÞ ¼ f ðxÞ. Often, shear stresses are of this nature. The presence of antisymmetric quantities alongside with symmetric ones introduces additional complications. For this reason, the behaviour of the material under shear is often left out by most users of UCs, usually without any reason provided, as if shear behaviour was irrelevant or unimportant which is obviously not the case. Proper treatment is to recognise antisymmetry when it is present and then derive the boundary conditions according to the antisymmetry as was elaborated in Section 2.5.1. It should be pointed out that in terms of an FE implementation, a symmetric loading condition and an antisymmetric loading condition for the same unit cell cannot be considered as two loading cases in the same job because they have to be analysed under different boundary conditions and hence have to be considered as two separate jobs.

5.3.2 Rotational symmetries Rotational symmetries are not as well-recognised type of symmetry as their reflectional counterparts, although their strict definition has been wellestablished mathematically and they are employed in some fields of physics, such as theory of crystals. Their use in formulation of unit cells has been much less popular than that of reflectional symmetries. Yet similar to misuse of reflectional symmetries when constructing unit cells, confusions and erroneous treatments can often be spotted in efforts to employ the rotational symmetries in the formulation of unit cells. One confusion is associated with variety of different shapes of the unit cells for the same micro/meso-structure. A typical example of the confused state can be found in relation to the unit cells for UD composites of hexagonal fibre packing, either as an idealisation of otherwise random distribution or deliberate

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design (Teply and Dvorak, 1998). As cited in (Li, 1999, 2001), various shapes of disparity could be defined as shown in Fig. 5.3. As if the state was not confusing yet, there has been a man-made and unnecessary addition to the list. In one account (De Kok and Meijer, 1999), a unit cell of curved side was employed. In fact, if the curve employed satisfies the conditions as will be presented later, there is nothing wrong with it, although its peculiar profile certainly does not add any clarity to the subject. The boundary conditions as provided in (De Kok and Meijer, 1999) were in fact correct. The justification offered for the curved side was to accommodate high fibre volume fractions. An earlier version of the justification as communicated at the conference where (Govaert et al., 1995) was published was to keep right angles at the two non-right angle corners in order to avoid stress singularities there. As was discussed in (Li, 2001), the highest volume fraction corresponded to the scenario when fibres came in contact with each other. Even then, the hexagonal unit cell could provide a straightforward solution without

Fig. 5.3 Various periodic elements and unit cells.

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truncating fibres. After taking account of the reflectional symmetries present in the hexagonal unit cell, one obtains a right trapezium. The curved side is unnecessary as far as fibre volume is concerned. The so-called stress singularities were physically impossible, given the fact that the two corners of the curved-sided unit cell were located in the middle of the matrix where there was not any material discontinuity whatsoever to justify any stress singularity. Fair to say that the boundary conditions as provided in (De Kok and Meijer, 1999) would not have resulted in any such stress singularity. In fact, the diversity can be unified as follows owing to the proper use of the rotational symmetry. Orthogonal translations lead to either of the large rectangular unit cells. Reflectional symmetries further bring them to a quarter as highlighted in red. The final rotational symmetry about the centre P of the quarter reduces the UC to those as shaded using various borderlines to partition the quarter UC. The requirements for the borderline is that it passes P and it is 180 rotationally symmetric about P. Special selections of this borderline have been illustrated in Fig. 5.3 using dash-double-dot chain to reproduce all shapes cited, including the curved-sided one and they are all equivalent as a unit cell to represent the material concerned. In terms of representing the material, there is absolutely no benefit in choosing the complex shapes of the unit cells over the simpler ones. To highlight the significance of rotational symmetries, it can be seen that the unification as achieved above amongst all differently shaped UCs is through the use a rotational symmetry. It should be noted that a rotational symmetry also reduces an infinite domain to a semi-infinite one. Just like the reflectional symmetries cannot be imposed twice about two parallel planes, one cannot apply rotational symmetries about two parallel axes in order to reduce the extent of the domain of interest from infinite to finite. Also similar to the reflectional symmetry, some fields are symmetric under a rotational symmetry whilst others are antisymmetric, in particular, shear stresses and strains. The concept of the so-called equivalent coordinate systems is equally wrong under rotational symmetries and the argument remains the same as provided for the reflectional symmetries in the previous subsection.

5.3.3 Translational symmetries Translational symmetries are the least understood type of symmetries in their applications to the formulations of UCs. However, it happens to be the most

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Common erroneous treatments and their conceptual sources of errors

Infinite

Relative displacement boundary conditions

Relative displacement boundary conditions

Infinite

Translational symmetry / Periodicity

Fig. 5.4 The use of translational symmetry to reduce from an infinite domain to finite.

important one as it is the only type of symmetry that can reduce an infinite domain to finite in a consistent manner, as has been elaborated in Section 2.2.3 and is illustrated in Fig. 5.4. Under this type of symmetry transformation, practical physical fields all keep their sense, i.e. antisymmetry does not arise. Although one could imagine a hypothetic antisymmetric case, it may not have many practical applications. In fact, translational symmetries have been noticed and mentioned often in publications involving UCs. However, they were mostly restricted to geometric aspects. Their implications on physical fields, in particular, displacement fields, have usually been bypassed. Sometimes, the reflectional or rotational symmetries are erroneously employed instead of translational symmetries by applying them more than once, as was pointed out in previous subsections. Geometrically, a translational symmetry is equivalent to periodicity. For this reason, almost every UC user acknowledges the so-called periodic boundary conditions, but mostly terminologically. Taking the word ‘periodic’ literally in the terminology of periodic boundary conditions constitutes a typical source of confusion. The term “periodic boundary conditions” was introduced by Suquet (1987) to refer to the boundary conditions for UCs, since, in presence of translational symmetries, periodicity is indeed present in stress and strain fields, but not in displacement field. This distinction was not clear to many users, who interpreted this term literally. However, when genuinely periodic displacement boundary conditions were imposed, one would realise straightaway that they could not be right as UCs would not

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deform at all. Different users would have to improvise subjectively and come up with fundamentally different boundary conditions for the same unit cell. Given the generic lack of uniqueness in the boundary conditions for UCs, even if one manages to end up with correct formulation by chance, such casual treatment does not offer any useful guidelines for others to follow in terms of appropriate definition and implementation of the boundary conditions. Whilst some could not be bothered to present the boundary conditions they obtained in such manner, some might have decided that the safest way forward with the boundary conditions they employed would be to keep them away from scrutiny. This is probably the reason why the detailed boundary conditions are missing in nine out of ten publications involving the use of UCs. In FEM, which is the common tool for the analysis of UCs, the essential boundary conditions are prescribed in displacements. Periodic strains and stress fields do not usually deliver periodic displacement field. Therefore, it will be less confusing to avoid attaching the word ‘periodic’ to the type of the boundary conditions which are not periodic. In fact, periodicity is available but only in the relative displacement field. To reflect that whilst avoiding any confusion, the authors advocate the use of the term ‘relative displacement boundary conditions’, which was introduced previously in Subsection 4.3.4, to denote the boundary conditions for UCs. Translational symmetries are partially responsible for the confusing variety of the appearances of UCs, in terms of the translations employed, when there exist redundant ones, as shown in Fig. 5.5(a). The rectangular UC has the simplest types of translations as it involves translations in orthogonal directions only. However, the price to pay is that the size of the UC is double that of the others because the period in the vertical direction is longer. The diamond-shaped and the hexagonal ones are the smallest sizes one can obtain using the translational symmetries alone, whilst the former employs translations in two directions (60 from the horizontal axis) and the latter in three directions (0 and 60 from the horizontal axis). The sizes can be further reduced if one is prepared to increase complication in formulation by taking advantage of available reflectional and rotational symmetries as argued in (Li, 1999). Another cause for the diversity in the appearances of UC is the fact that using the same types of translations one can extract different UCs as shown in Fig. 5.5(b). This can be explained using an analogy of the periodicity of a sinusoidal function. A complete period does not have to be from 0 to 2p. It can be from any x to xþ2p. It should be noted in this regard that the UCs in Fig. 5.5(a) and (b) should be considered to be the same,

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Fig. 5.5 Variations in appearances of rectangular, diamond-shaped and hexagonal unit cells where the respective unit cells are extracted in different ways in plots (a) and (b).

respectively, as each pair will share exactly the same set of boundary conditions (Li, 2008) whilst representing the same material. If any difference, it will be down to meshing the domain to be analysed using FE as their appearances are different. Rather on the contrary, seemingly identical UCs under different boundary conditions are different UCs and they represent different materials. An example is shown in Fig. 5.6 where two different microstructures lead to UCs of identical geometry. However, since they are obtained through the use of different symmetries due to the difference in the microstructures, the boundary conditions are different between these two UCs and they apparently represent different materials. This is also a good example where

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Fig. 5.6 Example of seemingly identical UCs corresponding to different microstructures: (a) simple square packing; (b) square packing with further reflectional symmetries.

boundary conditions define the nature of the problem, rather than serve as a decorative artefact as they are perceived to be by some people.

5.3.4 Redundant boundary conditions When relative displacement boundary conditions are imposed according to the translational symmetries, users are often hit by error messages suggesting conflicts in imposed boundary conditions, depending on the FE code employed. The reason is the redundancies in the boundary conditions. When periodic boundary conditions are imposed, redundant conditions are found on edges and vertices for 3D UCs and edges and corners for 2D UCs. Some FE codes, such as Abaqus/Standard, cannot tolerate such redundancies. In the imposition process, each boundary condition eliminates one degree of freedom. When a redundant boundary condition involves an eliminated degree of freedom, it is perceived as a constraint imposed on a nonexistent degree of freedom and hence the error. Identification of such

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redundancies requires some systematic analysis of geometry and kinematics. Filtering them out can only be done manually at the moment as shown in (Li, 2001; Li and Wongsto, 2004), unless one takes a step to automate the process (Li et al., 2015; Li, 2014). The sheer demand in this respect alone must have deterred many from adopting properly relative displacement boundary conditions which represent the translational symmetry conditions. Failing to apply translational symmetry conditions systematically, one may have to improvise and incorporate other manipulations not resulting from translational symmetry whilst still calling the obtained boundary conditions the ‘periodic boundary conditions’. This is believed to be the cause for the observation that there is fair amount of subjective interpretation of the so-called ‘periodic boundary conditions’ in the literature.

5.3.5 Incomplete use of available symmetries present in the microstructure For UD composites applications, the rectangular quarter-sized UC as shown in Fig. 5.7 has been the most popular one in use in the literature. In fact, the use of it exposes the user’s incompetence in formulating UCs. Having employed reflectional symmetries, one has lost the simplicity of a single set of boundary conditions for all loading cases. If so, why not take a step further to halve the size of it by taking advantage of the rotational symmetry about the centre as marked? Of course, the boundary conditions will be a little more complicated and hence require a slightly higher level of competence to formulate. The same can be said about the triangular UC as also shown in Fig. 5.7, where there is still a reflectional symmetry left unused. In this particular case, had the boundary conditions been formulated correctly, the

Fig. 5.7 Further simplification of shape of the unit cells using available symmetries.

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UCs of a further reduced size will be in fact simpler. The competence required is to spot the presence of a further symmetry and implement the symmetry considerations to derive relevant boundary conditions.

5.3.6 A unit cell as an assembly of multiple cells In the literature, it is not uncommon that an assembly of an array of identical cells is employed as the UC to be analysed, e.g. (Fang et al., 2005), where the authors even took the trouble to show the convergence of a predicted effective property as the number of cells included increased. With properly formulated boundary conditions, there will never be a need to do so, since the outcomes of the analyses of all models as those shown in Fig. 5.8 should produce identical results, the same as those obtained from a single unit cell, if correct boundary conditions have been employed. Any attempt of employing an assembly of cells shows a clear sign of incompetence in using UCs. The differences as found between UCs as assemblies of different number of cells, if any, clearly indicate the use of incorrect boundary conditions for the analysis. The convergence, even if it has been established, is nothing but to have errors from the boundary conditions diluted as discussed in Section 4.2.2 for RVEs. Another similar scenario that is even more frequently encountered in analysing UD composites using unit cells is to have a stack of slices as illustrated in Fig. 5.9 with a square unit cell containing a single fibre. Under uniform deformation at the upper length scale, there should not be any stress gradient in the longitudinal direction. If so, a single slice should be just as accurate as multiple ones. Often in the presented results, high stress gradients in the longitudinal direction in the case of stacked slices towards both ends

Fig. 5.8 Unit cells as assemblies of different numbers of identical cells.

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Fig. 5.9 Example of stacked slices as a smokescreen for incorrect boundary conditions employed.

had been hidden from the readers. They resulted from incorrect boundary conditions at both ends, if not elsewhere as well.

5.3.7 Essential and natural boundary conditions UCs, being a boundary value problem, are mainly analysed using FEM. In a boundary value problem, there are two basic types of boundary conditions: type I, or Dirichlet boundary conditions, and type II, or Neumann boundary conditions. In the problem of mechanics, the former involves displacements prescribed on the boundaries, whilst the latter has the spatial derivatives of displacements prescribed, which can be expressed in terms of traction. The FEM is variational principle based approach. In terms of variational principles (displacement-based and hence associated with conventional FEM formulation), type I boundary conditions are essential boundary conditions and type II are natural boundary conditions. Essential boundary conditions have to be imposed as they must be satisfied precisely for the displacement field employed in FEM to be permissible to allow a meaningful approximation. Natural boundary conditions in FEM are turned into the procedure of load application. They have to be prescribed in terms of loads applied to the structure but they will only be satisfied approximately as a result of energy minimisation, in line with the equilibrium conditions. In

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fact, natural boundary conditions are indeed equilibrium conditions at the boundary. These two types of boundary conditions are different in nature. They are treated differently in FEM and implemented through different procedures, viz. boundary conditions and loading conditions. Their differences have more fundamental implications as explained below. From the perspective of translational symmetries, the symmetry conditions should consist of displacement boundary conditions as well as traction boundary conditions. If one follows an analytical approach to solve the boundary value problem, these boundary conditions will have to be considered equally. However, with FEM, only the essential boundary conditions, i.e. relative displacement boundary conditions, must be imposed, whilst the periodic traction boundary conditions should be strictly left alone to ensure best approximation in FE analyses of UCs, as it can be proven (Li, 2012) that they are implied by the energy minimisation. Artificially imposing them to force the precise satisfaction of periodic traction boundary conditions would seem to have improved accuracy in terms of the satisfaction of traction boundary conditions. However, this is at the cost of worst off approximation elsewhere and, more critically, the fact that energy has not been minimised (Li, 2008), which compromises the very basis of the FE approximation. Fair to say, it is not easy to enforce the natural boundary conditions in the fashion of essential ones in an FE analysis of a UC . Nevertheless, there have been attempts in the literature doing exactly this, e.g. (Drago and Pindera, 2007; Yeh, 1992). In the latter reference, traction boundary conditions were termed as ‘equilibrium requirement’. Being unnecessary is an understatement for this type of exercise because it compromises the FEM approximation on one hand and it violates a basic rule of the variational principle which underlies FEM as a rigorous and mathematical study. The confusion of this kind was due to the lack of understanding of the natural boundary conditions as a basic concept in variational principles.

5.4 Post-processing The most common applications of UCs are for material characterization, through which effective properties of the material represented by the UC can be evaluated. In order to obtain effective properties as an important part of material characterisation, one has to evaluate the average stresses and strains. How to evaluate the average stresses and strains is often a detail most of the available publications employing UCs shy to clarify. Readers are lead to believe that average stresses and strains have to be calculated by averaging

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stresses and strains obtained from all elements involved in the model at their integration points or nodes. Although the averaging process will be proven not to be an effective approach, it is not wrong in principle. However, most of the results as available in the open literature are suspected to have been obtained simply as the arithmetic average of the stress and strain outputs at integration points or nodes, which would not be the right approach. A simple illustration can be given as follows. For the simplicity of argument, linear elements having only one integration point are employed and thus the stress obtained at the integration point gives the stress in the element. If the stress field in the UC has been correctly predicted using a mesh of two elements as shown in Fig. 5.10(a) with stress values specified, the arithmetic average would be obtained as 7 MPa. Now, imagine if one has the mesh of the UC refined by splitting the element on the top into two as shown in Fig. 5.10(b). This should affect neither the stress field, nor the average stress. However, if the average stress is calculated as the arithmetic average, it would become 8 MPa. If the mesh refinement is as shown in Fig. 5.10(c), the average stress would now become 6 MPa instead. There is apparent lack of certainty for the average stresses and strains if the arithmetic averaging process was adopted. This is fundamentally wrong. A proper averaging process for the stresses or strains should be conducted according to the weight of the integration point and the volume of the element. Such a process is not usually directly available from any commercial FE code. The coding required for post-processing the stresses and strains would not be far from the coding for the finite element concerned. Whilst this is not excessive for an experienced programmer, the required level of competence in coding is beyond the skill set of the most FE users nowadays. This is believed to be the reason why the details of post-processing have not been mentioned in many, if any, publications.

Fig. 5.10 Illustrations to reveal fundamental inconsistencies arising from employing the arithmetic stress averaging process in material characterisation using unit cells.

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However, such averaging process is completely unnecessary. A much more efficient and rigorous way is available if the UCs have been appropriately formulated, as will be shown later in this book.

5.5 Implementation issues As casual treatment has been the norm amongst many users in the employment and applications of UCs, errors introduced at the implementation level are not uncommon, in particular, those associated with the use of FEM as the solver. They are listed here as some warning signs for readers. (1) Dimensions of the UC and 2D idealisations Every physical object is three-dimensional. However, analysis of a 3D structure is usually time-consuming computationally. When a UC is deemed to be 3D, one is left no option but to go for a 3D analysis. Occasionally, some analyses can be idealized into a 2D problem, especially for microscopic modelling of unidirectional fibre reinforced composites. For mechanical applications, this is often based on a so-called generalised plane strain problem for the UC to be representative (Li, 2001). Use of a plane stress or a plane strain problem in the plane transverse to the fibres by mistakenly selecting finite elements from those categories constitutes a serious error because materials usually cannot be properly characterised under these stress states. Numerically, the reduction in computational cost from 3D to 2D is enormous, certainly substantially more than one-third. However, 2D idealisation could be a source of confusion for many. In terms of FEM implementation, it is by large the selection of appropriate type of 2D elements to be employed. Most commercial FE codes offer plane stress, plane strain and various types of so-called generalized plane strain elements. In general, unless the problem to be analysed is indeed a thin film, the plain stress condition is hardly encountered in the analysis of RVEs and UCs. The selection of plane strain element comes with a very restrictive implication. One can never obtain any meaningful uniaxial stress state under a plane strain condition. Effective in-plane properties of unidirectionally fibre reinforced composites cannot be obtained correctly through such an analysis. Assume that a stress s is applied in the x-direction whilst the y-direction is free from stress. The strain obtained in the x-direction is ε. Under a plain strain condition, this loading scenario will not reproduce a uniaxial stress state, since for a homogeneous and isotropic material of Young’s modulus E and Poisson’s ratio n, one has

Common erroneous treatments and their conceptual sources of errors

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s E sE: (5.1) ¼ ε 1  n2 Appropriate type of elements to employ are the so-called generalized plane strain elements. This type of elements has out-of-plane degrees of freedom. They are meant to be of values common for all elements involved in the mesh. A uniaxial in-plane stress state can be obtained by leaving the relevant degrees of freedom (dof) free so that the out-of-plane direction is free from constraints. There are cases where idealisations into 2D problems are possible but for practical consideration, 3D elements are employed. In these cases, one only needs one layer of elements in the direction which could otherwise be eliminated because there will be no variation in stress and strain distributions in this direction, as discussed in Subsection 5.3.6. (2) Effective elastic properties In order to obtain effective material properties, one ought to follow the definition of such properties and employ the UC as a means of virtual testing. It is not an uncommon mistake to determine Young’s modulus as stress divided by strain whilst turning a blind eye at the condition that the material is supposed to be at a uniaxial stress state at its upper length scale. (3) Mesh convergence In any FE analysis, it is the responsibility of the user to ensure that the mesh employed for serious analysis is a converged one. Before such convergence is reached, none of the results should be considered as meaningful. One should be aware that different quantities tend to converge differently. Usually effective elastic properties converge quickly, as well as the displacement. However, stresses converge at a much slower rate. (4) Constraints for rigid body translations In FEM, it is essential that the rigid body motions are constrained, in general. Appropriate boundary conditions for a UC may have some of them constrained already. Whilst insufficient constraints prevent the analysis from going ahead, over-constraining is just as erroneous. As far as rigid body motions are concerned, the constraints should be prescribed just right, no more and no less. For the benefit of serious users, some of the erroneous treatments are listed below. (i) Instead of constraining a specific dof at a node, e.g. to prevent the rigid body translation in the direction of the dof, all dofs at that node are constrained.

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(ii) Instead of constraining a dof at a single node, e.g. to prevent the rigid body translation in the direction of the dof, the dofs on a complete side of the RVE or UC are constrained. (5) Constraints for rigid body rotations In order to constrain a rigid body rotation, dofs at two different nodes have to be involved whilst employing solid elements where no rotational degrees of freedom are available, unlike in shell or beam elements. However, one cannot just use two arbitrary nodes although the combination of legitimate nodes is not unique. Users do not usually encounter problems if the domain is orthogonal, e.g. a rectangle. However, if a domain is as shown in Fig. 5.11, constraining rotations will become much less straightforward. Assume that the y-direction displacement at node A has already been constrained, i.e. vA ¼ 0, in order to eliminate rigid body translation in this direction. In order to constrain the rigid body rotation in the xy-plane, it is not uncommon that the y-direction displacement at node B is constrained as vB ¼ 0. This is unfortunately wrong and results from insufficient understanding of the implications of the rigid body rotation. Assume that node B is given an infinitesimal displacement, vB , in y-direction. If A and B lay on the same horizontal line, vB would result in an infinitesimal rigid body rotation about A, where the change in length AB is a higher order infinitesimal. It is straightforward to show that if A and B do not fall on the same horizontal line as depicted in Fig. 5.11, the change in length AB resulting from vB is no longer a higher order infinitesimal. Refer to Fig. 5.11(b) and denote the projection of AB to the horizontal line as h. As B displaces to B0 by vB , line AB moves to AB0 which is at an angle of d to AB. The original length of AB is h=cos a. After the displacement, it becomes h=cosða dÞ. The change in length can be given in a Taylor series as

Fig. 5.11 Illustration of constraints for rigid body rotation.

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h h cos a  cosða  dÞ  ¼ h cosða  dÞ cos a cosða  dÞcos a (5.2)   sin a 1 þ sin2 a 2 zh  2 d  d þ/ cos a 2 cos3 a If a ¼ 0 then   1 2 (5.3) Dzh  d þ // 2 It is indeed the second order of infinitesimal. However, at any finite value of a, the length change D is a same order of infinitesimal as d, which is of the same order of infinitesimal of vB , i.e. dzvB(cos2a)/h, and therefore the deformation cannot be a rigid body rotation. A correct way of constraining rigid body rotation can be vC ¼ vA ¼ 0 where C can be any node on the same horizontal line as A. For numerical accuracy, node C and node A should be as far apart as possible, i.e. H would offer a better choice if it is a node. Alternatively, one can constrain node C and D as uC ¼ uD ¼ 0. The lack of unique way of achieving the same outcome is undoubtedly one of the sources of confusion but the logic behind the variety should prevail. D ¼ AB0  AB ¼

5.6 Verification and the lack of ‘sanity checks’ It has been shown that there are so many aspects where users are likely to make mistakes in addition to a range of implementation issues, such that it is literally impossible to get it right on the first attempt. This has probably been one of the reasons why some users choose to follow an intuitive approach and create unit cells in a rather casual manner so that they could get some cases working fairly easily, in particular, cases involving uniaxial direct stresses. However, the reliable range of one’s intuition is usually limited, and the trend may turn sharply once the shear is involved. This is perhaps the reason why vast number of publications on this subject shied away from shear, as if shear was irrelevant. Before putting any UC developed to any meaningful application, it must be subjected to necessary verification. In relation to the development of UCs, one practice proves to be extremely helpful. It is referred to as ‘sanity checks’ where all phases involved are given the same material properties and the UC should behave as a monolithic material whose behaviour can therefore be assessed based on the common sense. ‘Sanity checks’ are thus to

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verify if common sense can be reproduced by the UC devised. Reproduction of some aspects of the common sense cannot guarantee the correctness of the UC, but any discrepancy will be sufficient to tell that there is definitely something wrong with the UC. The numerical results obtained from such checks are usually easy to process. With any mistake, either as genuine error in formulation or a typographic error in the input, stress/strain gradients will be shown, usually with excessive levels of concentration. The most common mistakes are with the boundary conditions. On the other hand, correct results are always boring, since absolutely uniform stress and strain fields should be obtained. It is therefore usually not exciting to report such an exercise. However, it is essential that the UC should pass these tests. There is no lack of examples in the literature where the UCs would not have been able to pass these ‘sanity checks’ in one respect or another, had these ‘sanity checks’ been conducted. Of course, in most cases, the boundary conditions employed were not shown in the sources where they were published, as if boundary conditions were irrelevant or automatic.

5.7 Concluding remarks RVEs and UCs are potentially very powerful means of virtual testing for the characterisation of materials of certain architecture at a lower scale. However, their applications have been littered with rough handling of crucial aspects, in particular, the boundary conditions. The consistency as required for the approach to become a useful tool is far from satisfactory as cited in this chapter. The lack of it as the state-of-the-art starts to cast a thick cloud of doubt over whether such an approach is scientific, let alone practical. Having identified the root sources for the inconsistencies and possible mistakes at implementation level as presented in this chapter, the authors are convinced that the necessary level of consistency and confidence are well within reach as is the main objective of this monograph. Correct formulation and implementation in a consistent manner will be presented systematically through the chapters to follow. As a measure of assurance, the approach as derived systematically and logically have been implemented through a piece of software called UnitCells© as a secondary development of Abaqus/CAE. It has been subjected to a reasonably wide range of verification and some basic levels of validation. It has been made available to the readers of the monograph via a special website provided and managed by the publisher whilst the copyright remains withheld by the authors. The achievement of this development serves at least as a positive sign that,

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once the loose and casual treatments, which are often wrong, have been ironed out, and once the tedious aspect of systematic implementation get automated, the multiscale approach using RVEs and UCs can be shown indeed to be a powerful tool. In particular, having established the logical rules for users to follow, construction of new RVE or UC to suit their specific applications will be straightforward if one is prepared to follow the rules.

References Abaqus Analysis User’s Guide, 2016. Abaqus 2016 HTML Documentation. De Carvalho, N.V., Pinho, S.T., Robinson, P., 2011. Reducing the domain in the mechanical analysis of periodic structures, with application to woven composites. Composites Science and Technology 71, 969e979. de Kok, J.M.M., Meijer, H.E.H., 1999. Deformation, yield and fracture of unidirectional composites in transverse loading: 1. Influence of fibre volume fraction and testtemperature. Composites Part A: Applied Science and Manufacturing 30, 905e916. Drago, A., Pindera, M.-J., 2007. Micro-macromechanical analysis of heterogeneous materials: macroscopically homogeneous vs periodic microstructures. Composites Science and Technology 67, 1243e1263. Fang, Z., Yan, C., Sun, W., Shokoufandeh, A., Regli, W., 2005. Homogenization of heterogeneous tissue scaffold: a comparison of mechanics, asymptotic homogenization, and finite element approach. Applied Bionics and Biomechanics 2 (1), 17e29. Govaert, L.E., Schellens, H.J., de Kok, J.M.M., Peijs, T., 1995. Micromechanical modelling of time dependent failure in transversely loaded composites. In: Proceedings of 3rd International Conference of Deformation and Fracture of Composites, 27e29 March 1995 University of Surrey. The Institute of Materials, pp. 77e85. Li, S., 1999. On the unit cell for micromechanical analysis of fibre-reinforced composites. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455, pp. 815e838. Li, S., 2001. General unit cells for micromechanical analyses of unidirectional composites. Composites Part A: Applied Science and Manufacturing 32, 815e826. Li, S., 2008. Boundary conditions for unit cells from periodic microstructures and their implications. Composites Science and Technology 68, 1962e1974. Li, S., 2012. On the nature of periodic traction boundary conditions in micromechanical FE analyses of unit cells. IMA Journal of Applied Mathematics 77, 441e450. Li, S., 2014. UnitCells© User Manual. Version 1.4. Li, S., Jeanmeure, L.F.C., Pan, Q., 2015. A composite material characterisation tool: UnitCells. Journal of Engineering Mathematics 95, 279e293. Li, S., Lim, S.-H., 2005. Variational principles for generalized plane strain problems and their applications. Composites Part A: Applied Science and Manufacturing 36, 353e365. Li, S., Reid, S.R., 1992. On the symmetry conditions for laminated fibre-reinforced composite structures. International Journal of Solids and Structures 29, 2867e2880. Li, S., Wongsto, A., 2004. Unit cells for micromechanical analyses of particle-reinforced composites. Mechanics of Materials 36, 543e572. Suquet, P.M., 1987. Elements of homogenization for inelastic solid mechanics. In: Sanchezpalencia, E., Zaoui, A. (Eds.), Homogenization Techniques for Composite Media. Lecture Notes in Physics, vol. 272. Springer, Berlin, Heidelberg. Teply, J.L., Dvorak, G.L., 1998. Bounds on overall instantaneous properties of elastic-plastic composites. Journal of the Mechanics and Physics of Solids 36, 29e58.

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Timoshenko, S., Goodier, J.N., 1969. Theory of Elasticity. McGraw-Hill. Whitcomb, J.D., Chapman, C.D., Tang, X., 2000. Derivation of boundary conditions for micromechanics analyses of plain and satin weave composites. Journal of Composite Materials 34, 724e747. Yeh, J.R., 1992. The effect of interface on the transverse properties of composites. International Journal of Solids and Structures 29, 2493e2502.

CHAPTER 6

Formulation of unit cells 6.1 Introduction Unit cells (UCs) have found wide applications in modelling and characterisation of materials with complex architectures in their lower length scales, in particular, composites. In the literature, UCs have been employed to facilitate various analyses, often in a casual manner unfortunately. Readers are easily left with an impression that UC modelling is simply a wellestablished routine exercise. However, once one has to put it in practice, all sorts of difficulties arise. It is in fact full of pitfalls if one intends to adopt it as a serious means of material characterisation. There are so many seemingly trivial issues, but, without putting them right, one cannot even apply the analysis successfully to an actually trivial problem! A comprehensive literature review on the subject is waived here. The reason for doing so is the fact that one often cannot find enough details in publications involving UCs. Roughly, in nine out of ten of them, one would not be able to reproduce the work due to the lack of information such as boundary conditions, load application and processing of the results. Amongst those which cared to include an account on boundary conditions, for instance, it was not uncommon to find statements such as that the boundary conditions were “proposed to be”, “assumed to be”, “approximated as” or simply “boundary conditions employed are”, as if it was meant to be an account as loose as it was presented, or a well-known practice and hence does not require any justification. With a critical and scientific attitude, it will become clear that few of them would stand scrutiny. Sometimes, simple anomalies could be identified straight away, if the readers cared to observe. Simple testimonies can be the so-called “sanity checks” as will be described fully later in the chapter. There is no lack of examples in the literature addressing hugely sophisticated issues using unit cells but not being able to pass a simple “sanity check”! Another simple observation readers might make is whether and how the analysis involves shear stress state. Majority of the analyses using UCs did not touch shear as if shear was either implied by the considerations made or was unimportant. The truth is of course far from that. Shear is just as important as any other Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00006-2

© 2020 Elsevier Ltd. All rights reserved.

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matrix-dominated behaviour of composites. Even for a macroscopically isotropic material, prediction of shear stiffness is important for checking if the isotropy has been preserved in the theoretical model. Specifically, one of conditions for isotropy is that there is a relationship amongst the Young’s modulus, E, shear modulus, G, and Poisson’s ratio, n, as G ¼ E=2ð1 þnÞ. Multiscale modelling using UCs can be a very effective tool for material characterisation, if it is carried out systematically and consistently. The objective of this chapter is therefore to provide a comprehensive account on how it can be done in a correct and reliable manner. Whilst establishing the methodology and following the logic behind each consideration, the myth over the subject will be dispelled, returning the subject a fair and crystal clear image as it deserves.

6.2 Relative displacement field and rigid body rotations A typical multiscale modelling involves two length scales, an upper one and a lower one. The objective of material characterisation using multiscale modelling is usually to evaluate the effective material properties in the upper scale based on the analyses conducted with the models at the lower scale. The basic assumptions made to the material at the upper scale are that it is effectively homogeneous and the stress and strain states prescribed to it are both uniform, same as during the experimental characterisation of the material where appropriate gauge length would have to be established to ensure the effects of uniformity. The homogeneity here can be justified either in a statistical sense or based on the periodicity in the structure at the lower scale. Correspondingly, UCs are resorted to in order to facilitate the analysis at the lower scale. To obtain the effective properties at the upper scale, one has to follow their definitions and to evaluate them from the uniform stresses and the uniform strains corresponding to a uniaxial or a pure shear stress state at the upper scale. Between the two sets of values, uniform stresses and uniform strains, if one of them is employed as the means of prescribing the loads to the UC to be analysed, the other will be obtained as a part of the results out of the analysis. This procedure is again very similar to what one would follow if these properties had to be measured experimentally. The material characterisation using UCs is therefore a genuine case of virtual testing. Once various relationships at the upper scale are established, the analysis has to be conducted at the lower scale. A crucial link across the two scales is

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the relationship between the lower scale displacement field and the upper scale strains. For problems where UCs are applicable, the regularity in the lower scale structure is assumed. The upper scale homogeneity is based on the existence of translational symmetries in three dimensions, not necessarily along the coordinate axes. As discussed in Subsection 2.5.4 of Chapter 2, the translational symmetry is present in the relative displacement field, i.e. 2 3 vU vU vU 6 7 9 8 9 6 vx vy vz 78 6 7 Dx > > < = < DU > = 6 vV vV vV 7> 6 7 DV ¼ 6 (6.1) 7 Dy : > > > vx vy vz 7> : ; : ; 6 6 7 DW 6 vW vW vW 7 Dz 4 5 vx vy vz For a given uniform deformation in the upper length scale, all components of the displacement gradient matrix are constants. In its general form, displacement gradient involved in (6.1) fully describes the strain state in the UC and the rigid body rotations of the UC. In most applications, one is interested primarily in the strain state, i.e. a part the displacement gradient, whilst the rigid body rotations can be left aside with the exception of finite deformations as will be addressed in Chapter 13. Alternatively, if one wishes to add some rigid body rotations on top of that as described by (6.1), it will not affect the strain state, since it will only be a matter of reference, i.e. how the deformation of the body is observed. To reveal this, the displacement gradient given in (6.1) can be conventionally partitioned into a symmetric and an antisymmetric part as follows: 2 3 vU vU vU 3 6 7 2 0 2 3 0 0 6 vx vy vz 7 ε ε ε xy xz 7 0 u0xy u0xz 6 7 6 x 6 vV vV vV 7 6 7 7 6 0 6 7 6 0 0 0 uxy 0 u0yz 7 þ6 6 7 ¼ 6 εxy εy εyz 7 4 5; 7 6 vx vy vz 7 4 5 6 7 0 0 uxz uyz 0 6 vW vW vW 7 ε0xz ε0yz ε0z 4 5 vx vy vz (6.2) where

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2

ε0x

 0 6 6 ε ¼ 6 ε0xy 4 ε0xz 2

ε0xy

ε0xz 3

ε0y

7 ε0yz 7 5

ε0yz

ε0z

 3 1 vU vW þ 6 2 vz vx 7 6 7 6 7 6    7 6 1 vV vU vV 1 vW vV 7 7 ¼6 þ 6 2 vx þ vy vy 2 vy vz 7 6 7 6  7    6 7 1 vW vV vW 4 1 vU vW 5 þ þ 2 vz vx 2 vy vz vz vU vx

  1 vV vU þ 2 vx vy

(6.3)

is the strain tensor under the assumption of small deformation (note that under finite deformation strain formulations involve higher order terms, as will be addressed in Chapter 13 specifically), which is uniform in the upper scale in the present problem, and 8 9 vW vV > > 8 9 > > > >  0 > > > > > > u vy vz > > > > yz > > > < = 1> < = vU vW 0 (6.4) uxz ¼  > > 2> vz vx > > > > > > > > > : u0 ; > > > vV vU > > > xy > >  : ; vx vy is the rigid body rotation vector, also being a part of the displacement field in the upper scale. If the strain part of (6.2) vanishes, the displacement gradient will only contain rigid body rotations. Since the rotation vector represents rigid rotations, its components can be assigned any value without altering the strain state, provided that it does not compromise the small deformation assumption. These rotations can be visualised as follows. Generate three straight lines, one on each of the three coordinate planes at 45 to the coordinate axes on both sides of it. The components of the rotation vector (6.4) define the rotation of each of these lines about the coordinate axis perpendicular to the respective line, e.g. the rotation of the 45 degrees line on the xy-plane about the z-axis, etc. Denoting the respective rotations with 45 in the superscript, the rotation vector (6.4) can be re-written as

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9 8 8 9 45 > 0 > > R > > > u x > > = = > < < yz > 45 45 0 R R ¼ ¼ uxz y > > > > > > > > > >  : R45 ; : u0xy ; z

(6.5)

For material characterisation, one is only interested in the strain part of the displacement gradient. Given the partitioning in (6.2), it is apparently a straightforward way of expressing the relative displacement field by dropping the rotation part, which is equivalent to letting the rotation vector to be zero. In this case, one obtains 2 3 0 0 0 8 9 8 9 ε ε ε 9 xy xz 78 > >u= > >u= 6 x < Dx = < < 6 7   0 0 0 7 v   v  ¼6 (6.6) 6 εxy εy εyz 7: Dy ;   > > > > : ; : ; 4 5 Dz w ðx0 ;y0 ;z0 Þ w ðx;y:zÞ ε0xz ε0yz ε0z Effectively, this eliminates the rigid body rotations by constraining the rotation of each of the three 45 degrees lines as described above. In other words, expressing the relative displacement in the form of (6.6) is equivalent to constraining rigid body rotations in this specific way. When the 45 degrees lines are so constrained, the shear strains are split into two equal halves. This corresponds to the rotations of both sides as sketched in Fig. 6.1(a), where a 2D view in the xy-plane is shown. However, this is by no means the only correct way of constraining rigid body rotations. If one wishes to have the rigid body rotations to be constrained differently, the relative displacement field can be expressed in general as (a)

(b)

(c)

Fig. 6.1 Different ways of constraining the rigid body rotation in the xy-plane.

112

02 8 9 ε0x > > B < DU = B6 B6 0 DV ¼ B6 6 εxy > > : ; B @4 0 DW εxz

Representative Volume Elements and Unit Cells

ε0xy ε0y ε0yz

3

ε0xz 7

2 0 7 4 R ε0yz 7 z 7þ 5 R y ε0z

Rz 0 Rx

1 9 3 8 Ry C C< Dx = Rx 5C C Dy ; C: 0 A Dz (6.7)

where Rx, Ry and Rz are the rotations of the aforementioned 45 lines about the x-, y- and z-axes, respectively. Any set of fixed values assigned to them defines a way of constraining the rigid body rotations. As a result, the displacement gradient may look different but the strain state will not be affected. For a special case considered above, rigid body rotations were constrained by setting to zero all three components in the rotation vector which prevented the rotations of 45 lines in the specific way as described. Consider a rotation vector defined as follows: 9 8 > > vW vV > > > > 9 8 þ > > > > 0 > 8 9 > > vy vz > > > ε > > > > yz > > > < Rx > = 1> = > = < < vU vW 0 R ¼ Ry ¼ ¼ εxz (6.8)   > > vx > : > ; 2> > > > > vz > > > > > Rz > > > : ε0xy ; > > > > vV þ vU > > > > > : vx vy ; where value of each component has the magnitude of the respective tensor shear strain. The 45 lines in this case are assigned rotations equal to the tensor shear strain in the plane of the rotation. This effectively constrains one side of the body from rotating as illustrated in Fig. 6.1(b) in a 2D view. Once again, whilst this eliminates the rigid body rotation in this plane, the strain state is not affected at all. In fact, there is no reason why the rigid body rotation cannot be constrained by assigning an arbitrary rotation to the 45 line as shown in Fig. 6.1(c) in order to constrain the rigid body rotation. When displacement gradient is partitioned according to (6.7) with rotation being defined via (6.8), the shear deformation will be due to rotation of only one of the two relevant sides whilst the other side remains stationary as a way of constraining the rigid body rotation. In this case, the relative displacement field (6.7) becomes

113

Formulation of unit cells

9 8 8 9 8 9  DU > > u < = < u = = <  v   v  ¼ DV : ; : ; > > ; : w ðx0 ;y0 ;z0 Þ w  DW ðx;y:zÞ 2 3 vU 0 0 7 2 6 vx 9 6 78 ε0x Dx > 6 7> < = 6 6 vV 7 vV 6 g0 0 7 ¼6 xy 6 vx 7> Dy > ¼ 6 vy ; 4 6 7: 0 6 7 Dz gxz 4 vW vW vW 5 vx vy vz

0 ε0y g0yz

3

8 9 7< Dx = 07 7 Dy ; 5: Dz ε0z 0

(6.9)

where g’s are the engineering shear strains as opposed to their tensorial counterparts in (6.7). This is equivalent to setting 9 8 9 8 > vV > > > > vW > 9 8 > > > > > > > 0 > 8 9 > > > > vz > > > > > > g vy > > > > > > yz > 0 > > > > > > > = = < < < vU = < = vW 0 (6.10) ¼ 0 and hence gxz ¼ > > > > > vz > vx > > > > > : > ; > > > > > > > > 0 > : g0 ; > > > > > vV > vU > > > > > xy > > > > ; > : > : vy ; vx The effect of (6.10) is that the rigid body rotations are eliminated by constraining the rotations of the fibres (imaginary directional entity of material) along the z-axis about the x- and y-axes, and the fibres along the y-axis about the z-axis as a rigid body, respectively. This does not violate any rules of deformation kinematics and therefore captures the same strain field as defined by (6.2). By constraining the rigid body rotations in different ways, one can obtain expressions different from (6.7). The lack of unique expression for relative displacement field can easily become a source of confusion if one is driven merely by his/her intuition without due respect of the basics of deformation kinematics. As long as the expression captures the same strain state, it is correct and one can choose from the abundance to achieve the best effect, in particular, convenience in implementation, as will be elaborated in the next section. Note that the rigid body rotation constraints as presented here have been re-arranged from (Li, 1999, 2001; Li and Wongsto, 2004; Li and Sitnikova, 2018) in order to keep the consistency as a single and complete account on the subject in this monograph as will be elaborated later.

114

Representative Volume Elements and Unit Cells

6.3 Relative displacement boundary conditions for unit cells Relative displacement field (6.9) provides the relationship between the displacements at corresponding points in different cells related by the translational symmetries present in the problem. If they are placed on opposite sides of the boundary of the cell, the increment in coordinates between them ð Dx Dy Dz ÞT can be obtained according to the translational symmetries chosen to shape the UC, as has been elaborated in Section 4.3.4. The relative displacement boundary conditions for the pair of pieces of the boundary can thus be obtained from Eq. (6.9). As discussed in Section 4.3.3, the boundary of the unit cell has to be partitioned into pieces in pairs with each pair being associated with a chosen translational symmetry, bearing in mind of the possibility of redundant translational symmetries. There should be neither gap nor overlap between these parts of the boundary. The preference of the term “relative displacement boundary conditions” to what have been conventionally called “periodic displacement boundary conditions” has been explained in Section 5.3.3, and it will be adopted henceforth to refer to the boundary conditions for UC. Relative displacement boundary conditions establish certain relations between displacements on two different parts of the boundary of the unit cell. The relationships are in fact the definition of relative displacements. In other words, the differences between the displacements on the two parts of the boundary are related through a translational symmetry of the relative displacement field. As established in the previous section, relative displacements can be related to the uniform strain field in the upper length scale. Such expression is not unique, and for subsequent applications the relative displacements defined by (6.9) is chosen, i.e. 3 2 0 8 09 8 9 8 9 0 0 ε x >

= >

= 6 7< Dx = 0 6 0 07 (6.11) v0  v ¼ 6 gxy εy 7 Dy > > > > ; 5: 4 : 0; : ; Dz w w g0xz g0yz ε0z This is the relative displacement boundary conditions in a general form with (u v w)T being the displacements on a part of the boundary as the original and ð u0 v 0 w 0 ÞT as the displacement on another part of the boundary as the image of the original under the translational symmetry defined by a translation of (Dx Dy Dz)T. The average strains involved in (6.11) will be treated

Formulation of unit cells

115

for the time being as some independent degrees of freedom to facilitate the formulation of the boundary conditions for UCs. Their full significance will be explained and the systematic treatment will be presented Section 6.6 of this chapter and a rigorous proof will be provided in Chapter 7. The reason for the choice of (6.9) over (6.6) is based purely on the implementation consideration where some of the equations imposing these conditions in the analysis will have fewer terms to be defined. For example, for the displacement in x-direction, using the above, one has u0  u ¼ ε0x Dx

(6.12)

whilst using (6.6), one would have 1 1 (6.13) u0  u ¼ ε0x Dx þ g0xy Dy þ g0xz Dz 2 2 The results in terms of obtained stress field in the lower length scale and the effective properties in the upper scale would be identical no matter which of the two expressions is used, provided that the respective expressions of boundary conditions have been applied and implemented correctly. Note that this will not be the case in problems of finite deformation, as will be elaborated in Chapter 13. A range of typical unit cells both in 2D and 3D cases have been established and published in (Li, 1999, 2001; Wongsto and Li, 2005; Li et al., 2009, 2011). With the underlying basic principles of their formulations being covered in previous chapters in this book, the relative displacement boundary conditions derived for the aforementioned types of unit cells are provided in the next section.

6.4 Typical unit cells and their boundary conditions in terms of relative displacements For each UCs considered, the relative displacement boundary conditions will be presented in a form of equations relating the displacements on one part of the boundary to those on another part. These two parts of the boundary are related through a certain translational symmetry.

6.4.1 2D unit cells 6.4.1.1 An introduction to 2D idealisation A 2D idealisation is usually a simplification of general 3D problem when relevant fields in one of the three dimensions exhibit certain simple features, e.g. being constant or vanishing in values, either as a nature of the problem

116

Representative Volume Elements and Unit Cells

or as an approximation. Applicability of 2D idealisation relies on the absence of variation of either stress or strain field in the third dimension out of the plane concerned. Typical examples in elasticity are plane stress and plane strain problems, and the anticlastic problem relevant to out-of-plane shear stress state as observed in the torsion of prisms, whose governing equation is shared by 2D heat conduction and many other physical problems (Bateman, 1932). These are physically different problems and should not be mixed up in application. In principle, the plane stress problem is applied to thin sheets with traction-free surfaces parallel to the plane and the justification is that the sheet is so thin that all three out of plane stresses (one direct and two shear components) do not have sufficient distance to build up to any significant magnitude and hence can be neglected as an approximation. The plane strain applies to problems over a domain which is of constant cross-section in the third direction and is infinite in extent in theory. There is another closely related problem, referred to as the generalised plane strain problem (Li and Lim, 2005), that shares the similar features, which is of relevance to composites applications, in particular, micromechanics of composites. The dividing line between them is that the plane strain problem should be in presence of constraints in the third direction such that every cross-section parallel to the plane remains stationary in the third direction, whilst the generalised plane strain problem allows expansion or contraction in the third direction. When such expansion or contraction is constrained, the generalised plane strain problem reproduces the plane strain problem. As discussed in Chapter 5, the plane strain problem is not directly applicable in material characterisation, whilst the plane stress problem can be useful to describe the in-plane behaviour of laminae in laminated composites. However, it should the noted that if the expansion or contraction in the third direction is set free in the generalised plane strain problem, it does not reproduce the plane stress problem because, in a plane stress problem, the out-of-plane stresses vanish everywhere whilst the generalised plane strain problem with free expansion or contraction in the third direction only requires zero stress resultants in this direction. For micromechanical analysis of UD composites, it is usually reasonable to assume that fibres are infinite in length and that both the material properties and the geometry remain constant in fibre direction. As a result, the problem can be fully described in the plane perpendicular to the fibres, hence can be idealised as 2D problem. Examples of applications of the generalised plane strain problem to UD composites were presented in (Li, 2001; Wongsto and Li, 2005). It is worth noting that the formulation of generalised plane strain elements available in existing commercial FE codes

117

Formulation of unit cells

has limitations. Namely, it allows an additional global out-of-plane displacement associated with the out-of-plane direct strain plus two rotations to represent global bending through an additionally defined reference node. The desirable generalisation requires an anticlastic part to incorporate the longitudinal shear behaviour, which has not been included. In order to address the more generalised plane strain problem as relevant to the behaviour of UD composites, there are two alternative ways of facilitating the analysis. 1) Whilst remaining within 2D formulation, the generalised plane strain elements can be employed for the evaluation of all in-plane effective properties as well as the Young’s modulus and Poisson’s ratio in the outof-plane direction. For the out-of-plane shear, an analogy to 2D steady state heat conduction problem can be resorted to. It is available in most commercial FE codes, such as Abaqus, as was demonstrated in (Wongsto and Li, 2005). The analogy is based on consideration that the present problem of out-of-plane shear deformation and the heat conduction problem are both governed by the same governing equation. For the heat conduction problem vqx vqy þ ¼0 vx vy where   qx qy

¼

(6.14)

k11

k12

k12

k22



Tx



Ty

are heat fluxes and T is the temperature field and 8 9 vT > > > > >   < vx > = Tx ¼ VT ¼ > vT > Ty > > > > : ; vy

(6.15)

(6.16)

with kij (i,j ¼ 1,2) being the heat conductivity of the material. In the problem of out-of-plane shear deformation, T is the out-of-plane displacement, kij are the shear stiffnesses and qx and qy the negative out-ofplane shear stresses. Because of the nature of the governing equation, the solution is often called a harmonic function. A function of this type for a displacement field often shows the appearance of some kind of saddle surface and this pattern of the deformation is referred to as anticlastic. Since this part of the generalised plane strain problem is not incorporated

118

Representative Volume Elements and Unit Cells

in Abaqus, the heat transfer elements in Abaqus can be used instead with the same meshes as were generated for the analysis in the Abaqus generalised plane strain problem. The shear moduli are input in place of heat conductivity and concentrated forces in lieu of concentrated heat fluxes. In the output, the heat flux components should be interpreted as negative longitudinal shear stresses. This option is computationally efficient but demanding on users’ understanding and manipulation skills. Given the physical needs for a more generalised plane strain problem, commercial FE code developers are encouraged to further generalise their generalised plane strain elements to incorporate the anticlastic behaviour as part of stress analysis. In absence of such generalisation, the analogy as described above has to be used. The active independent degrees of freedom for this analysis are Tx0 and Ty0 as temperature gradients in the upper length scale in the heat conduction problem and shear strains for the out-of-plane deformation problem. They are related to heat fluxes or negative shear stresses at the upper length scale as ( 0 0

( 0 ) qx Tx k11 k012 (6.17) ¼ 0 0 0 Ty0 k12 k22 qy where k0ij are the effective heat conduction coefficients or effective longitudinal shear stiffnesses in their respective problem. The determination of them is usually the objective of multiscale material characterisation. 2) The 2D UC is analysed using 3D brick elements by adding a third dimension to it to avoid splitting the analysis into two parts as described above. This option will be further elaborated in the subsections to follow. Effectively, this introduces another translational symmetry along the third coordinate axis by distance t. Given the 2D nature of the generic problem, t can be of an arbitrary positive value. It is advisable to select a value in relation to the characteristic in-plane element size so that a side length of a 3D brick elements in this direction would not be too disproportionate with respect to other sides. In this case, one only needs a single layer of elements in the out-of-plane direction, because correct results will not have any variation in stress field in this direction at all. Any stress gradient observed in this direction will indicate an error of some kind beyond doubt, most likely in the imposition of the boundary conditions. Such error can be easily diagnosed using the “sanity checks” as will be introduced in Section 6.9. It should and can be eliminated completely. Having more layers of elements in this direction, as

119

Formulation of unit cells

commented in the previous chapter, would make absolutely no difference, except for increasing the computational demands. In fact, the practice of building up layers in the thrid direction as adopted in the literature is typically a way of hiding errors by diluting their effects via the use of multiple layers. 6.4.1.2 2D unit cells with translational symmetries along coordinate axes The simplest unit cell for materials with translational symmetries along coordinate axes in a plane is a rectangle with its side lengths corresponding to the distances of translations associated with the symmetries, as shown in Fig. 6.2(a). A special case is the square unit cell (Li, 1999, 2001) when the lengths of both sides happen to be equal. There is no reason in theory why a complex-shaped UC, such as that in Fig. 6.2(b) cannot be used. However, with curved sides of the boundary, the geometric definition of the UC will become more complicated accordingly, unless the tessellation happens to correspond to the physical arrangement of geometric features at the lower length scale, such as the fish scale tiling as cited in Chapter 2. In a 2D space, the relative displacement boundary conditions can be reduced from (6.11) to ( ) ( ) " #( ) ε0x 0 u0 Dx u  v ¼ (6.18) Dy v0 g0xy ε0y The relative displacement boundary conditions are given as follows. For the sides of the boundary related through a translation in the x-direction, the relevant translation is     Dx 2a ¼ (6.19) Dy 0

(a)

y

(b)

2b

O 2a

x

2b

y

O

x

2a

Fig. 6.2 Different shapes of 2D unit cells with translational symmetries along coordinate axes: (a) rectangular 2D UC; (b) an example of a complex-shaped 2D UC.

120

Representative Volume Elements and Unit Cells

for which the relative displacement boundary conditions can be obtained from (6.18) as ujða;yÞ  ujða;yÞ ¼ 2ε0x a

(6.20)

vjða;yÞ  vjða;yÞ ¼ 2g0xy a Similarly, the translation relevant to the other pair of sides is     Dx 0 ¼ Dy 2b

(6.21)

and hence ujðx;bÞ  ujðx;bÞ ¼ 0

(6.22)

vjðx;bÞ  vjðx;bÞ ¼ 2ε0y b:

As a corner is shared by two segments of the boundary, it is involved in both sets of relative displacement boundary conditions as given above. Corners (a, b) and (a, b) are related through (6.20) and so are (a,b) and (a,b). Since (a, b) and (a,b) are also related through (6.22), the relationship between (a, b) and (a,b) has been implied. Therefore, if condition (6.22) is imposed between them, it will be redundant. In some solvers, e.g. Abaqus/Explicit, redundant boundary conditions are acceptable due to the explicit algorithm adopted. If the solver employed, such as Abaqus/ Standard, does not allow such redundancies in the boundary conditions, one will have to remove them manually. It is therefore a good practice to avoid them. In order to do so, the boundary conditions (6.20) and (6.22) are to be imposed only to the sides with the corners excluded. It is worth noting that exclusion of edges from faces is not because the conditions for the faces do not apply to the edges, but it is strictly to avoid redundant boundary conditions for the UC. This will apply throughout the book. A special set of boundary conditions imposed to the corners of the rectangular UC is as follows. The underlying translational symmetry considerations are that corners (a, b) and (a,b) are the images of (a,b) under the horizontal and vertical translations, respectively, whilst corner (a,b) is the image of (a, b) under a horizontal and a vertical translation combined, namely ujða;bÞ  ujða;bÞ ¼ 2ε0x a vjða;bÞ  vjða;bÞ ¼ 2g0xy a

 corresponding to translation

   Dx 2a ¼ Dy 0

121

Formulation of unit cells



ujða;bÞ  ujða;bÞ ¼ 0 vjða;bÞ  vjða;bÞ ¼ 2ε0y b

corresponding to translation

Dx Dy



  0 ¼ 2b

ujða;bÞ  ujða;bÞ ¼ 2ε0x a vjða;bÞ  vjða;bÞ ¼ 2g0xy a þ 2ε0y b ( corresponding to translation

Dx Dy

(

) ¼

2a 0

)

( þ

0

)

(6.23)

2b

These three sets of boundary conditions fully describe the relationships between the vertices through the translations, yet this combination of boundary conditions is not unique. One can easily come up with different combinations of translations to achieve the same. However, there are two important considerations to bear in mind when deriving boundary conditions at the corners, which explain why the form of boundary conditions as given by (6.23) represents a good practice. The first consideration is the independence of the boundary conditions. Indeed, in (6.23) all sets of equations are independent, which means that none of them can be reproduced as a linear combination of the rest. Furthermore, in presence of (6.23), boundary conditions for the corners in any other form can be reproduced as a linear combination of those in (6.23). For this simple case of UC it is easily seen that if one of the corners is selected as the original, the remaining corners can be reproduced as the images of the original through the available translations and their combinations. As a result, assuming there are N corners in the group related through available translational symmetries, these translational symmetries offer N-1 sets of independent relative displacement boundary conditions amongst the corners in this group. This will hold true for groups of corners in all the types of UCs as presented in this chapter. As has already been stated, the selection of independent boundary conditions is not unique, and mathematically it does not make a difference whether all the vertices in the group are related to the same one through the respective translations, or different pairs of vertices are related accordingly, as long as none of the conditions is redundant. However, the former method is more systematic, as it guarantees the independence of the

122

Representative Volume Elements and Unit Cells

boundary conditions obtained, whilst for the latter method their independence has to be verified. The second important consideration is related to the implementation of the boundary conditions in a finite element model. Namely, when imposing these relative displacement conditions at the corners, the conventions in the actual solver employed must be followed. For instance, in Abaqus/Standard, equation boundary conditions need to be prescribed in a certain form, where all terms on the right-hand side are moved to the left, after a change in their sense, of course. With a zero right-hand side, one only needs to specify the terms on the left-hand side. As have already been explained in Section 5.3.4, the actual constraint implemented inside Abaqus eliminates the first degree of freedom (dof) appearing on the lefthand side of the equation. Since some degrees of freedom tend to appear in multiple equations, the user must make sure that such a dof does not appear as the first term in those equations more than once. Otherwise, the execution of the analysis may abort in error as the dof eliminated through a previous equation cannot be eliminated again when it appears as the first dof the second time. A good practice is to have a different dof as the first in the dof list for each equation constraint, as was done in (6.23), in relation to dofs on node (-a,-b). This good practice will be followed throughout the chapter when presenting the constraints for all the UCs considered here. Namely, the dofs on the nodes that appear more than once in boundary condition equations will never be specified as a first term in any of the equations. Once again, the reason for doing so is purely to facilitate their practical implementation by providing the exact order in which the terms of the equations should be input, and thus avoiding errors resulting from referring to eliminated dof when processing an FE model internally in the FE code adopted. When generating the model, the boundary conditions should be supplemented with appropriate constraints to eliminate rigid body motions. The use of relative displacement field in the formulation of relative displacement boundary conditions has already implied the constraints of the UC from rigid body rotations. Therefore, only rigid body translations should be eliminated. It can done by constraining a single node which is free from any constraint yet. A convenient node could be corner (a, b), i.e. ujða;bÞ ¼ vjða;bÞ ¼ 0:

(6.24)

Formulation of unit cells

123

Note that in the practical analysis inside many commercial FE codes, equations in (6.23) are considered by convention as the constraints on the first dof appearing in the equation and hence node (a, b) is free from constraints even in presence of (6.23). It is almost certainly the case that, when one goes through the first few attempts of the analysis of the UC, high stress concentration is observed at this node. However, this does not necessarily mean that any wrong constraints have been imposed on this node. More likely, mistakes have been made somewhere else as it is almost impossible to correctly implement the UCs from the first attempt. It could be the user’s first reaction to avoid constraining this particular node and choose one inside the UC instead. In such case, any stress concentration present there might not even be visible. However, this would be a completely wrong practice, as it merely conceals the error rather than resolves the problem. In fact, it is advisable to keep the constraints at this corner as any modelling error present there could easily report a range of potential mistakes in the implementation of the UC. Having tens of glitches in the first attempt is not an exaggeration but personal experiences of the authors, given the fact that they have been FE users having well over 40 years of experience between them. These glitches must be ironed out rather than being hidden away. The “sanity checks” as described in Section 6.9 of this chapter will help greatly in this respect. The most popular applications of square unit cells are for the analysis of UD composites with fibres arranged or idealised in a square packing. One can easily work out that the maximum fibre volume fraction this packing allows is 78.54%, assuming fibres are of perfectly circular cross-sections of an equal size, although this is not practically achievable in reality. Readers are reminded that square fibre packing does not show effective transverse isotropy at the upper length scale (Li, 2001) and therefore is not a suitable candidate if it is meant to be an idealisation of otherwise transversely isotropic material. Given this understanding, if transverse isotropy is deemed to be unimportant, the square packing model can be employed for characterisation of UD composites as the simplest form of UCs. However, the 2D UC as formulated above is not directly applicable to the characterisation of UD composites in the plane transverse to the fibres. An appropriate UC formulation should be governed by a generalised plane strain problem instead. To achieve this, the above formulation should be supplemented with one more condition after using the so-called generalised plane strain elements as available in Abaqus. Such elements come with a

124

Representative Volume Elements and Unit Cells

common dof for all elements in the model at a reference node. Denote it as w0. It is related to the longitudinal direct strain ε0z as follows: w0 ¼ tε0z

(6.25)

where ε0z can be defined as one of the dofs at the reference node associated with the definition of the generalised plane strain elements. Assuming a unit length in the out-of-plane dimension, one has t ¼ 1, and the relevant boundary conditions for the anticlastic problem are wjða;yÞ  wjða;yÞ ¼ 2ag0xz wjðx;bÞ  wjðx;bÞ ¼ 2bg0yz

for sides; and

(6.26)

wjða;bÞ  wjða;bÞ ¼ 2ag0xz wjða;bÞ  wjða;bÞ ¼ 2bg0yz

for corners;

(6.27)

wjða;bÞ  wjða;bÞ ¼ 2ag0xz þ 2bg0yz where w is a variable out-of-plane displacement field as opposed to w0 introduced as a part of the generalised plane strain problem. Based on the analogy as described previously, it would be the temperature T in the heat conduction problem whilst g0xz and g0yz would be the temperature gradients in the x- and y-directions, Tx and Ty, respectively. Again, the rigid body displacement has to be constrained. In this case, without loss of generality, one can have wjða;bÞ ¼ 0

(6.28)

It can now be explained why in Eq. (6.11) the coefficient matrix is set to a lower triangular form instead of upper triangular form as in the authors’ previous publications, e.g. (Li, 1999, 2001; Li and Wongsto, 2004). With the latter, the out-of-plane shear strains would be the rotations of the z-axis relative to the xy-plane and there would not be out-of-plate displacement. This would make the anticlastic problem less obvious. With the lower triangular form, the out-of-plane shear strains are directly associated with the out-of-plane displacement which is analogous to temperature in the heat conduction problem. Employing 3D brick elements for the analysis of the problem adds a third dimension and effectively makes it a 3D problem. A meshed FE model of a square unit cell representing the UD composite is shown in Fig. 6.3 (Li and Wongsto, 2004; Li, 2014). Note that a single layer of elements in

125

Formulation of unit cells

t 2a

2a

x

y z

Fig. 6.3 FE model for 3D unit cell for UD composites idealised as square packing of fibres.

the out-of-plane direction is employed, the justification for which has already been given earlier. With a single layer of elements in the z-direction, having excluded the edges and vertices one will find that there are no nodes on any of the faces parallel to the z-direction. However, this does not undermine the significance of the boundary conditions for faces, as those for edges and vertices are all derived from them. The complete set of relative displacement boundary conditions for a square UC with added thickness in out-of-plane direction are ujða;y;zÞ  ujða;y;zÞ ¼ 2ε0x a vjða;y;zÞ  vjða;y;zÞ ¼ 2g0xy a wjða;y;zÞ  wjða;y;zÞ ¼

2g0xz a

8 9 8 9 < Dx = < 2a = corresponding to translation Dy ¼ 0 : ; : ; Dz 0

(6.29)

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Representative Volume Elements and Unit Cells

ujðx;a;zÞ  ujðx;a;zÞ ¼ 0 vjðx;a;zÞ  vjðx;a;zÞ ¼ 2ε0y a

(6.30)

wjðx;a;zÞ  wjðx;a;zÞ ¼ 2g0yz a

8 9 8 9 < Dx = < 0 = corresponding to translation Dy ¼ 2a : ; : ; Dz 0

ujðx;y;tÞ  ujðx;y;0Þ ¼ 0 vjðx;y;tÞ  vjðx;y;0Þ ¼ 0

(6.31)

wjðx;y;tÞ  wjðx;y;0Þ ¼ ε0z t

8 9 8 9 < Dx = < 0 = corresponding to translation Dy ¼ 0 : ; : ; Dz t

These boundary conditions are given between paired faces. Redundant boundary conditions are present at the edges and vertices if they have been included as parts of the faces. If one wishes to eliminate them, the following procedure can be taken. Edges must be excluded from the faces and the vertices from the edges. The boundary conditions (6.29)e(6.31) are to be imposed to such defined faces first. The twelve edges can be put into three groups, each having four edges parallel to one coordinate axis. In each group, the three edges are reproduced from the remaining one through translational symmetries available in the structure as have been employed to define the unit cell. Relating the displacements on one to those at the remaining three results in independent boundary conditions on such edges. The same justification for selection of independent boundary conditions applies here as was employed when deriving boundary conditions for the vertices of 2D UC above, namely, this method

127

Formulation of unit cells

ensures that no redundant boundary conditions are specified. These conditions can be given as: For the edges parallel to the x-axis: ujðx;a;0Þ  ujðx;a;0Þ ¼ 0 vjðx;a;0Þ  vjðx;a;0Þ ¼ 2aε0y wjðx;a;0Þ  wjðx;a;0Þ ¼ 2ag0yz corresponding to translation

(

Dx Dy Dz

)

( ¼

0 2a 0

)

ujðx;a;tÞ  ujðx;a;0Þ ¼ 0 vjðx;a;tÞ  vjðx;a;0Þ ¼ 2aε0y wjðx;a;tÞ  wjðx;a;0Þ ¼ 2ag0yz þ tε0z

8 9 8 9 < Dx = < 0 = corresponding to translation Dy ¼ 2a : ; : t ; Dz

ujðx;a;tÞ  ujðx;a;0Þ ¼ 0 vjðx;a;tÞ  vjðx;a;0Þ ¼ 0 wjðx;a;tÞ  wjðx;a;0Þ ¼ tε0z

8 9 8 9

= >

= corresponding to translation Dy ¼ 2a > > : ; > : > ; Dz 0

(6.34)

Note that the number of independent sets of boundary conditions for each group of edges related through available translational symmetries will always be one less than the number of edges in the group which, in line with the considerations given to the corners as in the planar case as presented earlier the current subsection, will hold true for all the UCs considered in this chapter. Derivations of boundary conditions in the planar case are accomplished after establishing those for the corners corresponding to the edges as addressed above. Adding the third dimension results in eight vertices. They form a single group where any one of them can reproduce the remaining seven through the translational symmetries available in the architecture at the lower length scale as have been employed to define the unit cell. This results in seven sets of boundary conditions given as follows.

130

Representative Volume Elements and Unit Cells

ujða;a;0Þ  ujða;a;0Þ ¼ 2aε0x vjxða;a;0Þ  vjða;a;0Þ ¼ 2ag0xy wjða;a;0Þ  wjða;a;0Þ ¼ 2ag0xz

8 9 8 9 < Dx = < 2a = corresponding to translation Dy ¼ 0 : ; : ; Dz 0

ujða;a;0Þ  ujða;a;0Þ ¼ 2aε0x vjxða;a;0Þ  vjða;a;0Þ ¼ 2ag0xy þ 2aε0y wjða;a;0Þ  wjða;a;0Þ ¼ 2ag0xz þ 2ag0yz 8 9 8 9 < Dx = < 2a = corresponding to translation Dy ¼ 2a : ; : ; Dz 0 ujða;a;tÞ  ujða;a;0Þ ¼ 0 vjða;a;tÞ  vjða;a;0Þ ¼ 0 wjða;a;tÞ  wjða;a;0Þ ¼ tε0z

8 9 8 9 < Dx = < 0 = corresponding to translation Dy ¼ 0 : ; : ; Dz t

ujða;a;tÞ  ujða;a;0Þ ¼ 2aε0x vjða;a;tÞ  vjða;a;0Þ ¼ 2ag0xy wjða;a;tÞ  wjða;a;0Þ ¼ 2ag0xz þ tε0z 8 9 8 9 < Dx = < 2a = corresponding to translation Dy ¼ 0 : ; : ; Dz t

(6.35)

Formulation of unit cells

131

ujða;a;tÞ  ujða;a;0Þ ¼ 2aε0x vjða;a;tÞ  vjða;a;0Þ ¼ 2ag0xy þ 2aε0y wjða;a;tÞ  wjða;a;0Þ ¼ 2ag0xz þ 2ag0yz þ tε0z 8 9 8 9 < Dx = < 2a = corresponding to translation Dy ¼ 2a : ; : ; Dz t ujða;a;tÞ  ujða;a;0Þ ¼ 0 vjða;a;tÞ  vjða;a;0Þ ¼ 2aε0y wjða;a;tÞ  wjða;a;0Þ ¼ 2ag0yz þ tε0z 8 9 8 9 < Dx = < 0 = corresponding to translation Dy ¼ 2a : ; : ; Dz t ujða;a;tÞ  ujða;a;0Þ ¼ 0 vjða;a;tÞ  vjða;a;0Þ ¼ 0 wjða;a;tÞ  wjða;a;0Þ ¼ þtε0z

8 9 8 9 < Dx = < 0 = corresponding to translation Dy ¼ 0 : ; : ; Dz t

As before, the rigid body translations are eliminated on the vertex that is free from constraints, which is node (a, a, 0) in the present case, as ujða;a;0Þ ¼ vjða;a;0Þ ¼ wjða;a;0Þ ¼ 0

(6.36)

Readers are reminded that the rigid body rotations have already been eliminated by employing the relative displacement field (6.11) to formulate relative displacement boundary conditions. Apparently, the way of filtering the redundant boundary conditions at corners is not unique, along with so many other sources of nonuniqueness in the presentation of unit cells in general. However, provided that the approach taken is logical, the outcomes in terms of obtained stresses and strains should be identical and provided the formulation followed is

132

Representative Volume Elements and Unit Cells

systematic, the level of confusion can be kept at minimum. This is the reason why a systematic account is advocated throughout this monograph. 6.4.1.3 2D unit cell with translational symmetries along two nonorthogonal directions When a 2D pattern shows translational symmetries along two nonorthogonal directions, a UC of parallelogramatic shape as shown in Fig. 6.4 (Li et al., 2009) can be formulated. Its side lengths are equal to the distances of translations associated with the symmetries. Unlike in the previous case, where coordinates of the nodes on the opposite faces/edges were related in a straightforward way, if translations are in directions different from the coordinate axis, or when sides of the UC are not orthogonal to coordinate axes, the relationship between the coordinates of the respective points becomes more complex. For instance, for parallelogramatic UC as considered here, the coordinates of corresponding points on opposite sides of the UC are related as follows. x' ¼ x þ ma cos a þ nb sin b

(6.37)

y' ¼ y þ ma sin a þ nb cos b

where a and b are the side lengths and a and b are the inclines of the sides to the x- and y-axes, respectively, as shown in Fig. 6.4; m ¼ 1 and n ¼ 0 for the sides inclined at a to the x-axis and m ¼ 0 and n ¼ 1 for the sides inclined at b to the y-axis. However, practically, it is not always necessary to explicitly define such a relationship, as at the FE implementation stage the correspondence of the points is ensured by employing identical tessellations on the opposite faces of the UC. Henceforth, for UCs of more complex shapes, the exact coordinates of the points will not be specified explicitly as the subscripts in mathematical expressions of boundary conditions. Instead, numerical or symbolic y a

‹4›

‹3›

C

β

A

B D ‹1›

α

b

‹2›

x

Fig. 6.4 Unit cell of parallelogramatic shape.

133

Formulation of unit cells

notations will be employed to denote different components of the boundary of the UC. The relative displacement boundary conditions on the sides (excluding corners) are ujA  ujB ¼ ε0x a cos a vjA  vjB ¼ g0xy a cos a þ ε0y a sin a



corresponding to translation

Dx Dy



 ¼

a cos a a sin a



(6.38)

ujC  ujD ¼ ε0x b sin b vjC  vjD ¼ g0xy b sin b þ ε0y b cos b  corresponding to translation

Dx Dy



 ¼

b sin b b cos b



(6.39)

whilst those for the corners are ujh2i  ujh1i ¼ ε0x a cos a vjh2i  vjh1i ¼ g0xy a cos a þ aε0y sin a     Dx a cos a corresponding to translation ¼ Dy a sin a ujh4i  ujh1i ¼ ε0x b sin b vjh4i  vjh1i ¼ g0xy b sin b þ ε0y b cos b     Dx b sin b corresponding to translation ¼ Dy b cos b

ujh3i  ujh1i ¼ ε0x ða cos a þ b sin bÞ vjh3i  vjh1i ¼ g0xy ða cos a þ b sin bÞ þ ε0y ða sin a þ b cos bÞ       Dx b sin b a cos a corresponding to translation ¼ þ Dy b cos b a sin a (6.40)

134

Representative Volume Elements and Unit Cells

The rigid body translations in this case are to be constrained at the corner h1i as follows: ujC1D ¼ vjC1D ¼ 0:

(6.41)

In a similar way to what has been presented in the previous section, a third dimension can be introduced to analyse generalised 2D problems by using a single layer of 3D elements in the out-of-plane direction. Alternatively, one can keep the problem in 2D by employing a generalised plane strain formulation, in which case out-of-plane shear behaviour is addressed in a separate analysis based on an analogy with heat conduction problem. In this respect, the additional actions, such as the treatment of the reference node for generalised plain strain elements and the companion heat conduction analogy for the out-of-plane deformation, should follow what have been presented in the previous subsection. 6.4.1.4 2D unit cells in presence of more than two translational symmetries The hexagonal unit cell (Li, 2001), Fig. 6.5, rectangular unit cell from staggered layout (Li et al., 2011), Fig. 6.6, and that for the fish scale pattern as shown in Fig. 4.1 all fall in the group where there exist more than two translational symmetries along different directions in the plane. Apparently, not every axis of translation can be aligned with a coordinate axis. As has been argued in Chapter 2, only two of the symmetries are independent and any additional ones can always be expressed as a combination of the two independent ones. In presence of more translational symmetries, there are a few considerations users have to make. (1) The first is how to tessellate the pattern. It would be fairly natural if the translations are orthogonal as was the case in the previous subsection. Even so, one might choose curved side to complicate the matter to an extent, at least the appearance. Whilst geometric tessellations could be a substantial mathematical subject in their own right, there are a few simple rules one can follow as will be demonstrated below. (2) If there is a natural tessellation, e.g. when the packing in Fig. 6.5(a) was naturally cascaded into hexagonal patches through manufacturing or purpose of application, one can follow the choice of the nature. Another example is the fish scale tiling as cited in Chapter 2. In cases like this,

135

Formulation of unit cells

Fig. 6.5 (a) Hexagonal packing and (b) a hexagonal unit cell.

y

ξ

η (x0,y0)

x

Fig. 6.6 Unit cell for a staggered layout with the directions of translations specified.

(3)

(4)

(5) (6)

what the user needs to do is to partition the boundary in relation to the translational symmetries as present in the pattern. This is usually straightforward enough and one cannot be misguided by his/her intuition. In the case of the hexagon, the translations relevant to the boundary of the UC are in x-, y-, x- and h-directions, understanding that the one in the x-direction by a distance of two side lengths is not useful and one of the remaining three can be reproduced by the other two. Voronoi tessellation is an effective approach to tessellate the pattern using the available features in the pattern, e.g. the centres of fibre crosssections, as the seeds. With exceptions, a UC can be obtained from the Voronoi cells generated and it is certainly the case if all the cells turn out to be identical. The hexagonal unit cell can be obtained this way. Voronoi cells can also be generated with some of the seeds omitted systematically. For instance, in the hexagonal packing, if one omits every other row of the seeds, a rectangular UC as shown in Fig. 5.3 can be obtained. The price for that is that the partitioning of the internal features of the UC may not necessarily be the most favourable one for the subsequent meshing. Once a UC has been generated, it can be shifted in any direction by any distance without spoiling its validity, although its internal partition may vary. Equally true, the profile of the sides can be altered as one wishes, provided that the opposite sides related through the associated translational

136

Representative Volume Elements and Unit Cells

symmetry are altered in exactly the same way. Because of this, the sides of a UC do not have to be straight. On the other hand, without any special reason, there will be no need to employ curved sides. (7) One can also distort an obtained Voronoi cell by relocating the corners to obtain a UC of a different shape, e.g. diamond in the case of hexagonal packing, in which case two opposite side shrink to two points, provided that the alteration has been done systematically without compromising the translational symmetries. All above result in UCs of different appearances. Therefore, the lack of uniqueness is a generic feature of UCs. Serious users are advised to focus on the considerations behind the variety of appearances rather than the appearances themselves, and the unchanging rule of constructing UCs using translational symmetries. If so, the variety is merely a superficial fact whilst the underlying principle remains rigid and universally applicable. There is also a decision the user has to make in terms of the selection of the independent translations. However, as long as the selected translations are independent, although the intermediate derivations will be different, the way of obtaining boundary conditions will not be affected, neither the mathematics nor the physics associated with the problem. Take the hexagonal UC as an example to start with. As shown in Fig. 6.5(b), there are four translational symmetries available, in x-direction by two pffiffiffi side lengths, i.e. 2b, b being the side length, in y-, x- and h-directions by 3b, respectively. There are three different pairs of segments of the boundary, each of which is associated with one of the translations. Expressed in terms of relative displacement, the boundary conditions between any pair of segments can be given universally as in (6.11). All one needs to do is to identify the translation associated with each pair of segments of the boundary before substituting it into (6.11). Between the segments related by the translation in the y-axis, one has     Dx 0 p ffiffi ffi ¼ (6.42) Dy 3b leading to relative displacement boundary conditions for the pair of segments ujy¼pffiffi3b 2  ujy¼pffiffi3b 2 ¼ 0 =

=

vjy¼pffiffi3b 2  vjy¼pffiffi3b 2 ¼

pffiffiffi 0 3bεy :

(6.43)

=

=

Between the segments of the boundary related by the translation along the x-axis (Fig. 6.5(b)), one has

137

Formulation of unit cells



 ¼

Dy

3b 2 pffiffiffi 3b 2

 (6.44)

=

Dx

=



giving =

=

=

3bg0xy 2 þ

=

2

 vjx¼pffiffi3b

2

¼

=

=

vjx¼pffiffi3b

2

pffiffiffi 0 3bεy 2:

(6.45)

=

ujx¼pffiffi3b 2  ujx¼pffiffi3b 2 ¼ 3bε0x

Between the sides related by the translation along the h-axis (Fig. 6.5(b)), one has

Dy



 ¼

3b 2 pffiffiffi  3b 2

 (6.46)

=

Dx

=



resulting in =

2

=

vjh¼pffiffi3b  vjh¼pffiffi3b ¼ 3bg0xy

2

pffiffiffi 0 3bεy 2:

(6.47)

=

ujh¼pffiffi3b  ujh¼pffiffi3b ¼ 3bε0x

Given the available translational symmetries, the 6 corners can be divided into two groups h1,3,5i and h2,4,6i. None of the corners will be translated from one group to another according the available symmetries. Since each group consists of three vertices, only two sets of boundary conditions will be independent. The sets are formed by relating the two vertices in each group to the remaining one through the respective translations, the reasons for which are the same as those given in Subsection 6.4.1.2. Choosing corner h1i from the first and h4i from the second group as the vertices which are related to the other two in the respective groups through translations, the relative displacement boundary conditions become uC3D  uC1D ¼ vC3D  vC1D ¼

3bε0x 2 3bg0xy 2

þ

pffiffiffi 0 3bεy 2

138

Representative Volume Elements and Unit Cells

uC5D  uC1D ¼ 0 pffiffiffi vC5D  vC1D ¼ 3bε0y

(6.48)

uC2D  uC4D ¼ 0 pffiffiffi vC2D  vC4D ¼ 3bε0y uC6D þ uC4D ¼ vC6D þ vC4D ¼

3bε0x 2 3bg0xy 2

þ

pffiffiffi 0 3bεy 2

:

Additionally, the displacements at the corner h1i are constrained as ujh1i ¼ vjh1i ¼ 0

(6.49)

to eliminate rigid body translations. Note that since the dofs at corner h1i and those at corner h4i appear in more than one set of the equations, neither of them is specified as the first term in any of the equations. They have not been eliminated and hence are available for constraints to eliminate rigid body translations. Other considerations, such as the generalised plane strain condition and the analogy for the out-of-plane deformation, have been given in Section 6.4.1.2. They can be applied to the hexagonal UC in a straightforward manner and hence will not be repeated here. Same as for the square UC, the most popular application of hexagonal unit cells is for the analysis of UD composites with fibres arranged or idealised in a hexagonal packing, which preserves the transverse isotropy perfectly as far as the elastic behaviour is concerned. It is also the closest packing one can obtain and the maximum fibre volume fraction this packing allows is 90.69%, although this is not practically achievable. Similar to the square UC, the hexagonal one can also be extended to its 3D form, if a third dimension is added by employing 3D brick elements as shown in Fig. 6.7 with a single layer of elements being used in fibre direction, sufficiency of which has already been explained earlier in the chapter. The relative displacement boundary conditions are

139

Formulation of unit cells

ujy¼pffiffi3b 2  ujy¼pffiffi3b 2 ¼ 0 =

pffiffiffi 0 3bεy pffiffiffi 0 3bgyz 2 2þ

2þ

pffiffiffi 0 3bgyz

2

=

=

=

=

2

=

=

=

=

=

=

=

wjh¼pffiffi3b 2  wjh¼pffiffi3b 2 ¼ 3bg0xz

pffiffiffi 0 3bεy

2

=

=

vjh¼pffiffi3b 2  vjh¼pffiffi3b 2 ¼ 3bg0xy

2 2

pffiffiffi 0 3bgyz

=

=

ujh¼pffiffi3b 2  ujh¼pffiffi3b 2 ¼ 3bε0x

(6.50)

2

=

=

wjx¼pffiffi3b 2  wjx¼pffiffi3b 2 ¼ 3bg0xz

pffiffiffi 0 3bεy

=

=

vjx¼pffiffi3b 2  vjx¼pffiffi3b 2 ¼ 3bg0xy

=

=

ujx¼pffiffi3b 2  ujx¼pffiffi3b 2 ¼ 3bε0x

=

=

wjy¼pffiffi3b 2  wjy¼pffiffi3b 2 ¼

=

=

vjy¼pffiffi3b 2  vjy¼pffiffi3b 2 ¼

2

=

=

ujz¼t  ujz¼0 ¼ 0 vjz¼t  vjz¼0 ¼ 0 wjz¼t  wjz¼0 ¼ tε0z :

t 2b

x

y z

Fig. 6.7 3D unit cell for UD composites of hexagonally packed fibres.

140

Representative Volume Elements and Unit Cells

To avoid redundant boundary conditions along edges and at vertices, one can put the eighteen edges into five groups: (1) The three edges parallel to the z-axis passing corners h1,3,5i as in the 2D case, respectively, denoted as edge I, III and V. Their translations can be described by two sets of independent relations. Relating the displacement at edge I to those at the remaining two, according to the procedure defined in Subsection 6.4.1.2, the following two independent sets of boundary conditions are obtained: 9 8 9 8 3b 2 > = > = < Dx > < pffiffiffi > for Dy ¼ 3b 2 > > > ; > ; : : Dz 0

=

2þ

2

pffiffiffi 0 3bgyz

=

=

=

wjIII  wjI ¼ 3bg0xz

=

pffiffiffi 0 3bεy

2þ

=

vjIII  vjI ¼ 3bg0xy

2

=

ujIII  ujI ¼ 3bε0x

2

ujV  ujI ¼ 0 pffiffiffi vjV  vjI ¼ 3bε0y pffiffiffi wjV  wjI ¼ 3bg0yz

8 9 8 9 < Dx = < p0ffiffiffi = for Dy ¼ 3b : ; : ; Dz 0

(6.51) (2) The three edges parallel to the z-axis passing corners h2,4,6i as in the 2D case, respectively, denoted as edge II, IV and VI. The respective sets of boundary conditions in this case are ujIV  ujII ¼ 0 pffiffiffi vjIV  vjII ¼ 3bε0y pffiffiffi wjIV  wjII ¼ 3bg0yz

8 9 8 9 < Dx = < p0ffiffiffi = for Dy ¼ 3b : ; : ; Dz 0

ujVI þ ujII ¼ 3bε0x

9 8 9 8 3b 2 > Dx > > > < = < pffiffiffi = for Dy ¼  3b 2 > > > ; : ; > : Dz 0

=

2

pffiffiffi 0 3bgyz

=

=

2

=

=

wjVI þ wjII ¼ 3bg0xz

pffiffiffi 0 3bεy

=

2

=

vjVI þ vjII ¼ 3bg0xy

2

2

(6.52) (3) A group of four edges, two in the front and two at the back, related through the translation in the y-direction. Independent boundary

141

Formulation of unit cells

conditions for this group are defined by three sets of equations, where translations of three of the edges are related to one, e.g. ujy¼pffiffi3b 2;z¼0  ujy¼pffiffi3b 2;z¼0 ¼ 0

8 9 8 9 > < Dx > = > < p0ffiffiffi > = for Dy ¼ 3b > > > : ; > : ; Dz 0

=

=

vjy¼pffiffi3b 2;z¼0  vjy¼pffiffi3b 2;z¼0 ¼

pffiffiffi 0 3bεy

=

=

wjy¼pffiffi3b 2;z¼0  wjy¼pffiffi3b 2;z¼0 ¼

pffiffiffi 0 3bgyz

=

=

ujy¼pffiffi3b 2;z¼t  ujy¼pffiffi3b 2;z¼0 ¼ 0

8 9 8 9 Dx 0 > > > < = < pffiffiffi > = pffiffiffi 0 ffiffi ffiffi p p vjy¼ 3b 2;z¼t  vjy¼ 3b 2;z¼0 ¼ 3bεy for Dy ¼ 3b > > > > : ; : ; pffiffiffi 0 Dz t 0 ffiffi ffiffi p p wjy¼ 3b 2;z¼t  wjy¼ 3b 2;z¼0 ¼ 3bgyz þ tεz =

=

=

=

=

=

ujy¼pffiffi3b 2;z¼t  ujy¼pffiffi3b 2;z¼0 ¼ 0

8 9 8 9 > < Dx > = >

= for Dy ¼ 0 > > : ; > : > ; Dz t

=

=

vjy¼pffiffi3b 2;z¼t  vjy¼pffiffi3b 2;z¼0 ¼ 0 =

=

wjy¼pffiffi3b 2;z¼t  wjy¼pffiffi3b 2;z¼0 ¼ tε0z =

=

(6.53) (4) A group of four edges, two in the front and two at the back, related through the translation in the x-direction and the 3 independent relationships can be chosen as =

2

pffiffiffi 0 3bgyz 2 9 8 9 8 3b 2 > = < Dx > = > < pffiffiffi > 3b 2 for Dy ¼ > > > ; : ; > : Dz 0 =

=

2þ

=

=

=

=

wjx¼pffiffi3b 2;z¼0  wjx¼pffiffi3b 2;z¼0 ¼ 3bg0xz

pffiffiffi 0 3bεy

=

2þ

=

=

vjx¼pffiffi3b 2;z¼0  vjx¼pffiffi3b 2;z¼0 ¼ 3bg0xy

2

=

ujx¼pffiffi3b 2;z¼0  ujx¼pffiffi3b 2;z¼0 ¼ 3bε0x

=

=

142

Representative Volume Elements and Unit Cells

ujx¼pffiffi3b 2;z¼t  ujx¼pffiffi3b 2;z¼0 ¼ 3bε0x 2 =

2þ

=

2

pffiffiffi 0 3bgyz 2 þ tε0z 9 8 9 8 3b 2 > = < Dx > = > < pffiffiffi > for Dy ¼ 3b 2 > > > ; : ; > : Dz t

2þ

=

=

=

=

=

wjx¼pffiffi3b 2;z¼t  wjx¼pffiffi3b 2;z¼0 ¼ 3bg0xz

pffiffiffi 0 3bεy

=

=

vjx¼pffiffi3b 2;z¼t  vjx¼pffiffi3b 2;z¼0 ¼ 3bg0xy =

=

=

=

ujx¼pffiffi3b 2;z¼t  ujx¼pffiffi3b 2;z¼0 ¼ 0

8 9 8 9 Dx > > >0> > > < = > < > = for Dy ¼ 0 > > > > > > : ; > : > ; Dz t

=

=

vjx¼pffiffi3b 2;z¼t  vjx¼pffiffi3b 2;z¼0 ¼ 0 =

=

wjx¼pffiffi3b 2;z¼t  wjx¼pffiffi3b 2;z¼0 ¼ tε0z

(6.54)

=

=

(5) A group of four edges related through the translation in the h-direction give the following three independent relationships =

2

pffiffiffi 0 3bgyz 2 8 9 8 3b 2 > < Dx > = > < pffiffiffi for Dy ¼  3b > > : ; > : Dz 0

9 > =

=

2

=

=

=

=

=

wjh¼pffiffi3b 2;z¼0  wjh¼pffiffi3b 2;z¼0 ¼ 3bg0xz

pffiffiffi 0 3bεy

=

2

=

=

vjh¼pffiffi3b 2;z¼0  vjh¼pffiffi3b 2;z¼0 ¼ 3bg0xy

2

=

ujh¼pffiffi3b 2;z¼0  ujh¼pffiffi3b 2;z¼0 ¼ 3bε0x

2

> ;

=

=

143

Formulation of unit cells

=

2 2

2

pffiffiffi 0 3bgyz 2 þ tε0z 8 9 8 3b 2 > < Dx > = > < pffiffiffi for Dy ¼  3b > > : ; > : Dz t

2

=

=

=

=

wjh¼pffiffi3b 2;z¼t  wjh¼pffiffi3b 2;z¼0 ¼ 3bg0xz

pffiffiffi 0 3bεy

=

=

=

vjh¼pffiffi3b 2;z¼t  vjh¼pffiffi3b 2;z¼0 ¼ 3bg0xy

=

ujh¼pffiffi3b 2;z¼t  ujh¼pffiffi3b 2;z¼0 ¼ 3bε0x

9 > =

=

=

=

9 ( ) 8 < Dx = 0 for Dy ¼ 0 ; : t Dz

=

=

vjh¼pffiffi3b 2;z¼t  vjh¼pffiffi3b 2;z¼0 ¼ 0 =

=

wjh¼pffiffi3b 2;z¼t  wjh¼pffiffi3b 2;z¼0 ¼ tε0z

=

ujh¼pffiffi3b 2;z¼t  ujh¼pffiffi3b 2;z¼0 ¼ 0

2

> ;

(6.55)

=

=

There are 12 vertices, at the ends of edges I w VI. They can be put in two groups, 6 associated with edge I, III, and V and 6 with II, IV and VI. For the similar considerations, there are 5 sets of independent relationships in each group. For the first group 8 9 8 9 < Dx = < 0 = for Dy ¼ 0 : ; : ; Dz t

ujI;z¼t  ujI;z¼0 ¼ 0 vjI;z¼t  vjI;z¼0 ¼ 0 wjI;z¼t  wjI;z¼0 ¼ tε0x

9 9 8 8 < p3bffiffiffi 2 > = < Dx = > for Dy ¼ 3b 2 ; > : : 0 > ; Dz

=

2þ

2

pffiffiffi 0 3bgyz

=

=

=

wjIII;z¼0  wjI;z¼0 ¼ 3bg0xz

pffiffiffi 0 3bεy

=

2þ

=

vjIII;z¼0  vjI;z¼0 ¼ 3bg0xy

2

=

ujIII;z¼0  ujI;z¼0 ¼ 3bε0x

2

144

=

2þ

=

wjIII;z¼t  wjI;z¼0 ¼ 3bg0xz

pffiffiffi 0 3bεy

2

pffiffiffi 0 3bgyz

2 þ tεz

=

2þ

=

vjIII;z¼t  vjI;z¼0 ¼ 3bg0xy

2

=

ujIII;z¼t  ujI;z¼0 ¼ 3bε0x

Representative Volume Elements and Unit Cells

0

9 8 9 8 3b 2 > = < Dx > = > < pffiffiffi > for Dy ¼ 3b 2 > > > ; : ; > : Dz t =

=

ujV ;z¼0  ujI;z¼0 ¼ 0 pffiffiffi vjV ;z¼0  vjI;z¼0 ¼ 3bε0y pffiffiffi wjV ;z¼0  wjI;z¼0 ¼ 3bg0yz

8 9 8 9 < Dx = < p0ffiffiffi = for Dy ¼ 3b : ; : ; Dz 0

ujV ;z¼t  ujI;z¼0 ¼ 0 8 9 8 9 < Dx = < p0ffiffiffi = pffiffiffi 0 vjV ;z¼t  vjI;z¼0 ¼ 3bεy for Dy ¼ 3b : ; : ; pffiffiffi 0 Dz t 0 wjV ;z¼t  wjI;z¼0 ¼ 3bgyz þ tεz and for the second group ujII;z¼t  ujII;z¼0 ¼ 0 vjII;z¼t  vjII;z¼0 ¼ 0 wjII;z¼t  wjII;z¼0 ¼ tε0x ujIV ;z¼0  ujII;z¼0 ¼ 0 pffiffiffi vjIV ;z¼0  vjII;z¼0 ¼ 3bε0y pffiffiffi wjIV ;z¼0  wjII;z¼0 ¼ 3bg0yz

8 9 8 9 < Dx = < 0 = for Dy ¼ 0 : ; : ; Dz t 8 9 8 9 < Dx = < p0ffiffiffi = for Dy ¼ 3b : ; : ; Dz 0

(6.56)

145

Formulation of unit cells

ujIV ;z¼t  ujII;z¼0 ¼ 0 pffiffiffi vjIV ;z¼t  vjII;z¼0 ¼ 3bε0y pffiffiffi wjIV ;z¼t  wjII;z¼0 ¼ 3bg0yz þ tε0z

8 9 8 9 < Dx = < p0ffiffiffi = for Dy ¼ 3b : ; : ; Dz t

ujVI;z¼0 þ ujII;z¼0 ¼ 3bε0x 2 =

pffiffiffi 0 3bεy 2 =

=

wjVI;z¼0 þ wjII;z¼0 ¼ 3bg0xz 2 

pffiffiffi 0 3bgyz 2 =

vjVI;z¼0 þ vjII;z¼0 ¼ 3bg0xy 2 

8 9 8 3bffiffiffi2 < Dx = > < p for Dy ¼  3b : ; > : Dz 0

9 > =

=

=

=

2

> ;

ujVI;z¼t þ ujII;z¼0 ¼ 3bε0x 2 =

pffiffiffi 0 3bεy 2 pffiffiffi 0 3bgyz 2 þ tε0z =

=

wjVI;z¼t þ wjII;z¼0 ¼ 3bg0xz 2 

=

vjVI;z¼t þ vjII;z¼0 ¼ 3bg0xy 2 

9 8 9 8 > > 2 3b Dx = < = < pffiffiffi for Dy ¼  3b 2:5 > : ; > ; : Dz t

=

=

=

(6.57) The rigid body translations are eliminated by setting ujI;z¼0 ¼ vjI;z¼0 ¼ wjI;z¼0 ¼ 0

(6.58)

As was stated previously, the correct imposition of the above equation constraints on the faces, edges and vertices requires that edges and vertices are excluded from faces and vertices excluded from the edges. For the three examples as cited at the beginning of this subsection, namely, the hexagonal UC, rectangular UC from a staggered layout and

146

Representative Volume Elements and Unit Cells

ξ

y

η

‹3›

‹2›

‹1›

e

h

c

a 44

‹4›

‹5›

x ‹6›

b Fig. 6.8 The unit cell from the staggered pattern.

the fish scale pattern, although their appearances are rather different, topologically they are identical, all being 6-sided polygons, if top and bottom sides the rectangular UC from a staggered layout are split into two each because the top and bottom sides are associated with two different translational symmetries. The same formulation can be adapted for the staggered pattern as shown in Fig. 6.6 if the unit cell is defined as in Fig. 6.8. The relevant translations are modified respectively to (

( ) b The x  direction : ¼ 0 by a distance of b; (6.59) 0 ( ) ( ) Dx a pffiffiffiffiffiffiffiffiffiffiffiffiffiffi The x  direction : Dy ¼ h by a distance of c ¼ h2 þ a2 Dz 0 (6.60) Dx Dy Dz

)

147

Formulation of unit cells

(

) ( ) Dx ab The h  direction : Dy ¼ h by a distance of e; Dz 0 9 8 9 8 < Dx = < 0 = The z  direction : Dy ¼ 0 ; : ; : Dz t

(6.61)

(6.62)

by a distance of t; as required for its 3D extension. With these translations and their appropriate combinations, Eqs. (6.51) e(6.57) can be systematically reproduced for this particular UC, without duplicating the presentation and justifications.

6.4.2 3D unit cells 6.4.2.1 Introduction The UCs presented in previous subsections are 2D in nature as the mathematical problems they aim to solve are 2D, although a third dimension may be incorporated to facilitate the associated analysis, given the shortage of generalisation of 2D problems in existing commercial FE codes. As a result, only one layer of elements is required in the third dimension due to the absence of variation of stress and strain fields involved in these problems. There are a wide range of practical problems which cannot be simplified to 2D reasonably and they have to be dealt with in a 3D space. Applications cover lattice structures, porous materials and composites with certain types of reinforcements, such as particulates and textile preforms. In some cases, e.g. lattice structures and textile preforms, the regular packing represents a reasonable approximation if manufacturing variability can be neglected, whilst in other cases, it is based on statistic equivalence, e.g. for particulate reinforced composites and porous materials. There is always a need to characterise them theoretically so that they can be designed to deliver desired properties. In classifying various packing arrangements, the study on crystals, e.g. (Nye, 1985), is found to be of great assistance. Many packing systems can be interpreted from crystal structures in a straightforward manner, such as simple cubic (SC), face centred cubic (FCC), body centred cubic (BCC) and close packed hexagonal (CPH) packing. In these idealised packing systems, there exist a range of geometric symmetries, in particular, translational ones. Having established a systematic account on 2D UCs as presented

148

Representative Volume Elements and Unit Cells

earlier in this chapter, UCs of a 3D nature will be the subject of this subsection. For particulate-reinforced composites or porous materials, isotropy is one of the statistical properties in general resulting from the random distribution of the reinforcing particles or pores in the matrix. Regrettably, none of the available crystal structures of regular packing shows their macroscopic isotropy naturally. In fact, packed layout as an idealisation should be employed with caution if the isotropy is an important property to preserve. With isotropic spherical particulate reinforcements in isotropic matrix, it was found in (Li and Wongsto, 2004) that FCC, BCC and CPH packing systems could be taken as isotropic if one can put up with minor discrepancies, whilst SC packing exhibits significant anisotropy. All three cubic packing systems, namely, SC, FCC and BCC, remain cubic symmetric where there are only 3 independent elastic properties as opposed to 2 for isotropic materials, whilst CPH is transversely isotropic requiring 5 independent elastic properties to characterise. There have been some attempts in the literature in recent years to take account of effects of the randomly distributed particulates in matrix, but usually subjected to restrictive assumptions which undermine to an extent the objective of taking account of the randomness of particulate distribution. The way of incorporating the randomness of distribution is elaborated in Chapter 9 where formulation of RVEs is addressed. As was discussed multiple times previously in this book, the shape of the unit cell for a given system is not unique. In the discussion to follow, 3D Voronoi tessellations (Ahuja and Schachter, 1983) will be used to tessellate the packing systems involved, which can be generated from a set of seeds, such as the centres of distributed particles. A Voronoi cell for a seed can be described as an envelope generated from all bisecting planes perpendicular to the segments connecting this seed to all other surrounding seeds. Taking particulate reinforced composites as an example and assuming that the particulates are spheres of equal size, the advantage of the Voronoi cell is that each cell contains a single and complete particulate. Some of the cells obtained in this way can be developed into the relevant UCs although some cannot as will be discussed later. Even so, the tessellations will be helpful to understand the structures of the packing systems concerned.

149

Formulation of unit cells

6.4.2.2 3D unit cell with translational symmetries along three noncoplanar axes For materials of translational symmetries along three non-coplanar axes, the simplest shape for the unit cell is a parallelepiped with its side lengths corresponding to the distances of translations associated with the symmetries. A special case is a cuboidal unit cell (Li and Wongsto, 2004) as shown in Fig. 6.9 which will be taken as an example to illustrate the relative displacement boundary conditions. For 3D unit cell, in order to avoid prescribing redundant conditions, the faces (excluding edges), edges (excluding vertices) and vertices need to be addressed separately, as was argued previously. The three translations along the coordinate axes are given as follows. 8 9 8 9 > < Dx > = > < 2a > = The x  direction : Dy ¼ 0 by a distance of 2a; (6.63) > > : ; > : > ; Dz 0 9 8 9 8 > = > < Dx > =

The y  direction : Dy ¼ 2b by a distance of 2b; (6.64) > > ; > : ; : > Dz 0 8 9 8 9 > < Dx > = >

= The z  direction : Dy ¼ 0 by a distance of 2c. (6.65) > > : ; > : > ; Dz 2c For faces x ¼ a and x ¼ a (excluding edges)

z

2c

y

O x

2a

2b Fig. 6.9 Cuboidal unit cell.

150

Representative Volume Elements and Unit Cells

ujða;y;zÞ  ujða;y;zÞ ¼ 2aε0x vjða;y;zÞ  vjða;y;zÞ ¼ 2ag0xy

(6.66)

wjða;y;zÞ  wjða;y;zÞ ¼ 2ag0xz For faces y ¼ b and y ¼ b (excluding edges) ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 2bε0y

(6.67)

wjðx;b;zÞ  wjðx;b;zÞ ¼ 2bg0yz For faces z ¼ c and z ¼ c (excluding edges) ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

(6.68)

wjðx;y;cÞ  wjðx;y;cÞ ¼ 2cε0z For edges parallel to the x-axis (excluding vertices) ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 vjðx;b;cÞ  vjðx;b;cÞ ¼ 2bε0y wjðx;b;cÞ  wjðx;b;cÞ ¼ 2bg0yz ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 vjðx;b;cÞ  vjðx;b;cÞ ¼ 2bε0y

(6.69)

wjðx;b;cÞ  wjðx;b;cÞ ¼ 2bg0yz þ 2cε0z ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 vjyðx;b;cÞ  vjðx;b;cÞ ¼ 0 wjðx;b;cÞ  wjðx;b;cÞ ¼ 2cε0z For edges parallel to the y-axis (excluding vertices), after eliminating any redundant conditions

Formulation of unit cells

151

ujða;y;cÞ  ujða;y;cÞ ¼ 2aε0x vjða;y;cÞ  vjða;y;cÞ ¼ 2ag0xy wjða;y;cÞ  wjða;y;cÞ ¼ 2ag0xz ujða;y;cÞ  ujða;y;cÞ ¼ 2aε0x vjða;y;cÞ  vjða;y;cÞ ¼ 2ag0xy

(6.70)

wjða;y;cÞ  wjða;y;cÞ ¼ 2ag0xz þ 2cε0z ujða;y;cÞ  ujða;y;cÞ ¼ 0 vjða;y;cÞ  vjða;y;cÞ ¼ 0 wjða;y;cÞ  wjða;y;cÞ ¼ 2cε0z For edges parallel to the z-axis (excluding vertices), after eliminating any redundant conditions ujða;b;zÞ  ujða;b;zÞ ¼ 2aε0x vjða;b;zÞ  vjða;b;zÞ ¼ 2ag0xy wjða;b;zÞ  wjða;b;zÞ ¼ 2ag0xz ujða;b;zÞ  ujða;b;zÞ ¼ 2aε0x vjða;b;zÞ  vjða;b;zÞ ¼ 2ag0xy þ 2bε0y wjða;b;zÞ  wjða;b;zÞ ¼ 2ag0xz þ 2bg0yz ujða;b;zÞ  ujða;b;zÞ ¼ 0 vjða;b;zÞ  vjða;b;zÞ ¼ 2bε0y wjða;b;zÞ  wjða;b;zÞ ¼ 2bg0yz For the vertices, after eliminating any redundant conditions, one has

(6.71)

152

Representative Volume Elements and Unit Cells

ujða;b;cÞ  ujða;b;cÞ ¼ 0 vjða;b;cÞ  vjða;b;cÞ ¼ 0 wjða;b;cÞ  wjða;b;cÞ ¼ 2cε0z ujða;b;cÞ  ujða;b;cÞ ¼ 2aε0x vjða;b;cÞ  vjða;b;cÞ ¼ 2ag0xy wjða;b;cÞ  wjða;b;cÞ ¼ 2ag0xz ujða;b;cÞ  ujða;b;cÞ ¼ 2aε0x

(6.72)

vjða;b;cÞ  vjða;b;cÞ ¼ 2ag0xy þ 2bε0y wjða;b;cÞ  wjða;b;cÞ ¼ 2ag0xz þ 2bg0yz ujða;b;cÞ  ujða;b;cÞ ¼ 0 vjða;b;cÞ  vjða;b;cÞ ¼ 2bε0y wjða;b;cÞ  wjða;b;cÞ ¼ 2bg0yz ujða;b;cÞ  ujða;b;cÞ ¼ 2aε0x vjða;b;cÞ  vjða;b;cÞ ¼ 2ag0xy wjða;b;cÞ  wjða;b;cÞ ¼ 2ag0xz þ 2cε0z ujða;b;cÞ  ujða;b;cÞ ¼ 0 vjða;b;cÞ  vjða;b;cÞ ¼ 2bε0y wjða;b;cÞ  wjða;b;cÞ ¼ 2bg0yz þ 2cε0z ujða;b;cÞ  ujða;b;cÞ ¼ 2aε0x vjða;b;cÞ  vjða;b;cÞ ¼ 2ag0xy þ 2bε0y wjða;b;cÞ  wjða;b;cÞ ¼ 2ag0xz þ 2bg0yz þ 2cε0z Rigid body translations are eliminated by applying the constraints the vertex (a, b, c) as ujða;b;cÞ ¼ vjða;b;cÞ ¼ wjða;b;cÞ ¼ 0

(6.73)

Formulation of unit cells

153

It is a reasonably straightforward exercise to extend the formulation of boundary conditions obtained for a cuboidal UC to a parallelepiped UC as shown in Fig. 6.10, for which three translational symmetries are noncoplanar but non-orthogonal either. The extension will be similar to that from the formulation of the rectangular UC in Section 6.4.1.2 to the formulation of parallelogramatic UC in Section 6.4.1.3. The expressions of the relative displacement boundary conditions will become slightly more complicated but without fundamental difference. 6.4.2.3 3D unit cell for SC packing The cuboidal UC as presented above is by large the most popular UC employed in various micromechanical analyses for material characterisation and architectural design. A special case of the cuboidal UC is the so-called simple cubic (SC) packing, which is a typical type of crystal structures as illustrated in Fig. 6.10, where spheres have been employed to help with the visualisation. With the idealised packing system shown in Fig. 6.10, the maximum volume fraction of the spheres as reinforcements is 52.36%. The Voronoi cell for the SC packing is a right cube which is a natural choice of UC for materials of such an architecture. The boundary conditions for the cubic UC can be simply obtained by assigning a common side length in all three dimensions to the cuboidal unit cell. Nevertheless, the boundary conditions for SC UC will be reproduced here by taking a more systematic approach in terms of filtering out the redundancies and description of topology. This is to facilitate the subsequent derivation of boundary conditions for FCC and BCC cases which have substantially more complex geometry. Before deriving boundary conditions for UCs of complex geometry, it is essential to correctly determine the number of the faces, edges and the vertices, respectively, in the polyhedron forming the UC boundary. To

Fig. 6.10 A parallelepiped unit cell for the triclinic architecture.

154

Representative Volume Elements and Unit Cells

verify that the respective numbers were counted properly, Euler’s polyhedron formula (Euler, 1758, Richeson, 2008) can be of use: Faces þ Vertices ¼ Edges þ 2. (6.74) Note that the Voronoi cells are always convex polyhedra, hence Eq. (6.74) is always applicable for verification of the geometry of the UCs. In particular, it is easy to check that it is satisfied for the UC corresponding to SC packing, which has 3 pairs of faces, 12 edges and 8 vertices. Readers are reminded that an edge is the intersection of two faces and a vertex is the intersection of at least three faces. As has been explained previously, because an edge is shared by two faces and a vertex by at least three faces, boundary conditions as prescribed to faces are duplicated at these locations to give rise to redundancies. Filtering out the redundant ones tends to require a significant effort, especially for the FCC and BCC UCs. The coordinate system employed is shown in Fig. 6.11(b), as well as the exposed faces that are labelled with the capital letters. The faces that are not shown are paired with exposed ones, i.e. face A (not shown) is paired with B (labelled), C (not shown) with D (labelled), and E (labelled) with F (not shown). The translations employed in the definition of the UC are from A to B, C to D, and E to F, respectively. These are three independent translations present in the packing system, along x, y and z, respectively, all by a distance of 2b which is a diameter of the spheres as shown on Fig. 6.11(a). The translation in the SC packing system from one Voronoi cell to another can be generally expressed as follows. 8 9 8 9 Dx 2bi > > > > < = < = Dy ¼ 2bj (6.75) > > > : ; > : ; Dz 2bk

Fig. 6.11 Unit cell for SC packing: (a) packing visualisation, (b) Voronoi cell as the unit cell with faces labelled and dimensions specified and (c) labels for the edges and vertices.

155

Formulation of unit cells

where i, j and k are the number of cells translated in the x-, y- and z-directions, respectively. Any two cells are related through a set of integers i, j and k. Apparently, face A translates to B by i ¼ 1, j ¼ k ¼ 0, C to D by i ¼ 0, j ¼ 1, k ¼ 0 and E to F by i ¼ j ¼ 0, k ¼ 1. As a result, the translation between these pairs of faces can be given as 8 9 8 9 8 8 9 8 9 9 8 9 > > > > > < Dx > = < 2b > = >

=

= < Dx > = < Dx > = Dy ¼ 0 ; Dy ¼ 2b and Dy ¼ 0 > > > > > > > > : ; : > ; > : > ; : > ; : ; : ; Dz AB 0 0 2b Dz CD Dz EF (6.76) Substituting them into the relative displacement field (6.11), one obtains the relative displacement boundary conditions as 8 9 8 9 8 2bε0 9 x > > > >

= =

= < 0 v  v ¼ 2bgxy denoted as UB  UA ¼ FAB (6.77a) > > > > : > ; ; : > ; : w B w A 2bg0xz corresponding to the translation from face A to face B and similarly 8 9 8 9 8 0 9 > > > >

=

= < = 0 2bε v  v ¼ denoted as UD  UC ¼ FCD ; y > > > > : > ; : > ; : ; w D w C 2bg0yz (6.77b) 8 9 9 8 9 8 0 > u u > > > > > < = = < = < 0 v denoted as UF  UE ¼ FEF :  v ¼ > > > > > : > ; ; : ; : w F w E 2bε0z

(6.77c)

Note that the expressions of FAB, FCD and FEF are of slightly different forms from those in (Li and Wongsto, 2004) because of the different expression of the relative displacement field (6.11) where the upper triangle of the coefficient matrix is set to zero as opposed to the lower triangle as in (Li and Wongsto, 2004). The preference for employing lower triangular form has been already explained in Subsection 6.4.1.2. Also, the typos in the labelling of the faces in (Li and Wongsto, 2004) have been corrected, with the right labelling being used in Fig. 6.11(b). The numbering of edges and vertices in SC packing UC is shown in Fig. 6.11(c). Four edges parallel to each coordinate axes form a group in

156

Representative Volume Elements and Unit Cells

which the edges can be related through three sets of independent translations. For example, consider edges parallel to z-axis with edge I being the independent one. The remaining three edges can be obtained through appropriate translations of edge I, by setting i ¼ 1, j ¼ k ¼ 0 to reproduce edge II, i ¼ j ¼ 1, k ¼ 0 to recover edge III and i ¼ 0, j ¼ 1, k ¼ 0 for edge IV. Substituting these integers into (6.75) before employing expression of the relative displacement field (6.11), one obtains a set of independent boundary conditions for this set of edges as follows. UII  UI ¼ FAB UIII  UI ¼ FAB þ FCD

(6.78a)

UIV  UI ¼ FCD By the similar argument, the independent boundary conditions for the set of edges parallel to the y-axis and for those to the x-axis can be obtained respectively as UVI  UV ¼ FEF UVII  UV ¼ FEF þ FAB

(6.78b)

UVIII  UV ¼ FAB UX  UIX ¼ FCD UXI  UIX ¼ FCD þ FEF

(6.78c)

UXII  UIX ¼ FEF In much the same way, there is only one independent vertex, whilst the remaining ones can be obtained by appropriate translations. Choosing vertex 1 to be an independent one, the translations for the vertices should be defined as: Vertex 2 by i ¼ 1; j ¼ k ¼ 0; giving U2  U1 ¼ FAB Vertex 3 by i ¼ j ¼ 1; k ¼ 0; giving U3  U1 ¼ FAB þ FCD Vertex 4 by i ¼ 0; j ¼ 1; k ¼ 0; giving U4  U1 ¼ FCD Vertex 5 by i ¼ 0; j ¼ 0; k ¼ 1; giving U5  U1 ¼ FEF

(6.79)

Vertex 6 by i ¼ 1; j ¼ 0; k ¼ 1; giving U6  U1 ¼ FAB þ FEF Vertex 7 by i ¼ j ¼ k ¼ 1 giving U7  U1 ¼ FAB þ FCD þ FEF and Vertex 8 by i ¼ 0; j ¼ 1; k ¼ 0 giving U8  U1 ¼ FCD þ FEF :

Formulation of unit cells

157

By now, 6 faces, 12 edges and 8 vertices, that were all counted correctly as can be verified by Euler’s polyhedron formula (6.74), have been assigned appropriate boundary conditions without any redundancy. Constraints employed to eliminate rigid body translations in this case are u1 ¼ v1 ¼ w1 ¼ 0; denoted as U1 ¼ 0 (6.80) The above should have reproduced the results from the previous section if one sets all side lengths to 2b, but the format of presentation has been modified so that it can be followed in subsequent subsections. 6.4.2.4 3D unit cell for FCC packing A face centred cubic packing system is visualised in Fig. 6.12 where views from different perspectives are shown. It is a closely packed system which offers the maximum particulate volume fraction of 74.05%. The Voronoi cell for FCC packing can be obtained as shown in Fig. 6.13(a). It has 6 pairs of faces, 24 edges and 14 vertices, which satisfies Euler’s polyhedron formula (6.74). The coordinate systems employed are shown in Fig. 6.13(b), where the labels on the exposed faces are also specified. The six faces that are not shown are paired with the exposed ones parallel to them following the same rule in terms of notations as was employed for SC packing UC. In coordinate system x-y-z, the equations for the faces can be given as follows. pffiffiffi x þ y ¼  2b ð‘ þ ’ for face B and ‘  ’ for face AÞ pffiffiffi x  y ¼  2b ð‘ þ ’ for face D and ‘  ’ for face CÞ pffiffiffi y þ z ¼  2b ð‘ þ ’ for face F and ‘  ’ for face EÞ pffiffiffi y þ z ¼  2b ð‘ þ ’ for face H and ‘  ’ for face GÞ (6.81) pffiffiffi x þ z ¼  2b ð‘ þ ’ for face N and ‘  ’ for face MÞ pffiffiffi x  z ¼  2b ð‘ þ ’ for face L and ‘  ’ for face KÞ

Fig. 6.12 An FCC packing viewed from different perspectives.

158

Representative Volume Elements and Unit Cells

Fig. 6.13 The Voronoi cell as a unit cell for the FCC packing: (a) the Voronoi cell (rhombic dodecahedron); (b) the coordinate system and face labelling for the unit cell.

Note that letters I and J were not used when labelling the surfaces to avoid confusion with notations of edge labels. There are three independent translations present in the packing system, all by a distance of 2b (diameter of the spheres in Fig. 6.12) along the respective axes x, h and 2 indicated in Fig. 6.13(b) as opposed to the translations being along the coordinate axes in the SC case. Since axes x, h and 2 are all in the directions of translational symmetry, given the UC is a Voronoi cell, they are perpendicular to their respective faces, x to the A and B, h to G and H and 2 to E and F. The translation in the FCC packing system from one Voronoi cell to another can be generally expressed in the x-h-2 coordinate system, which is unfortunately non-orthogonal, as follows. 8 9 8 9 > > < Dx > =

= Dh ¼ 2b j (6.82) > > > : ; : > ; D2 k where i, j and k are the number of cells translated along x-, h- and 2-directions, respectively. Any two cells are related through a set of integers i, j and k. In order to make practical sense out of these translations, one has to transform these non-orthogonal coordinate system into the conventional xy-z coordinate system. It is relatively straightforward linear transformation defined as 9 9 8 38 2 Dx > 1 0 0 > > = < = < Dx > 1 6 7 (6.83) Dy ¼ pffiffiffi 4 1 1 1 5 Dh > > > > 2 ; : ; : D2 0 1 1 Dz

159

Formulation of unit cells

The 6 pairs of faces are related through a set of integers i, j and k each as follows. B translates to A by i ¼ 1; j ¼ k ¼ 0; D to C by i ¼ 1; j ¼ k ¼ 1; F to E by i ¼ j ¼ 0; k ¼ 1; H to G by i ¼ 0; j ¼ 1; k ¼ 0;

(6.84)

N to N by i ¼ 1; j ¼ 1; k ¼ 0; L to K by i ¼ 1; j ¼ 0; k ¼ 1 Substituting these combinations into (6.82) and then to (6.83) produces the expression for translations in the x-y-z coordinate system, substituting which to (6.11) yields the relative displacement boundary conditions for the faces as follows. 9 8 0 8 9 8 9 > > ε > > x > > > >

= =

= pffiffiffi < 0 0 v  v ¼ 2b gxy þ εy denoted as UB  UA ¼ FAB > > > > : > ; > : > ; > > w B w A ; : g0 þ g0 > xz yz (6.85a) 8 8 9 8 9 > ε0x > > > >

=
> > > > : ; > : ; > w C w D : g0  g0 xz

yz

9 > > > = > > > ;

denoted as UD  UC ¼ FCD

(6.85b) 9 8 8 9 8 9 > > 0 > > > > > > =

=
> > > > : > ; > : > ; > > w E w F : g0 þ ε0 ; yz z (6.85c)

160

Representative Volume Elements and Unit Cells

9 8 8 9 8 9 > > 0 > > > > > >

= =

= pffiffiffi < 0 ε v  v ¼ 2b denoted as UH  UG ¼ FGH y > > > > : > ; > : > ; > > > w H w G : g0  ε 0 ; yz z 8 9 8 9 > >

=

= v  v > > : > ; : > ; w N w M

8 > > > pffiffiffi < ¼ 2b > > > :

(6.85d) 9 > ε0x > > = 0 gxy denoted as UN  UM ¼ FMN > > > g0xz þ ε0z ; (6.85e)

9 8 8 9 8 9 0 > > ε > > x u u > > > > < > = = < > = pffiffiffi < 0 g v  v ¼ 2b denoted as UL  UK ¼ FKL xy > > > > : > ; > : > ; > > > w L w K : g0  ε0 ; xz

z

(6.85f) Given the translational symmetries available, any edge can reproduce two others, putting three of them in a group. There are 8 groups in total and they are mutually independent. Within each group, the dofs on two of the edges can be eliminated as follows. UXXIII þ UI ¼ FKL &  UXVIII þ UI ¼ FAB

(6.86a)

UXVII þ UII ¼ FAB &  UXX þ UII ¼ FMN

(6.86b)

UXXII þ UIII ¼ FCD &  UXIX þ UIII ¼ FMN

(6.86c)

UXXIV þ UIV ¼ FKL &  UXXI þ UIV ¼ FCD

(6.86d)

UXV  UV ¼ FKL & UX  UV ¼ FAB

(6.86e)

UIX  UVI ¼ FAB & &UXII  UVI ¼ FMN

(6.86f)

UXIV  UVII ¼ FCD & UXI  UVII ¼ FMN

(6.86g)

UXVI  UVIII ¼ FKL & UXIII  UVIII ¼ FCD

(6.86h)

where the notations of the edges are specified in Fig. 6.14(a).

Formulation of unit cells

161

There are two types of vertices involved in the UC, one as the intersections of three faces and the other of four faces. They are naturally grouped according to their types. First group includes vertices 1e6 and the second contains vertices 7e12, as specified in Fig.14(b), and the two groups are mutually independent. Within each group, the relationships among the vertices are obvious in the boundary conditions presented below. U2 þ U1 ¼ FAB þ FCD

(6.87a)

U3 þ U1 ¼ FCD

(6.87b)

U4 þ U1 ¼ FKL

(6.87c)

U5 þ U1 ¼ FAB

(6.87d)

U6 þ U1 ¼ FMN

(6.87e)

within the second group, there are two mutually independent subgroups, one associated with vertex 7 and the other with 8. Thus U12 þ U7 ¼ FMN

(6.87f)

U13 þ U7 ¼ FAB

(6.87g)

U14 þ U7 ¼ FEF

(6.87h)

U9  U8 ¼ FMN

(6.87i)

U10  U8 ¼ FAB

(6.87j)

Fig. 6.14 Unit cell for FCC packing: (a) labelling of edges (b) labelling for vertices.

162

Representative Volume Elements and Unit Cells

U11  U8 ¼ FEF (6.87k) The rigid body translations are eliminated by constraining vertex 1 as follows: u1 ¼ v1 ¼ w1 ¼ 0 denoted as U1 ¼ 0 (6.88) This concludes the derivation of boundary conditions for the FCC UC. However, the actual implementation of an FE analysis of such a UC is still a substantial challenge and meshing alone can take a significant amount of efforts. In this respect, a hint from the authors’ experience might be of help. The UC can be partitioned into four geometrically identical parts, one of which is shown in Fig. 6.15(a). It is an equilateral parallelepiped, i.e. its faces are all of identical geometry and edges are of equal length, hence it is relatively easy to mesh. The top view of the part as shown in Fig. 6.15(a) covers an angular section of 120 and three of them form a polyhedron as shown in Fig. 6.15(b). The fourth part sits on the top of the assembly of the other three, as shown in Fig. 6.15(c) when viewed from the side in the direction as indicated, to recover the complete UC. If the internal structure has the

Fig. 6.15 Construction of FCC UC: (a) one of the four identical parts after suggested partition of the FCC UC; (b) assembly of three identical parts; (c) completing the FCC UC by adding the fourth identical part.

Formulation of unit cells

163

symmetry with respect to this partition, the three internal faces exposed by the partition can be tessellated identically to ensure seamless assembly after meshing the part. One only needs to mesh one of the four parts as the remaining three can be copied before being rotated and translated to position to form the complete UC sharing the common nodes on the interfaces. 6.4.2.5 3D unit cell for body centred cubic packing (BCC) A body centred cubic packing system is depicted in Fig. 6.16(a). It is apparently not as closely packed as FCC counterpart but more packed than the SC one. The maximum particulate volume fraction achievable with this packing is 68.02%. The Voronoi cell is shown in Fig. 6.16(b) It has 7 pairs of faces, 36 edges and 20 vertices, which satisfies (6.74). There are 4 independent pffiffiffi translational symmetries in x-, y-, and z-directions by a distance of 4b 3 and in r-direction by 2b, where axis r is at equal angle to x-, y- and z-axes. The translation in the BCC packing system from one Voronoi cell to another can be generally expressed as a combination of those in the x-, y-, z- and r-directions as 8 pffiffiffi 9 8 9 dx > 2i= 3 > > > > > > > > > = < 2j=pffiffi3ffi > < dy > = pffiffiffi ¼ 2b (6.89) > > > > dz 2k= 3 > > > > > > > ; > : : ; dr l where i, j, k and l are the number of cells translated along x-, y-, z- and rdirections, respectively. Any two cells are related through a set of integers i, j,

Fig. 6.16 (a) BCC packing and (b) The Voronoi cell (tetrakaidecahedron) corresponding to BCC packing with faces and dimensions labelled.

164

Representative Volume Elements and Unit Cells

k and l. As the space is three-dimensional, r can be expressed in terms of x, y and z, giving the actual translation in terms of coordinate increments as follows. 8 9 8 9 3> dx > 2 > > > < > = < Dx > = 1 2 0 0 1 > 7 dy 6 Dy ¼ 4 0 2 0 1 5 (6.90) > > > > dz > > : ; 2 > ; > Dz 0 0 2 1 : dr The 7 pairs of faces are related through a set of integers i, j, k and l each as follows. A translates to B by i ¼ 1; j ¼ k ¼ l ¼ 0; C to D by i ¼ 0; j ¼ 1; k ¼ l ¼ 0; E to F by i ¼ j ¼ 0; k ¼ 1; l ¼ 0; G to H by i ¼ j ¼ k ¼ 0; l ¼ 1;

(6.91)

P to Q by i ¼ 0; j ¼ 1; k ¼ 0; l ¼ 1; K to L by i ¼ 0; j ¼ k ¼ 1; l ¼ 1; M to N by i ¼ j ¼ 0; k ¼ 1; l ¼ 1 Substituting these combinations into (6.89), and after using (6.11), one obtains the relative displacement boundary conditions for the faces as follows: 8 9 8 9 8 9 0 > > ε > > x > > > > =

= 4b < 0 = v  v ¼ pffiffiffi gxy denoted as UB  UA ¼ FAB > > > 3> > : > ; > : > ; > 0 ; > w B w A : gxz (6.92a) 8 8 9 8 9 > 0 > > > >

=

= < 4b 0  v ¼ pffiffiffi εy v > > 3> : > ; : > ; > > w C w D : g0

yz

9 > > > = > > > ;

denoted as UD  UC ¼ FCD

(6.92b)

165

Formulation of unit cells

8 9 8 9 8 9 > u u > > > > > >0> < = < = 4b < = v  v ¼ pffiffiffi 0 denoted as UF  UE ¼ FEF > > > > 3> : > ; > : ; > ; : w F w E ε0 z

(6.92c) 9 8 0 8 9 8 9 > > ε > > x > > > >

= =

= < 2b 0 0 g þ ε v  v ¼ pffiffiffi xy y > > > > 3> : > ; > : ; > > w H w G ; : g0 þ g0 þ ε0 > xz yz z (6.92d) denoted as UH  UG ¼ FGH 8 8 9 8 9 > > > > >

=

= 2b < v  v ¼ pffiffiffi > > 3> : > ; : > ; > > w w : Q

P

ε0x g0xy  ε0y

9 > > > =

> > > g0xz  g0yz þ ε0z ; (6.92e)

denoted as UQ  UO ¼ FPQ 9 8 0 8 9 8 9 > > ε > > x > > > > =

=

= < 2b 0 0 gxy  εy  v ¼ pffiffiffi v > > > 3> > : > ; : > ; > > w K w L ; : g0  g0  ε0 > xz yz z (6.92f) denoted as UL  UK ¼ FKL 8 8 9 8 9 > > > > >

=

= 2b < v  v ¼ pffiffiffi > > 3> > : > ; : > ; > w w : N

M

ε0x g0xy þ ε0y

9 > > > =

> > > g0xz þ g0yz  ε0z ; (6.92g)

denoted as UN  UM ¼ FMN

166

Representative Volume Elements and Unit Cells

With the relative displacement boundary conditions for the faces of the UC established above, one can proceed with the edges and vertices. Since edges and vertices are also parts of faces, the boundary conditions for them can always be derived from those for the faces, as has been demonstrated in previous subsections. Without duplicating similar arguments, these boundary conditions are simply provided below, from which it should be reasonably straightforward for readers to reveal the detailed relationships amongst edges and vertices based on translational symmetries. The underlying principle is that the boundary conditions for edges and vertices should be complete yet without any redudancy. Readers are encouraged to fill the gap as a test of their competence in dealing with this type of problems. Bear in mind that the presentation of these boundary conditions is not unique. If one ended up with some which look different from those provided below, it would not necessarily be that they were wrong. Correct outcomes are ensured by correct logic and derivation, not their appearances, at least for this class of problems. With reference to the edge labelling as shown in Fig. 6.17(a), the boundary conditions for edges can be obtained as: UXIII þ UI ¼ FAB UXXXV þ UI ¼ FGH UXIV þ UII ¼ FAB UXXXVI þ UII ¼ FPQ UXV þ UIII ¼ FAB UXXXIII þ UIII ¼ FKL

(6.93a)

(6.93b)

(6.93c)

Fig. 6.17 The unit cell for BCC packing (a) edges labelled and (b) vertices labelled.

Formulation of unit cells

UXVI þ UIV ¼ FAB UXXXIV þ UIV ¼ FMN UXVII þ UV ¼ FCD UXXXII þ UV ¼ FGH UXVIII þ UVI ¼ FCD UXXX þ UVI ¼ FMN UXIX þ UVII ¼ FCD UXXVI  UVII ¼ FPQ UXX þ UVIII ¼ FCD UXXVIII  UVIII ¼ FKL UXXI þ UIX ¼ FEF UXXXI þ UIX ¼ FGH UXXII þ UX ¼ FEF UXXVII  UX ¼ FKL UXXIII þ UXI ¼ FEF UXXV  UXI ¼ FMN UXXIV þ UXII ¼ FEF UXXIX þ UXII ¼ FPQ

167

(6.93d)

(6.93e)

(6.93f)

(6.93g)

(6.93h)

(6.93i)

(6.93j)

(6.93k)

(6.93l)

Accordingly, the boundary condition for vertices are: U11 þ U1 ¼ FMN U14 þ U1 ¼ FGH

(6.94a)

U21 þ U1 ¼ FAB U13 þ U2 ¼ FGH U16 þ U2 ¼ FPQ U22 þ U2 ¼ FAB

(6.94b)

U15 þ U3 ¼ FPQ U10 þ U3 ¼ FKL U23 þ U3 ¼ FAB

(6.94c)

168

Representative Volume Elements and Unit Cells

U9 þ U4 ¼ FKL U12 þ U4 ¼ FMN

(6.94d)

U24 þ U4 ¼ FAB U7 þ U5 ¼ FCD U18 þ U5 ¼ FMN

(6.94e)

U20 þ U5 ¼ FGH U8 þ U6 ¼ FEF U17 þ U6 ¼ FPQ

(6.94f)

U19 þ U6 ¼ FGH Rigid body translations are constrained by (6.95) u1 ¼ v1 ¼ w1 ¼ 0 denoted as U1 ¼ 0 With the boundary conditions formulated above, the analysis can proceed in theory. However, this UC is apparently more sophisticated topologically than any other UCs that have been presented in this book. One of the features is that there are 8 faces of a hexagonal shape. Most of the available meshing tools do not allow such a polygon and one has to partition them appropriately first before these faces can be tessellated. A particular care needs to be exercised regarding the identical tessellations for paired faces, especially when tetrahedral elements are employed, as then triangles will feature in the surface tessellation. Opposite faces are usually not visible simultaneously, hence when the UC rotated by 180 to view from the opposite side, seemingly identical tessellations could be exactly wrong way around since the opposite sides are supposed to match by translation, not rotation, as will be elaborated in Section 6.5. 6.4.2.6 3D unit cell for close packed hexagonal packing (CPH) Close packed hexagonal packing illustrated in Fig. 6.18(a) is a closely packed system. Its packing density is in fact identical to that of the FCC packing. However, when viewed from a specific perspective, one in every three layers of spheres employed to illustrate the packing system is offset by one space as compared with the FCC packing. As a result, the Voronoi cell shown in Fig. 6.18(b) looks slightly different from that of the FCC, although they are of the same volume. It shares the same number of edges and vertices with that of FCC but the topology is slightly different. The diamond-shaped faces do not have opposite faces which are translationally symmetric to them.

Formulation of unit cells

169

Fig. 6.18 (a) CPH packing and (b) The Voronoi cell for CPH packing.

Because of this very feature, the Voronoi cell is not a UC for CPH packing in the context of the present chapter. Apparently, the Voronoi cell cannot fill up the space simply by translations. It can be fitted into the gaps but only after a 180 rotation. Additional symmetries, such as reflections and rotations, which will be the subject of Chapter 8, can be used in the construction of such UCs. A compromise is to use a hexagonal prism as the simplest geometry one can obtain for this packing. Referring to Fig. 6.19(b), if the six faces parallel to the x-axis are extended in the x-direction, they form a hexagonal cylindrical surface which intersects the three particulates immediately above the one in the Voronoi cell concerned and three immediately below it. If the hexagonal cylinder is truncated by two planes perpendicular to the x-axis through the centres of the three particulates immediately pffiffiffi above and below, respectively, a hexagonal prism of a height of 4b 6 forms. It contains a complete particulate plus six fragments at six of its vertices, one from each of the three particulates immediately above and below, respectively. Each fragment is of 1/6 the volume of the particulate if the geometry of the particulate is assumed to be sufficiently symmetric. The total volume of the prism is twice that of the Voronoi cell. The prism can fill up the space through translations in the yz-plane and along the x-axis and therefore offers as a perfect UC for this packing system. Having achieved the above understanding and alternative, the formulation of the UC can be waived as it can be readily adapted from that in Section 6.4.1.4. The only difference is that one will need substantial meshing in the axial direction of the prism, instead of a single layer of elements, as the variation in this direction will be an important characteristic in the present context.

170

Representative Volume Elements and Unit Cells

(a)

(b)

Fig. 6.19 (a) Unit cells for CPH packing system; (b) FCC equivalence.

In this particular case, the boundary conditions will be identical to those as given in Section 6.4.1.4 if the same coordinate system is adopted, although many elements will have to be employed in the x-direction in this case in general. 6.4.2.7 A unit cell for laminated composites Laminates are by far the most popular form of applications of composites in engineering. They are widely used as a replacement of sheet metal, for instance, as aircraft skin. Often, they have been treated as if they were a “material” in design instead of a structure if one acknowledges the layup construction of the laminate. For design and manufacturing practices, it is essential to have the effective properties of such “materials”, such as the elastic constants and thermal expansion characteristics. A conventional way of obtaining them is through the use of the so-called classic laminate theory (CLT) to evaluate the effective stiffness matrices, commonly denoted as A, B and D (Jones, 1998). It should be stated that the elements of these stiffness matrices are not the conventional elastic constants themselves, although they are closely related. The relationship can be described briefly as follows. One needs to find the inverse of the laminate stiffness matrix to obtain the compliance matrix first. Then the inverse of the elements of the compliance matrix will lead to effective elastic constants according to the definitions of these material properties. It is clear that the manipulation is slightly beyond what one can efficiently handle manually. A small piece of computer code will be essential using whatever platform, Matlab, Excel, etc. Although this is meant to be a simple code, it is not always available to most users directly, unless one purchases specialised codes for composites analysis, which are a lot cheaper, but still not trivial in cost, than structural FE codes, such as Abaqus. Given the fact that most designers should have been equipped with an FE code, it is definitely handy to build such a facility into the available FE platform.

Formulation of unit cells

171

With the justification above, the easiest way to fulfil the task would be through a UC. This is the subject of this subsection. If a laminate is characterised as a material, e.g. as a sheet metal, one needs to ensure a macroscopically uniaxial or pure shear stress state. Experimentally, it is usually requires a certain treatment since the “material” is heterogeneous and anisotropic in addition to restrictions brought in by the physical loading mechanisms. For testing under uniaxial stress, the specimen will have to allow sufficient length for the end effects at loading section to decay before reaching the testing section within the gauge length. Sideways, the interfaces between laminae are of different elastic characteristics due to different fibre orientations on either side of it, which can result in the so-called free edge effects. Because of this, one has to incorporate sufficient breadth of the specimen to dilute the free edge effects. These experimental challenges can be easily avoided in a virtual testing. A block of material of the full thickness of the laminate can be taken from a laminate under the desired macroscopic stress state well within the gauge length in the sense of a free body diagram. If the relative displacement boundary conditions are prescribed to the block, it will be free from the end effects and the free edge effects. It can be perfectly representative of a block taken from the gauge length of the specimen under a physical test. It is therefore an ideal UC for the laminate. In the part of a laminates away from edges, laminar stresses remain constant in the plane of the lamina. Their distribution in the thickness direction within each laminar is either constant or up to a linear variation in the case of bending when the flexural stiffnesses are to be evaluated. Linear brick elements can capture such variation perfectly. Therefore, the UC can be constructed with a single column of linear brick elements, one element per lamina. The dofs involved in the model are very limited and the actual analysis can be as quick as if one uses other means, e.g. Matlab or Excel, provided that the problem has been formulated and coded properly. In the authors’ implementation, as will be introduced briefly in Chapter 14, a GUI interface has been created, through which the application of the UC has been made effortless. The formulation of this UC is mostly the implementation of relative displacement boundary conditions, as the architecture is trivial. Unlike other UCs presented in this chapter, where the strain field in the upper length scale is uniform, for laminate applications, a constant gradient in the thickness direction should be allowed to reflect the effects of bending. With bending incorporated, the UC will be able to deliver the flexural stiffness as well as the in-plane effective elastic constants. Under the condition of uniform distribution of generalised strains at the length scale of the laminate, the

172

Representative Volume Elements and Unit Cells

displacement field can be expressed in terms of generalised strains for laminates as (Li et al., 1994) 1 u ¼ xε0x þ xzkx þ yzcxy 2 1 (6.96) v ¼ xg0xy þ yε0y þ yzky þ xzcxy 2 1 1 1 w ¼  x2 kx  y2 ky  xycxy 2 2 2

   where ε0x ; ε0y ; g0xy are in-plane strains and kx ; ky ; cxy are curvatures associated with bending deformation of the laminate. Apparently, expressions (6.96) are only unique up to the rigid body motions and hence a different scheme of constraining rigid body motions will result in a slightly different presentation. On the other hand, generalised strains at the laminate length scale can be defined through the relative displacements between reference points P0 in a cell and P0’ in a different cell. Such definition can be employed to express relative displacements between reference points P0 and P0’ in terms of generalised strain field which is assumed to be uniform.      1 u00  u0 ¼ x00  x0 ε0x þ x00 z00  x0 z0 kx þ y00 z00  y0 z0 cxy 2       v00  v0 ¼ x00  x0 g0xy þ y00  y0 ε0y þ y00 z00  y0 z0 ky  1 0 0 x0 z0  x0 z0 cxy 2    1 2 1 2 1 w00  w0 ¼  x0 0  x20 kx  y0 0  y20 ky  x00 y00  x0 y0 cxy 2 2 2 þ

(6.97) The relative displacements between a point P in a cell and a selected reference point P0 in the same cell are periodic from cell to cell throughout, just like the relative coordinates, i.e. u  u0 ¼ u0  u00 v  v0 ¼ v0  v00 w  w0 ¼ w 0  w00

x  x0 ¼ x0  x00 and

y  y0 ¼ y0  y00

z  z0 ¼ z

0

 z00

(6.98)

Formulation of unit cells

173

From (6.97) and (6.98) the relative displacements between corresponding points in different cell can be obtained in terms of macroscopic strains as follows 1 u0  u ¼ ðx0  xÞε0x þ ðx0  xÞzkx þ ðy0  yÞzcxy 2   1 v 0  v ¼ x00  x0 g0xy þ ðy0  yÞε0y þ ðy0  yÞzky þ ðx0  xÞzcxy 2 1 w 0  w ¼ xðx0  xÞkx  yðy0  yÞky  ðyðx0  xÞ þ xðy0  yÞÞcxy 2 (6.99) 1 1 1 where x ¼ ðx0 þ xÞ; y ¼ ðy0 þ yÞ and z ¼ ðz þ z0 Þ ¼ z ¼ z0 2 2 2 (6.100) If the centre of the UC is taken as the origin of the coordinate system, then x¼y ¼ 0 (6.101) Setting the in-plane side lengths to be 2a in both x- and y-directions, the relative displacement boundary conditions for the UC can be obtained as ujða;y;zÞ  ujða;y;zÞ ¼ 2aε0x þ 2azkx vjða;y;zÞ  vjða;y;zÞ ¼ 2ag0xy þ azcxy for faces perpendicular to the x-axis wjða;y;zÞ  wjða;y;zÞ ¼ 0 (6.102) ujðx;a;zÞ  ujðx;a;zÞ ¼ 2azkx þ azcxy vjðx;a;zÞ  vjðx;a;zÞ ¼ 2aε0y þ 2azky wjðx;a;zÞ  wjðx;a;zÞ ¼ 0

(6.103)

for faces perpendicular to the y-axis They involve the generalised strains for laminates which are in an equivalent position to the upper scale strains as in other UCs derived previously. These boundary conditions are to be imposed to the sides perpendicular to the

174

Representative Volume Elements and Unit Cells

x-axis and those perpendicular to the y-axis, respectively, whilst the top and bottom surfaces are left free as in the CLT analysis. 6.4.2.8 A unit cell from Cn rotational symmetry (Li et al., 2014) The Cn rotational symmetry can be best presented in a cylindrical coordinate system. However, most FE codes are formulated in a rectangular coordinate system. The symmetry conditions obtained in the former coordinate system have to be transformed to the latter before they can be imposed in an FE analysis. For displacements, the transformation between the two coordinate systems (Fig. 6.20(a)) is given as follows. ux ¼ u ur ¼ v cos q þ w sin q

(6.104)

uq ¼ v sin q þ w cos q where x, r and q are the coordinates in the longitudinal, radial and circumferential directions, respectively, with the displacements in both coordinate systems sketched in Fig. 6.20(a). The cylindrical and rectangular coordinate systems are so arranged that they share the common origin. It is assumed that the geometry of the structure possesses a periodic appearance in the longitudinal and circumferential directions. When it is subjected to macroscopically uniform loading, such as a temperature change, axial tension/compression and twist, or loading of the distribution with the same periodic characteristics as the geometry, the internal stress and strain fields are expected to be periodic, i.e. sij ðx þ b; r; q þ aÞ ¼ sij ðx  b; r; q  aÞ εij ðx þ b; r; q þ aÞ ¼ εij ðx  b; r; q  aÞ

(6.105)

Fig. 6.20 (a) Relationship between displacement components ux, ur and uq in cylindrical x-r-q coordinate system, and displacement components u, v and w in Cartesian (x-y-z) coordinate system; (b) UC formed treating the rotation about x-axis as a type of translation in circumferential direction.

Formulation of unit cells

175

where 2a is the period in the circumferential direction and 2b is the period in the longitudinal direction. A unit cell representing a full period in both the longitudinal and circumferential directions can be defined without loss of generality in the domain b  x  b 0  r  RðqÞ

(6.106)

a  q  a where RðqÞ defines the external profile of the domain in the circumferential direction. If it is a constant, the domain will be a right circular cylinder. In a cylindrical coordinate system, the kinematic equations associated with the deformation in a polar plane are as follows. vur vr ur 1 vuq εq ¼ þ r vq r 1 vur vuq uq grq ¼ þ þ r vq vr r

εr ¼

(6.107)

Apparently, with a strain field periodic in the circumferential direction, the in-plane displacements ur and uq will have to be of the same periodic characteristics, unlike in the situation in a rectangular coordinate system in which the displacement field corresponding to a periodic strain field is not periodic in general. Thus, the circumferential periodicity results in the following periodic displacement boundary conditions ux jq¼a ¼ ux jq¼a ur jq¼a ¼ ur jq¼a

(6.108)

uq jq¼a ¼ uq jq¼a In the circumferential direction, the paired parts of the boundary are q ¼ a with a ¼ p=n and the symmetry relating them is Cxn . The longitudinal coordinate is linear. The periodicity or the translational symmetry of the strain field in the longitudinal direction leads to the following relative displacement boundary conditions

176

Representative Volume Elements and Unit Cells

ux jx¼b  ux jx¼b ¼ 2bε0x ur jx¼b  ur jx¼b ¼ 0

(6.109)

uq jx¼b  uq jx¼b ¼ 2fr where f is the relative angle of twist about x-axis. The boundary conditions in Eqs. (6.108) and (6.109) must be imposed upon the UC before it provides a truthful representation of the complete structure. The radial and circumferential displacements must be transformed to the rectangular coordinate system for their applications in FE analysis. Temperature loading, axial loading and torsion can be applied, at the same time if desired. The symmetry involved in this subsection is apparently rotational in nature. To help to draw a similarity with the other UCs as presented in this chapter which all based on translational symmetries, one could imagine the Cn rotational symmetry as a kind of translational one along the curvilinear circumferential direction as marked by an arrow in Fig. 6.20(a) where the shape of the UC is as sketched in Fig. 20(b).

6.5 Requirements on meshing UCs are analysed using FEM after imposing the boundary conditions as derived in the previous section under various considerations, which is by large the most challenging part in implementing a UC. However, before the boundary conditions can be properly imposed, the UC to be analysed has to be meshed appropriately. Conventional requirements on meshing as in any FE analysis will have to be met for UCs as well. In addition, a crucial requirement for the correct implementation of the UCs is that the parts of the boundary paired through the associated translational symmetry are tessellated in exactly the same way so that if one of these parts translates to the position of the other according to the associated symmetry, there should be perfect match. This requirement will not necessarily be satisfied if the nodes on both sides are positioned identically, since identical node locations do not guarantee identical tessellation (Li and Wongsto, 2004). An illustration is shown in Fig. 6.21, where the top and bottom surface share the same tessellation, while the front and back faces do not, although the involved nodes match perfectly. Readers are reminded that symmetry conditions have been established based on the concept of free body diagrams, which implies the

Formulation of unit cells

177

Fig. 6.21 Compatibility of tessellations on opposite sides of a unit cell.

continuity between adjacent unit cells over the part of boundary they share. Continuity can only be maintained if the adjacent sides of the neighbouring unit cells are tessellated identically. This continuity requirement also appears in conventional meshing. When different parts that are meshed separately are joined together, the meshes on both sides have to be compatible. Simply sharing nodes may not be enough, especially if tetrahedral elements have been employed and the surface is tessellated into triangular areas. There can be a further implementation issue associated with the relative displacement boundary conditions which are always expressed in the form that displacements on a part of the boundary of the UC are related to those on another part of the boundary. They are often prescribed as equation boundary conditions. They need to be imposed in the way precisely in the format as required by the FE code, that is, node by node. In some FE codes, e.g. Abaqus/Standard, nodes are allowed to be grouped so that the equation boundary conditions can be imposed between groups. In this case, the user has to ensure that the correct one to one correspondence between the two groups as some FE codes, e.g. Abaqus/Standard again, have a default node number sorting scheme in ascending order, which would upset the correspondence if not manually suppressed.

6.6 Key degrees of freedom and average strains The effective, or average, strains in the upper scale involved in the formulation of relative displacement boundary conditions offer extremely

178

Representative Volume Elements and Unit Cells

useful handles for the subsequent analysis at the lower length scale and provide convenient links between the two scales involved. They are not any part of the unit cell directly, but are introduced into the lower scale analysis through the boundary conditions for the unit cell as extra degrees of freedom. These extra dofs are effective strains, or the generalised strains in the case of the UC for laminates, at the upper length scale, but involved in the boundary conditions for the analysis at the lower length scale. They are therefore the key links between the two scales involved in the problem and this is why they will be referred to as the key degrees of freedom (Kdofs). In terms of FE implementation of the boundary conditions, Kdofs can be introduced through a single node with all effective strains involved as its effective dofs. Alternatively, one can employ several nodes, each having one dof associated with an effective strain. They can be prescribed the values of desirable effective strains as prescribed “displacements”. Out of the analysis, reactions at Kdofs can be obtained. It will be shown in Chapter 7 that these “reaction forces” are associated with the effective, or average, stresses corresponding to the effective strains prescribed. Here, “displacements” and “reaction forces” are put inside quotation marks because they are not exactly displacement and forces, at least in terms of their dimensions. The nodal displacements at the Kdofs are strains and hence dimensionless and the reaction forces are of a dimension of forcelength. They are therefore displacements and reaction forces in a generalised sense. They are nevertheless appropriate energy conjugates. Alternatively, concentrated “forces” can be prescribed to Kdofs, which is equivalent to prescribing effective stresses. Out of the analysis, the nodal “displacements” can be obtained at Kdofs, thus giving the effective strains directly. Both approaches described above in terms of the use of the Kdofs have their advantages, depending on the outcomes required. If one is interested in the stiffness matrix of the material, it can be obtained by prescribing unit effective strains at Kdofs. The appropriate procedure that should be followed in order to obtain the stiffness matrix is described in detail in Section 7.5. However, if one is interested in the effective elastic constants, such as Young’s moduli, Poisson’s ratios and shear moduli, applying effective uniaxial stress or pure shear stress will help to obtain these material properties more directly. This method is elaborated in Section 7.6. It should be stressed here that the Kdofs behave in the same manner as ordinary dofs in many ways. If a value, zero or non-zero, of “displacement” is prescribed to any of them, it introduces an essential boundary condition which will be strictly observed in the analysis. However, if any of them is left free without prescribing a “displacement” or a “force”, intentionally

179

Formulation of unit cells

or unintentionally, it will be understood by the FE code as a natural boundary condition which will only be satisfied approximately in the sense of energy minimisation. Effectively, it is equivalent to prescribing a zero “force” at the dof. If one intends to find a column of the stiffness matrix, a unit effective strain needs to be prescribed at the corresponding Kdof, whilst the remaining Kdofs will have to be constrained, i.e. prescribed a zero “displacement”, in order to ensure a uniaxial effective strain state under which the stiffness is defined. If the remaining Kdofs are left free, it is effectively to prescribe a uniaxial effective stress state, with the loading being expressed in terms of applied strain. It is now clear that effective or average strains in the upper scale are either known a priori as prescribed values or obtained as the direct output from the FE analysis in terms of “nodal displacements”. Those familiar with FEM theory are aware that as “nodal displacements”, they are free from numerical errors introduced by any FE post-processing. Typically, stresses and strains obtained from FEM as a part of post-processing lose an order of accuracy as compared with nodal displacements. Obtaining effective straining as “nodal displacements” helps to preserve the highest order of numerical accuracy. This restates the fact that best solution is often the simplest solution!

6.7 Average stresses and effective material properties The average stresses over the unit cell obtained from the lower length scale analysis give the effective stresses in the upper length scale. Without the concept of the Kdofs, one would have to devise a procedure of averaging stresses over the unit cell. However, the details of stress averaging methods used are hardly ever reported in the literature. These averages should not be arithmetic averages of the stresses obtained at the integration points, as was elaborated in Section 5.4. A correct but by no means the best approach to calculate the effective stress should be 1 s0 ¼ V

All elements X e¼1

Z Ue

1 sdU ¼ V

All integration points X in element All elements X e¼1

Wi si jdet Jji

i¼1

(6.110) 0

where s is any of the stress components whilst s is its effective counterpart, V is the volume of the UC, Wi is the weight associated with the integration

180

Representative Volume Elements and Unit Cells

point and jdetJji is the determinant of the Jacobian matrix associated with the integration point. The evaluation of the Jacobian matrix requires the access to the coordinates of all nodes in the element. In terms of accuracy, the stresses obtained at integration points have suffered a loss already as it is an inevitable sacrifice to make when using FEM due to differentiation of the displacement field, which is the direct outcome of FEM, in order to obtain strains before stresses can be evaluated. Yet, integrating them again will result in further numerical approximation due to the numerical interpolation within each element involved in order to facilitate the numerical integration. Although the mathematical manipulations associated with (6.110) are not excessively complicated, without the access to the source code when using any commercial FE code the above poses a substantial post-processing challenge. It is certainly beyond the competence of any inexperienced user of FE. However, the description of post-processing procedure is very rarely provided in publications involving the use of unit cells, thus casting a thick cloud over the issue. Without the use of Kdofs, the effective strains would have to be obtained in the same manner, hence strain averaging procedure would have to suffer the same drawbacks. Use of Kdofs substantially simplifies the procedure of determining the effective strains, which are obtained as direct output from the FE analysis free from any post-processing numerical errors. Furthermore, one of the most important contributions of the Kdofs is that they offer an effortless way of obtaining effective stresses that eliminates any need for postprocessing. The concentrated forces at them, whether calculated as reactions to the prescribed effective strains or as the prescribed loads, are associated with the effective stresses as follows. Fx ¼ V s0x Fy ¼ V s0y Fz ¼ V s0z Fyz ¼ V s0yz

(6.111)

Fxz ¼ V s0xz Fxy ¼ V s0xy where V is the volume of the unit cell, or, for 2D unit cells, the area of the unit cell as previously defined. One might have doubts about the volume involved in the expression above. It is nevertheless correct, given the

181

Formulation of unit cells

generalised sense of these “forces” and their unconventional dimension of forcelength. If these concentrated forces are prescribed the values equal to the volume of the UC one after another in a series of analyses, it is equivalent to prescribing unit effective stresses, and the set of the effective strains obtained in each case defines a column of the compliance matrix of the material from which effective elastic properties can be evaluated easily. In fact, the inverse of the effective strain corresponding to the applied unit effective stress gives the effective Young’s modulus directly without need for any substantial post-processing. In general, the effective properties can be evaluated from concentrated “forces” and the “nodal displacements” at the respective Kdofs as follows. Consider Kdof associated with ε0x . The loading at this Kdof can be defined by either prescribing concentrated force Fx, in which case the value of ε0x is obtained as the nodal displacement at this Kdof, or by prescribing a value of ε0x , then Fx is to be obtained as the nodal reaction force at this Kdof. All other Kdofs must be left free from constraints to ensure a uniaxial or pure shear stress state. The relevant effective properties are then calculated as Ex0 ¼ s0x =ε0x ¼ Fx =V ε0x n0xy ¼  ε0y =ε0x

(6.112)

n0xz ¼  ε0z =ε0x whilst s0y ¼ s0z ¼ s0yz ¼ s0xz ¼ s0xy ¼ 0

or

Fy ¼ Fz ¼ Fyz ¼ Fzx ¼ Fxy ¼ 0 Applying the same reasoning when dealing with the remaining Kdofs one after another, one obtains Ey0 ¼ s0y =ε0y ¼ Fy =V ε0y n0yx ¼  ε0x =ε0y

(6.113)

n0yz ¼  ε0z =ε0y whilst

s0x ¼ s0z ¼ s0yz ¼ s0xz ¼ s0xy ¼ 0

or

Fx ¼ Fz ¼ Fyz ¼ Fzx ¼ Fxy ¼ 0; Ez0 ¼ s0z =ε0z ¼ Fz =V ε0z

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Representative Volume Elements and Unit Cells

n0zx ¼  ε0x =ε0z

(6.114)

n0zy ¼  ε0y =ε0z s0x ¼ s0y ¼ s0yz ¼ s0xz ¼ s0xy ¼ 0

whilst

or

0 Fx ¼ Fy ¼ Fyz ¼ Fzx ¼ Fxy ¼ 0; Gyz ¼ s0yz =g0yz ¼ Fyz =V g0yz

(6.115) whilst

s0x ¼ s0y ¼ s0z ¼ s0xz ¼ s0xy ¼ 0 Fx ¼ Fy ¼ Fz ¼ Fzx ¼ Fxy ¼

or

0 0; Gzx

whilst s0x ¼ s0y ¼ s0z ¼ s0yz ¼ s0xy ¼ 0

¼ s0zx =g0zx ¼ Fzx =V g0zx (6.116)

or

0 Fx ¼ Fy ¼ Fz ¼ Fyz ¼ Fxy ¼ 0; Gxy ¼ s0xy =g0xy ¼ Fxy =V g0xy

(6.117) whilst

s0x ¼ s0y ¼ s0z ¼ s0yz ¼ s0xz ¼ 0

or

Fx ¼ Fy ¼ Fz ¼ Fyz ¼ Fxz ¼ 0 Note that it is essential that each of the relations (6.112)e(6.117) are applied strictly under the conditions specified. By definition, Young’s moduli and Poisson’s ratios should be obtained under respective uniaxial stress state and shear modulus under pure shear stress state. Any violation of the conditions will inadvertently result in misinterpretation of the outcomes and hence of the obtained values of these material properties. The discussion and the associated analyses presented so far do not involve temperature loading, i.e. temperature was assumed to be constant. Temperature considerations will have to be incorporated if one is interested in evaluating the effective thermal expansion coefficients. This will be the subject of discussion in the next section.

6.8 Thermal expansion coefficients Thermal expansion coefficients are often of significance in many engineering applications, in particular, for composites reinforced with carbon fibres, since carbon fibres exhibit rather unconventional thermal expansion characteristics which tend to cause significant problems sometimes whilst

183

Formulation of unit cells

offering beneficial properties in some other applications. In order to evaluate effective thermal expansion coefficients, all one needs to do is to prescribe a temperature loading, i.e. a temperature change. The temperature change is usually defined with respect to a reference temperature. Thermal loading is a well-established module in most commercial FE codes. To facilitate such an analysis, users will have to input the thermal expansion coefficients of the material at the lower length scale, e.g. those of the constituent materials for a composite, along with necessary conventional elastic properties. Meshing and definition of boundary conditions should be exactly the same as in the analyses for obtaining effective elastic properties, and so should be the interpretation of the output at the Kdofs. The difference is in the type of applied loading. Namely, instead of a force or a displacement prescribed at a Kdof, a unit temperature change is prescribed whilst leaving all Kdofs free from any constraints. The effective strain at these Kdofs obtained as nodal displacements in response to the unit temperature change gives the effective thermal expansion coefficients directly. If the temperature change DT is prescribed a value different from unity, thermal expansion coefficients are determined according to a0x ¼ ε0x =DT ; a0y ¼ ε0y =DT ; a0z ¼ ε0z =DT

(6.118a)

a0yz ¼ g0yz =DT ; a0xz ¼ g0xz =DT ; a0xy ¼ g0xy =DT

(6.118b)

whilst

s0x ¼ s0y ¼ s0z ¼ s0yz ¼ s0xz ¼ s0xy ¼ 0

or

Fx ¼ Fy ¼ Fz ¼ Fyz ¼ Fzx ¼ Fxy ¼ 0 In general, thermal expansion coefficients form a symmetric tensor corresponding to the strain tensor. For anisotropic materials, coupling between direct and shear components is present. The coupling terms are rarely mentioned because orthotropic materials are usually dealt with in engineering. With the development of modern materials of complicated internal architectures, e.g. textile composites, the orthotropy cannot always be assumed. At least, one should have the ability to identify the presence of anisotropy, if any, before taking orthotropy for granted.

6.9 “Sanity checks” as basic verifications The procedure of formulating unit cells as presented in previous sections of this chapter is systematic. For any serious micromechanical material

184

Representative Volume Elements and Unit Cells

characterisation, it is the way forward. Being systematic, it can be relatively easily programmed into some fixed templates, but it could be tedious to implement. There are many places where mistakes could easily be made, either as oversights or as typographical errors. Ironing out such mistakes is not always a straightforward task, certainly not an automatic one. Whether the analysis is to be carried out manually or automatically, it is crucial that means are available to verify that all the measures have been implemented correctly. In this respect, unit cell users often tend to validate their results against the experimental results. However, the reality is that experimental data are always limited, and there are only limited aspects that can be examined and measured experimentally. Fitting to one aspect of the experiment can hardly be considered as validation, especially when there are obvious anomalies where the predicted results are against common sense in other aspects. As far as the development of the micromechanical material characterisation tools is concerned, one needs verifications more than the experimental validations and well before any serious experimental validation. As a rigorous procedure, rushing into experimental validation without extensive verification is not a scientific attitude. One should at the very least exhaust the range of “sanity checks” as will be described in detail below as basic verifications. If any unit cell formulated and implemented fails to pass any of the “sanity checks”, it is incorrect, no matter how well some of the results agree with experimental data. The first set of “sanity checks” can be set up as follows. A single set of material properties are assigned to all phases of the constituents involved in the UC, so that the unit cell would essentially represent a homogeneous material. The analyses are carried out under all loading conditions. In each case, perfectly uniform stress and strain fields should be obtained for each loading case. In terms of FE analysis output, all stress and strain contour plots should show as dull mono-coloured images, an example of which for hexagonal UC is shown in Fig. 6.22(a). Any multi-coloured fringes are a sign to alert unless the legend for the contour plot shows trivial variation range over a single value, as illustrated by the contour plot in Fig. 6.22(b), where such single value in the legend happens to be numerical zero. Otherwise, the unit cell has not been either formulated or implemented correctly. Typical erroneous stress concentrations are found around the faces, edges and vertices which signify the incorrect boundary conditions there in most cases. Having achieved uniform stress and strain field, a check on their values is simple but also essential, as they must correspond to the prescribed loading identically. The stresses and strains should be related according to the

Formulation of unit cells

185

Fig. 6.22 pical contour plots output from a “sanity check” analysis: (a) uniform stress field in over a hexagonal UC; (b) trivial variation of stress over the UC.

material properties as assumed for these analyses. One also needs to check the ratios between the strains, as they should coincide with the Poisson’s ratios as assigned to the material to facilitate the analyses. Finally, the data processing phase of the analyses should be conducted. The predicted effective properties must be identical to those assigned to the material to facilitate the analyses. Passing these “sanity checks” suggests that one must have eliminated at least 90% of the errors made in the implementation of a unit cell one way or another. In other words, it is fair to say that the endeavour to pass these checks may consume most of the efforts in implementing the UCs. Therefore, once the UC model have passed the checks, one gains substantial assurance in the correctness of the implementation of UCs.

6.10 Concluding remarks Traditionally, material characterization has been associated with material testing to obtain desired material properties. The establishment of UCs is to facilitate a computational means as an alternative to the physical testing, and it is often referred to as virtual testing. Whilst virtual testing has never meant to be introduced to replace physical testing completely, and it will never do, it can help to minimise the demand on physical testing, which is usually expensive and time-consuming. In many of the modern materials, in particular, in fibre reinforced composites, heterogeneity and anisotropy are often the key features. In presence of symmetries in materials of regular structure at the lower length scale, they can be used to establish UCs to drastically reduce the demand on the material characterization by the virtual testing means. In this

186

Representative Volume Elements and Unit Cells

respect, translations are the most important type of symmetries and the UCs formulated in this chapter are exclusively based on this type of symmetries, except a single case involving a cylindrical coordinate system. Identifying the geometric presence of such symmetries is a relatively easy step. Interpreting their implications on the UC to be formulated, in particular, the derivation of precise boundary conditions for the analysis of the UC, requires a new concept, i.e. the relative displacement field. With it, the formulation of the UC will then rest on a firm ground, whilst most mistakes found in this field were due to the lack of this concept as a starting point. Further reflectional and rotational symmetries, if present in the UC identified, can be taken advantage of to reduce the size of the UC. This subject will be elaborated in Chapter 8. Properly formulated boundary conditions bring the effective strains at the upper length scale (they are in fact average strains at lower length scale) into the UC concerned. They have been referred to as the Kdofs, which are the crucial part of the UC formulation, and also offer a profound convenience to the process of material characterization. Whilst the “displacement” at each Kdof gives the corresponding effective strain directly, the “nodal force” at this Kdof is simply related to the effective stress or average stress in the UC. They do not only simplify the post-processing greatly, but can also be seen vividly as the link between the two length scales involved, whilst avoiding the accumulation of undue numerical errors. The significance of “sanity checks” simply cannot be overstated. In fact, for any newly created UC, passing these checks is the most demanding task. The credibility of any UC not subjected to these checks should never be accepted. Failing any check signifies beyond any doubt at least a mistake made somewhere. Resorting to experimental validations without these checks is deemed dodgy and futile. The rules of formulating UCs can readily be applied to modern textile composites, both 2D and 3D as will be attended in Chapter 12. Their extension to other physical field, e.g. a wide range of physical processes classified under the diffusion problems, such as heat/electric conduction, fluid permeability in porous medium, etc. is straightforward, as will be presented in Chapter 10. The UCs as formulated in this chapter can be adapted to characterise the relevant diffusion coefficients effectively. The formulations of boundary conditions for UCs as presented in this chapter often appear to be tedious. However, they are systematic and hence suitable for programming. Readers are reminded that the finite element method was unthinkable to apply manually, but, once coded appropriately,

Formulation of unit cells

187

it has become a universally applicable tool, without which modern engineering can hardly sustain itself. The claimed systematic nature of the UCs formulated here has been fully demonstrated through a code, UnitCells©, developed at the University of Nottingham, as a secondary development of Abaqus/CAE (Li et al., 2015; Li, 2014). It is highly automated and literally requires no user’s interference in setting up the boundary conditions and processing the results. The detailed discussion of the code will be presented in Chapter 14 of the book.

References Ahuja, N., Schachter, B.J., 1983. Pattern Models. Wiley, New York. Bateman, H., 1932. Partial Differential Equations of Mathematical Physics. Cambridge University Press, Cambridge, UK. Euler, L., 1758. Demonstratio nonnullarum insignium proprieatatum, quibus solida hedris planis inclusa sunt praedita. Novi Commentarii Academiae Scientiarum Petropolitanae 4, 72e93 (in Latin). Jones, R.M., 1998. Mechanics of Composite Materials. CRC Press, Boca Raton. Li, S., 1999. On the unit cell for micromechanical analysis of fibre-reinforced composites. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455, 815e838. Li, S., 2001. General unit cells for micromechanical analyses of unidirectional composites. Composites Part A: Applied Science and Manufacturing 32, 815e826. Li, S., 2014. UnitCells© User Manual, Version 1.4. Li, S., Jeanmeure, L.F.C., Pan, Q., 2015. A composite material characterisation tool: UnitCells. Journal of Engineering Mathematics 95, 279e293. Li, S., Kyaw, S., Jones, A., 2014. Boundary conditions resulting from cylindrical and longitudinal periodicities. Computers & Structures 133, 122e130. Li, S., Lim, S.-H., 2005. Variational principles for generalized plane strain problems and their applications. Composites Part A: Applied Science and Manufacturing 36, 353e365. Li, S., Reid, S.R., Soden, P.D., 1994. A finite strip analysis of cracked laminates. Mechanics of Materials 18, 289e311. Li, S., Singh, C.V., Talreja, R., 2009. A representative volume element based on translational symmetries for FE analysis of cracked laminates with two arrays of cracks. International Journal of Solids and Structures 46, 1793e1804. Li, S., Sitnikova, E., 2018. An excursion into representative volume elements and unit cells. In: Beaumont, P.W.R., Zweben, C.H. (Eds.), Comprehensive Composite Materials II. Elsevier, Oxford. Li, S., Warrior, N., Zou, Z., Almaskari, F., 2011. A unit cell for FE analysis of materials with the microstructure of a staggered pattern. Composites Part A: Applied Science and Manufacturing 42, 801e811. Li, S., Wongsto, A., 2004. Unit cells for micromechanical analyses of particle-reinforced composites. Mechanics of Materials 36, 543e572. Nye, J.F., 1985. Physical Properties of Crystals. Clarendon Press, Oxford. Richeson, D.S., 2008. Euler’s Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press. Wongsto, A., Li, S., 2005. Micromechanical FE analysis of UD fibre-reinforced composites with fibres distributed at random over the transverse cross-section. Composites Part A: Applied Science and Manufacturing 36, 1246e1266.

CHAPTER 7

Periodic traction boundary conditions and the key degrees of freedom for unit cells 7.1 Introduction Unit cell analysis is typically conducted employing finite element method. As most, if not all, FE codes are formulated from a displacement-based variational principle, such as the minimum total potential energy or virtual displacement, all the discussion in this chapter will therefore be based on variational principles of this nature, unless otherwise specified, so that the conclusions obtained will be directly applicable to FE analyses of unit cells, although this chapter itself does not involve any FE analysis explicitly. In Chapter 5, it was suggested to call the ‘periodic boundary conditions for displacements’ the ‘relative displacement boundary conditions’, in order to avoid any undue confusion as displacement field is not periodic. Readers are also reminded that the two key considerations in constructing UCs are free body diagram and symmetry. A free body diagram represents material continuity. When applied to displacements, it eventually leads to the relative displacement boundary conditions after incorporating the symmetry consideration. Another aspect of the material continuity is expressed in terms of tractions according to Newton’s third law, i.e. T1 ¼ T2 and T3 ¼ T4

(7.1)

where tractions involved in a scenario like UCs are illustrated in Fig. 7.1. In presence of a translational symmetry in the horizontal direction by a distance of Dx, T1 and T2 map to T3 and T4, respectively, hence T1

T2

'x Unit cell

T3

T4

Fig. 7.1 Tractions involved when a finite segment is taken out as a unit cell. Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00007-4

© 2020 Elsevier Ltd. All rights reserved.

189

j

190

Representative Volume Elements and Unit Cells

T1 ¼ T3 and T2 ¼ T4 (7.2) Eqs. (7.1) and (7.2) lead to the traction boundary conditions for the UC as T2 ¼ T3 (7.3) As the traction field is periodic as implied in Eq. (7.2), given the effective homogeneity in the upper length scale, it is perfectly legitimate to call (7.3) a periodic traction boundary condition. Mathematically, periodic traction boundary conditions complement the relative displacement boundary conditions in defining a complete set of boundary conditions for the physical problem as a boundary value problem. So far, the formulation of UCs addressed the relative displacement boundary conditions only. It is the objective of the present chapter to settle the periodic traction boundary conditions. In the context of variational principles, displacement and traction boundary conditions are of different natures, hence their treatments in FE analyses are completely different. In a displacement-based variational principle, displacement boundary conditions are essential boundary conditions. They have to be imposed before trial functions (displacement fields) become admissible for the variational principle. In the context of FE practices, this corresponds to imposing boundary conditions to nodal degrees of freedom. On the other hand, traction boundary conditions are natural boundary conditions (Nemat-Nasser and Hori, 1999; Fung and Tong, 2001), which are not to be imposed but will be satisfied, usually approximately, as the energy is minimised in the same way as equilibrium equations are satisfied. Natural boundary conditions should not be imposed to the trial functions of the energy functional, since doing so would restrict the domain of admissible trial functions and hence result in a solution with the total potential energy functional not being minimised. As an illustration, a simplistic example of a bar under linearly distributed axial force is quoted from (Li, 2008), to give an indication on the possible error resulting from the imposition of a natural boundary condition as if it was an essential boundary condition. The schematic drawing of the problem analysed is shown in Fig. 7.2 and the Natural boundary condition V=0

Displacement boundary condition u=0 p=kx

E, A O

L

x, u

Fig. 7.2 A straight bar under a linearly distributed axial force.

Periodic traction boundary conditions and the key degrees of freedom for unit cells

191

outcomes of analysis are summarised in Table 7.1. It can be seen that, using a quadratic trial function to approximate a cubic exact solution, the approximation can be obtained to give an approximate total potential energy 4/5760 above its minimum (which corresponds to exact solution), bearing in mind the negative sense of the total potential energy. Having the natural boundary condition imposed to the trial function, the solution is further 5/5760 worse off, although the boundary conditions have been completely satisfied. Variational principles state that natural boundary conditions are satisfied as a part of variation process, e.g. as a part of the stationary value condition for the functional concerned, in this case, the total potential energy. In the context of FE practices, traction boundary conditions, as the natural boundary conditions, will be taken care of as loads are applied, requiring nothing more than discretising the prescribed traction into nodal forces {f} and incorporating them into the stiffness equation [K]{u} ¼ {f}, which, whilst representing the equilibrium conditions, has incorporated the considerations of deformation kinematics and materials constitutive relationship as preconditions to the FE analysis. When traction takes zero values, as is often the case, no action is actually required at all for the user to take in FE analyses (in contrast, zero displacement boundary conditions can never be left unattended!). Apparently, essential boundary conditions and natural boundary conditions are treated completely differently in applications of a variational principle or FE analysis. The outcomes are different as well in the way how these two types of boundary conditions are satisfied. Essential boundary conditions are satisfied strictly whilst the natural boundary conditions are only satisfied approximately in an approximate solution, as can be seen in Table 7.1. Imposing natural boundary conditions so that they would be satisfied strictly does not help the accuracy of the approximation numerically as shown in the example above, in general. Mathematically, it is fundamentally wrong to do so. It is conceivable that relative displacement boundary conditions are essential boundary conditions since they have to be satisfied for a displacement field to become admissible. However, the nature of the periodic traction boundary conditions, namely, whether they are the natural boundary conditions or not, is not clear. Reflected in the actual treatment in FE analyses, it is uncertain whether these boundary conditions should be imposed as essential ones after expressing them in terms of displacements, or left alone in the same way as natural boundary conditions are treated. It should be pointed out that boundary conditions are natural boundary conditions not

192 Table 7.1 Comparison of different solutions (x ¼ x=L). Displacement field Stress at free end Stress at fixed end Total potential energy 2A 2A Trial function u 36EA s 3kL s 3kL P 3kEA 2 2 2 L5 kL3 1 6

1 56

144 5760 140 5760

0

34

135 5760

0

Representative Volume Elements and Unit Cells

Exact solution 1  x3 Approximate solution ða þbxÞð1  xÞ 12 ð2 þ3xÞð1  xÞ without imposing natural boundary condition Wrong solution with natural að1 þxÞð1  xÞ 38 ð1 þ xÞð1  xÞ boundary condition imposed

Periodic traction boundary conditions and the key degrees of freedom for unit cells

193

because they are given in terms of tractions. Boundary conditions are not natural boundary conditions until they have been proven to be so from the variational calculus and a proof will be provided below to eliminate any doubt about the fact that the periodic traction boundary conditions are indeed natural boundary conditions complementing the relative displacement boundary conditions. It should be noted that in analyses based on methods other than variational principles, e.g. finite difference and analytical solutions using series, etc., both displacement and traction boundary conditions should be imposed.

7.2 Boundaries and boundary conditions for unit cells resulting from translational symmetries Assume that the spatial domain occupied by the unit cell is U and its boundary is denoted as vU. Given the periodic appearance of the architecture at the lower length scale in the material represented by the unit cell, its boundary vU can be split into two parts, vUþ and vU, which are related through translational symmetry transformations, often in a piecewise manner. The unit cell under discussion here is in a general sense and it can be 2D as well as 3D and of any appropriate shape. Take a 2D square unit cell (Li, 2001), also presented in subsection 6.4.1.2 in Chapter 6 for example, as illustrated in Fig. 7.3(a). It occupies a square domain U in the xy-plane (2D space) with its boundary vU being the square frame ABCD; vUþ consists of segments BC and DC while vU of segments AD and AB. The symmetry between vUþ and vU is piecewise-defined, namely, segments BC on vUþ and AD on vU are related through translation in the x-direction by a distance of 2b and DC and AB through a translation in the y-direction by a distance of 2b. Similar descriptions apply to 2D hexagonal unit cells (Li, 2001) as shown in Fig. 7.3(b) and 3D unit cells as shown in Fig. 7.3(c) where vU has not been marked in the drawing but is right opposite to vUþ. The relative displacements between those on vUþ and vU can in general be given as follows  0 uþ i  ui ¼ εij Dxj

(7.4)

þ   0 where uþ i and ui are the displacements on vU and vU , respectively, εij the average strains in the unit cell and Dxi the increments of coordinates from a point on vU to its corresponding point on vUþ. The average strains

194

Representative Volume Elements and Unit Cells

n+

(a)

y

w:+

y=b

D

C

n

n+

S1 S2

x x=b

: x=b

A

B

w:

y=b

n y

(b)

S2

w:+

S1 x

S3 :

w: z

(c)

S1 y S3

w:+

x S2

Fig. 7.3 Unit cells with their boundaries specified: (a) a square unit cell; (b) a hexagonal unit cell; (c) a cubic unit cell.

Periodic traction boundary conditions and the key degrees of freedom for unit cells

195

in (7.4) plays a pivotal role in the formulation of the UC and they will be referred to as the key degrees of freedom (Kdofs) as have been introduced in Section 6.6 of Chapter 6. Eq. (7.4) is in fact identical to (6.6) of Chapter 6 but is presented in tensor notation. It will be employed instead of (6.9) of Chapter 6 for convenience in the context of the present chapter which is primarily analytical derivations and hence the ease in implementation does not enter the consideration here. As fully elaborated there, the difference between Eqs. (6.6) and (6.9) in Chapter 6 is a matter of rigid body rotations. Constraints (7.4) are the relative displacement boundary conditions for the UC. If the displacements had been given prescribed values individually all over the boundary, they would be sufficient to guarantee a unique solution for the boundary value problem. However, since they are all given as relationships between values on one part of the boundary and those on another, they only provide a part of the boundary conditions required to formulate an appropriate boundary value problem for the UC mathematically. The boundary value problem should be complemented with some more boundary conditions. The translational symmetries also result in periodic traction boundary conditions as argued for (7.3). Given in general form according to the definition of tractions they can be written as follows:


 
 þ    þ þ þ þ Tiþ ¼ sþ þ s ¼0 ij nj ¼ sij nj ¼ Ti or sij x nj x ij x nj x (7.5) þ   where sþ ij and sij are the stresses and ni and ni the outward normals on þ  þ  vU and vU , respectively, x and x are the coordinates of corresponding points on vUþ and vU surfaces, respectively, presented as vectors using the boldface notation instead of indices. Given the translational symmetry between vUþ and vU (in a piecewise manner), the outward normals on them at the corresponding points are related as

þ n ¼  nþ (7.6) j x j x .

Eqs. (7.4) and (7.5) together provide a complete set of boundary conditions for the boundary value problem, three relating displacements on opposite faces and three equating tractions on opposite faces. The uniqueness of the solution is guaranteed for this type of problems up to rigid body translations whilst rigid body rotations have been eliminated once (7.4) is imposed.  Sometimes in the literature, conditions (7.5) are given as sþ ij ¼ sij . For a general 3D problem, stress has six independent components whilst traction

196

Representative Volume Elements and Unit Cells

has only three. Using stresses directly would give too many independent conditions, in which case a solution does not exist in general, mathematically. Whilst these conditions exist as a consequence of the translational symmetry transformation, they cannot be used as boundary conditions for the boundary value problem of unit cells because some of them are not exposed on the surface of the boundary and hence their values cannot be prescribed. Given the periodic microstructure, boundary vU of the unit cell can be split into vUþ and vU as mentioned previously. As they are both piecewise-defined, in general, one can have vUþ ¼

K X

vUþ m

m¼1

vU ¼

K X

(7.7) vU m

m¼1 þ  where vUþ and vU, m and vUm , m ¼ 1,2,...K, are segments of vU respectively, related to each other through one of the translational symmetries employed to define the unit cell. Because of this, both can be mapped to a common surface Sm, say, the mid-surface between vUþ m and vU , as illustrated in Fig. 7.3, as follows. m

vUþ m /Sm vU m /Sm

with

xþ m /xm

on Sm

x m /xm

on Sm

(7.8)

Surface Sm can then be considered as the common domain in which quantities including stresses, displacements and unit outward normals on vUþ and vU, respectively, are defined so that
þ
 þ  s sþ ij xm ¼ sij ðxm Þ; ij xm ¼ sij ðxm Þ
þ þ nþ j xm ¼ nj ðxm Þ;
þ þ duþ i xm ¼ dui ðxm Þ;


  n j xm ¼ nj ðxm Þ

ðm ¼ 1; 2; ...KÞ.

(7.9)


  du i xm ¼ dui ðxm Þ

The existence of a common domain is a natural consequence of the translational symmetries employed to define UCs. This will be of significance in the subsequent derivations.

197

Periodic traction boundary conditions and the key degrees of freedom for unit cells

7.3 Total potential energy and variational principle for unit cells under prescribed average strains Assume that linear elasticity applies to the unit cell under consideration. The minimum total potential energy principle can be employed as the governing variational principle which also underlies FEM, although alternative approaches to the formulation of the FEM may be known under other names, such as virtual displacement, weighted residual or other weak forms of boundary value problems. As in conventional practice of micromechanical analysis, body forces are neglected. Under prescribed macroscopic strains ε0ij , with a bar on the top to signify a prescribed value, as the loading condition, the total potential energy for the present problem is the same as the strain energy since there is no prescribed force to contribute to the term of potential of external forces. Adopting Einstein’s repeated subscripts convention, the total potential energy in the unit cell can be written as Z 1 P¼ sij εij dU (7.10) 2 U

where U is the domain in which the problem is defined, the unit cell in the context of the present discussion, sij the stresses and εij the strains. A trial displacement field, ui, has to satisfy boundary conditions (7.4) in order for it to be considered admissible for the variational problem, in addition to the satisfaction of the kinematic equations of elasticity εij ¼

 1
ui;j þ uj;i : 2

(7.11)

Stationary value condition of the total potential energy (7.10) requires Z Z Z 1 (7.12) dP ¼ d sij εij dU ¼ sij dεij dU ¼ sij dui;j dU ¼ 0: 2 U

U

U

To manipulate the above, one may find the Gauss theorem useful. Z Z gi;i dU ¼ gi ni dS; (7.13) U

vU

where ni is the unit outward normal to vU. Let gi ¼ sij duj .

(7.14)

198

Representative Volume Elements and Unit Cells

Given the symmetry of the stress tensor, i.e. sji ¼ sij , one has gi;i ¼ ðsij duj Þ;i ¼ sij;i duj þ sij duj;i ¼ sij;i duj þ sij dui;j .

(7.15)

Condition (7.12) can then be re-written as follows Z Z dP ¼ sij dui;j dU ¼ dH  sij;j dui dU ¼ 0

(7.16)

U

where

U

Z

dH ¼

sij nj dui dS:

(7.17)

vU

Because of the arbitrariness of dui in U, its coefficient sij;j in Eq. (7.16) has to vanish to ensure that the integral vanishes as a necessary condition for Eq. (7.16) to be satisfied. This leads to the equilibrium equations of linear elasticity as the Euler’s equations of the variational problem sij;j ¼ 0 in U.

(7.18)

To describe verbally, the equilibrium equations do not have to be satisfied when introducing trial functions but they will be satisfied as a part of the stationary value condition of the total potential energy. Similarly, dH has to vanish as the natural boundary condition in order to satisfy the stationary value condition of the total potential energy, leading to dH ¼ 0 (7.19) This will be further manipulated in the next section to deliver the desired conclusion.

7.4 Periodic traction boundary conditions as the natural boundary conditions for unit cells With the partition of the boundary into vUþ and vU due to the way in which the essential boundary conditions (7.4) are imposed, dH as defined in (7.17) can be expressed into Z Z Z dH ¼ sij nj dui dS ¼ sij nj dui dS þ sij nj dui dS; (7.20) vU

vU

which can be rewritten for clarity as

vUþ

199

Periodic traction boundary conditions and the key degrees of freedom for unit cells

Z dH ¼

sþ ij





þ þ
þ x nþ j x dui x dS þ þ

Z


 
 
 s ij x nj x dui x dS;

vU

vUþ

(7.21) where the domain for each quantity is defined. As parts of the boundary, surfaces vUþ and vU are different. However, as the domains for the two integrals in (7.21), they are identical and can be unified. Without loss of generality, Sm as defined in (7.8) will be considered hereafter as the common domain for the two segments related through a translation symmetry, one on vUþ and another vU, provided that the integrands þ þ þ þ þ       sþ ij ðx Þnj ðx Þdui ðx Þ and sij ðx Þnj ðx Þdui ðx Þ are evaluated on þ  vU and vU , respectively. Therefore, (7.21) becomes 0 K XB Z
þ þ
þ þ
þ sþ dH ¼ @ ij xm nj xm dui xm dS m¼1

vUþ m

Z þ

1
 
 
 C s ij xm nj xm dui xm dS A

vU m

¼

K Z  X m¼1

 þ þ    ðx Þn ðx Þdu ðx Þ þ s ðx Þn ðx Þdu ðx Þ dS sþ m m m m m m ij j i ij j i

Sm

(7.22) For 2D unit cells, the domain of the integrals in (7.21) will be 1D and the similar argument can be made for 3D UCs. From (7.4), one has  0  þ 0 duþ i  dui ¼ dεij Dxj or dui ¼ dui  dεij Dxj

(7.23)

where dε0ij vanishes identically, i.e. dε0ij h0, because any prescribed value will not allow a non-vanishing variation. However, the symbol of dε0ij is kept for the time being as it will be helpful to identify the reactions as will be shown later. Making use of (7.23) and (7.6), (7.22) can be written as follows

200

Representative Volume Elements and Unit Cells

K Z   X þ   dH ¼ ðx Þn ðx Þ þ s ðx Þn ðx Þ du sþ m m m m ij j ij j i ðxm Þ m¼1

Sm

 þ 0 þ sþ ðx Þn ðx ÞDx dε m j m k ik dS ij K Z    X þ    0 ðx Þn ðx Þ þ s ðx Þn ðx Þ du ðx Þ þ R dε sþ ¼ m j m m j m m ij ij dS ij ij i m¼1

Sm

(7.24) where Rij ¼

K Z X m¼1

þ sþ ik ðxm Þnk ðxm ÞDxj dS

Sm

 1 0
þ   sik ðxm Þnþ k ðxm Þ þ sik ðxm Þnk ðxm Þ K Z X 1 C B ¼ ADxj dS @ 2 m¼1
þ  þ   Sm þ sik ðxm Þnk ðxm Þ  sik ðxm Þnk ðxm Þ

(7.25)

are the reactions at the Kdofs to the prescribed ‘nodal displacement’ ε0ij . In the expression of dH given by (7.24) the term associated with Rij vanishes identically in the case of prescribed Kdofs, because dε0ik h0. For the first term in (7.24) to vanish for arbitrary du i ðxm Þ, a necessary condition is þ   sþ ij ðxm Þnj ðxm Þ þ sij ðxm Þnj ðxm Þ ¼ 0

(7.26)

Mapping back from Sm to vUþ and vU, respectively, (7.26) becomes
þ þ
þ

     (7.27) sþ ik xm nk xm þ sik xm nk xm ¼ 0 ðm ¼ 1; 2; ...KÞ This reproduces (7.5) identically, which was obtained in Section 7.2 from the translational symmetry consideration, yet it was not clear there whether it should be imposed in an FE analysis. It has now been proven that (7.27), which is the same as (7.5), can be derived as a necessary condition for the total potential energy to take its stationary value, i.e. dP ¼ 0, as given in (7.12) under the relative displacement boundary conditions (7.4). Boundary conditions associated with traction obtained as a necessary condition for the total potential energy functional to take a stationary value are by definition natural boundary conditions in the theory of

Periodic traction boundary conditions and the key degrees of freedom for unit cells

201

variational principles (Washizu, 1982). Natural boundary conditions are the by-product of the Euler’s equation in the variational problem. Mechanically, the Euler’s equation of the total potential energy is the equilibrium equations of elasticity. The natural boundary conditions are satisfied in exactly the same sense as the Euler’s equation is satisfied, i.e. as the conditions for the total potential energy to take its stationary value, representing the equilibrium conditions on the relevant part of the boundary. This concludes the proof that the periodic traction boundary conditions as given in (7.5) are indeed natural boundary conditions of the variational problem under the relative displacement boundary conditions as the essential boundary conditions. To apply any approach based on the minimum total potential energy principle, such as FEM, to unit cells, it is the user’s responsibility to make sure that the trial displacement field satisfies kinematic (7.11) and relative displacement boundary conditions (7.4), whilst equilibrium (7.18) and periodic traction boundary conditions (7.5) will be satisfied by the stationary value condition of the total potential energy functional, viz., the solution of stiffness equations in FEM. Therefore, the user should not impose natural boundary conditions as if they were essential boundary conditions, as this would be conceptually wrong and it does not help the numerical accuracy in general, either, as illustrated in Table 7.1. Having established periodic traction boundary conditions as natural boundary conditions, it is then reassuring to treat them in the way they should be treated, provided that the solver is based on the minimum total potential energy principle or its variants. The correct treatment is no treatment! All natural boundary conditions can be dealt with in the same way. When symmetries, such as reflections and rotations, are used, similar situations arise where displacement boundary conditions reside with traction boundary conditions. The correct way of applying boundary conditions is to impose displacement boundary conditions whilst leaving traction boundary conditions alone. The present formulation is displacement-based as are most FE codes, if not all. Should a stress-based approach be followed through using the minimum total complementary potential energy principle, e.g. as in (Li and Hafeez, 2009), by going through a similar process, one would expect that relative displacement boundary conditions become the natural boundary conditions, whilst the periodic traction boundary conditions have to be imposed as essential boundary conditions before trial stress fields become admissible. When a generalised variational principle such as that of Hellinger-

202

Representative Volume Elements and Unit Cells

Reissner (Washizu, 1982) is employed, both the relative displacement and periodic traction boundary conditions will become natural boundary conditions. However, there are not many, if any, FE codes developed to date based on the minimum total complementary potential energy principle or the Hellinger-Reissner variational principle.

7.5 The nature of the reactions at the prescribed key degrees of freedom Out of micromechanical FE analyses of unit cells, the way to evaluate effective/average stresses and strains in them had been made obscure to a great extent in the literature. Most authors appear to be reluctant to mention any details of post-processing the results they conducted. The messages transmitted seem to suggest either (i) it was not important or (ii) it was well-established and there was a standard procedure everybody was taking. The reality is however far from either of these. Average stresses had been evaluated in some existing analyses by averaging the stresses obtained from the FE results either over a section (Xia et al., 2003, 2006) or the complete volume of the unit cell. Either way, it involves a lot of elements. A demanding aspect is that such a postprocessing requirement cannot be fulfilled by any of the existing commercial FE codes from within them and a significant post-processing effort would therefore have to be made by the users. As described in Chapter 5, in order to incorporate different weights of the stresses from different elements or integration points, one would have to write the program for the type of finite elements employed in the analysis. This is a very demanding task, often beyond the skill set of ordinary FEM users. Even if one has desirable skills, it still takes time and efforts, rendering the use of unit cells a formidable task for many, whilst the lack of a clear instruction in this respect mystifies this aspect of the applications of UCs in general. As have already been explained, average strains can be prescribed at the Kdofs as a loading mechanism for the UCs. When relative displacement boundary conditions are employed to prescribe boundary conditions for UCs, there should never be a need to average strains, since they should be readily obtained as the nodal ‘displacements’ at these Kdofs. When the Kdofs are prescribed with known values, effectively, the UCs is subjected to a known combination of average strains. In this section, it will be proven below with mathematical rigour that the reactions obtained

203

Periodic traction boundary conditions and the key degrees of freedom for unit cells

at these Kdofs are related to the average stresses in a straightforward manner. Making use of (7.24), the reactions at the Kdofs ε0ij defined by (7.25) can be re-presented on their respective parts of the boundary as K Z
 1X þ   Rij ¼ sþ ik ðxm Þnk ðxm Þ  sik ðxm Þnk ðxm Þ Dxj dS 2 m¼1 Sm

0 1B ¼ @ 2

Z


þ þ
þ sþ ik x nk x Dxj dS 

1

Z


 
 C s ik x nk x Dxj dS A:

vU

vUþ

(7.28)  Given Dxj ¼ xþ j  xj as defined by the relevant translational symmetry, the above can be further manipulated as follows. 0 Z 
þ  þ
þ  þ 1B Rij ¼ @ sþ xj  x j dS ik x nk x 2 vUþ

1  
 
 þ C s xj  x ik x nk x j dS A

Z  vU

0 1B ¼ @ 2

Z


þ þ
þ þ sþ ik x nk x xj dS 

1B  @ 2


 
 þ C s ik x nk x xj dS A

vU

vUþ

0

1

Z

Z

vUþ


þ þ
þ  sþ ik x nk x xj dS 

Z

1
 
  C s ik x nk x xj dS A:

vU

(7.29) In the expression above, the second term in the first bracket and the first term in the second bracket can be transformed as follows, given the periodic traction boundary conditions (7.5).

204

Z vU

Z

Representative Volume Elements and Unit Cells

s ik


 
 þ x nk x xj dS ¼ 

Z

vU


þ þ
þ  sþ ik x nk x xj dS ¼ 

vUþ

Z


þ þ
þ þ sþ ik x nk x xj dS
 
  s ik x nk x xj dS:

(7.30)

vUþ

The domains, vUþ and vU are both equivalent to the sum of Sm, i.e. S14S24/4Sm and hence are interchangeable, therefore Z Z
 
 
 
  s dS ¼ s x n x x ik k j ik x nk x xj dS vUþ

Z

sþ ik


þ þ
þ þ x nk x xj dS ¼

vU

vU

Z

sþ ik




(7.31)


þ þ x nþ k x xj dS þ

vUþ

Making use of (7.30) and (7.31), expression (7.29) becomes Z Z
þ þ
þ þ
 
  Rij ¼ sþ dS þ s x n x x j ik x nk x xj dS ik k vU

vUþ

Z

(7.32)

ðsik xj Þnk dS

¼ vU

In order to reveal the identity of Rij, the divergence theorem (7.13) is applied to the integral on the right hand side of (7.32) which yields Z Z Z Z Rij ¼ ðsik xj Þnk dS ¼ ðsik xj Þ;kdU ¼ sik;k xj dU þ sik djk dU U

vU

Z

¼

U

U

sij dU U

(7.33) where sik;k vanishes because of the equilibrium Eq. (7.18) and also use has been made of xi;j ¼ dij with dij being Kroneker delta. Rij can then be expressed as Rij ¼ V s0ij .

(7.34)

Periodic traction boundary conditions and the key degrees of freedom for unit cells

where V is the volume of U, the domain of the unit cell and Z 1 0 sij ¼ sij dU. V

205

(7.35)

U

This is exactly the definition of the average stresses. Therefore, after obtaining the reactions from the FE analysis of a UC, the average stresses in the UC can be calculated directly as 1 s0ij ¼ Rij ; V

(7.36)

waiving any need of sophisticated post-processing. With average strains being the imposed ‘displacements’ at Kdofs, and average stresses obtained according to (7.36), the procedure for obtaining material effective stiffness matrix is straightforward. Average/effective stresses and strains are related through the effective stiffness as in the following effective stress-strain relationship for the composite. 8 09 2 38 ε0 9 0 0 0 0 0 0 sx > > > x > c c c c c c > > > 11 12 13 14 15 16 > > > > > > > > > 7 > 0> > > 6 0 > 0 > > 0 0 0 0 0 7> ε > s 6 > > > y y c c c c c c > > > > 21 22 23 24 25 26 7 6 > > > > > > > 6 > > > > > 7 > > > 0 0 < sz = 6 c 0 c 0 c 0 c 0 c 0 c 0 7< εz > = 6 31 32 33 34 35 36 7 ¼ ; (7.37) 7 6 0 > 6 0 0 0 0 0 0 7> g 0 > > s > > > > c c c c c c yz yz > > > 6 41 42 43 44 45 46 7> > > > > > > > 7> 6 > > > > 0 0 0 0 0 0 > > > 7 6 0 0 >s > >g > > 4 c51 c52 c53 c54 c55 c56 5> > > > xz xz > > > > > > > > > > > > 0 0 0 0 0 0 : 0 ; : ; 0 c61 c62 c63 c64 c65 c66 sxy gxy where engineering shear strains have been employed for the contracted presentation of stress and strain tensors. Six unit average strain states should then be prescribed one by one, as individual loading cases, as follows 8 9 8 9 8 9 8 9 8 9 8 9 1> >0> >0> >0> >0> 0> > > > > > > > > > > > > > > > > > > > > > > > > > > > >0> > > >0> >0> > > > >1> >0> > > > >0> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

> 0> 1> 0> 0> 0> 0> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > 0> 0> 0> 1> 0> 0> > > > > > > > > > > > > > > > > > > > > > > > ; : ; : ; : ; : ; : ; : > 0 0 0 0 0 1

206

Representative Volume Elements and Unit Cells

It should be pointed out that the use of unit strains as given above is purely for their numerical conveniences rather than their practical magnitude. Their validity is strictly limited to linear problem where responses of the material are all proportional. Otherwise, such magnitude of strains would have certainly pushed the material behaviour into nonlinearity due to finite deformation. A proper account on the applications of UCs to the finite deformation problem will be delivered in Chapter 13. For each loading case, six average stress components are obtained which provide one column of the effective stiffness matrix as involved in (7.37). After obtaining the complete stiffness matrix, one can extract effective material properties, e.g. in terms of engineering elastic constants. It should be noted that the components of the stiffness matrix are not elastic constants themselves. To calculate the elastic properties, compliance matrix, S0, should be determined first, which is the inverse of a stiffness matrix: 3 2 0 s11 s012 s013 s014 s015 s016 7 6 6 s0 s0 s0 s0 s0 s0 7 6 21 22 23 24 25 26 7 7 6 7 6 0 0 0 0 0 0 7  0 6 6 s31 s32 s33 s34 s35 s36 7 S ¼6 7 6 s0 s0 s0 s0 s0 s0 7 6 41 42 43 44 45 46 7 7 6 7 6 0 6 s51 s052 s053 s054 s055 s056 7 5 4 s061 s062 s063 s064 s065 s066 2 0 0 0 0 0 0 31 c11 c12 c13 c14 c15 c16 7 6 6 c0 c0 c0 c0 c0 c0 7 6 21 22 23 24 25 26 7 7 6 7 6 0 0 0 0 0 0 6 c31 c32 c33 c34 c35 c36 7 7 6 ¼6 7 6 c0 c0 c0 c0 c0 c0 7 6 41 42 43 44 45 46 7 7 6 7 6 0 0 0 0 0 0 6 c51 c52 c53 c54 c55 c56 7 5 4 0 c61

0 c62

0 c63

0 c64

0 c65

(7.39)

0 c66

In general, the compliance matrix of a completely anisotropic material can be expressed in terms of effective elastic constants as follows.

Periodic traction boundary conditions and the key degrees of freedom for unit cells

2

1 6 E0 6 1 6 6 0 6 n12 6 0 6 E 1 6 6 6 0 6 n13 6 0 6  0  6 E1 6 S ¼6 6 h014 6 6 E0 6 1 6 6 0 6 h 6 15 6 0 6 E1 6 6 4 h0 16 E10

n0  210 E2

n0  310 E3

h041 0 G23

h051 0 G13

1 E20

n0  320 E3

h042 0 G23

h052 0 G13

1 E30

h043 0 G23

h053 0 G13



n023 E20

h024 E20

h034 E30

1 0 G23

m054 0 G13

h025 E20

h035 E30

m045 0 G23

1 0 G13

h026 E20

h036 E30

m046 0 G23

m056 0 G13

3 h061 0 7 G12 7 7 0 7 h62 7 7 0 7 G12 7 7 0 7 h63 7 7 0 7 G12 7 7. 0 7 m64 7 7 0 7 G12 7 7 7 0 7 m65 7 0 7 G12 7 7 7 1 5 0 G12

207

(7.40)

0 , G 0 and G 0 are effective Young’s moduli and shear where E10 , E20 , E30 , G23 13 12 0 0 moduli and n23 , n13 and n012 are the effective Poisson’s ratios. They are familiar material properties to users. h0ij (with values of i and j as shown in the matrix (7.40) are the coupling coefficients between respective shear strains and direct stresses. m0ij are the coupling coefficients between respective shear strains and other shear stresses. The remaining constants are determined by the symmetric nature of the compliance matrix, e.g.

n021 n012 ¼ 0 E20 E1 h041 h014 0 ¼ 0 G23 E1 m054 m045 ¼ 0 0 ; G13 G23

(7.41) .; etc:

The two set of effective material constants h0ij and m0ij are not quite familiar to most users as they appear only in generally anisotropic materials (Lekhnitskii, 1977). Whilst they can be mathematically defined, there has been absolutely no practical means of measuring them, in general.

208

Representative Volume Elements and Unit Cells

In the case of an orthotropic material, compliance matrix can simplified to 2

1 6 E0 6 1 6 6 0 6 n12 6 6 E0 6 1 6 6 0 6 n13 6 0  0 6 6 E1 S ¼6 6 6 6 0 6 6 6 6 6 6 0 6 6 6 4 0

n0  210 E2

3

n0  310 E3

0

0

n032 E30

0

0

1 E30

0

0

0

0

1 0 G23

0

0

0

0

1 0 G13

0

0

0

0

1 E20 



n023 E20

0 7 7 7 7 7 0 7 7 7 7 7 7 0 7 7 7 7. 7 7 0 7 7 7 7 7 7 0 7 7 7 7 1 5 0 G12

(7.42)

which yields the expressions for the effective elastic properties as E10 ¼

1 s011

E20 ¼

1 s022

s0 n023 ¼  32 s022

s0 n013 ¼  31 s011

s0 n012 ¼  21 s011

0 G13 ¼

1 s055

E30 ¼

1 s033

s0 n012 ¼  21 ; s011 0 G12 ¼

(7.43)

1 s066

These nine effective elastic constants form the full set of elastic material properties for an orthotropic material. The number of material properties will vary depending on the extent of orthotropy in the material, which is to be determined through the material categorisation as advocated in Chapter 3 of this book. It is clear that without appropriate categorisation, effective material properties produced by blindly applying characterisation procedure could be misleading and the whole such exercise could be out of context. In general, if one is interested in the effective stiffness matrix only, prescribing average strains will be the most direct means to achieve the goal.

Periodic traction boundary conditions and the key degrees of freedom for unit cells

209

However, if one is interested in determining the effective elastic constants, the method described above is not the most efficient, as it requires finding the inverse of a 66 matrix. Though this task is not formidable, there is a substantially simpler alternative that will be presented in the next section.

7.6 Prescribed concentrated ‘forces’ at the key degrees of freedom The Kdofs as introduced to an FE model through the relative displacement boundary conditions are dofs of the FE model for a UC, like the nodal displacements. Each of them can be constrained by prescribing either a zero value, or a non-vanishing value, e.g. 1, as shown in the previous section. However, this is not the only way of making use of the Kdofs. As an alternative, a concentrated ‘force’ can be applied at each of these Kdofs as the load. The objective of this section is to reveal the relationship between such concentrated ‘forces’ applied at the Kdofs and the average stresses in the UC. The term ‘forces’ is placed inside a pair of quotation marks because they are in fact generalised forces having the dimension of forcelength. In Section 7.3, the minimum total potential energy principle has been presented under the relative displacement boundary conditions when average strains were prescribed. When concentrated ‘forces’ are prescribed at these discrete Kdofs, as is the subject of the present section, they will contribute to the potential of external forces in the total potential energy, which will then become Z 1 P¼ sij εij dU  F ij ε0ij ; (7.44) 2 U

where F ij , a bar on the top for a prescribed value as the notation employed previously, are prescribed concentrated ‘forces’ at the Kdofs and ε0ij are the average strains as the Kdofs in the UC. Following the same variation process with the necessary manipulations as in Section 7.4, the variation of the functional gives Z dP ¼  sij;j dui dU þ dQ. (7.45) U

where the first the term vanishes due to the equilibrium condition (7.18) and dQ ¼ dH  F ij dε0ij

(7.46)

210

Representative Volume Elements and Unit Cells

with dH as defined in (7.17). Taking advantage of the mathematical manipulations provided in (7.21)e(7.25) after accepting the periodic traction boundary conditions and mapping back from Sm to vUþ and vU, respectively, 0 Z 
þ  þ
þ  þ 1B  sþ  x dH ¼ @ x n x x j j dS ik k 2 vUþ 1 (7.47) Z  
 
 þ C 0  s xj  x ik x nk x j dS Adεij vU

Employing mathematical manipulations of the coefficient of dε0ij on the right hand side of the equation similar to those performed in Eqs. (7.29)e(7.34) will lead to 0 1 Z dH ¼ @ sij dUAdε0ij ¼ V s0ij dε0ij (7.48) U

Substituting (7.48) back into (7.46), one obtains   dQ ¼ V s0ij  F ij dε0ij

(7.49)

For dP as given in (7.45) to vanish in order for the total potential energy to take its stationary value, given the arbitrariness of dε0ij , a necessary condition for dQ ¼ 0 is F ij ¼ V s0ij

(7.50)

or 1 s0ij ¼ F ij V

(7.51)

i.e. the prescribed concentrated ‘forces’ at the Kdofs as the loads are proportional to the average stresses in the UC with a proportion factor being the volume of the UC. The above outcome can be intuitively verified from an energy balance perspective. Consider the unit cell as a structure; if it is only loaded through the 6 Kdofs, the energy put into the structure is from these 6 ‘forces’. Given the ‘nodal displacements’ at these Kdofs ε0ij and the elasticity of the structure, the energy should be 1 U1 ¼ F ij ε0ij 2

(7.52)

Periodic traction boundary conditions and the key degrees of freedom for unit cells

211

If the UC is considered as having been homogenised, then the energy in it should be a product of strain energy density and its volume, i.e. 1 U2 ¼ V s0ij ε0ij (7.53) 2 Since U1 and U2 must be equal for any strain state ε0ij , from equality of the right hand sides of the two expressions above, (7.51) is reproduced. The two methods of prescribing the loading at Kdofs can now be summarised as follows. (1) When average strains ε0ij are prescribed through the Kdofs, their values, ε0ij , are known. Out of the solution, the reactions Rij at these Kdofs can be obtained. Average stresses s0ij can be obtained from to the reactions Rij according to (7.36). (2) When concentrated ‘forces’, F ij , are prescribed at the Kdofs, the average strains ε0ij can be obtained as the ‘nodal displacements’ at these Kdofs out of the solution. Average stresses s0ij are applied through prescribing concentrated forces F ij at the Kdofs as loads, where s0ij and F ij are related according to (7.50). Unit average stress is equivalent to a concentrated ‘force’ of the magnitude of the volume of the UC. An important fact involved in the loading mechanism by prescribing a concentrated ‘force’ at one of the Kdofs is that it produces an effective uniaxial or pure shear stress state, in which most effective material properties are defined, as described in Chapter 3 through (3.19) to (3.23) there. Correspondingly, prescribing average strains through Kdofs as described in the previous section gives rise to uniaxial strain states which are not the conditions under which effective elastic properties are measured.

7.7 Examples With methods of prescribing the load to a UC being established and justified, a number of practical examples are presented below demonstrating how the developed methodology is to be applied to analyse a range of UCs.

7.7.1 A 2D square unit cell Boundary conditions for 2D rectangular UCs have been derived in Section 6.4.1.2 of Chapter 6. Taking half side lengths a ¼ b ¼ 1 mm, the volume can be obtained as V ¼ 2  2  1 ¼ 4 mm2, if a unit thickness is assumed. At the fibre volume fraction of 60%, the fibre diameter is 1.7491 mm. Assumed constituent properties are provided in Table 7.2 as quoted from

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Representative Volume Elements and Unit Cells

Table 7.2 Material properties of the constituent fibre and matrix. E (GPa) v a (10L6/ C)

Volume fraction

Fibre Matrix

60% 40%

10 1

0.2 0.3

5 50

(Li, 2001). Abaqus was employed as the FE solver. Using the generalised plane strain elements to generate the mesh, the Kdof ε0z in z-direction is readily available as the out of plane displacement. This displacement is common for all generalised plane strain elements in Abaqus along with two more rotational dofs which should be suppressed as they are irrelevant to the present problem. Other Kdofs are ε0x , ε0y and g0xy . The two ways of using the Kdofs are illustrated below. 7.7.1.1 Prescribed average strains As has been stated earlier, this method offers straightforward means of calculating the stiffness matrix of the material. According to the outline of method given in Section 7.5, in order to obtain the first column of a stiffness matrix, appropriate Kdofs are prescribed values as follows ε0xx ¼ 1 and ε0yy ¼ ε0zz ¼ g0xy ¼ 0: which, essentially, corresponds to a unidirectional strain state being applied. It should be noted that zero values have to be prescribed for other displacements, because they are essential boundary conditions. The reactions obtained at the respective Kdofs from the FE solution are Rxx ¼ 14:580 kNm

Ryy ¼ 3:8619 kNm

Rzz ¼ 4:2849 kNm

and Rxy ¼ 0:

The average stresses can be worked out as s0xx ¼ Rxx =V ¼ 3.6450 GPa s0yy ¼ Ryy =V ¼ 0.9655 GPa s0zz ¼ Rzz =V ¼ 1.0712 GPa and s0xy ¼ Rxy =V ¼ 0: Notice the differences in dimensions and units of various quantities adopted here for the ease of presentation, which have to be unified in any actual FE analysis.

Periodic traction boundary conditions and the key degrees of freedom for unit cells

213

These average stresses are in fact the elements of the stiffness matrix of the composite. It is not straightforward to obtain effective engineering elastic constants from these stresses and strains since the material is not under an effectively uniaxial stress state. Therefore, it is not the most convenient way of making use of Kdofs and this way of using the Kdofs will therefore not be pursued in any further examples. Of course, nothing stops one following the procedure as provided in Section 7.5 to obtain the elastic constants after finding the compliance matrix first, except that more calculations are involved in the post-processing. 7.7.1.2 Prescribed average stresses A concentrated force is prescribed at the appropriate Kdof as F zz ¼ 4 Nm. It is worth noting that zero forces, including F xx ¼ F yy ¼ F xy ¼ 0 at remaining Kdofs, are not to be applied but to be implied, as they are natural boundary conditions. Effectively, this represents an effective uniaxial stress state in the UC and the prescribed average stresses are s0zz ¼ F zz =V ¼ 1 MPa and s0xx ¼ s0yy ¼ s0xy ¼ 0: The average strains as nodal displacements at these Kdofs are obtained directly from the nodal displacements at the Kdofs from the FE solution as ε0zz ¼ 1:5617  104

ε0xx ¼ ε0yy ¼ 3:6284  105

ε0xy ¼ 0:

It is worth noting that the uniaxial stress state can also be reproduced by applying only one displacement at the relevant Kdof, whilst leaving the remaining Kdofs free of any constraint. In this case, equivalent output to be processed would be the reaction force at the Kdof where displacement was applied, and the nodal displacements at all other Kdofs. The reaction force at the Kdof where displacement was applied is associated with the effective uniaxial or pure shear stress whilst the prescribed displacement at it is associated with the corresponding effective strain. The nodal displacements at all other Kdofs give other effective strains whilst their corresponding effective stresses vanish under this loading condition, given a uniaxial or pure shear stress state. Since the material represented by the unit cell is under an effective uniaxial stress state, relevant effective elastic constants can be evaluated in a straightforward manner as follows 0 Ezz ¼ s0zz =ε0zz ¼ 6:4033 GPa n0zx ¼ n0zy ¼ ε0xx =ε0zz ¼ 0:2323:

214

Representative Volume Elements and Unit Cells

Similarly, if a concentrated force is applied to another Kdof, e.g. F xx ¼ 4 Nm; it produces the following effective uniaxial stress state s0xx ¼ F xx =V ¼ 1 MPa and s0yy ¼ s0zz ¼ s0xy ¼ 0; and the average strains obtained are ε0xx ¼ 3:0348  104 ε0zz ¼ 3:6284  105

ε0yy ¼ 6:9721  105 ε0xy ¼ 0:

Relevant effective elastic constants can be evaluated as 0 ¼ s0xx =ε0xx ¼ 3:2951 GPa n0xy ¼ ε0yy =ε0xx ¼ 0:2297 Exx

n0xz ¼ ε0zz =ε0xx ¼ 0:1196: To apply shear loading, one has to prescribe F xy ¼ 4 Nm; which generates an effective pure shear stress state. The average strains are output as ε0xy ¼ 1.0936  103 and ε0xx ¼ ε0yy ¼ ε0zz ¼ 0; and the relevant effective shear modulus is calculated according to 0 ¼ s0xy =ε0xy ¼ 0:9144 GPa. Gxy

Given the square orthotropic nature of the composite with squarely packed fibres, there are altogether six independent effective material prop0 , n0 ¼ n0 , E 0 ¼ E 0 , n0 ¼ n0 and G 0 ¼ erties, five of which, Ezz zx zy xx yy xy yx xy 0 Gyx , have been determined out of three loading cases analysed above. 0 ¼ The remaining effective property is longitudinal shear modulus Gxz 0 Gyz . This cannot be obtained from a generalised plane strain problem. Instead, an anticlastic problem has to be analysed within Abaqus using the steady state heat conduction problem as an analogy, as described in Section 6.4.1.1 of Chapter 6. The results obtained are summarised in Table 7.3 along with those generated using the theory of Hashin and Rosen (1964). It should be pointed out that the bounds obtained by Hashin and Rosen (1964), whilst serving as relevant references, should not be understood too

Hexagonal unit cell

Lower bounds

Upper bounds

E0z (GPa) E0x ¼ E0y (GPa) G0xy (GPa) G0xz ¼ G0yz (GPa) n0zx ¼ v0zy n0xy a0z (106/ C) a0x ¼ a0y (106/ C)

6.4029 2.8552 1.0824 1.1533 0.23337 0.31896 8.0815 23.804

6.4029 2.5381 0.91055 1.1495 0.2334 0.2798 8.0785 23.848

6.4029 3.0152 1.1780 1.1495 0.2334 0.3937 8.0785 23.848

1 2

6.4033 3.2951 (2.5623)1 0.91441 (1.3398)2 1.1948 0.23234 0.22974 (0.40105)1 8.1233 23.195

Hashin and Rosen’s theory (Hashin and Rosen, 1964)

Value in brackets was obtained if the coordinate system had been rotated by 45 about the x-axis Value obtained according to G0xy ¼ G0yx ¼ E0xx/2(1 þ n0xy)

Periodic traction boundary conditions and the key degrees of freedom for unit cells

Table 7.3 Effective material properties. Square unit cell (transverse properties in 45 direction)

215

216

Representative Volume Elements and Unit Cells

literally. There were a number of cases where the upper and lower bounds became identical. In theory, this would mean that an exact solution had been obtained. Similarly, when different upper and lower bounds were obtained, the exact solution would have to lie between these bounds. Whilst the statements as provided above were all correct, they were meant to be correct to the problem as was analysed in (Hashin and Rosen, 1964), i.e. based on cylinder-in-cylinder model. A more realistic model could have more accurate solution whilst sitting outside their bounds.

7.7.2 A 2D hexagonal unit cell Boundary conditions for 2D hexagonal UCs shown in Fig. 7.3b have been derived in Section 6.4.1.4 of Chapter 6. Taking pffiffiffi b ¼ 1 mm with a unit thickness, the volume is obtained as V ¼ 2  3  1 ¼ 3.4641 mm2 . At the fibre volume fraction of 60%, the fibre diameter will be 1.6268 mm. The same constituent properties from Table 7.2 are used as in the previous example. A uniaxial effective stress state is applied by prescribing F zz ¼ 3.4641 Nm, i.e. the same magnitude of the volume of the UC. The values of the average stresses are s0zz ¼ F zz =V ¼ 1 MPa

and s0xx ¼ s0yy ¼ s0xy ¼ 0:

The nodal displacements output at Kdofs give average strains as ε0zz ¼ 1:5618  104

ε0xx ¼ ε0yy ¼ 3:6448  105

ε0xy ¼ 0

Since the material represented by the unit cell is under uniaxial stress state macroscopically, relevant effective elastic constants are evaluated in a straightforward manner as follows 0 Ezz ¼ s0zz =ε0zz ¼ 6:4029 GPa n0zx ¼ n0zy ¼ ε0xx =ε0zz ¼ 0:2334

Similarly, if a concentrated force is applied to another Kdof as F xx ¼ 3.4641 Nm; it produces the following effective uniaxial stress state s0xx ¼ F xx =V ¼ 1 MPa

and s0yy ¼ s0zz ¼ s0xy ¼ 0

with the average strains being ε0xx ¼ 3:5023  104 ε0zz ¼ 3:6448  105

ε0yy ¼ 1:1171  104 ε0xy ¼ 0

Periodic traction boundary conditions and the key degrees of freedom for unit cells

217

Relevant effective elastic constants are evaluated as 0 ¼ s0xx =ε0xx ¼ 2:8553 GPa Exx

n0xy ¼ ε0yy =ε0xx ¼ 0:3190

n0xz ¼ ε0zz =ε0xx ¼ 0:1041 To apply pure shear loading, one can prescribe F xy ¼ 3:4641 Nm which generates an effective pure shear stress state with average stresses being s0xy ¼ F xy =V ¼ 1 MPa and s0xx ¼ s0yy ¼ s0zz ¼ 0 The average strains obtained as nodal displacements at the Kdofs are ε0xy ¼ 9.2389  104 and ε0xx ¼ ε0yy ¼ ε0zz ¼ 0 The relevant effective shear modulus is calculated as 0 ¼ s0xy =ε0xy ¼ 1:0824 GPa Gxy

Since composite with the hexagonal fibre packing is transversely isotropic, there are altogether five independent effective material properties, 0 , n0 ¼ n0 , E 0 ¼ E 0 and n0 ¼ n0 , have been found four of which, Ezz zx zy xx yy xy yx 0 ¼ G 0 is out of first two loading cases. The transverse shear modulus Gxy yx obtained from the third loading case which is in fact unnecessary as it can be obtained from  the  relationship of transverse isotropy 0 ¼ G 0 ¼ E 0 2 1 þn0 . However, it is still worthwhile to analyse Gxy yx xx xy such a loading case because the result may serve as an excellent ‘sanity check’ on the correct imposition of appropriate boundary conditions which is usually not a trivial task. Same as in the previous example, the remaining effective property, the 0 ¼ G 0 , cannot be obtained from a generlongitudinal shear modulus Gxz yz alised plane strain problem and one would have to resort to a steady state heat transfer analysis as an analogy as mentioned before. All obtained results are presented in Table 7.3 along with those from the square packed composite.

7.7.3 A 3D rhombic dodecahedron unit cell for FCC packing A series of 3D UCs were formulated in subsection 6.4.2 following Li and Wongsto (2004) for particlulate reinforced composites based on various particle packing idealisations. Boundary conditions were derived based on translational symmetries alone. From the perspective of implementation of

218

Representative Volume Elements and Unit Cells

Table 7.4 Material properties of the constituent particulate and matrix. E (GPa) v a (10L6/ C) Volume fraction

Particulate Matrix

76 3.01

0.230 0.394

4.9 60

30% 70%

the Kdofs, they bear the same characteristics as the 2D examples as presented in the previous subsections. The rhombic dodecahedron UC for face centerd cubic packing, complete set of boundary conditions for which are given in Section 6.4.2.4 of Chapter 6, is taken as an example to illustrate the use of Kdofs in the case of 3D unit cells. The material properties of the constituents are listed in Table 7.4 as quoted from Li and Wongsto (2004). Assuming half inter-particle spacing b ¼ 1 mm, the volume of the unit cell and the radius of the spherical particle are pffiffiffi V ¼ 4 2b3 ¼ 5:6569 mm3 and a ¼ 0.73995mm Apply the following concentrated force at the Kdofs F xx ¼ 5:6569 Nm. The average stresses are obtained as s0xx ¼ Fxx =V ¼ 1 MPa and s0yy ¼ s0zz ¼ s0yz ¼ s0xz ¼ s0xy ¼ 0 The average strains obtained as nodal displacements at these Kdofs are ε0xx ¼ 1:9417  104

ε0yy ¼ ε0zz ¼ 7:2401  105

ε0yz ¼ ε0xz ¼ ε0xy ¼ 0 Since the material represented by the unit cell is under uniaxial stress state macroscopically, relevant effective elastic constants can be evaluated in a straightforward manner as follows 0 Exx ¼ s0xx =ε0xx ¼ 5:1501 GPa n0xy ¼ n0xz ¼ ε0yy =ε0xx ¼ 0:3729

To apply pure shear loading, one can prescribe F yz ¼ 5:6569 Nm which generates an effective pure shear stress state s0yz ¼ Fyx =V ¼ 1 MPa

and s0xx ¼ s0yy ¼ s0zz ¼ 0

Periodic traction boundary conditions and the key degrees of freedom for unit cells

219

The average strains can be obtained as nodal displacements at the Kdofs from the FE solution ε0yz ¼ 4.8744  104 and ε0xx ¼ ε0yy ¼ ε0zz ¼ gxz ¼ gxy ¼ 0 One can then obtain the relevant effective shear modulus 0 Gyz ¼ s0yz =ε0yz ¼ 2:0515 GPa

Because of the cubic orthotropy of the material represented by this unit 0 ¼ cell, there are only three independent effective material properties, Exx 0 0 0 0 0 0 0 0 Eyy ¼ Ezz , nxy ¼ nxz ¼ nyz and Gyz ¼ Gxz ¼ Gxy , which have all been found out of two loading cases, which is fewer than in two previous examples. However, they took much longer to run given the 3D nature of this problem. If one takes advantage of the effective orthotropy of the material, understanding that there is no interactions between direct stresses and shear strains, and vice versa, the two loading cases can be merged, saving one loading case to be analysed. However, the local stresses in the unit cell will be superimposed, which may not always be desirable. It should be noted that the material represented by the rhombic dodeca0 calculated hedral UC is not isotropic because the value of shear stiffness Gxy using the relationship of isotropy is not the same as that obtained from the FE analysis of UC, namely   0 0 Exx . = 2 1 þ n0xy ¼ 1.8756 GPas2.0515 GPa ¼ Gxy For practical applications, one may wish to neglect this disparity and approximate the material as an isotropic one. It is worth noting that for the material represented by a simple cubic UC as presented in Section 0 would be 6.4.2.3 of Chapter 6, the two respective values of Gxy 2.554 GPa and 1.839 GPa. Therefore, rhombic dodecahedral UC provides closer approximation to the material isotropy than the cubic one.

7.8 Conclusions By resorting to rigorous mathematics and mechanics, it has been proven that the periodic traction boundary conditions for unit cells resulting from the consideration of translational symmetries are natural boundary conditions. They have been derived from the variational principle based on the total potential energy when the unit cell is subjected to relative displacement boundary conditions as the essential boundary conditions for the variational problem. Natural boundary conditions in FE analyses should not be imposed

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Representative Volume Elements and Unit Cells

and they are satisfied as a part of the stationary value condition of the energy functional. In application to UCs, the correct treatment of the periodic traction boundary conditions is not to impose them if the FE solver is formulated on the displacement-based energy approaches, including the minimum total potential energy principle, virtual displacement principle, etc. Most, if not all, of the commercial FE codes are indeed of this category. Imposing natural boundary conditions in an FE analysis as if they were essential ones is conceptually wrong and numerically less accurate. The relative displacement boundary conditions introduce Kdofs to the UC. They are not a part of the mesh for the UC, but they become a part of the UC, in fact, a very important and useful part. Once incorporated, they are dofs like other nodal displacements in the sense that they can be prescribed fixed values of displacements or loads can be applied to them as concentrated forces. Under prescribed displacements, reactions can be obtained at these Kdofs and under applied forces nodal displacements can be output at Kdofs. The nodal displacements at Kdofs, either prescribed or obtained as outcomes of the analysis, give the average strains in the UC directly without any post-processing. The forces at these Kdofs, either as the reactions to the prescribed displacements or as the applied loads, are related to the average stresses in the UC directly through a constant factor, the volume of the UC. If one absorbs value of the volume into the loads to be prescribed, average stresses can be applied directly without having to post-process the results from the FE analysis. Again, this simple relationship between the nodal forces at the Kdofs and the average stresses has been rigorously proven in this chapter. As the outcomes of the present chapter, no further action has been introduced in order to implement the UCs formulated in previous chapter. Rather on the contrary, they provided perfect assurances for not imposing periodic traction boundary conditions and for waiving the requirement on the sophisticated post-processing for obtaining average stresses and average strains. Practical conveniences have been found to be rest on profound mathematic rigour. Several examples have been shown. The results obtained demonstrated the applications of the UCs. They may well serve as effective verification cases in users’ future practices.

References Fung, Y.C., Tong, P., 2001. Classical and Computational Solid Mechanics. World Scientific, London.

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Hashin, Z., Rosen, B.W., 1964. The elastic moduli of fiber-reinforced materials. Journal of Applied Mechanics 31, 223e232. Lekhnitskii, S.G., 1977. Theory of Elasticity of an Anisotropic Body. Mir Publications, Moscow. Li, S., 2001. General unit cells for micromechanical analyses of unidirectional composites. Composites Part A: Applied Science and Manufacturing 32, 815e826. Li, S., 2008. Boundary conditions for unit cells from periodic microstructures and their implications. Composites Science and Technology 68, 1962e1974. Li, S., Hafeez, F., 2009. Variation-based cracked laminate analysis revisited and fundamentally extended. International Journal of Solids and Structures 46, 3505e3515. Li, S., Wongsto, A., 2004. Unit cells for micromechanical analyses of particle-reinforced composites. Mechanics of Materials 36, 543e572. Nemat-Nasser, S., Hori, M., 1999. Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, Oxford. Washizu, K., 1982. Variational Methods in Elasticity and Plasticity. Pergamon, Oxford. Xia, Z., Zhang, Y., Ellyin, F., 2003. A unified periodical boundary conditions for representative volume elements of composites and applications. International Journal of Solids and Structures 40, 1907e1921. Xia, Z., Zhou, C., Yong, Q., Wang, X., 2006. On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites. International Journal of Solids and Structures 43, 266e278.

CHAPTER 8

Further symmetries within a UC 8.1 Introduction Amongst three types of symmetries, translation, reflection and rotation, as introduced in Chapter 2, the key role of the translational symmetry in the formulation of UCs has been demonstrated in Chapter 6. If one was not interested in the computational efficiency, the development of UCs could have stopped there, signaling the message that the translational symmetry alone is sufficient for the characterisation of materials homogeneous in their upper length scale whilst having regular architectures in their lower length scale. It is the only type of symmetry offering a rigorous basis to reduce an infinite medium to a finite domain without losing any important information of the material. Unfortunately, it is this type of symmetry that has attracted the least amount of attention in the literature resulting in various misperceptions as reviewed in Chapter 5. Having formulated UCs using the translational symmetry alone as presented in Chapter 6, careful observers would easily spot the presence of other symmetries in many of the UCs formulated, typically reflectional and rotational ones, and sometimes, their combinations. One does not have to use them in order to apply these UCs to material characterisation. However, any use of them will result in a substantial reduction in the size of the UC to be analyzed. Such reduction will not make much difference in an analytical study, either in formulation of the problem or in the demands for solving the problem. However, when FEM is employed to analyze a UC, the computational cost will vary with the size of the domain to be analyzed. The larger the size of the UC, the more elements and nodes are involved and the more dofs there are. The computational cost usually follows a power law with the number of dofs, roughly speaking, to the power of three. Doubling to size of the problem would therefore lead to almost eight times increase of computational cost. Therefore, it makes perfect sense to minimise the size of UCs. Additional symmetries present in UCs can reduce the size of the UCs to make the analysis more efficient. However, saving computational cost usually comes at a price.

Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00008-6

© 2020 Elsevier Ltd. All rights reserved.

223

j

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Representative Volume Elements and Unit Cells

Translational symmetries keep the sense of all physical fields concerned. As a result, a common set of boundary conditions suits all loading conditions. Reflectional and rotational symmetries, on the other hand, reverse the sense of some of the components of displacements and shear stresses and strains. The boundary conditions obtained under one loading condition may not be applicable to another loading condition. The complications in the presentation of boundary conditions caused by use of additional symmetries will be resolved in this chapter. Just like the shapes of the UCs defined through translational symmetries lack uniqueness, the applications of further symmetries present in the UC add a substantial variety to the subject. Without thorough understanding, due care and systematic approach, one can easily get confused. When using translational symmetries alone to construct UCs, any valid periodic tessellation can be shifted by any distance in any direction without affecting the validity of the boundary conditions. The only difference will be in the subsequent meshing of the UC since the features within the UC may have been truncated differently. However, in order to take advantage of additional symmetries as available in the structure, care needs to be exercised when selecting the initial tessellation. For instance, each of cells A and B as shown in Fig. 8.1 makes a perfect UC based on translational symmetries in horizontal and vertical directions. Apparently, there are quite a few reflectional and rotational symmetries within A, but none in B. To make use of additional symmetries

B A

Fig. 8.1 Examples of choices of UCs for plain weave textile composite.

Further symmetries within a UC

225

available in a UC to be analyzed in order to further reduce its size, one needs to be mindful of such symmetries when carrying out the initial tessellation. It is also advisable that tessellations giving the simplest cell geometries should be employed, preferably, based on translations along orthogonal directions, whenever possible, in order to keep the subsequent development as manageable as possible. This guideline will be followed throughout this chapter. This example reiterates the fact that seemingly different UCs could represent the same material. On the other hand, it has been elaborated in Subsection 5.3.3 that seemingly identical UCs could be extracted from rather different materials and hence lead to different effective properties (Li, 2008). Boundary conditions for UCs are based on relative displacements. Further symmetries will have to rest on the same basis but on top of the UCs already established by using translational symmetries alone. A UC established that way has left rigid body translations free, which will have to be constrained before an FE analysis can be conducted. As far as such a UC is concerned, the constraints for eliminating rigid body translations do not have to be in any particular format, provided that they are imposed consistently. Given the subject of the present chapter, where there exist additional symmetries, there is no reason why such constraints cannot be imposed at a node which lies on the symmetry plane of a reflection, or the intersection of such planes when there are more symmetries than one. Alternatively, if the symmetries concerned are of a rotational sense, the node can be selected on the axis of rotational symmetry or the intersection of such axes. With such a node being fixed, all relative displacements can be taken with respect to this point. If the origin of the coordinate system is placed at this point, the relative displacements are all turned to their absolute sense. This would simplify the argument greatly whenever such a point can be taken. The contents of this chapter will be presented under this assumption as far as possible. Exceptions will be encountered later in the chapter, in Subsection 8.3.3 and Section 8.4, where further symmetries about other planes or axes will be taken advantage of when the relative displacements will have to be employed as will be formulated accordingly. Before then, reader can ignore the relative nature of the displacements for the time being on the assumption as stated above.

8.2 Further reflectional symmetries to existing translational symmetries Consider the cuboidal cell defined in Subsection 6.4.2.2, which results from translational symmetries along coordinate axes. When available, the

226

Representative Volume Elements and Unit Cells

use of a reflectional symmetry will halve the size of the UC to be analyzed. The proper use of the reflectional symmetry, as elaborated in Chapter 2, involves two underlying considerations, namely, continuity in displacement field and symmetry in the relative displacement field.

8.2.1 One reflectional symmetry Without loss of generality, assume that there exists a further reflectional symmetry in the cell about the plane perpendicular to the x-axis. By default, the symmetry plane must pass the center of the cuboid where the origin of the coordinate system is placed to waive the specific consideration of relativeness of the displacement, so that the absolute displacements would be the same as their relative counterpart, being relative to the origin of the coordinate system. Introducing a reflectional symmetry about the plane perpendicular to the x-axis will not affect the pairing of the faces of the cell parallel to the x-axis. The boundary conditions on them should not be altered, except those at the edges and vertices newly created by the symmetry plane. For the faces perpendicular to the x-axis, as the cell is halved, the surviving one loses its counterpart as a pair. Instead, a new face through the symmetry plane has been created. The focus will be on the formulation of the boundary conditions on these two faces. The discussion will have to discriminate the loading cases as some are symmetric whilst others antisymmetric. With respect to the reflectional symmetry about a plane perpendicular to the x-axis, the symmetric and antisymmetric components of effective stresses, strains and displacements are listed in Table 8.1. For the simplicity of the presentation without loss of generality, the loading as the stimulus will be specified in terms of effective stresses at the upper length scale. The effective strains as well as the displacements, strains and stresses in the lower scale will be considered as the responses to the loading. Table 8.1 Nature of symmetry of the effective stresses as loads and displacements as deformation with respect to the reflectional symmetry transformation about xplane. Stimuli/responses x-plane reflection

Loading cases (effective stresses) Deformation (displacements)

s0xx ; s0yy ; s0zz and s0yz s0xz &s0xy u v&w

Symmetric Antisymmetric Antisymmetric Symmetric

Further symmetries within a UC

227

It is worth noting that direct stresses are always symmetric as well as one of the shear stresses whilst the remaining two shear stresses are antisymmetric. Instead of memorising which is which, the position can be understood as follows. Consider stress as a tensor, either in its effective sense at the upper length scale or local sense at its lower length scale. Each component of it is associated with two base vectors, i.e. unit vectors i, j and k in the direction of coordinate axes x, y and z, respectively, as indicated by the subscripts in their tensor notations. A reflectional symmetry transformation about a coordinate plane reverses the direction of the base vector normal to the plane, i in the present case, resulting in a change in the sense associated with the direction. A direct stress is associated with the same base vector either twice or none and therefore the sense will change either twice or none, and effectively will be kept unchanged. A shear stress component, on the other hand, is associated with two different base vectors and the reversed base vector under the reflectional transformation is involved either only once, giving rise to the change in sense, or none, i.e. keeping its sense. The same argument applies to strains. The sense of displacements can be determined in a slightly simpler way, by considering displacements as a vector any component of which is associated with one of the base vectors. Its sense changes if the associated base vector reverses its direction under the reflectional symmetry. Otherwise, it keeps its sense. To take advantage of the reflectional symmetry, one has to follow the principle of symmetry which is stated as follows. A structure responds in a symmetric manner in terms of internal displacement, strain and stress fields to a symmetric stimulus (loading condition) and vice versa. In terms of material categorisation, the material having a reflectional symmetry falls in the category of monoclinic. The interactive components in the stiffness and the compliance matrices between the symmetric effective stresses, s0xx ; s0yy ; s0zz &s0yz in this case, and antisymmetric effective stresses, s0xz &s0xy in this case, vanish as a result. This will be duly reflected in the boundary conditions to be derived, as well as the results from analyses of this kind. Another essential consideration is the use of free body diagrams, as fully elaborated in Chapter 2. To recapitulate briefly, if a body is separated into two parts as free bodies, the continuity of the body under investigation requires the two faces created by the separation to share the common displacements, so that the separation only takes place imaginarily to allow inspection of the internal state of the body without compromising the physical continuity of the body.

228

Representative Volume Elements and Unit Cells

The continuity condition as stated above also has another perspective in terms of the interacting forces. The tractions exposed on both sides of the surface created by the separation are action and reaction. According to Newton’s third law, actions and reactions are equal in magnitude, opposite in direction and, most importantly, act on different bodies. This very law of Newton’s underlines the principle of free body diagrams in terms of forces. A special topic related to tractions has been delivered in the previous chapter. For the cuboidal unit cell under consideration, if it is separated into two halves by the symmetry plane in a free body sense, the displacements on both sides of the symmetry plane should meet two considerations simultaneously. They are (i) common on the symmetry plane to ensure the continuity and (ii) symmetric or antisymmetric about the symmetry plane depending on the nature of symmetry of the loading. These two requirements will give rise to the boundary conditions. In order to implement single reflectional symmetry, two sets of boundary conditions will have to be derived, one corresponding to the loading involving symmetric stress components and another one accounting for antisymmetric components, as specified in Table 8.1. 8.2.1.1 Boundary conditions under a symmetric loading (any of s0x ; s0y ; s0z and s0yz or their combination) Given the close relationship between the stresses and strains at the upper length scale, they can be used interchangeably to define the loading conditions under discussion as will be exercised in this manner throughout this chapter. Because the material is effectively monoclinic, antisymmetric effective strains g0xz and g0xy will vanish under a symmetric loading. On the symmetry plane, the two tangential displacements meet the requirements of the two considerations, continuity and symmetry, whatever values they take. However, the normal displacement will violate the symmetry consideration if it does not vanish. As a result, it will have to be zero in order to maintain the continuity and to exhibit symmetric deformation. Vanishing normal displacement results in the symmetric displacement boundary condition on the symmetry plane as follows: ujð0;y;zÞ ¼ 0 symm

denoted as

symm

Ujð0;y;zÞ ¼ Fx0

(8.1)

where Fx0 does not constrain displacements v and w and hence these two displacements are free from any constraint as argued in Chapter 2. Implied in

229

Further symmetries within a UC

(8.1) is the elimination of the rigid body translation in the x-direction for the UC as a whole, which should no longer be constrained separately when the UC is analyzed using FE. There is also another face perpendicular to the x-axis to look after. The full UC used to have two opposite faces perpendicular to the x-axis as a pair. Referring to Subsection 6.4.2.2, the relative displacement boundary conditions (6.66) between this pair of faces are ujða;y;zÞ  ujða;y;zÞ ¼ 2aε0x vjða;y;zÞ  vjða;y;zÞ ¼ 0

denoted as

Ujx¼a  Ujx¼a ¼ Fxsymm

wjða;y;zÞ  wjða;y;zÞ ¼ 0 (8.2) where the antisymmetric effective strains g0xz and g0xy have been left out from the right hand side, comparing with Eq. (6.66), due to the symmetric loading condition. When the cell is halved, the surviving face loses its paired counterpart and only a half of the original UC, taking that with x  0 without loss of generality, will be considered as the new UC. It will then be desirable to eliminate the displacements on the x¼ -a face. The symmetry consideration requires ujða;y;zÞ ¼ ujða;y;zÞ vjða;y;zÞ ¼ vjða;y;zÞ

(8.3)

wjða;y;zÞ ¼ wjða;y;zÞ : The boundary condition for the UC on the face of x ¼a can be obtained from (8.2) and (8.3) as ujða;y;zÞ ¼ aε0x

denoted as

symm Ujða;y;zÞ ¼ Fxa

(8.4)

symm does not constrain displacements v and w and hence these where again Fxa two displacements on the face are not subjected to any constraint due to the same reason as for (8.1). For a half sized UC, the boundary conditions on the faces can be summarised as follows. Given the fact that faces parallel to the x-axis are not affected, Eqs. (6.67) and (6.68) in Subsection 4.2.2 of Chapter 6, which happen to be independent of the antisymmetric effective strains g0xz and g0xy , give the boundary conditions on these faces. These effective strains as the Kdofs will not appear in the problem. Effectively it is equivalent

230

Representative Volume Elements and Unit Cells

to having them constrained. Complete set of boundary conditions on the faces of the halved UC under symmetric loading is therefore as follows. For face x ¼ 0 (excluding edges) ujð0;y;zÞ ¼ 0

i.e.

symm

Ujð0;y;zÞ ¼ Fx0 :

(8.1 repeat)

For face x ¼ a (excluding edges) ujða;y;zÞ ¼ aε0x

symm i.e. Ujða;y;zÞ ¼ Fxa :

(8.4 repeat)

For faces y ¼ -b and y ¼ b (excluding edges), as in (6.67) but for the current loading condition ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 2bε0y

i.e. Ujðx;b;zÞ  Ujðx;b;zÞ ¼ Fysymm :

wjðx;b;zÞ  wjðx;b;zÞ ¼ 2bg0yz (8.5) For faces z ¼ -c and z ¼ c (excluding edges), as in (6.68) but for the current loading condition ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

i.e.

Ujðx;y;cÞ  Ujðx;y;cÞ ¼ Fzsymm :

wjðx;y;cÞ  wjðx;y;cÞ ¼ 2cε0z (8.6) As has been elaborated in Chapter 6, boundary conditions on the faces are perfectly applicable to the edges associated with the faces. However, the conditions at the edges, which are intersections of the two faces, have to be imposed separately from those on the faces because otherwise redundant conditions would arise compromising the appropriate definition of the problem when implemented as an FE code. The same applies to the vertices, which are the intersections of the edges. Therefore, throughout the present chapter, three sets of boundary conditions, one for the faces, one for edges and one for vertices, will be presented for each case considered. For edges parallel to the x-axis (excluding vertices), they are not affected by the reflectional symmetry and hence Eq. (6.69) as obtained in Subsection 6.4.2.2 apply directly. Therefore, the boundary conditions are

Further symmetries within a UC

231

Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fysymm Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fysymm þ Fzsymm

(8.7)

Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fzsymm : For edges parallel to the y-axis (excluding vertices), the boundary conditions (6.70) obtained for the whole cell in Subsection 6.4.2.2 can be denoted as follows under the present loading condition Ujða;y;cÞ  Ujða;y;cÞ ¼ Fxsymm Ujða;y;cÞ  Ujða;y;cÞ ¼ Fxsymm þ Fzsymm

(8.8)

Ujða;y;cÞ  Ujða;y;cÞ ¼ Fzsymm : The third equation in (8.8) is irrelevant since edges x ¼-a and z ¼c are no longer involved in the half cell and can be dropped. The term Uj(a,y,c) can be eliminated from the first two equations, hence the boundary conditions for edges parallel to the y-axis at the x ¼ a end of the UC reduce to Ujða;y;cÞ  Ujða;y;cÞ ¼ Fzsymm :

(8.9)

If the boundary conditions given by Eq. (8.9) are considered as constraints, they eliminate the displacements at edge x¼a and z¼c as represented by the first term in the equation, then the dofs at the edge x¼a and z¼-c, i.e. Uj(a,y,c), will be left free as far as (8.9) is concerned. A separate constraint can be found for this edge from symmetry condition (8.4). Given the other symmetry condition (8.1), the boundary conditions for all edges parallel to the y-axis can be summarised as symm

Ujð0;y;cÞ ¼ Fx0

Ujða;y;cÞ  Ujða;y;cÞ ¼ Fzsymm

(8.10)

symm Ujða;y;cÞ ¼ Fxa :

One can use the third equation to simplify the second. However, this will spoil the generality of the presentation, given that the third equation, as well as the first, applies to only one of the displacements, whilst the second does to all three displacements. This step will therefore be waived. For edges parallel to the z-axis (excluding vertices), similar argument will lead to symm

Ujð0;b;zÞ ¼ Fx0

Ujða;b;zÞ  Ujða;b;zÞ ¼ Fysymm symm Ujða;b;zÞ ¼ Fxa :

(8.11)

232

Representative Volume Elements and Unit Cells

Vertices are intersections, or overlaps, of edges and hence are the logical products of the intersecting edges. The boundary conditions at them should the logical sums of the boundary conditions at the intersecting edges because the vertices will have to satisfy the boundary conditions for all edges intersecting at them. By applying the logical sum, the redundancy in the boundary conditions has been eliminated. The first and the third equations from (8.10) and (8.11), respectively, thus lead to symm

Ujð0;b;cÞ ¼ Fx0

symm Ujða;b;cÞ ¼ Fxa

(8.12)

whilst Eq. (8.7), and the second from (8.10) and (8.11), respectively, lead to Ujxða;b;cÞ  Ujða;b;cÞ ¼ Fysymm Ujða;b;cÞ  Ujða;b;cÞ ¼ Fzsymm

(8.13)

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fysymm þ Fzsymm : Eqs. (8.12) and (8.13) collectively define the boundary conditions at vertices. The constraints for faces, edges and vertices should be topped up with necessary constraints against rigid body translations in the y- and z-directions as vjð0;0;0Þ ¼ wjð0;0;0Þ ¼ 0;

(8.14)

whilst that in the x-direction has been eliminated by (8.1). For the correct use of displacements instead of relative displacements, the rigid body translations have to be constrained at the origin. It is important to place the origin at a node on the symmetry plane. 8.2.1.2 Boundary conditions under an antisymmetric loading (any of s0xz and s0xy or their combination) Given the monoclinic nature of the material, symmetric effective strains ε0x ; ε0y ; ε0z and g0yz will vanish under the current antisymmetric loading. If the ‘vice versa’ part of the principle of symmetry as presented early on within the current subsection is put explicitly, the principle of symmetry states: A structure responds in an antisymmetric manner in terms of internal displacement, strain and stress fields to an antisymmetric stimulus in terms of loading condition. By following the similar argument on the continuity but under antisymmetric condition, the boundary conditions on the faces of the half-sized UC can be obtained and summarised as follows, bearing in mind that symmetric effective strains vanish.

233

Further symmetries within a UC

For face x ¼ 0 (excluding edges) vjð0;y;zÞ ¼ wjð0;y;zÞ ¼ 0

denoted as

anti Ujð0;y;zÞ ¼ Fx0 :

(8.15)

The rigid body translations in the y- and z-directions for the UC as a whole have thus been eliminated and no further constraint should be imposed separately in this respect. For face x ¼ a (excluding edges), from (6.66) and the antisymmetric reflection vjða;y;zÞ ¼ ag0xy denoted as wjða;y;zÞ ¼ ag0xz anti None of Fx0

anti Ujða;y;zÞ ¼ Fxa :

(8.16)

anti constrains displacement u. and Fxa Faces y ¼ -b and y ¼ b are not affected by the symmetry. Adapting respective boundary conditions formulated for the whole cell, Eq. (6.67), under the current antisymmetric loading condition results in

ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 0

denoted as

Ujðx;b;zÞ  Ujðx;b;zÞ ¼ Fyanti :

wjðx;b;zÞ  wjðx;b;zÞ ¼ 0 (8.17) Similarly, Eqs. (6.68) lead to the boundary conditions for faces z ¼ -c and z ¼ c (excluding edges) ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

denoted as

Ujz¼c  Ujz¼c ¼ Fzanti :

wjðx;y;cÞ  wjðx;y;cÞ ¼ 0 (8.18) Following the similar argument as in the previous subsection, the boundary conditions at edges and vertices can be obtained. For the edges parallel to the x-axis (excluding vertices) Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fyanti Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fyanti þ Fzanti

(8.19)

Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fzanti : For edges parallel to the y-axis (excluding vertices), from (8.15e8.18)

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Representative Volume Elements and Unit Cells

anti Ujð0;y;cÞ ¼ Fx0

Ujða;y;cÞ  Ujða;y;cÞ ¼ Fzanti

(8.20)

anti Ujða;y;cÞ ¼ Fxa :

For edges parallel to the z-axis (excluding vertices), from (8.15e8.17) anti Ujð0;b;zÞ ¼ Fx0

Ujða;b;zÞ  Ujða;b;zÞ ¼ Fyanti

(8.21)

anti Ujða;b;zÞ ¼ Fxa :

For the similar consideration as in the previous subsection, the boundary conditions at the vertices can be obtained as the logical sums of those at the edges as follows. anti Ujð0;b;cÞ ¼ Fx0 anti Ujða;b;cÞ ¼ Fxa

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fyanti

(8.22)

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fzanti Ujða;b;cÞ  Ujða;b;cÞ ¼ Fyanti þ Fzanti : Another constraint should be imposed to eliminate rigid body translation in the x-direction at the origin ujð0;0;0Þ ¼ 0:

(8.23)

It is worth noting that since symmetric and antisymmetric loading have to be analyzed under different boundary conditions, extra efforts will have to be made in implementing the UC. This is the price to pay for the reduced size of the UC and hence reduced computational cost of the analysis. As far as the material characterisation is concerned, this does not pose any problem because the UC will have to be analyzed under uniaxial or pure shear loading cases individually anyway. The actual difference computationally is that after employing the symmetry, the two analyses involve inverting or decomposing slightly different stiffness matrices due to the different constraints imposed separately. These stiffness matrices are smaller than that of the full size UC. One of the analyses involves four loading cases whilst the other involves two loading cases. Computing cost for any additional loading case on top

Further symmetries within a UC

235

of the first loading case is insignificant. With the full size UC, the stiffness matrix only needs to be inverted or decomposed once, although its size is almost twice as big. Assuming the half sized stiffness matrix takes 1/8 of the computational efforts required for the full sized ones, using the symmetry could reduce the computing time to 1/4. Given this as an idealistic position, as the size of the stiffness matrix of the half UC is in fact a bit larger than half of stiffness matrix of the full size UC, practical computing time for the half UC should be around 1/3 to 1/2 of the full UC. If one is interested in the actual stress distributions under combined loading conditions, the use of such symmetry will prohibit any combination between a symmetric load and an antisymmetric one. This does not prevent the user from employing superposition law to add them together. However, this would place a significant demand on the post-processing of the results. The use of a reflectional symmetry breaks the relationship between one pair of opposite faces, in the present case, those perpendicular to the x-axis. As a result, the newly created face and the existing face in this direction do not have to be subject to any restriction on the matching tessellation when meshing the UC, provided that the mesh generated is reasonably fine, whilst the remaining pairs of faces are still subject to this restriction. This position will change when dealing with rotational symmetries in the next section.

8.2.1.3 Unification of formulation of the boundary conditions for single reflectional symmetry The two types of loading cases are of different natures, yet the presentation of boundary conditions as obtained in the previous subsections can be uni* , F * , F * and F * have been defined for fied as shown in Table 8.2. Terms Fx0 xa y z each loading case in previous subsections individually in terms of the average strains associated with the specific loading case. Symbol ‘*’ in the superscripts can be replaced by ‘symm’ and ‘anti’, respectively, to reproduce the boundary conditions for each of the two loading cases.

8.2.2 Two reflectional symmetries If one can identify yet another reflectional symmetry, e.g. about the plane perpendicular to the y-axis, further reduction of the size of the UC is possible. For the reasons as were elaborated in Section 8.1, the coordinate system will be set with the two symmetry planes as two of the coordinate planes having the origin on the intersection. With two symmetries, the material is effectively orthotropic. There will be neither interaction between effective direct stresses and any effective shear stress, nor between the

236

Representative Volume Elements and Unit Cells

Table 8.2 Unified boundary conditions for the case of single reflectional symmetry about the x-plane Part of the boundary Location Boundary conditions

Faces (excluding edges)

Edges (excluding vertices)

x¼0 x¼a y¼b z¼c parallel to x-axis

* Ujð0;y;zÞ ¼ Fx0 * Ujða;y;zÞ ¼ Fxa

Ujðx;b;zÞ  Ujðx;b;zÞ ¼ Fy* Ujz¼c  Ujz¼c ¼ Fz* Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fy* Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fy* þ Fz* Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fz*

parallel to y-axis

* Ujð0;y;cÞ ¼ Fx0

Ujða;y;cÞ  Ujða;y;cÞ ¼ Fz* * Ujða;y;cÞ ¼ Fxa

parallel to z-axis

* Ujð0;b;zÞ ¼ Fx0

Ujða;b;zÞ  Ujða;b;zÞ ¼ Fy* * Ujða;b;zÞ ¼ Fxa

Vertices

all

* Ujð0;b;cÞ ¼ Fx0 * Ujða;b;cÞ ¼ Fxa

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fy* Ujða;b;cÞ  Ujða;b;cÞ ¼ Fz* (0,0,0)

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fy* þ Fz* vjð0;0;0Þ ¼ wjð0;0;0Þ ¼ 0 if *¼symm ujð0;0;0Þ ¼ 0 if *¼anti

effective shear stresses themselves. In this case, effective stresses will have different symmetry characteristics as specified in Table 8.3. Whilst the size of the UC has been reduced again to a quarter of the full UC, one will have to deal with these four sets of boundary conditions, one for the direct effective stresses and one for each of the three effective shear stresses as independent loading cases, and therefore four analyses involving inverting or decomposing four stiffness matrices will have to be conducted. Despite this, the computational savings should still be sufficiently significant.

237

Further symmetries within a UC

Table 8.3 Nature of symmetry of the effective stresses as loads and displacements as deformation with respect to the two reflectional symmetry transformations. Stimuli/responses x-plane reflection y-plane reflection

Loading cases (effective stresses)

Deformation (displacements)

s0xx ; s0yy ; s0zz s0yz s0xz s0xy u v w

Symmetric Symmetric Antisymmetric Antisymmetric Antisymmetric Symmetric Symmetric

Symmetric Antisymmetric Symmetric Antisymmetric Symmetric Antisymmetric Symmetric

The principle underlying the derivation remains the same as was employed in the previous section, hence only the outcomes will be provided below. 8.2.2.1 Boundary conditions under s0x ; s0y and s0z This loading condition is symmetric for both reflectional symmetries. The symmetry nature of the displacements for each reflectional symmetry can be found in Table 8.3. For face x ¼ 0 (excluding edges), reflection about x-plane gives ujð0;y;zÞ ¼ 0 denoted as

s Ujð0;y;zÞ ¼ Fx0

(8.24)

For face x ¼ a (excluding edges), translation in the x-direction as given in Eqs. (6.66) and reflection about the x-plane lead to ujða;y;zÞ ¼ aε0x

denoted as

s Ujða;y;zÞ ¼ Fxa

(8.25)

s and F s offers any constraint on displacements v As before, none of Fx0 xa and w. Similarly, the presence of the second reflectional symmetry about the yplane, the symmetry conditions on faces y ¼ 0 and y ¼ b and translational symmetry in the y-direction, Eqs. (6.67), lead to the boundary conditions on y ¼ 0 and y ¼ b (excluding edges) as

vjðx;0;zÞ ¼ 0 vjðx;b;zÞ ¼ bε0y

denoted as denoted as

s Ujðx;0;zÞ ¼ Fy0 s Ujðx;b;zÞ ¼ Fyb

s and F s do not constrain displacements u and w. where Fx0 xa

(8.26) (8.27)

238

Representative Volume Elements and Unit Cells

Since faces z ¼ -c and z ¼ c are not affected by the symmetries, the boundary conditions on them (excluding edges) remain the same as (6.68), not affected by the loading condition ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

denoted as

Ujðx;y;cÞ  Ujðx;y;cÞ ¼ Fzs

wjðx;y;cÞ  wjðx;y;cÞ ¼ 2cε0z (8.28) Given the current loading condition, all right hand sides involve direct effective strains only. The boundary conditions for the edges parallel to the x-axis (excluding vertices) reduce from (8.26e8.28) to s Ujðx;0;cÞ ¼ Fy0

Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fzs

(8.29)

s Ujðx;b;cÞ ¼ Fyb

for edges parallel to the y-axis (excluding vertices), from (8.24e8.28) s Ujð0;y;cÞ ¼ Fx0

Ujða;y;cÞ  Ujða;y;cÞ ¼ Fzs Ujða;y;cÞ ¼

(8.30)

s Fxa

and for edges parallel to the z-axis (excluding vertices), from (8.24) e(8.27) s s Ujð0;0;zÞ ¼ Fx0 þ Fy0 s s Ujð0;b;zÞ ¼ Fx0 þ Fyb s s Ujða;b;zÞ ¼ Fxa þ Fyb

(8.31)

s s Ujða;0;zÞ ¼ Fxa þ Fy0

Employing similar considerations, the boundary conditions for the vertices as the logical sums of those at edges, Eqs. (8.28)e(8.30), are obtained as

239

Further symmetries within a UC

s s Ujð0;0;cÞ ¼ Fx0 þ Fy0 s s Ujð0;b;cÞ ¼ Fx0 þ Fyb s s Ujða;0;cÞ ¼ Fxa þ Fy0

(8.32)

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fzs s s Ujða;b;cÞ ¼ Fxa þ Fyb :

The rigid body translations in the x- and y-directions have been eliminated through (8.24) and (8.26), respectively, and the above boundary conditions should only be topped up with one in the z-direction wjð0;0;0Þ ¼ 0:

(8.33)

8.2.2.2 Boundary conditions under s0yz This loading condition is symmetric for reflectional symmetry about the x-plane whilst antisymmetric for reflectional symmetry about the yplane. Refer to Table 8.3 for the corresponding nature of the displacements. From the x-plane reflection (symmetric) and the translation in the x-direction as given in Eq. (6.66), the boundary conditions for faces x ¼ 0 and x ¼ a (excluding edges) are obtained as ujð0;y;zÞ ¼ ujða;y;zÞ ¼ 0

denoted as

yz

Ujð0;y;zÞ ¼ Fx0

and

Ujða;y;zÞ

yz ¼ Fxa ;

(8.34) for faces y ¼ 0 and y ¼ b (excluding edges) from translation in the y-direction and the y-plane reflection (antisymmetric) ujðx;0;zÞ ¼ wjðx;0;zÞ ¼ 0

denoted as

ujðx;b;zÞ ¼ 0 denoted as wjðx;b;zÞ ¼

bg0yz

yz

Ujðx;0;zÞ ¼ Fy0 ;

(8.35)

yz

(8.36)

Ujðx;b;zÞ ¼ Fyb ;

and for faces z ¼ -c and z ¼ c (excluding edges) as given in (6.68) ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

denoted as

Ujðx;y;cÞ  Ujðx;y;cÞ ¼ Fzyz :

wjðx;y;cÞ  wjðx;y;cÞ ¼ 0 (8.37)

240

Representative Volume Elements and Unit Cells

Boundary conditions for the edges parallel to the x-axis (excluding vertices), obtained as the logical sums of boundary conditions on faces y ¼ 0&b in Eqs. (8.35) and (8.36) and z ¼ c in Eq. (8.37), are yz

Ujðx;0;cÞ ¼ Fy0

Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fzyz . Ujðx;b;cÞ ¼

(8.38)

yz Fyb :

Similarly, boundary conditions for edges parallel to the y-axis (excluding vertices) as the logical sums of boundary conditions on faces x ¼ 0&a in Eq. (8.34) and z ¼ c in Eq. (8.37) become yz

Ujð0;y;cÞ ¼ Fx0 ; Ujða;y;cÞ  Ujða;y;cÞ ¼ Fzyz ;

(8.39)

yz Ujða;y;cÞ ¼ Fxa ;

and for edges parallel to the z-axis (excluding vertices) as the logical products of faces x ¼ 0&a in Eq. (8.34) and y ¼ 0&b in Eqs. (8.35) and (8.36) yz

yz

yz

yz

Ujð0;0;zÞ ¼ Fx0 þ Fy0 Ujð0;y;zÞ ¼ Fx0 þ Fyb

(8.40)

yz

yz Ujða;0;zÞ ¼ Fxa þ Fy0

yz

yz Ujða;b;zÞ ¼ Fxa þ Fyb :

Boundary conditions for the vertices as the logical sums of boundary conditions for intersecting edges, from Eqs. (8.38)e(8.40), are as follows yz

yz

yz

yz

Ujð0;0;cÞ ¼ Fx0 þ Fy0 Ujð0;b;cÞ ¼ Fx0 þ Fyb

yz

yz Ujða;0;cÞ ¼ Fxa þ Fy0

(8.41)

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fzyz yz

yz Ujða;b;cÞ ¼ Fxa þ Fyb : yz

yz

It is worth reminding that at x ¼ 0, Fy0 implies Fx0 according to yz yz yz yz Eqs. (8.34) and (8.35). Their logical sum is Fy0 , i.e. Fy0 þ Fx0 ¼ Fy0 . However, for the unification as will be presented in Subsection

241

Further symmetries within a UC

8.2.2.5, they are kept in this form without further simplification in presentation. The rigid body translations in the x- and z-directions have been eliminated through (8.35) and the remaining translation, the one in the y-direction, has to be constrained as vjð0;0;0Þ ¼ 0:

(8.42)

8.2.2.3 Boundary conditions under s0xz This loading condition is antisymmetric for reflectional symmetry about the x-plane whilst symmetric for reflectional symmetry about the y-plane, as shown in Table 8.3, where the symmetry nature of each displacement is also indicated. For face x ¼ 0 (excluding edges) resulting from reflection about x-plane (antisymmetric) vjð0;y;zÞ ¼ wjð0;y;zÞ ¼ 0

denoted as

xz Ujð0;y;zÞ ¼ Fx0 :

(8.43)

For face x ¼ a (excluding edges) from combined translation, Eq. (6.66), and reflection about the x-plane vjða;y;zÞ ¼ 0 denoted as wjða;y;zÞ ¼

ag0xz

xz Ujða;y;zÞ ¼ Fxa :

(8.44)

For faces y ¼ 0 and y ¼ b (excluding edges), the reflection is antisymmetric vjðx;0;zÞ ¼ vjðx;b;zÞ ¼ 0 ¼

denoted as

xz Ujðx;0;zÞ ¼ Fy0

and

Ujðx;b;zÞ

xz Fyb :

(8.45) For faces z ¼ -c and z ¼ c (excluding edges) as directly from (6.68), given the current loading condition ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

denoted as

Ujz¼c  Ujz¼c ¼ Fzxz :

wjðx;y;cÞ  wjðx;y;cÞ ¼ 0 (8.46) For edges parallel to the x-axis (excluding vertices) from Eqs. (8.45) and (8.46)

242

Representative Volume Elements and Unit Cells

xz Ujðx;0;cÞ ¼ Fy0

Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fzxz

(8.47)

xz Ujðx;b;cÞ ¼ Fyb

for edges parallel to the y-axis (excluding vertices) from Eqs. (8.43)e(8.46) xz Ujð0;y;cÞ ¼ Fx0

Ujða;y;cÞ  Ujða;y;cÞ ¼ Fzxz

(8.48)

xz Ujða;y;cÞ ¼ Fxa

and for edges parallel to the z-axis (excluding vertices) from Eqs. (8.44) and (8.45) xz xz Ujð0;0;zÞ ¼ Fx0 þ Fy0 xz xz Ujð0;b;zÞ ¼ Fx0 þ Fyb xz xz Ujða;0;zÞ ¼ Fxa þ Fy0

(8.49)

xz xz Ujða;b;zÞ ¼ Fxa þ Fyb xz and F xz at y ¼ 0 and y¼b imply F xz and F xz , respectively, where both Fx0 xa y0 yb but are nevertheless kept in the expressions above for the subsequent unification. For the vertices as the logical products of the edges, one has xz xz Ujð0;0;cÞ ¼ Fx0 þ Fy0 xz xz Ujð0;b;cÞ ¼ Fx0 þ Fyb xz xz Ujða;0;cÞ ¼ Fxa þ Fy0

(8.50)

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fzxz xz xz Ujða;b;cÞ ¼ Fxa þ Fyb :

The rigid body translations in the y- and z-directions have been eliminated through (8.43) and hence only the translation in the x-direction should be constrained ujð0;0;0Þ ¼ 0:

(8.51)

8.2.2.4 Boundary conditions under s0xy This loading condition is antisymmetric for both reflections concerned, as shown in Table 8.3.

243

Further symmetries within a UC

For face x ¼ 0 (excluding edges) vjð0;y;zÞ ¼ wjð0;y;zÞ ¼ 0

denoted as

xy

Ujð0;y;zÞ ¼ Fx0 :

(8.52)

For face x ¼ a (excluding edges) vjða;y;zÞ ¼ ag0xy denoted as wjða;y;zÞ ¼ 0

xy Ujða;y;zÞ ¼ Fxa :

(8.53)

For face y ¼ 0 (excluding edges) ujðx;0;zÞ ¼ wjðx;0;zÞ ¼ 0;

denoted as

xy

(8.54)

xy

(8.55)

Ujðx;0;zÞ ¼ Fy0 :

For face y ¼ b (excluding edges) ujðx;b;zÞ ¼ wjðx;b;zÞ ¼ 0;

denoted as

Ujðx;b;zÞ ¼ Fyb :

For faces z ¼ -c and z ¼ c (excluding edges) ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

denoted as

Ujðx;y;cÞ  Ujðx;y;cÞ ¼ Fzxy :

wjðx;y;cÞ  wjðx;y;cÞ ¼ 0 (8.56) For edges parallel to the x-axis (excluding vertices) xy

Ujðx;0;cÞ ¼ Fy0

Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fzxy Ujðx;b;cÞ ¼

(8.57)

xy Fyb ;

for edges parallel to the y-axis (excluding vertices) xy

Ujð0;y;cÞ ¼ Fx0

Ujða;y;cÞ  Ujða;y;cÞ ¼ Fzxy

(8.58)

xy Ujða;y;cÞ ¼ Fxa

and for edges parallel to the z-axis (excluding vertices) xy

xy

xy

xy

Ujð0;0;zÞ ¼ Fx0 þ Fy0 Ujð0;b;zÞ ¼ Fx0 þ Fyb

xy

xy Ujða;0;zÞ ¼ Fxa þ Fy0 xy

xy Ujða;b;zÞ ¼ Fxa þ Fyb

(8.59)

244

Representative Volume Elements and Unit Cells

where each of the two terms on the right hand side of the above equations requires w ¼ 0 and their logical sum is not affected by the duplication. The respective boundary conditions for the vertices are obtained as logical sums of those at the edges, which yields xy

xy

xy

xy

Ujð0;0;cÞ ¼ Fx0 þ Fy0 Ujð0;b;cÞ ¼ Fx0 þ Fyb

xy

xy Ujða;0;cÞ ¼ Fxa þ Fy0

(8.60)

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fzxy xy

xy Ujða;b;cÞ ¼ Fxa þ Fyb :

The rigid body translations have all been eliminated through (8.52) and (8.54). The boundary conditions formulated above, including those in the previous subsection can be reduced to 2D problem in a straightforward manner simply by dropping everything associated with the third dimension, i.e. anything to do with w and faces perpendicular to the z-axis.

8.2.2.5 Unification of formulation of the boundary conditions for two reflectional symmetries Similarly to unification of boundary conditions for single reflectional symmetry that was presented in Subsection 8.2.1.3, four sets of boundary conditions obtained in case of two reflectional symmetries can be unified as * , F * , F * , F * and F * in the right hand sides shown in Table 8.4. Terms Fx0 xa y0 yb z of the expressions of boundary conditions have been defined individually for each loading case in Subsections 8.2.2.1-8.2.2.4 in terms of the average strains associated with the specific loading case. Symbol ‘*’ can be replaced by ‘s’, ‘yz’, ‘xz’ and ‘xy’, respectively, to reproduce the boundary conditions for each of the four loading cases,

8.2.3 Three reflectional symmetries With a reflectional symmetry about each of the coordinate planes, the boundary conditions can be obtained as follows, whilst the rigid body translations have been completely eliminated and there is no need to do anything specific about them anymore.

245

Further symmetries within a UC

Table 8.4 Unified boundary conditions in case of two reflectional symmetries about x- and y-planes respectively Part of the boundary Location Boundary conditions

Faces (excluding edges)

Edges (excluding vertices)

x¼0 x¼a y¼0 y¼b z¼c parallel to x-axis

* Ujð0;y;zÞ ¼ Fx0 * Ujða;y;zÞ ¼ Fxa

* Ujðx;0;zÞ ¼ Fy0 * Ujðx;b;zÞ ¼ Fyb Ujðx;y;cÞ  Ujðx;y;cÞ ¼ Fz* * Ujðx;0;cÞ ¼ Fy0

Ujðx;b;cÞ  Ujðx;b;cÞ ¼ Fz* * Ujðx;b;cÞ ¼ Fyb

parallel to y-axis

* Ujð0;y;cÞ ¼ Fx0

Ujða;y;cÞ  Ujða;y;cÞ ¼ Fz* * Ujða;y;cÞ ¼ Fxa

parallel to z-axis

* * Ujð0;0;zÞ ¼ Fx0 þ Fy0 * * Ujð0;b;zÞ ¼ Fx0 þ Fyb * * Ujða;0;zÞ ¼ Fxa þ Fy0 * * Ujða;b;zÞ ¼ Fxa þ Fyb

Vertices

all

* * Ujð0;0;cÞ ¼ Fx0 þ Fy0 * * Ujð0;b;cÞ ¼ Fx0 þ Fyb * * Ujða;0;cÞ ¼ Fxa þ Fy0

Ujða;b;cÞ  Ujða;b;cÞ ¼ Fz* (0,0,0)

* * Ujða;b;cÞ ¼ Fxa þ Fyb : wjð0;0;0Þ ¼ 0 if *¼s; vjð0;0;0Þ ¼ 0 if *¼yz; ujð0;0;0Þ ¼ 0 if *¼xz; none if *¼xy.

8.2.3.1 Boundary conditions under s0x ; s0y and s0z For faces x ¼ 0 and x ¼ a (including edges and vertices) ujð0;y;zÞ ¼ 0

(8.61)

246

Representative Volume Elements and Unit Cells

ujða;y;zÞ ¼ aε0x :

(8.62)

For faces y ¼ 0 and y ¼ b (including edges and vertices) vjðx;0;zÞ ¼ 0

(8.63)

vjðx;b;zÞ ¼ bε0y :

(8.64)

For faces z ¼ 0 and z ¼ c (including edges and vertices) wjðx;y;0Þ ¼ 0

(8.65)

wjðx;y;cÞ ¼ cε0z :

(8.66)

The edges and vertices can be absorbed into the faces without causing any conflict. This represents the simplest case to deal with. It is also a case where one’s intuition is usually sufficient to warrant a correct solution and has naturally been the most popular amongst users. 8.2.3.2 Boundary conditions under s0yz For faces x ¼ 0 and x ¼ a (excluding edges), being symmetric ujð0;y;zÞ ¼ ujða;y;zÞ ¼ 0;

(8.67)

for faces y ¼ 0 and y ¼ b (excluding edges), being antisymmetric ujðx;0;zÞ ¼ wjðx;0;zÞ ¼ 0 ujðx;b;zÞ ¼ 0 wjðx;b;zÞ ¼ bg0yz

(8.68) (8.69)

and for faces z ¼ 0 and z ¼ c (excluding edges), being antisymmetric ujðx;y;0&cÞ ¼ vjðx;y;0&cÞ ¼ 0:

(8.70)

For edges parallel to the x-axis (excluding vertices) ujðx;0;0&cÞ ¼ vjðx;0;0&cÞ ¼ wjðx;0;0&cÞ ¼ 0 ujðx;b;0&cÞ ¼ vjðx;b;0&cÞ ¼ 0

(8.71)

wjðx;b;0&cÞ ¼ bg0yz ; for edges parallel to the y-axis (excluding vertices) ujð0&a;y;0&cÞ ¼ vjð0&a;y;0&cÞ ¼ 0; and for edges parallel to the z-axis (excluding vertices)

(8.72)

Further symmetries within a UC

247

ujð0&a;0;zÞ ¼ wjð0&a;0;zÞ ¼ 0 ujð0&a;b;zÞ ¼ 0

(8.73)

wjð0&a;b;zÞ ¼ bg0yz : For the vertices, one has ujð0&a;0;0&cÞ ¼ vjð0&a;0;0&cÞ ¼ wjð0&a;0;0&cÞ ¼ 0 ujð0&a;b;0&cÞ ¼ vjð0&a;b;0&cÞ ¼ 0

(8.74)

wjð0&a;b;0&cÞ ¼ bg0yz 8.2.3.3 Boundary conditions under s0xz For faces x ¼ 0 and x ¼ a (excluding edges), being antisymmetric vjð0;y;zÞ ¼ wjð0;y;zÞ ¼ 0 vjða;y;zÞ ¼ 0 wjða;y;zÞ ¼ ag0xz :

(8.75) (8.76)

For faces y ¼ 0 and y ¼ b (excluding edges), being symmetric vjðx;0;zÞ ¼ vjðx;b;zÞ ¼ 0:

(8.77)

For faces z ¼ 0 and z ¼ c (excluding edges), being antisymmetric ujðx;y;0&cÞ ¼ vjðx;y;0&cÞ ¼ 0;

(8.78)

For edges parallel to the x-axis (excluding vertices) ujðx;0&b;0&cÞ ¼ vjðx;0&b;0&cÞ ¼ 0;

(8.79)

for edges parallel to the y-axis (excluding vertices) ujð0&a;y;0&cÞ ¼ vjð0&a;y;0&cÞ ¼ 0 wjð0;y;0&cÞ ¼ 0

(8.80)

wjða;y;0&cÞ ¼ ag0xz and for edges parallel to the z-axis (excluding vertices) vjð0&a;0&b;zÞ ¼ 0 wjð0;0&b;zÞ ¼ 0 wjða;0&b;zÞ ¼ ag0xz :

(8.81)

248

Representative Volume Elements and Unit Cells

For the vertices, one has ujð0&a;0&b;0&cÞ ¼ vjð0&a;0&b;0&cÞ ¼ 0 wjð0;0&b;0&cÞ ¼ 0

(8.82)

wjða;0&b;0&cÞ ¼ ag0xz : 8.2.3.4 Boundary conditions under s0xy For faces x ¼ 0 and x ¼ a (excluding edges), being antisymmetric vjð0;y;zÞ ¼ wjð0;y;zÞ ¼ 0 vjða;y;zÞ ¼ ag0xy wjða;y;zÞ ¼ 0:

(8.83) (8.84)

For faces y ¼ 0 and y ¼ b (excluding edges), being antisymmetric ujðx;0&b;zÞ ¼ wjðx;0&b;zÞ ¼ 0:

(8.85)

For faces z ¼ 0 and z ¼ c (excluding edges), being symmetric wjðx;y;0Þ ¼ wjðx;y;cÞ ¼ 0:

(8.86)

For edges parallel to the x-axis (excluding vertices) ujðx;0&b;0&cÞ ¼ wjðx;0&b;0&cÞ ¼ 0;

(8.87)

for edges parallel to the y-axis (excluding vertices) vjð0;y;0&cÞ ¼ 0 vjða;y;0&cÞ ¼ ag0xy

(8.88)

wjð0&a;y;0&cÞ ¼ 0 and for edges parallel to the z-axis (excluding vertices) ujð0&a;0&b;zÞ ¼ 0 vjð0;0&b;zÞ ¼ 0 vjða;0&b;zÞ ¼ ag0xy wjð0&a;0&b;zÞ ¼ 0: For the vertices, one has

(8.89)

Further symmetries within a UC

249

ujð0&a;0&b;0&cÞ ¼ 0 vjð0;0&b;0&cÞ ¼ 0 vjða;0&b;0&cÞ ¼ ag0xy

(8.90)

wjð0&a;0&b;0&cÞ ¼ 0: It can be seen that substantial efforts will have to be made to filter out redundant boundary conditions at edges and vertices when employing three reflectional symmetries in order to reduce the size of the UC. It happens to be that no such redundancy arises under the loading condition of direct stresses at the upper length scale even if the edges and vertices are absorbed into the faces. However, this is true only under this loading condition. When shear loading is involved, redundancies are inevitable if edges and vertices are not dealt with separately. If the FE solver cannot tolerate the redundant boundary conditions, analyses will be prohibited. This must be the reason why shear loading has been avoided in most of the attempts in the literature. With the treatments as presented above, nothing would stop appropriate analyses, provided that the boundary conditions are formulated and implemented systematically. If the notations of various symbols employed to prescribe average strains to the faces of the UC were adopted, the boundary conditions for different loading cases could also be unified as was done previously, but it is waived here for the readers to fill the gap as an exercise.

8.2.4 Various examples of application 8.2.4.1 Application to the 2D UC for square packed UD composites After reducing the formulation in Subsection 8.2.2 from its general 3D form to its 2D counterpart and setting b¼a, it becomes directly applicable to the UC for square packed UD composites as shown in Fig. 8.2 (Li, 1999). It is a quarter of its original size and the computational cost will be reduced to a fraction of its full size model. However, different boundary conditions will have to be imposed under different loading conditions. In the UC as shown in Fig. 8.2, there is in fact a further reflectional symmetry about the dash-dot chain line. Whilst it make perfect geometric sense, the mechanical significance is limited. After imposing a symmetry of this kind, the applicable loading conditions will be limited to equal biaxial stresses (symmetric) and equal biaxial tension and compression (antisymmetric). One might use these to reconstruct a number of different loading cases using the superposition law, but the post-processing can be demanding.

250

Representative Volume Elements and Unit Cells

y

a Matrix

Fibre

x

O Fig. 8.2 The UC for square packed UD composite after the use of reflectional symmetries.

A more practical use of this particular symmetry could to facilitate meshing, as one can partition the domain along the axis and mesh up one part of it, e.g. that shaded part, before copying the mesh to the other part. 8.2.4.2 Application to the UC (UC) for the simple cubic packing (SC) The formulation in Subsection 8.2.3 can be applied in a straightforward manner after setting b¼c¼a to the UC for the simple cubic packing as established in the previous chapter (Subsection 6.4.2.3), provided that the particulate in the UC has the desired reflectional symmetries about the coordinate planes. Assuming it is a perfect sphere, the UC of reduced size will be as shown in Fig. 8.3, where mesh is shown on the surfaces in order to distinguish between faces of different characteristics.

z

y x O Fig. 8.3 The UC for SC packing after the use of reflectional symmetries.

Further symmetries within a UC

251

It should be pointed out that the analyses of UCs of the reduced sizes and that of full-sized UC should produce identical results. For instance, the outcomes from analyses of the UCs as cited above should be the same as those presented in Chapter 7 for their full size counterparts. The differences are strictly in the computational costs at the price of sophistication in implementation. Practically, the employment of a UC of a reduced size as a one-off exercise is usually not advisable because it is not worthwhile to take the trouble unless the UC is very large in size. However, if a UC has been generated for serious applications and is meant to be used repeatedly, for instance, as a means of virtual testing, where thousands of cases need to be tested in a single batch job, any reduction of computational costs will have to be explored.

8.3 Further rotational symmetries to existing translational symmetries Given the relative simplicity of reflectional symmetries as explored in the previous section, they would be the preferred type of symmetries to be employed to reduce the size of the UC whenever they are available. However, in some modern materials, e.g. lattices and textile preforms, as well as in macroscopic structures, such as a turbine rotor, there might be lack of reflectional symmetries whilst obvious symmetries in rotations are present (Li and Reid, 1992). Making use of such a symmetry can also reduce the size of the UC to be analyzed. However, this is a relatively less explored type of symmetry in UCs and hence mistakes can easily be made as reviewed in Chapter 5. Whilst the benefit of reducing computational costs due to reduced size of the model comes at the same price of different boundary conditions for different loading cases, as in the case of reflectional symmetry, it imposes another restriction on the tessellation of the mesh on some of the faces associated with this symmetry. As these faces map back to themselves under the rotational symmetry as the faces of the UC of reduced size, the tessellation on these faces should exhibit the symmetry as defined by the rotation. Whilst this is not excessively demanding in terms of meshing, it is one more issue on top of the challenges of the rest.

8.3.1 One rotational symmetry Take the cuboidal UC as established in Subsection 6.4.2.2 again as a starting point. Assume that there exist a 180 rotational symmetry about the z-axis, which also sets the origin appropriately so that the displacements in this context will be relative to the origin, making them effectively the same as

252

Representative Volume Elements and Unit Cells

absolute ones for the ease of presentation. The material will exhibit a monoclinic behavior with the z-axis as its principal axis (refer to Chapter 3). The UC can then be halved. The way of partitioning is not unique unlike in the case of a reflectional symmetry, and the surface employed for partitioning does not have to be a plane, provided that the surface is 180 rotationally symmetric about the same axis as well. Without loss of generality, the x-plane is selected to fulfill partitioning, and the partition with x  0 will be taken as the UC of reduced size, as sketched in Fig. 8.4a. The rotational symmetry transformation reverses the directions of axes x and y. Therefore, following the similar argument as provided under reflectional symmetry, the symmetric and antisymmetric components of effective stresses, strains and displacements are as listed in Table 8.5. The boundary conditions on the four faces parallel to the x-axis remain the same as defined by Eqs. (6.67) and (6.68) in terms of relative displacement boundary conditions. This is because the rotational symmetry tends to relate the displacements on the x > 0 side of a face to the x < 0 side of the same face or another face. The x < 0 part of the UC will be dropped

y

(a)

(b)

y

P

P O P'

C

x

O P'

x

P''

Fig. 8.4 Rotational symmetry about the z-axis (top view): (a) selection of the partitioning plane and the half of UC to be retained (shaded) in the example considered; (b) schematic illustrating mapping of the corresponding points on the boundary. Table 8.5 Nature of symmetry of the effective stresses as loads and displacements as deformation with respect to the rotational symmetry transformation about the z-axis. Stimuli/responses Rotation about z-axis

Loading cases (effective stresses) Deformation (displacements)

s0xx ; s0yy ; s0zz and s0xy s0xz &s0yz u&v w

Symmetric Antisymmetric Antisymmetric Symmetric

253

Further symmetries within a UC

off the UC after the application of the symmetry. Therefore, the rotational symmetry will not provide any constraints to the these faces of the new UC of a reduced size. Attention will be focused on the newly created face x ¼ 0 and the x¼a face which has lost its paired counterpart. On the x ¼ 0 face, the symmetry transformation maps the y  0 half to the y  0 half, i.e. Pð0; y; zÞ /

P 0 ð0;  y; zÞ

(8.91)

It is therefore essential that these two halves of face x ¼ 0 are tessellated identically according to the above mapping before boundary conditions can be prescribed. The x¼a face is the image of the x¼-a face under the rotational symmetry, referring to Fig. 8.4b, as defined through mapping Pða; y; zÞ

/

P 0 ða;  y; zÞ:

(8.92)

Given the translational symmetry in the x-direction P 0 ða;  y; zÞ /

P 00 ða;  y; zÞ;

(8.93)

it is therefore also essential that the tessellation on face x ¼ a is identical between the y  0 half and the y  0 half satisfying the following mapping. Pða; y; zÞ

/

P 00 ða;  y; zÞ:

(8.94)

8.3.1.1 Boundary conditions under a symmetric loading (any of s0x ; s0y ; s0z and s0xy or their combination) To satisfy the rotational symmetry condition, given the symmetric nature of the loading condition and the nature of corresponding quantities as shown in Table 8.5, according to the derivation in Section 2.5.2, the boundary conditions on the facex ¼ 0 excluding the edges and vertices will have to be ujð0;y;zÞ ¼ ujð0;y;zÞ vjð0;y;zÞ ¼ vjð0;y;zÞ wjð0;y;zÞ ¼ wjð0;y;zÞ

ujð0;y;zÞ þ ujð0;y;zÞ ¼ 0 or

vjð0;y;zÞ þ vjð0;y;zÞ ¼ 0

ðy > 0Þ

wjð0;y;zÞ  wjð0;y;zÞ ¼ 0 (8.95)

with boundary conditions on the line along the axis of rotation being

254

ujð0;0;zÞ ¼ 0 vjð0;0;zÞ ¼ 0

Representative Volume Elements and Unit Cells

ðexcluding endsÞ.

(8.96)

Implied in (8.96) is the elimination of the rigid body translation in the x- and y-directions, and hence no further constraint should be imposed separately in this respect. Although (8.96) is a special case of (8.95) when y ¼ 0, the boundary conditions on line x¼ y¼ 0 are expressed differently from those on face x ¼ 0 as far as FE implementation is concerned, because both sides of y¼0 on face x¼0 are rotationally symmetric about the z-axis and on the axis (i.e. x¼y¼0) a point maps to itself. Line x¼ y ¼ 0 will therefore be considered as an extra edge with its ends as extra vertices. Conditions (8.95) will be imposed to the face excluding edges. Although conditions (8.95) apply for y > 0 as well as y < 0, the outcomes from these two scenarios reproduce each other and hence only one of them is independent. This explains the condition y > 0 for (8.95). After the imposition of boundary conditions (8.95), the dofs on the half of the x ¼ 0 face with y > 0 are eliminated. Since the material is effectively monoclinic, antisymmetric effective strains g0yz and g0xz will vanish in the absence of coupling between the symmetric and antisymmetric deformations under the current symmetric loading. The original boundary conditions between x¼ -a and x¼ a given in Eq. (6.66) become ujða;y;zÞ  ujða;y;zÞ ¼ 2aε0x vjða;y;zÞ  vjða;y;zÞ ¼ 2ag0xy

(8.97)

wjða;y;zÞ  wjða;y;zÞ ¼ 0: Bearing in mind that coordinate y can be positive as well as negative in (8.97), it is clear that (8.97) are equally true if y is replaced by ey. The rotational symmetry requires ujða;y;zÞ ¼ ujða;y;zÞ vjða;y;zÞ ¼ vjða;y;zÞ wjða;y;zÞ ¼ wjða;y;zÞ

ujða;y;zÞ þ ujða;y;zÞ ¼ 0 or

vjða;y;zÞ þ vjða;y;zÞ ¼ 0

(8.98)

wjða;y;zÞ  wjða;y;zÞ ¼ 0:

Eliminating the displacement on the x ¼ -a face from (8.97) and (8.98), one obtains the boundary condition on x¼ a excluding edges and vertices

255

Further symmetries within a UC

ujða;y;zÞ þ ujða;y;zÞ ¼ 2aε0x vjða;y;zÞ þ vjða;y;zÞ ¼ 2ag0xy

ðy > 0Þ

(8.99)

wjða;y;zÞ  wjða;y;zÞ ¼ 0 with ujða;0;zÞ ¼ aε0x

ðexcluding endsÞ.

(8.100)

vjða;0;zÞ ¼ ag0xy Again, line x¼a and y ¼ 0 has also to be considered as an edge with its both ends as vertices. When meshing, the surface tessellation on both sides of these extra edges should be identical, satisfying the respective rotational symmetries. The boundary conditions on the remaining faces are not affected by the rotational symmetry and they are obtained in (6.67) and (6.68) and summarised under the current loading condition as follows. For faces y ¼ -b and y ¼ b (excluding edges) ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 2bε0y

(8.101)

wjðx;b;zÞ  wjðx;b;zÞ ¼ 0 For faces z ¼ -c and z ¼ c (excluding edges) ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

(8.102)

wjðx;y;cÞ  wjðx;y;cÞ ¼ 2cε0z For the edges parallel to the x-axis (excluding vertices) that are not affected by the rotational symmetry, after omitting the average shear strains not involved in the present load case, the boundary conditions are obtained from (6.69) as follows.

256

Representative Volume Elements and Unit Cells

ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 vjðx;b;cÞ  vjðx;b;cÞ ¼ 2bε0y wjðx;b;cÞ  wjðx;b;cÞ ¼ 0 ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 vjðx;b;cÞ  vjðx;b;cÞ ¼ 2bε0y

(8.103)

wjðx;b;cÞ  wjðx;b;cÞ ¼ 2cε0z ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 vjðx;b;cÞ  vjðx;b;cÞ ¼ 0 wjðx;b;cÞ  wjðx;b;cÞ ¼ 2cε0z The four edges parallel to the y -axis, two being the x ¼ 0 face and two on the x¼ a face, are split centrally by the partition plane chosen for the implementation of the rotational symmetry, with the y > 0 half mapping to the y < 0 half for each of the edges. They will therefore be subjected to conditions (8.95) and (8.99), respectively, as a result of the rotational symmetry. In the meantime, two are also found on each of the z¼ -c and z¼ c faces. They should also satisfy (8.102). After eliminating the redundant ones, the boundary conditions for this set of edges (excluding vertices) can be given as follows, bearing in mind that their presentation is not unique. ujð0;y;cÞ þ ujð0;y;cÞ ¼ 0 vjð0;y;cÞ þ vjð0;y;cÞ ¼ 0 wjð0;y;cÞ  wjð0;y;cÞ ¼ 0 ujð0;y;cÞ þ ujð0;y;cÞ ¼ 0 vjð0;y;cÞ þ vjð0;y;cÞ ¼ 0 wjð0;y;cÞ  wjð0;y;cÞ ¼ 2cε0z ujð0;y;cÞ  ujð0;y;cÞ ¼ 0 vjð0;y;cÞ  vjð0;y;cÞ ¼ 0 wjð0;y;cÞ  wjð0;y;cÞ ¼ 2cε0z

ðy > 0Þ

(8.104a)

257

Further symmetries within a UC

where the relationship between edges (0, y, c) and (0, -y, -c) is established from the combined translation from (0, -y, -c) to (0, -y, c) and rotation from (0, -y, c) to (0, y, c); and ujða;y;cÞ þ ujða;y;cÞ ¼ 2aε0x vjða;y;cÞ þ vjða;y;cÞ ¼ 2ag0xy wjða;y;cÞ  wjða;y;cÞ ¼ 0 ujða;y;cÞ þ ujða;y;cÞ ¼ 2aε0x vjða;y;cÞ þ vjða;y;cÞ ¼ 2ag0xy

ðy > 0Þ

(8.104b)

wjða;y;cÞ  wjða;y;cÞ ¼ 2cε0z ujða;y;cÞ  ujða;y;cÞ ¼ 0 vjða;y;cÞ  vjða;y;cÞ ¼ 0 wjða;y;cÞ  wjða;y;cÞ ¼ 2cε0z where combined symmetry transformations have been applied between edges (a, y, c) and (a, -y, -c). The four conventional edges parallel to the z-axis, two on the x ¼ 0 face and two on the x¼ a face, are subjected to conditions (8.95) and (8.99), respectively, as a result of the rotational symmetry, and, as they are also found on each of the y¼ -b and y¼ b faces, they should also satisfy (8.101). After eliminating the redundant conditions, the boundary conditions for the two pairs of conventional edges (excluding vertices) can be given as follows in addition to (8.96) and (8.100), bearing in mind that their presentation is not unique. From Eq. (8.95) for face x ¼ 0 and (8.101) for faces y ¼ b, one obtains ujð0;b;zÞ ¼ 0 vjð0;b;zÞ ¼ bε0y

(8.105a)

wjð0;b;zÞ  wjð0;b;zÞ ¼ 0 whilst from Eq. (8.97) for face x ¼ a and (8.101) for faces y ¼ b, one obtains

258

Representative Volume Elements and Unit Cells

ujða;b;zÞ ¼ aε0x vjða;b;zÞ ¼ ag0xy  bε0y

(8.105b)

wjða;b;zÞ  wjða;b;zÞ ¼ 0: In addition to the four conventional edges parallel to the z-axis, there are two extra edges parallel to the z-axis across the centre of the two faces at x¼0 and x¼a, for which the boundary conditions have already been given in (8.96) and (8.100), respectively. They are nevertheless provided again to be incorporated in the group of edges as follows. ujð0;0;zÞ ¼ 0 vjð0;0;zÞ ¼ 0

ðexcluding endsÞ

ujða;0;zÞ ¼ aε0x vjða;0;zÞ ¼ ag0xy

ðexcluding endsÞ

(8.106)

(8.107)

Each vertex is the intersection of edges in the three directions, hence the boundary conditions for the vertices can be obtained from those for the edges. However, a vertex is also the intersection of three orthogonal faces, and it is slightly easier to derive the boundary conditions for vertices from those of the three intersecting faces. For the vertices as the logical products of the edges ujð0;b;cÞ ¼ 0 vjð0;b;cÞ ¼ bε0y and vjð0;b;cÞ ¼ bε0y wjð0;b;cÞ  wjð0;b;cÞ ¼ 0

(8.108a)

wjð0;b;cÞ  wjð0;b;cÞ ¼ 2cε0z wjð0;b;cÞ  wjð0;b;cÞ ¼ 2cε0z ujða;b;cÞ ¼ aε0x vjða;b;cÞ ¼ ag0xy  bε0y and vjða;b;cÞ ¼ ag0xy þ bε0y wjða;b;cÞ  wjða;b;cÞ ¼ 0 wjða;b;cÞ  wjða;b;cÞ ¼ 2cε0z wjða;b;cÞ  wjða;b;cÞ ¼ 2cε0z

(8.108b)

259

Further symmetries within a UC

ujð0;0;cÞ ¼ 0 vjð0;0;cÞ ¼ 0

(8.108c)

wjð0;0;cÞ  wjð0;0;cÞ ¼ 2cε0z ujða;0;cÞ ¼ aε0x vjða;0;cÞ ¼ ag0xy

(8.108d)

wjða;0;cÞ  wjða;0;cÞ ¼ 2cε0z : The above conditions should be topped up with necessary constraints against rigid body translation in the z-direction wjð0;0;0Þ ¼ 0:

(8.109)

8.3.1.2 Boundary conditions under an antisymmetric loading (any of s0yz and s0xz or their combination) By following the similar argument but under the antisymmetric condition, and bearing in mind the monoclinic nature of the material that implies the absence of coupling between the symmetric and antisymmetric deformations under the current antisymmetric loading, as specified in Table 8.5, the boundary conditions can be obtained as follows. For face x ¼ 0 (excluding edges), being antisymmetric ujð0;y;zÞ  ujð0;y;zÞ ¼ 0 vjð0;y;zÞ  vjð0;y;zÞ ¼ 0

ðy > 0Þ;

(8.110)

wjð0;y;zÞ þ wjð0;y;zÞ ¼ 0 with an extra edge at x ¼ 0 and y ¼ 0 under the boundary condition wjð0;0;zÞ ¼ 0

ðexcluding endsÞ.

(8.111)

For face x ¼ a (excluding edges), being antisymmetric, in conjunction with Eq. (6.66)

260

Representative Volume Elements and Unit Cells

ujða;y;zÞ  ujða;y;zÞ ¼ 0 vjða;y;zÞ  vjða;y;zÞ ¼ 0

ðy > 0Þ;

(8.112)

wjða;y;zÞ þ wjða;y;zÞ ¼ 2ag0xz with the following on the extra edge wjða;0;zÞ ¼ ag0xz

ðexcluding endsÞ.

(8.113)

For faces y ¼ -b and y ¼ b (excluding edges), Eq. (6.67) hold as ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 0

(8.114)

wjðx;b;zÞ  wjðx;b;zÞ ¼ 2bg0yz : For faces z ¼ -c and z ¼ c (excluding edges), Eq. (6.68) hold as ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

(8.115)

wjðx;y;cÞ  wjðx;y;cÞ ¼ 0: Condition (8.111) has eliminated rigid body translation in the zdirection, and hence no further constraint should be imposed in this respect. For edges parallel to the x-axis (excluding vertices) the boundary conditions can be obtained from Eq. (8.114) for y ¼ b and (8.115) for z ¼ c as their logical sums ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 vjðx;b;cÞ  vjðx;b;cÞ ¼ 0 vjðx;b;cÞ  vjðx;b;cÞ ¼ 0

(8.116)

wjðx;b;cÞ  wjðx;b;cÞ ¼ 0 wjðx;b;cÞ  wjðx;b;cÞ ¼ 2bg0yz : For edges parallel to the y-axis (excluding vertices) as intersections of faces x ¼ 0 (8.110) and for z ¼ c (8.115)

261

Further symmetries within a UC

ujð0;y;cÞ  ujð0;y;cÞ ¼ 0 vjð0;y;cÞ  vjð0;y;cÞ ¼ 0 wjð0;y;cÞ þ wjð0;y;cÞ ¼ 0 ujð0;y;cÞ  ujð0;y;cÞ ¼ 0 vjð0;y;cÞ  vjð0;y;cÞ ¼ 0

ðy > 0Þ

(8.117a)

wjð0;y;cÞ þ wjð0;y;cÞ ¼ 0 ujð0;y;cÞ  ujð0;y;cÞ ¼ 0 vjð0;y;cÞ  vjð0;y;cÞ ¼ 0 wjð0;y;cÞ  wjð0;y;cÞ ¼ 0 where combined symmetry transformations have been applied between edges (x ¼ 0, y, z¼ c) and (x ¼ 0, -y, z¼ -c). Those as intersections of x ¼ a (8.112) and for z ¼ c (8.115) are ujða;y;cÞ  ujða;y;cÞ ¼ 0 vjða;y;cÞ  vjða;y;cÞ ¼ 0 wjða;y;cÞ þ wjða;y;cÞ ¼ 0 ujða;y;cÞ  ujða;y;cÞ ¼ 0 vjða;y;cÞ  vjða;y;cÞ ¼ 0

ðy > 0Þ

(8.117b)

wjða;y;cÞ þ wjða;y;cÞ ¼ 0 ujða;y;cÞ  ujða;y;cÞ ¼ 0 vjða;y;cÞ  vjða;y;cÞ ¼ 0 wjða;y;cÞ  wjða;y;cÞ ¼ 0 where combined symmetry transformations have been applied between edges (a, y, c) and (a, -y,-c). For edges (excluding vertices) parallel to the z-axis, in addition to (8.111) and (8.113), the boundary conditions can be obtained from Eq. (8.110), for face x ¼ 0 and (8.114), for faces y ¼ b, as

262

Representative Volume Elements and Unit Cells

ujð0;b;zÞ  ujð0;b;zÞ ¼ 0 vjð0;b;zÞ  vjð0;b;zÞ ¼ 0

(8.118a)

wjð0;b;zÞ ¼ wjð0;b;zÞ ¼ bg0yz whilst from Eq. (8.112) for face x ¼ a and (8.114) for faces y ¼ b, one obtains ujða;b;zÞ  ujða;b;zÞ ¼ 0 vjða;b;zÞ  vjða;b;zÞ ¼ 0

(8.118b)

wjða;b;zÞ ¼ ag0xz  bg0yz : For the vertices as the logical products of intersecting edges ujð0;b;cÞ  ujð0;b;cÞ ¼ 0 ujð0;b;cÞ  ujð0;b;cÞ ¼ 0 vjð0;b;cÞ  vjð0;b;cÞ ¼ 0 vjð0;b;cÞ  vjð0;b;cÞ ¼ 0

(8.119a)

wjð0;b;cÞ ¼ bg0yz wjð0;b;cÞ ¼ bg0yz ujða;b;cÞ  ujða;b;cÞ ¼ 0 ujða;b;cÞ  ujða;b;cÞ ¼ 0 vjða;b;cÞ  vjða;b;cÞ ¼ 0 vjða;b;cÞ  vjða;b;cÞ ¼ 0

(8.119b)

wjða;b;cÞ ¼ ag0xz þ bg0yz wjða;b;cÞ ¼ ag0xz  bg0yz ujð0;0;cÞ  ujð0;0;cÞ ¼ 0 vjð0;0;cÞ  vjð0;0;cÞ ¼ 0 wjð0;0;cÞ ¼ 0

(8.119c)

Further symmetries within a UC

263

ujða;0;cÞ  ujða;0;cÞ ¼ 0 vjða;0;cÞ  vjða;0;cÞ ¼ 0

(8.119d)

wjða;0;cÞ ¼ ag0xz : The above boundary conditions should be topped up with necessary constraints against rigid body translations in the x- and y-directions, respectively ujð0;0;0Þ ¼ vjð0;0;0Þ ¼ 0:

(8.120)

The boundary conditions formulated above can be reduced to 2D problem in a straightforward manner simply by dropping everything associated with the third dimension, i.e. anything to do with w and faces perpendicular to the z-axis.

8.3.2 Two rotational symmetries If there exists another 180 rotational symmetry about the x-axis without loss of generality, in addition to the one about the z-axis as addressed in the previous subsection, the size of the UC can be further reduced. With the origin of the coordinate system placed at the intersection of the two axes of rotational symmetries, relative and absolute displacements are virtually the same. Take a further partition at z ¼ 0, as shown in Fig. 8.5(a). With two symmetries, the material is effectively orthotropic and all interactions between effective direct stresses and any shear stress, and between the effective shear stresses themselves disappear. The symmetric and antisymmetric components of effective stresses and displacements are listed in Table 8.6. Displacement boundary conditions can be derived based on the mapping as shown in Fig. 8.5(a) and (b) in the similar fashion as elaborated in the previous section. The outcomes are presented below. 8.3.2.1 Boundary conditions under s0x ; s0y and s0z As was for the rotational symmetry about the z-axis investigated previously, the direct stresses at the upper length scale as the loading condition are always symmetric. Face x ¼ 0 was created when dealing with the rotational symmetry about the z-axis. When partition is made using the z ¼ 0 plane to introduce the rotational symmetry about the x-axis, the boundary conditions on the x ¼ 0 face (excluding edges) will not be affected whilst losing the half of the volume with z < 0, except at the edge at z ¼ 0 as will be dealt with later.

264

Representative Volume Elements and Unit Cells

z

(a)

(b)

z

P

P''

P O

y

y

O

P' P' Fig. 8.5 A further rotational symmetry about the x-axis (viewed from the positive x-direction) to an existing one about the z-axis and an UC of further reduced size: (a) selection of the partitioning plane and the half of UC to be retained (shaded) in the example considered; (b) schematic illustrating mapping of the corresponding points on the boundary. Table 8.6 Nature of symmetry of the effective stresses as loads and displacements as deformation with respect to the two reflectional symmetry transformations. Stimuli/responses x-axis rotation z- axis rotation

Loading cases (effective stresses) Deformation (displacements)

s0xx ; s0yy ; s0zz s0yz s0xz s0xy u v w

Symmetric Symmetric Antisymmetric Antisymmetric Symmetric Antisymmetric Antisymmetric

Symmetric Antisymmetric Antisymmetric Symmetric Antisymmetric Antisymmetric Symmetric

ujð0;y;zÞ þ ujð0;y;zÞ ¼ 0 vjð0;y;zÞ þ vjð0;y;zÞ ¼ 0

ðy > 0Þ

(8.121)

wjð0;y;zÞ  wjð0;y;zÞ ¼ 0 with an extra edge (parallel to the z-axis) ujð0;0;zÞ ¼ 0 vjð0;0;zÞ ¼ 0

ðexcluding endsÞ.

(8.122)

265

Further symmetries within a UC

Similar argument applies to face x ¼ a (excluding edges), resulting in the following boundary conditions: ujða;y;zÞ þ ujða;y;zÞ ¼ 2aε0x vjða;y;zÞ þ vjða;y;zÞ ¼ 0

ðy > 0Þ

(8.123)

wjða;y;zÞ  wjða;y;zÞ ¼ 0 with an extra edge (parallel to the z-axis) ujða;0;zÞ ¼ aε0x

ðexcluding endsÞ.

(8.124)

vjða;0;zÞ ¼ 0 The faces y ¼ -b and y ¼ b are both retained in the UC of reduced size after employing the two rotational symmetry transformations as considered here. These two faces are not affected by either of the rotational symmetries. Therefore, the boundary conditions (6.67) apply in this case (excluding edges) after reflecting the current loading condition. ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 2bε0y

(8.125)

wjðx;b;zÞ  wjðx;b;zÞ ¼ 0: Face z ¼ 0 is created to apply the new rotational symmetry about the xaxis. The boundary conditions on this face (excluding edges) are as follows: ujðx;y;0Þ  ujðx;y;0Þ ¼ 0 vjðx;y;0Þ þ vjðx;y;0Þ ¼ 0

ðy > 0Þ.

(8.126)

wjðx;y;0Þ þ wjðx;y;0Þ ¼ 0 with an extra edge at (x, 0, 0) (parallel to the x-axis) where the boundary conditions are vjðx;0;0Þ ¼ 0 wjðx;0;0Þ ¼ 0

ðexcluding endsÞ.

(8.127)

For face z ¼ c (excluding edges), following the consideration as depicted in Fig. 8.5(a) along the same line as when Eqs. (8.99) were obtained previously, one has

266

Representative Volume Elements and Unit Cells

ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ þ vjðx;y;cÞ ¼ 0

ðy > 0Þ

(8.128)

wjðx;y;cÞ þ wjðx;y;cÞ ¼ 2cε0z with an extra edge at (x, 0, c) (parallel to the x-axis), where the boundary conditions are vjðx;0;cÞ ¼ 0 ðexcluding the endsÞ. wjðx;0;cÞ ¼

(8.129)

cε0z

The rigid body translations have all been eliminated. Apparently, in order to implement the rotational symmetry about the x-axis with the partitioning plane as chosen, the z ¼ 0 and z¼ c faces should be tessellated symmetrically about axes y¼z¼0 and y¼0 and z¼c as was the case for the symmetry about the z-axis previously. For the edges parallel to the x-axis (excluding vertices), in addition to (8.127) and (8.129) for the extra edges, combining Eq. (8.125) for faces y ¼ b and (8.126) for face z ¼ 0, one obtains ujðx;b;0Þ  ujðx;b;0Þ ¼ 0 vjðx;b;0Þ ¼ bε0y

(8.130a)

wjðx;b;0Þ ¼ 0 whilst from Eq. (8.125) for faces y ¼ b and (8.128) for face z ¼ c, the boundary conditions are ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 vjðx;b;cÞ ¼ bε0y

(8.130b)

wjðx;b;cÞ ¼ cε0z : For the edges parallel to the y-axis (excluding vertices and intersections with y ¼ 0), combining Eq. (8.121) for face x ¼ 0 and (8.126) for z ¼ 0, the boundary conditions are obtained as ujð0;y;0Þ ¼ ujð0;y;0Þ ¼ 0 vjð0;y;0Þ þ vjð0;y;0Þ ¼ 0 wjð0;y;0Þ ¼ wjð0;y;0Þ ¼ 0

ðy > 0Þ

(8.131a)

267

Further symmetries within a UC

from Eq. (8.121) for face x ¼ 0 and (8.128) for face z ¼ c ujð0;y;cÞ ¼ ujð0;y;cÞ ¼ 0 vjð0;y;cÞ þ vjð0;y;cÞ ¼ 0

ðy > 0Þ

(8.131b)

wjð0;y;cÞ ¼ wjð0;y;cÞ ¼ cε0z from Eq. (8.123) for face x¼ a and (8.126) for face z ¼ 0 ujða;y;cÞ ¼ ujða;y;0Þ ¼ aε0x vjða;y;cÞ þ vjða;y;0Þ ¼ 0

ðy > 0Þ

(8.131c)

wjða;y;cÞ ¼ wjða;y;0Þ ¼ 0 and from Eq. (8.121) for face x¼ a and (8.128) for face z ¼ c ujða;y;cÞ ¼ ujða;y;cÞ ¼ aε0x vjða;y;cÞ þ vjða;y;cÞ ¼ 0

ðy > 0Þ

(8.131d)

wjða;y;cÞ ¼ wjða;y;cÞ ¼ cε0z : For the edges parallel to the z-axis (excluding vertices), in addition to extra edges where the boundary conditions are given in (8.122) and (8.124), from Eq. (8.121) for face x ¼ 0 and (8.125) for faces y ¼ b, the boundary conditions are obtained as ujð0;b;zÞ ¼ 0 vjð0;b;zÞ ¼ bε0y

(8.132a)

wjð0;b;zÞ  wjð0;b;zÞ ¼ 0 from Eq. (8.123) for face x ¼ a and (8.125) for faces y ¼ b ujða;b;zÞ ¼ aε0x vjða;b;zÞ ¼ bε0y

(8.132b)

wjða;b;zÞ  wjða;b;zÞ ¼ 0: For the conventional eight vertices as intersections of three orthogonal faces, the boundary conditions as the logical sums of those for the edges are

268

Representative Volume Elements and Unit Cells

ujð0;b;0Þ ¼ 0 vjð0;b;0Þ ¼ bε0y

(8.133a)

wjð0;b;0Þ ¼ 0 ujð0;b;cÞ ¼ 0 vjð0;b;cÞ ¼ bε0z

(8.133b)

wjð0;b;cÞ ¼ cε0z ujða;b;0Þ ¼ aε0x vjða;b;0Þ ¼ bε0y

(8.133c)

wjða;b;0Þ ¼ 0 ujða;b;cÞ ¼ aε0x vjða;b;cÞ ¼ bε0y

(8.133d)

wjða;b;cÞ ¼ cε0z : The four extra edges as introduced through (8.121), (8.123), (8.126) and (8.128) form a frame in the y ¼ 0 plane and the boundary conditions for the four vertices are ujð0;0;0Þ ¼ 0 vjð0;0;0Þ ¼ 0

(8.133e)

wjð0;0;0Þ ¼ 0 ujð0;0;cÞ ¼ 0 vjð0;0;cÞ ¼ 0

(8.133f)

wjð0;0;cÞ ¼ cε0z ujða;0;0Þ ¼ aε0x vjða;0;0Þ ¼ 0 wjða;0;0Þ ¼ 0

(8.133g)

269

Further symmetries within a UC

ujða;0;cÞ ¼ aε0x vjða;0;cÞ ¼ 0

(8.133h)

wjða;0;cÞ ¼ cε0z The rigid body translations have all been eliminated. 8.3.2.2 Boundary conditions under s0yz According to Table 8.6, loading case s0yz is antisymmetric for the rotation about the z-axis as has been addressed previously when face x ¼ 0 was created. It is symmetric for the rotation about the x-axis. Face z ¼ 0 has been generated to take advantage of this symmetry. For face x ¼ 0, according to Subsection 2.5.2, bearing in mind the symmetry nature of the displacements under this loading condition as given in Table 8.6, the antisymmetric symmetry conditions about the z-axis are ujð0;y;zÞ  ujð0;y;zÞ ¼ 0 vjð0;y;zÞ  vjð0;y;zÞ ¼ 0

ðy > 0Þ

(8.134)

wjð0;y;zÞ þ wjð0;y;zÞ ¼ 0 which apply to this face with the edges excluded. In addition, the extra edge at x ¼ 0 and y ¼ 0 should be constrained as wjð0;0;zÞ ¼ 0

ðexcluding the endsÞ.

(8.135)

Boundary conditions on face x ¼ a (excluding edges) incorporate both the rotational symmetry about the z-axis and translational symmetry in the x-direction, with the latter yielding boundary conditions given by Eq. (6.66), resulting in ujða;y;zÞ  ujða;y;zÞ ¼ 0 vjða;y;zÞ  vjða;y;zÞ ¼ 0

ðy > 0Þ

(8.136)

wjða;y;zÞ þ wjða;y;zÞ ¼ 0 with constraint on the extra edge being wjða;0;zÞ ¼ 0 ðexcluding the endsÞ.

(8.137)

Faces y ¼ -b and y ¼ b are not affected by the rotational symmetries and hence boundary conditions on these faces (excluding edges) are obtained from Eq. (6.67) as follows:

270

Representative Volume Elements and Unit Cells

ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 0

(8.138)

wjðx;b;zÞ  wjðx;b;zÞ ¼ 2bg0yz : Face z ¼ 0 was created through the rotational symmetry about the x-axis hence the loading case s0yz is symmetric under the symmetry transformation, which results in boundary conditions on this face (excluding edges) as follows: ujðx;y;0Þ  ujðx;y;0Þ ¼ 0 vjðx;y;0Þ þ vjðx;y;0Þ ¼ 0

(8.139)

wjðx;y;0Þ þ wjðx;y;0Þ ¼ 0 with the following constraints on the extra edges: vjðx;0;0Þ ¼ 0 wjðx;0;0Þ ¼ 0

ðexcluding the endsÞ.

(8.140)

Boundary conditions on face z ¼ c (excluding edges) require incorporating of both the rotational symmetry about the x-axis and translational symmetry in the z-direction as given in Eq. (6.68), which results in ujðx;y;cÞ  ujðx;y;cÞ ¼ 0 vjðx;y;cÞ þ vjðx;y;cÞ ¼ 0

(8.141)

wjðx;y;cÞ þ wjðx;y;cÞ ¼ 0 with vjðx;0;cÞ ¼ 0 wjðx;0;cÞ ¼ 0

ðexcluding the endsÞ.

(8.142)

For the edges parallel to the x-axis (excluding vertices), in addition to (8.140) and (8.142) for the extra edges, bearing in mind the symmetry nature of the displacements under this loading condition as given in Table 8.6, the boundary conditions for the conventional edges are obtained from Eqs. (8.138) and(8.139) as ujðx;b;0Þ  ujðx;b;0Þ ¼ 0 vjðx;b;0Þ ¼ 0 wjðx;b;0Þ ¼ bg0yz and from (8.138) and (8.141) as

(8.143a)

271

Further symmetries within a UC

ujðx;b;cÞ  ujðx;b;cÞ ¼ 0 vjðx;b;cÞ ¼ 0

(8.143b)

wjðx;b;cÞ ¼ bg0yz : For the edges parallel to the y-axis (excluding vertices and intersections with y ¼ 0), from Eqs. (8.134) and (8.139) ujð0;y;0Þ  ujð0;y;0Þ ¼ 0 vjð0;y;0Þ ¼ vjð0;y;0Þ ¼ 0

ðy > 0Þ

(8.144a)

wjð0;y;0Þ þ wjð0;y;0Þ ¼ 0 from Eqs. (8.134) and (8.141) ujð0;y;cÞ  ujð0;y;cÞ ¼ 0 vjð0;y;cÞ ¼ vjð0;y;cÞ ¼ 0

ðy > 0Þ

(8.144b)

wjð0;y;cÞ þ wjð0;y;cÞ ¼ 0 from Eqs. (8.136) and (8.139) ujða;y;0Þ  ujða;y;0Þ ¼ 0 vjða;y;0Þ ¼ vja;y;0 ¼ 0

ðy > 0Þ

(8.144c)

wjða;y;0Þ þ wjða;y;0Þ ¼ 0 and from Eqs. (8.136) and (8.141) ujða;y;cÞ  ujða;y;cÞ ¼ 0 vjða;y;cÞ ¼ vjða;y;cÞ ¼ 0

ðy > 0Þ.

(8.144d)

wjða;y;cÞ þ wjða;y;cÞ ¼ 0 For the edges parallel to the z-axis (excluding vertices), in addition to those for extra edges (8.135) and (8.137), the boundary conditions for the four conventional edges are obtained and from Eqs. (8.134) and (8.138) as ujð0;b;zÞ  ujð0;b;zÞ ¼ 0 vjð0;b;zÞ  vjð0;b;zÞ ¼ 0 wjð0;b;zÞ ¼ bg0yz and from Eqs. (8.136) and (8.138)

(8.145a)

272

Representative Volume Elements and Unit Cells

ujða;b;zÞ  ujða;b;zÞ ¼ 0 vjða;b;zÞ  vjða;b;zÞ ¼ 0

(8.145b)

wjða;b;zÞ ¼ bg0yz : For the conventional eight vertices, the boundary conditions can be obtained as the logical sums of those of the intersecting edges ujð0;b;0Þ  ujð0;b;0Þ ¼ 0 vjð0;b;0Þ ¼ 0

(8.146a)

wjð0;b;0Þ ¼ bg0yz ujða;b;0Þ  ujða;b;0Þ ¼ 0 vjða;b;0Þ ¼ 0

(8.146b)

wjða;b;0Þ ¼ bg0yz ujð0;b;cÞ  ujð0;b;cÞ ¼ 0 vjð0;b;cÞ ¼ 0

(8.146c)

wjð0;b;cÞ ¼ bg0yz ujða;b;cÞ  ujða;b;cÞ ¼ 0 vjða;b;cÞ ¼ 0

(8.146d)

wjða;b;cÞ ¼ bg0yz : The boundary conditions for the four vertices as a part of the four extra edges as introduced through (8.135), (8.137), (8.140) and (8.142) are vjð0;0;0Þ ¼ 0 wjð0;0;0Þ ¼ 0 vjða;0;0Þ ¼ 0 wjða;0;0Þ ¼ 0 vjð0;0;cÞ ¼ 0 wjð0;0;cÞ ¼ 0 vjða;0;cÞ ¼ 0 wjða;0;cÞ ¼ 0:

(8.146e)

273

Further symmetries within a UC

The above boundary conditions have to be topped up with a constraint to eliminate rigid body translation in the x-direction ujð0;0;0Þ ¼ 0:

(8.147)

8.3.2.3 Boundary conditions under s0xz This loading condition is antisymmetric for both rotational symmetries concerned. Face x ¼ 0 was created through the rotational symmetry about the z-axis and the loading case s0xz is antisymmetric. This results in boundary conditions as follows, bearing in mind the symmetry nature of the displacements under this loading condition as given in Table 8.6. ujð0;y;zÞ  ujð0;y;zÞ ¼ 0 vjð0;y;zÞ  vjð0;y;zÞ ¼ 0

ðy > 0Þ

(8.148)

wjð0;y;zÞ þ wjð0;y;zÞ ¼ 0 which apply to this face with the edges excluded. Extra edge should be constrained as wjð0;0;zÞ ¼ 0

ðexcluding the endsÞ.

(8.149)

Boundary conditions imposed on face x ¼ a (excluding edges) incorporate both the rotational symmetry about the z-axis and translational symmetry in the x-direction as given in Eq. (6.66), hence ujða;y;zÞ  ujða;y;zÞ ¼ 0 vjða;y;zÞ  vjða;y;zÞ ¼ 0

ðy > 0Þ

(8.150)

wjða;y;zÞ þ wjða;y;zÞ ¼ 2ag0xz with wjða;0;zÞ ¼ ag0xz

ðexcluding the endsÞ

(8.151)

For faces y ¼ -b and y ¼ b (excluding edges), boundary conditions are obtained from Eq. (6.67), after reflecting the current loading condition, as ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 0 wjðx;b;zÞ  wjðx;b;zÞ ¼ 0:

(8.152)

274

Representative Volume Elements and Unit Cells

Face z ¼ 0 (excluding edges) was created through the rotational symmetry about the x-axis and the loading case s0xz is antisymmetric under the symmetry transformation, hence ujðx;y;0Þ þ ujðx;y;0Þ ¼ 0 vjðx;y;0Þ  vjðx;y;0Þ ¼ 0

ðy > 0Þ

(8.153)

ðexcluding the endsÞ.

(8.154)

wjðx;y;0Þ  wjðx;y;0Þ ¼ 0 with ujðx;0;0Þ ¼ 0

Face z ¼ c (excluding edges) have both the rotational symmetry about the x-axis and the translational symmetry in the z-direction as given in Eq. (6.68) incorporated as ujðx;y;cÞ þ ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

ðy > 0Þ

(8.155)

ujðx;0;cÞ ¼ 0 ðexcluding the endsÞ.

(8.156)

wjðx;y;cÞ  wjðx;y;cÞ ¼ 0 with

For the conventional edges parallel to the x-axis (excluding vertices), in addition to (8.154) and (8.156) for the extra edges, the boundary conditions are the logical sums of Eq. (8.152) for faces y ¼ b and (8.153) for face z ¼ 0. ujðx;b;0Þ ¼ 0 vjðx;b;0Þ  vjðx;b;0Þ ¼ 0

(8.157a)

wjðx;b;0Þ  wjðx;b;0Þ ¼ 0 and of conditions given by Eq. (8.152) for faces y ¼ b and (8.155) for face z¼c ujðx;b;cÞ ¼ 0 vjðx;b;cÞ  vjðx;b;cÞ ¼ 0 wjðx;b;cÞ  wjðx;b;cÞ ¼ 0:

(8.157b)

275

Further symmetries within a UC

For the edges parallel to the y-axis (excluding vertices and the intersections with y ¼ 0), the boundary conditions can be obtained from Eq. (8.148) for face x ¼ 0, (8.153) for faces z ¼ 0 and (8.155) for z ¼ c as ujð0;y;0Þ ¼ ujð0;y;0Þ ¼ 0 vjð0;y;0Þ  vjð0;y;0Þ ¼ 0 wjð0;y;0Þ ¼ wjð0;y;0Þ ¼ 0 ujð0;y;cÞ ¼ ujð0;y;cÞ ¼ 0

(8.158a)

vjð0;y;cÞ  vjð0;y;cÞ ¼ 0 wjð0;y;cÞ ¼ wjð0;y;cÞ ¼ 0 and from Eq. (8.150) for face x ¼ a, (8.153) for faces z ¼ 0 and (8.155) for z ¼ c as ujða;y;0Þ ¼ ujða;y;0Þ ¼ 0 vjða;y;0Þ  vjða;y;0Þ ¼ 0 wjða;y;0Þ ¼ wjða;y;0Þ ¼ ag0xz ujða;y;cÞ ¼ ujða;y;cÞ ¼ 0

(8.158b)

vjða;y;cÞ  vjða;y;cÞ ¼ 0 wjða;y;cÞ ¼ wjða;y;cÞ ¼ ag0xz : For the conventional edges parallel to the z-axis (excluding vertices), in addition to (8.148) and (8.150), the boundary conditions are obtained from Eqs. (8.149) and (8.151) for faces x ¼ 0 and x ¼ a, and (8.152) for faces y ¼ b as follows ujð0;b;zÞ  ujð0;b;zÞ ¼ 0 vjð0;b;zÞ  vjð0;b;zÞ ¼ 0

(8.159a)

wjð0;b;zÞ ¼ 0 ujða;b;zÞ  ujða;b;zÞ ¼ 0 vjða;b;zÞ  vjða;b;zÞ ¼ 0

(8.159b)

wjða;b;zÞ ¼ ag0xz : For the vertices as the logical products of the intersecting edges

276

Representative Volume Elements and Unit Cells

ujð0;b;0Þ ¼ 0 vjð0;b;0Þ  vjð0;b;0Þ ¼ 0

(8.160a)

wjð0;b;0Þ ¼ 0 ujð0;b;cÞ ¼ 0 vjð0;b;cÞ  vjð0;b;cÞ ¼ 0

(8.160b)

wjð0;b;cÞ ¼ 0 ujða;b;0Þ ¼ 0 vjða;b;0Þ  vjða;b;0Þ ¼ 0

(8.160c)

wjða;b;0Þ ¼ ag0xz ujða;b;cÞ ¼ 0 vjða;b;cÞ  vjða;b;cÞ ¼ 0

(8.160d)

wjða;b;cÞ ¼ ag0xz : The boundary conditions for the four vertices as a part of the four extra edges as introduced through (8.149), (8.151), (8.154) and (8.156) are ujð0;0;0Þ ¼ 0 wjð0;0;0Þ ¼ 0 ujð0;0;cÞ ¼ 0 wjð0;0;cÞ ¼ 0 ujða;0;0Þ ¼ 0

(8.160e)

wjða;0;0Þ ¼ ag0xz ujða;0;cÞ ¼ 0 wjða;0;cÞ ¼ ag0xz : The above boundary conditions have to be topped up with a constraint to eliminate rigid body translation in the y-direction vjð0;0;0Þ ¼ 0:

(8.161)

277

Further symmetries within a UC

8.3.2.4 Boundary conditions under s0xy As specified in Table 8.6, loading case s0xy is symmetric for the rotation about the z-axis and antisymmetric for the rotation about the x-axis. For face x ¼ 0, which is associated with the rotation about the z-axis, bearing in mind the symmetry nature of the displacements under this loading condition as given in Table 8.6, the symmetry boundary conditions on this face (excluding edges) are as follows ujð0;y;zÞ þ ujð0;y;zÞ ¼ 0 vjð0;y;zÞ þ vjð0;y;zÞ ¼ 0

ðy > 0Þ

(8.162)

wjð0;y;zÞ  wjð0;y;zÞ ¼ 0 with the constraints on the extra edge being ujð0;0;zÞ ¼ 0 vjð0;0;zÞ ¼ 0

ðexcluding the endsÞ.

(8.163)

Boundary conditions on face x ¼ a (excluding edges) have both the rotational symmetry about the z-axis and translational symmetry in the x-direction (6.66) incorporated as ujða;y;zÞ ¼ ujða;y;zÞ ¼ 0 vjða;y;zÞ ¼ vjða;y;zÞ ¼ ag0xy

ðy > 0Þ

(8.164)

wjða;y;zÞ  wjða;y;zÞ ¼ 0 with ujða;0;zÞ ¼ 0 ðexcluding the endsÞ.

(8.165)

vjða;0;zÞ ¼ ag0xy Faces y ¼ -b and y ¼ b (excluding edges) are not affected by either of the rotational symmetries, and hence (6.67) remains valid after reflecting the current loading condition ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 0 wjðx;b;zÞ  wjðx;b;zÞ ¼ 0:

(8.166)

278

Representative Volume Elements and Unit Cells

Face z ¼ 0 was created through the rotational symmetry about the xaxis, hence the loading case s0xy is antisymmetric under the symmetry transformation. Referring to the symmetry nature of displacements as specified in Table 8.6, the boundary conditions on this face (excluding edges) become ujðx;y;0Þ þ ujðx;y;0Þ ¼ 0 vjðx;y;0Þ  vjðx;y;0Þ ¼ 0

ðy > 0Þ

(8.167)

wjðx;y;0Þ  wjðx;y;0Þ ¼ 0 with ujðx;0;0Þ ¼ 0

ðexcluding the endsÞ

(8.168)

Boundary conditions on face z ¼ c (excluding edges) have both the rotational symmetry about the x-axis and translational symmetry in the z-direction (6.68) incorporated, which yields ujðx;y;cÞ ¼ ujðx;y;cÞ ¼ 0 vjðx;y;cÞ  vjðx;y;cÞ ¼ 0

ðy > 0Þ

(8.169)

wjðx;y;cÞ  wjðx;y;cÞ ¼ 0 with that on the extra edge being ujðx;0;cÞ ¼ 0 ðincluding the endsÞ.

(8.170)

For the conventional edges parallel to the x-axis (excluding vertices), in addition to (8.167) and (8.170) for the extra edges in this direction, the boundary conditions are obtained from Eqs. (8.166) and (8.167) as ujðx;b;0Þ ¼ 0 vjðx;b;0Þ  vjðx;b;0Þ ¼ 0

(8.171a)

wjðx;b;0Þ  wjðx;b;0Þ ¼ 0 and from Eqs. (8.166) and (8.168) as ujðx;b;cÞ ¼ 0 vjðx;b;cÞ  vjðx;b;cÞ ¼ 0 wjðx;b;cÞ  wjðx;b;cÞ ¼ 0:

(8.171b)

279

Further symmetries within a UC

For the edges parallel to the y-axis (excluding vertices and the intersections with y ¼ 0) boundary conditions are obtained from Eqs. (8.162), (8.167) and (8.169) as ujð0;y;0Þ þ ujð0;y;0Þ ¼ 0 vjð0;y;0Þ ¼ vjð0;y;0Þ ¼ 0 wjð0;y;0Þ  wjð0;y;0Þ ¼ 0 ujð0;y;cÞ ¼ ujð0;y;cÞ ¼ 0

ðy > 0Þ

(8.172a)

vjð0;y;cÞ ¼ vjð0;y;cÞ ¼ 0 wjð0;y;cÞ  wjð0;y;cÞ ¼ 0 and from Eqs. (8.164), (8.167) and (8.169) as ujða;y;0Þ ¼ ujða;y;0Þ ¼ 0 vjða;y;0Þ ¼ vjða;y;0Þ ¼ ag0xy wjða;y;0Þ  wjða;y;0Þ ¼ 0 ujða;y;cÞ ¼ ujða;y;cÞ ¼ 0

ðy > 0Þ.

(8.172b)

vjða;y;cÞ ¼ vjða;y;cÞ ¼ ag0xy wjða;y;cÞ  wjða;y;cÞ ¼ 0 For the conventional edges parallel to the z-axis (excluding vertices), in addition to (8.163) and (8.165), the boundary conditions are obtained from Eqs. (8.162), (8.164) and (8.166) as ujð0;b;zÞ ¼ 0 vjð0;b;zÞ ¼ 0

(8.173a)

wjð0;b;zÞ  wjð0;b;zÞ ¼ 0 ujða;b;zÞ ¼ 0 vjða;b;zÞ ¼ ag0xy wjða;b;zÞ  wjða;b;zÞ ¼ 0:

(8.173b)

280

Representative Volume Elements and Unit Cells

For the vertices, the boundary conditions are obtained as the logical sums of those for intersecting the edges as ujð0;b;0Þ ¼ 0 vjð0;b;0Þ ¼ 0

(8.174a)

wjð0;b;0Þ  wjð0;b;0Þ ¼ 0 ujð0;b;cÞ ¼ 0 vjð0;b;cÞ ¼ 0

(8.174b)

wjð0;b;cÞ  wjð0;b;cÞ ¼ 0 ujða;b;0Þ ¼ 0 vjða;b;0Þ ¼ ag0xy

(8.174c)

wjða;b;0Þ  wjða;b;0Þ ¼ 0 ujða;b;cÞ ¼ 0 vjða;b;cÞ ¼ ag0xy

(8.174d)

wjða;b;cÞ  wjða;b;cÞ ¼ 0: The boundary conditions for the four vertices as a part of the four extra edges as introduced through (8.163), (8.165), (8.168) and (8.170) are ujð0;0;0Þ ¼ 0 vjð0;0;0Þ ¼ 0 ujð0;0;cÞ ¼ 0 vjð0;0;cÞ ¼ 0 ujða;0;0Þ ¼ 0

(8.174e)

vjða;0;0Þ ¼ ag0xy ujða;0;cÞ ¼ 0 vjða;0;cÞ ¼ ag0xy where the conditions resulting from (8.168) and (8.170) are implied by those from (8.163) and (8.165) at the relevant ends of the edges.

281

Further symmetries within a UC

The above boundary conditions have to be topped up with a constraint to eliminate rigid body translation in the z-direction wjð0;0;0Þ ¼ 0:

(8.175)

It should be noted that unlike the case of reflectional symmetries, a third rotational symmetry about the third axis in the same coordinate system is not to give any new information, since the combination of such three rotations results in an identity transformation, i.e. one configuration transforming back to itself. Therefore, it will be of no interest for practical applications. However, the presence of further rotational symmetries about other axes may offer some advantages in this respect. An example will be provided in the next subsection.

8.3.3 Application to 3D 4-axial braided composites where more symmetries are present Textile composites represent an active development in structural applications of composites. Braided preforms have often found favor in some sectors of industry. Specifically, the 3D 4-axial braids are a generic type of such preforms. Plenty of symmetries are available in such an architecture. By using translational symmetries alone, a solid model of a UC can be obtained as shown in Fig. 8.6(a), a simplistic sketch of which is given in Fig. 8.6(b), where strokes represent fiber tows and blobs on the edges represent the truncated fragments of fiber tows as can be seen in Fig. 8.6(a). The UC of this kind is essentially a cuboid, and can be readily formulated employing the boundary conditions derived in Subsection 6.4.2.2 with a¼ b and c¼ h. One can reduce the size of the UC to be analyzed to a half (Fig. 8.6(c)) and then a quarter (Fig. 8.6(d)) of UC, by making use of the rotational symmetries about the z- and x-axes, respectively, as discussed in previous subsections. The UC formulated in the previous subsection should be sufficient to facilitate the analysis if one stays at this position. Practically, the saving in computational cost by using quarter sized UC relative to the full sized UC is limited, as in order to characterise the material, four analyses will have to be conducted for the former, whilst only one for the latter, which is four times larger. It is obvious that there exists one more rotational symmetry about axis y1 in the quarter model. Using it, the UC can be further reduced to that shown in Fig. 8.6(e), which is the smallest size of the UC one can obtain, namely, one-eighth of the full size UC. This will bring significant savings in computational cost even though the analysis will still have to be run four times.

282

Representative Volume Elements and Unit Cells

Fig. 8.6 UCs for 4-axial braided composite after using various symmetries: (a) A solid model (b) Full size UC (c) Half size UC (d) Quarter size UC (e) 1/8 UC.

Unfortunately, this further reduction comes at an additional cost on formulation. The quarter-sized UC has an original set of coordinates associated with it. The y1-axis for the further rotational symmetry does not pass through the origin of this coordinate system, and new coordinate system x1-y1-z1, will have to be introduced with its axes being parallel to x-y-z, respectively, as shown in Fig. 8.6(d) and (e). Therefore, the problem will

283

Further symmetries within a UC

have to be formulated in terms of relative displacements, rather than with respect to the origin of the original coordinate system. In fact, within the smallest UC, there are more geometric symmetries present. Specifically, by a combined reflection and a 90 rotation, the y  0 half as shown in Fig. 8.6(e) can reproduce the y  0 half. Furthermore, within any of such halves, a 180 rotational symmetry about the z1-axis is present. The final geometrically representative volume consists of only one fiber tow and a fragment of another tow. Having a geometrically representative volume of the smallest size is beneficial in terms of mesh generation as one can concentrate on a single tow, provided that it does not overlap with the fragment from another tow within this volume. One might also change the cross-section of the tow within this volume in order to maximise the tow volume fraction of the composite, which is often a difficult objective to achieve otherwise. However, as far as the analysis of such model is concerned, it would not be of any mechanical significance, as it would not be able to reproduce any loading case. Therefore, any further consideration along this line will not be pursued. Attention will be focused on the third rotational symmetry about the y1-axis as shown in Fig. 8.7. The quarter-size cuboidal UC was formulated in the previous section. Assume it is rotationally symmetric about axis y1, which is pointing into the plane of the image and is passing through the point R in Fig. 8.7(a). Use of the symmetry halves the size of the quarter-sized UC. To facilitate this, the quarter-size UC can be partitioned using any plane containing y1. Without loss of generality, the plane is chosen to be perpendicular to the z-axis and the shaded part shown in Fig. 8.7(a) is chosen as the UC to be established, which 1/8 of the full size model, with the interior as sketched in Fig. 8.6(e).

(a)

z

(b)

E

z

P'

P' R O

P

R x

O

x P

Fig. 8.7 Rotational symmetry about the y1-axis (side view): (a) selection of the partitioning plane and the half of UC to be retained (shaded) in the example considered; (b) schematic illustrating mapping of the points on the boundary.

284

Representative Volume Elements and Unit Cells

With dimensions as indicated in Fig. 8.6(b), the 1/8 UC is defined in the domain 0  x  b;

b  y  b

and

0  z  h=2:

(8.176)

Considering an arbitrary point P(x,y,z) in the 1/8 UC, it is transformed to P 0 (b-x,y,h-z) under this symmetry as shown in Fig. 8.7(a). The displacements of these points relative to R(b/2,y,h/2), the center of the quarter-size UC, as shown in Fig. 8.7(a), through which the y1-axis passes, are related according to the symmetry concerned (symmetric for the top sign and antisymmetric loading for the bottom sign as in  and H, respectively) as follows uP 0  uR ¼ HðuP  uR Þ vP 0  vR ¼ ðvP  vR Þ

uP  uP 0 ¼ uR  uR or

wP 0  wR ¼ HðwP  wR Þ

vP HvP 0 ¼ vR HvR

(8.177)

wP  wP 0 ¼ wR  wR :

To express the boundary conditions for the UC, the displacements at point R are required. They can be obtained by placing point P at the origin of the global coordinate system O(0,0,0), in which case point P0 coincides with E(b,0,h), as shown in Fig. 8.7a. Under symmetric loading conditions, uE þ uO ¼ uR þ uR ¼ 2uR

uE þ uO ¼ 2uR

vE  vO ¼ vR  vR ¼ 0

or

wE þ wO ¼ wR þ wR ¼ 2wR

vE ¼ vO

(8.178)

wE þ wO ¼ 2wR

whilst under antisymmetric loading conditions uE  uO ¼ uR  uR ¼ 0 vE þ vO ¼ vR þ vR ¼ 2vR wE  wO ¼ wR  wR ¼ 0

uE ¼ uO or

vE þ vO ¼ 2vR

(8.179)

wE ¼ wO :

It can be noted that point E at (x¼b, y¼0, z¼h) is in fact one of the extra vertices at (x¼ a, y¼ 0, z¼ c) defined in Subsection 8.3.2, where the displacements are known and expressed in terms of different combinations of the Kdofs depending on the loading cases. This relationship can be employed to determine the displacements at point R, as will be shown below, given that origin O has been constrained, i.e. uO ¼ vO ¼ wO ¼ 0:

(8.180)

285

Further symmetries within a UC

8.3.3.1 Boundary conditions under s0x , s0y , s0z or any combination of them (symmetric) Under the current loading condition, the displacements at E can be obtained from (8.133h) in the dimensions relevant to the present problem as uE ¼ bε0x ;

vE ¼ 0

& wE ¼ hε0z :

(8.181)

In conjunction with (8.178), one obtains 1 1 uR ¼ bε0x and wR ¼ hε0z : (8.182) 2 2 The boundary conditions for this UC under the current loading condition can be obtained as follows. Face z ¼ h/2 lies in the new partitioning plane. Placing P on this face and referring to Fig. 8.7(a), Equations (8.177) yield uP þ uP 0 ¼ ujðx;y;h=2Þ þ ujðbx;y;h=2Þ ¼ uR þ uR ¼ 2uR ¼ bε0x vP  vP 0 ¼ vjðx;y;h=2Þ  vjðbx;y;h=2Þ ¼ vR  vR ¼ 0

0  x  b=2 b  y  b:

wP þ wP 0 ¼ wjðx;y;h=2Þ þ wjðbx;y;h=2Þ ¼ wR þ wR ¼ 2wR ¼ hε0z (8.183)

Face z ¼ 0 is not related to any other face in the quarter-size UC, hence the boundary conditions on this face are not affected by the partition and remain as previously obtained in (8.126). ujðx;y;0Þ  ujðx;y;0Þ ¼ 0 vjðx;y;0Þ þ vjðx;y;0Þ ¼ 0 wjðx;y;0Þ þ wjðx;y;0Þ ¼ 0

0xb b  y  b:

(8.184)

Faces y ¼ b: The boundary conditions on these faces are obtained previously in (8.125) with the range of z reduced. ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 2bε0y

0xb 0  z  h=2:

(8.185)

wjðx;b;zÞ  wjðx;b;zÞ ¼ 0 Faces x ¼ 0 & b: The boundary conditions on these faces also remain as they are as previously obtained in (8.121) on x ¼ 0 and (8.123) on x ¼ b with the domain in the z-direction reduced.

286

Representative Volume Elements and Unit Cells

ujð0;y;zÞ þ ujð0;y;zÞ ¼ 0 vjð0;y;zÞ þ vjð0;y;zÞ ¼ 0 wjð0;y;zÞ  wjð0;y;zÞ ¼ 0

0yb 0  z  h=2

(8.186)

ujðb;y;zÞ þ ujðb;y;zÞ ¼ 2bε0x vjðb;y;zÞ þ vjðb;y;zÞ ¼ 0

0yb 0  z  h=2:

(8.187)

wjðb;y;zÞ  wjðb;y;zÞ ¼ 0 Reference can be made to Fig. 8.8 when applying the above boundary conditions. It shows a developed view of the 1/8 UC, where faces requiring appropriate partitioning are shaded. The mesh should be generated in such a way to ensure that the surface tessellations on faces x ¼ 0 and x ¼ b are symmetric about their respective central axes, shown in dash-double-dots lines, and so are those on faces z ¼ 0 and z ¼ h/2, noticing that the central axis for face z¼ h/2 is horizontal in Fig. 8.8, whilst the tessellations on opposite faces y ¼ b and y ¼ -b should be identical, same as before. To help relating the current UC back to where it was from, axis C as marked in Fig. 8.8 refers back to Fig. 8.4 where it was shown as a point in the xy-plane. 8.3.3.2 Boundary conditions under s0yz (antisymmetric) The displacements at point R are obtained in a similar manner as in the previous subsection. Under the current loading condition, from (8.146e) vE ¼ 0 and wE ¼ 0: In conjunction with (8.179), one obtains

(8.188)

vR ¼ 0:

(8.189)

Fig. 8.8 Developed view of the faces of the 1/8 UC.

287

Further symmetries within a UC

The boundary conditions for this UC under the current loading condition can be obtained as follows. Face z ¼ h/2: ujðx;y;h=2Þ  ujðbx;y;h=2Þ ¼ 0 vjðx;y;h=2Þ þ vjðbx;y;h=2Þ ¼ 0 wjðx;y;h=2Þ  wjðbx;y;h=2Þ ¼ 0

0  x  b=2 0  y  b:

(8.190)

Face z ¼ 0: The boundary conditions on face z ¼ 0 remain as they are as previously obtained in (8.139). ujðx;y;0Þ  ujðx;y;0Þ ¼ 0 vjðx;y;0Þ þ vjðx;y;0Þ ¼ 0 wjðx;y;0Þ þ wjðx;y;0Þ ¼ 0

0xb 0  y  b:

(8.191)

Faces y ¼ b: The boundary conditions on these faces also remain as they are as previously obtained, i.e. on y ¼ b, (8.138) with reduced z range. ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 0xb

vjðx;b;zÞ  vjðx;b;zÞ ¼ 0

0  z  h=2:

(8.192)

wjðx;b;zÞ  wjðx;b;zÞ ¼ 2bg0yz Faces x ¼ 0 & b: The boundary conditions on these faces also remain as they are as previously obtained, i.e. (8.134) on x ¼ 0 and (8.136) on x ¼ b with reduced z range. ujð0;y;zÞ  ujð0;y;zÞ ¼ 0 vjð0;y;zÞ  vjð0;y;zÞ ¼ 0 wjð0;y;zÞ þ wjð0;y;zÞ ¼ 0 ujðb;y;zÞ  ujðb;y;zÞ ¼ 0 vjðb;y;zÞ  vjðb;y;zÞ ¼ 0 wjðb;y;zÞ þ wjðb;y;zÞ ¼ 0

0yb 0  z  h=2

0yb 0  z  h=2:

(8.193)

(8.194)

288

Representative Volume Elements and Unit Cells

8.3.3.3 Boundary conditions under s0xz (symmetric) Under the current loading condition, from (8.160e) uE ¼ 0

and

wE ¼ bg0xz

(8.195)

in conjunction with (8.178), one obtains uR ¼ 0

and

1 wR ¼ bg0xz : 2

(8.196)

The boundary conditions for this UC under the current loading condition can be obtained as follows. Face z ¼ h/2: From (8.196) and (8.177), given the symmetric loading in this case, one has ujðx;y;h=2Þ þ ujðbx;y;h=2Þ ¼ 2uR ¼ 0 0  x  b=2

vjðx;y;h=2Þ  vjðbx;y;h=2Þ ¼ 0

b  y  b:

(8.197)

wjðx;y;h=2Þ þ wjðbx;y;h=2Þ ¼ 2wR ¼ bg0xz Face z ¼ 0: For the same reason as stated in the previous subsection, the boundary conditions on face z ¼ 0 remain as they are as previously obtained in (8.153). ujðx;y;0Þ þ ujðx;y;0Þ ¼ 0 vjðx;y;0Þ  vjðx;y;0Þ ¼ 0 wjðx;y;0Þ  wjðx;y;0Þ ¼ 0

0xb 0  y  b:

(8.198)

Faces y ¼ b: The boundary conditions also remain as they are as (8.152) on y ¼ b with reduced z range. ujðx;b;zÞ  ujðx;b;zÞ ¼ 0 vjðx;b;zÞ  vjðx;b;zÞ ¼ 0 wjðx;b;zÞ  wjðx;b;zÞ ¼ 0

0xb 0  z  h=2:

(8.199)

Faces x ¼ 0 & b: The boundary conditions remain as (8.148) on x ¼ 0 and (8.150) on x ¼ b with reduced z range.

289

Further symmetries within a UC

ujð0;y;zÞ  ujð0;y;zÞ ¼ 0 vjð0;y;zÞ  vjð0;y;zÞ ¼ 0 wjð0;y;zÞ þ wjð0;y;zÞ ¼ 0

0yb

(8.200)

0  z  h=2

ujðb;y;zÞ  ujðb;y;zÞ ¼ 0 vjðb;y;zÞ  vjðb;y;zÞ ¼ 0

0yb (8.201)

0  z  h=2:

wjðb;y;zÞ þ wjðb;y;zÞ ¼ 2bg0xz 8.3.3.4 Boundary conditions under s0xy (antisymmetric) Under the current loading condition, from (8.174e) uE ¼ 0 and

vE ¼ 0

(8.202)

in conjunction with (8.179), one obtains (8.203) vR ¼ 0: The boundary conditions for this UC under the current loading condition can be obtained as follows. Face z ¼ h/2: From (8.178) ujðx;y;h=2Þ  ujðbx;y;h=2Þ ¼ uR  uR ¼ 0 vjðx;y;h=2Þ þ vjðbx;y;h=2Þ ¼ vR þ vR ¼ 2vR ¼ 0 wjðx;y;h=2Þ  wjðbx;y;h=2Þ ¼ wR  wR ¼ 0

0  x  b=2 b  y  b: (8.204)

Face z ¼ 0: Since face z ¼ 0 is not related to other faces, the boundary conditions on it remain as previously obtained in (8.167) ujðx;y;0Þ þ ujðx;y;0Þ ¼ 0 vjðx;y;0Þ  vjðx;y;0Þ ¼ 0 wjðx;y;0Þ  wjðx;y;0Þ ¼ 0 .

0xb 0  y  b:

(8.205)

Faces y ¼ b: The boundary conditions on these faces remain the same as (8.166) with reduced z range.

290

Representative Volume Elements and Unit Cells

ujðx;b;zÞ  ujðx;b;zÞ ¼ 2bg0xy vjðx;b;zÞ  vjðx;b;zÞ ¼ 0

0xb 0  z  h=2:

(8.206)

wjðx;b;zÞ  wjðx;b;zÞ ¼ 0 Faces x ¼ 0 & b: The boundary conditions on these faces also remain as previously obtained, i.e. (8.162) on x ¼ 0 and (8.164) on x ¼ b. ujð0;y;zÞ þ ujð0;y;zÞ ¼ 0 vjð0;y;zÞ þ vjð0;y;zÞ ¼ 0 wjð0;y;zÞ  wjð0;y;zÞ ¼ 0

0yb 0  z  h=2

(8.207)

ujðb;y;zÞ þ ujðb;y;zÞ ¼ 0 vjðb;y;zÞ þ vjðb;y;zÞ ¼ 2ag0xy

0yb b  z  b:

(8.208)

wjðb;y;zÞ  wjðb;y;zÞ ¼ 0 The UC obtained in this section is of 1/8 of the full-size UC, but it remains capable of characterising the composite fully. However, macroscopic direct stresses and each macroscopic shear stress will have to be treated separately as the boundary conditions are different for each case. The implementation of the boundary conditions as presented above will require the mesh of the UC so created that opposite faces y ¼ b are tessellated in exactly the same form, whilst faces x ¼ 0 and x¼ b show symmetry about their respective centrelines parallel to the z-axis and face z ¼ 0 and z¼ h about their respective centrelines parallel to the x-axis. Similar to the full-size UC, edges and vertices of the UC need to be treated as required. Sufficient guidance can be found from previous subsections and the underlying considerations remain the same. Instead of providing them in the text, details and templates for the imposition of the boundary conditions are offered through the designated website associated with this monograph in a digital form as input files of Abaqus (2016) for interested readers to consult.

Further symmetries within a UC

291

8.4 Examples of mixed reflectional and rotational symmetries 8.4.1 Hexagonal packing A typical use of additional symmetries to UCs for hexagonally packed UD composites was elaborated in (Li, 1999). Employing orthogonal translational symmetries alone, a UC can be obtained as shown in Fig. 8.9, which has often been employed in the literature. The benefits of it are its rectangular shape and, most importantly, that one common set of boundary conditions out of translational symmetries applies to all loading cases. Within it, the presence of reflectional symmetries about central axes in horizontal and vertical directions can be employed to reduce the size to a quarter of it. Use of such a quarter sized UC should be regarded as an incompetent application since it leaves one more symmetry unused, namely, a further 180 rotational symmetry about point P, yet it requires imposition of different boundary conditions for different loading conditions whilst using the quarter sized UC. Having taken the trouble of employing different boundary conditions under different loading conditions, one does not take full advantage of saving the computational costs since this is not the smallest size of the model one can achieve. Apart from it being rectangular, there is hardly any benefit from using it, although it is the most cited UC in the literature for applications to this type of composites. The most efficient one is a UC of the size reduced to 1/8 of the original one obtained after applying the 180 rotational symmetry about point R. Another significance of realising this 180 rotational symmetry lies in the unification of UCs of various appearances for the same hexagonally packed UD composites. Depending on the borderline chosen to partition the quarter sized UC using the rotational symmetry in order to obtain the 1/8 sized one, a range of UCs of various shapes as can be found in the literature, as has been elaborated in Chapter 5. If anything, the most significant differences between them turn out to be their friendliness to FE meshing. Given the dimensions specified in Fig. 8.9, the boundary conditions for the quarter-sized UC obtained after employing the two reflectional symmetries can be readily found in Subsection 8.2.2 after reducing the presentation from 3D there to 2D as is relevant to the problem here. Further application of the rotational symmetry about point R results in the UC of smallest size, which is half of the quarter-sized one. For the meshing consideration as is necessary for its subsequent application, a straight line perpendicular to the segment connecting the

292

Representative Volume Elements and Unit Cells

Fig. 8.9 Reduction of UC sizes using additional symmetries.

centers of the two truncated fibers present in the quarter-sized UC is chosen. The boundary conditions can be obtained as follows. Since point R is away from the origin of the coordinate system employed, the rotational symmetry applies to displacements relative to that point. The absolute displacements do not show any symmetry about point R. After the use of the two reflectional symmetries, the displacement at the upper right corner E will be known and can be expressed in terms of the Kdofs active in the loading case concerned, as will be referred to loading case by loading case below. Similar to the arguments employed in the previous section in relation to (8.177) except that the rotation now is about the z-axis, the relative displacements show their symmetry nature as follows, i.e. in-plane ones being antisymmetric and out-of-plane one symmetric. Under a symmetric loading, one has uP 0  uR ¼ ðuP  uR Þ vP 0  vR ¼ ðvP  vR Þ wP 0  wR ¼ ðwP  wR Þ

uP þ uP 0 ¼ uR þ uR ¼ 2uR or

vP þ vP 0 ¼ vR þ vR ¼ 2vR

(8.209)

wP  wP 0 ¼ wR  wR ¼ 0:

Placing P at point E, with P0 will be at O which has been fully constrained, one has

293

Further symmetries within a UC

uE þ uO ¼ 2uR vE þ vO ¼ 2vR

uE ¼ 2uR or

wE  wO ¼ 0

vE ¼ 2vR

(8.210)

wE ¼ 0

whilst under antisymmetric loading conditions, accordingly, one has uP 0  uR ¼ ðuP  uR Þ

uP  uP 0 ¼ uR  uR ¼ 0

vP 0  vR ¼ ðvP  vR Þ

or

wP 0  wR ¼ ðwP  wR Þ

vP  vP 0 ¼ vR  vR ¼ 0 wP þ wP 0 ¼ wR þ wR ¼ 2wR (8.211)

uE  uO ¼ 0 vE  vO ¼ 0 wE þ wO ¼ 2wR

uE ¼ 0 or

vE ¼ 0

(8.212)

wE ¼ 2wR :

8.4.1.1 Boundary conditions under s0x ; s0y and s0z The 2D formulation of the UC under consideration is treated as a generalised plane strain problem in which a constant out-of-plane direct strain ε0z is present for the whole UC. This makes the out-of-plane displacement w redundant. The displacements at E under this loading case have been given in final equation in (8.32) where the z-coordinate is irrelevant, resulting in uE ¼ bε0x pffiffiffi vE ¼ 3bε0y :

(8.213)

Given zero displacements at the origin, applying the rotational symmetry about R under the present loading case, which is symmetric, whilst the inplane relative displacements are antisymmetric, one has 1 1 uR ¼ uE ¼ bε0x 2 2 pffiffiffi 1 3 0 vR ¼ vE ¼ bε : 2 2 y For side x ¼ 0 (excluding ends), according to (8.24)

(8.214)

294

Representative Volume Elements and Unit Cells

ujð0;yÞ ¼ 0:

(8.215)

For side x ¼ b (excluding ends), according to (8.25) ujða;yÞ ¼ aε0x :

(8.216)

For side y ¼ 0 (excluding ends), referred to (8.26) vjðx;0Þ ¼ 0:

(8.217) pffiffiffi For side x þ 3y ¼ 2b (excluding ends and point R), owing to the rotational symmetry ujG þ ujG0 ¼ 2uR ¼ bε0x pffiffiffi vjG þ vjG0 ¼ 2vR ¼ 3bε0y :

(8.218)

For the corners, in addition to point R, where the boundary conditions are given in (8.214) ujð0;0Þ ¼ vjð0;0Þ ¼ 0 ujð0;2b=pffiffi3 Þ ¼ 0 ujðb;0Þ ¼ ujðb;b=pffiffi3 Þ ¼ bε0x vjð0;2b=pffiffi3 Þ þ vjðb;b=pffiffi3 Þ ¼

(8.219) pffiffiffi 0 3bεy :

The rigid body translations have all been eliminated. 8.4.1.2 Boundary conditions under s0yz This particular 2D problem is a so-called anticlastic problem as previously described in Subsection 6.4.1, where both in-plane displacements are absent. There is only displacement w to be considered, which is out-of-plane. Its value at E under this loading case has been given in final equation in (8.41) as pffiffiffi wE ¼ 3bg0yz : (8.220) Applying the rotational symmetry about R under the present loading case, which is antisymmetric, one has pffiffiffi 3 0 wR ¼ (8.221) bgyz : 2 For face x ¼ 0 and x ¼ b are both free from any constraint according to (8.34).

Further symmetries within a UC

295

For face y ¼ 0 (excluding edges), according to (8.35) wjðx;0Þ ¼ 0:

(8.222)

pffiffiffi For face x þ 3y ¼ 2b (excluding edges), given the antisymmetry of the loading case and symmetry of displacement w, pffiffiffi (8.223) wjG þ wjG0 ¼ 3bg0yz : For the vertices, in addition to (8.221), wjð0;0Þ ¼ wjðb;0Þ ¼ 0

pffiffiffi wjð0;2b=pffiffi3 Þ þ wjðb;b=pffiffi3 Þ ¼ 3bg0yz : The rigid body translation has been constrained.

(8.224)

8.4.1.3 Boundary conditions under s0xz This is also an anticlastic problem. The displacement at E under this loading case have been given in (8.50), rewritten as wE ¼ bg0xz :

(8.225)

Applying the rotational symmetry about R under the present loading case, which is antisymmetric, one has 1 wR ¼ bg0yz : (8.226) 2 For face x ¼ 0 and x ¼ b (excluding edges), according to (8.43) and (8.44), wjð0;yÞ ¼ 0 wjðb;yÞ ¼ bg0xz

(8.227)

whilst face y ¼ 0pisffiffiffi free from constraint according to (8.45). For face x þ 3y ¼ 2b (excluding ends and point R), given the antisymmetry of the loading case and the symmetry of the relative out-of-plane displacement wjG þ wjG0 ¼ bg0xz : For the vertices, in addition to (8.226),

(8.228)

296

Representative Volume Elements and Unit Cells

wjð0;0Þ ¼ 0 wjð0;b=pffiffi3 Þ ¼ 0 wjðb;0Þ ¼ bg0xz

(8.229)

wjðb;b=pffiffi3 Þ ¼ bg0xz : The rigid body translations have all been constrained. 8.4.1.4 Boundary conditions under s0xy This a generalised plane strain problem as in Subsection 4.1.1. The displacements at E under this loading case have been given in final equation in (8.60). uE ¼ 0 vE ¼ bg0xy :

(8.230)

Due to the rotational symmetry about R under the present loading case, which is symmetric, from (8.210), one has uR ¼ 0 (8.231) 1 vR ¼ bg0xy : 2 For face x ¼ 0 and x ¼ b (excluding edges), according to (8.52) and (8.53), vjð0;yÞ ¼ 0 vjðb;yÞ ¼ bg0xy :

(8.232)

For face y ¼ 0 (excluding edges), according to (8.54), ujðx;0Þ ¼ 0:

(8.233) pffiffiffi For face x þ 3y ¼ 2b (excluding ends and point R), according to (8.209), ujG þ ujG0 ¼ 0 vjG þ vjG0 ¼ bg0xy : For the vertices

(8.234)

297

Further symmetries within a UC

ujð0;0Þ ¼ vjð0;0Þ ¼ 0 vjð0;2b=pffiffi3 Þ ¼ 0 ujðb;0Þ ¼ 0

(8.235)

vjðb;b=pffiffi3 Þ ¼ bg0xy ujð0;2b=pffiffi3 Þ þ ujðb;b=pffiffi3 Þ ¼ 0: Rigid body translations have been eliminated.

8.4.2 Plain weave Another example is with plain weave composites as shown in Fig. 8.10(a). A UC can be formulated based on the translational symmetries alone as indicated by the largest square in Fig. 8.10(a). If one is content with that, the relevant formulation in Chapter 6 should be sufficient. One can of course use the available reflectional symmetries to reduce the size of to a quarter, as shown in the first quadrant in Fig. 8.10(b). Furthermore, using the rotational symmetries about the horizontal and vertical axes passing the center of the quarter UC, the final size can be reduced to 1/16 of the original size, as shown by the red square. Again, because of the use of reflectional and rotational symmetries, different boundary conditions will have to be imposed for different loading conditions. The boundary conditions will be derived in this section, assuming that the plain weave as seen is a layer out of a stack of

(a)

(b)

y

y O' 2by

R'

x

O

G

P'

C

H

P

2bx O

R

x

Fig. 8.10 (a) UCs selection for plane weave composite use of various symmetries; (b) rotational symmetries present in a quarter-sized UC.

298

Representative Volume Elements and Unit Cells

numerous identical layers and geometric periodicity is present in the thickness direction. The boundary conditions for the quarter-sized UC can be found in Subsection 8.2.2 of this chapter. Further application of the two rotational symmetries reduces the size of this UC to a quarter of it. When formulating the boundary conditions for this UC, the least confusing approach is to consider it as an eight-faced object topologically, as shown in Fig. 8.11, instead of six-faced as it appears to be, with both the right and front sides being split into two separate faces. The border between each pair of faces forms an edge. Therefore, there are 15 edges altogether. In Fig. 8.11, edges have been numbered in Roman numerals up to hXXi, with hIXi, hXIIi and hXVIIi being omitted for the ease of associating the edge numbers with their locations. There are 11 vertices where three or four faces meet. Again, in Fig. 8.11, the vertices have been numbered to 12th with the ninth being void for the same consideration as skipping certain numbers at the edges. Vertex 11 corresponds to point C as in Fig. 8.10(b). The side lengths of the complete unit cell formed by employing translational symmetries only are 2bx and 2by in weft and warp directions, respectively, and the thickness of the layer of the composite is 2bz. After application of further symmetries, the respective dimension in the UC of reduced size become to bx/2, bx/2 and 2bz accordingly.

z

Back

⑤

⑧

‹VIII›

‹XX›

Left

‹V›

⑫

Top

‹XIII›

‹VII›

‹XVI›

‹IV›

①

⑥ ‹XVIII›

‹I›

‹XIX›

Front-upper

⑩ ‹XIV›

⑦

‹VI›

⑪

‹X›

②

Front-lower Bottom

④ Right-upper

‹III›

G

Right-lower

‹XV›

‹II›

x

‹XI›

H

③

Fig. 8.11 The faces, edges and vertices of the UC.

y

299

Further symmetries within a UC

8.4.2.1 Boundary conditions under s0x , s0y , s0z or any combination of them Face x ¼ 0 is not affected by the rotational symmetry and the boundary condition is as given in (8.24): ujx¼0 ¼ 0:

(8.236)

On faces x¼ bx/2, the rotational symmetry about axis G results in boundary conditions which are based on the concept of relative displacements as fully elaborated in previous subsections. uj

vj

 þ uj

 ¼ bx εx

  vj

¼0

1 b ;y;z 2 x

1 b ;y;z 2 x

wj

1 b ;y;z 2 x

0

1 b ;y;z 2 x

1 b ;y;z 2 x

 þ wj

1 b ;y;z 2 x

ðz > 0Þ

(8.237)

 ¼ 2w11

where subscript 11 to w refers to vertex 11, which is the intersection of both axes G and H. This vertex serves the same role as point R in the previous section. The difference is that the relevant displacements at R could be determined completely one way or another as there were two points related by the rotational symmetry, where displacements were known, but the position does not apply now, unfortunately. As a result, some of the displacements at vertex 11, whilst being involved in boundary conditions as in (8.237), will be left not prescribed, as is the case for w11 under this particular loading condition. On face y ¼ 0, again, previously employed reflectional symmetry dictates that vjðx;0;zÞ ¼ 0:

(8.238)

Face y¼ by/2 is governed by the rotational symmetry about the H axis.

300

Representative Volume Elements and Unit Cells

uj

x;

1 b ;z 2 y

x;

1 b ;z 2 y

vj

wj

x;

  uj

 þ vj

1 b ;z 2 y

x;

1 b ;z 2 y

x;

1 b ;z 2 y

¼0

 ¼ by εy

0

 þ wj

x;

ðz > 0Þ

(8.239)

 ¼ 2w11 :

1 b ;z 2 y

The top and bottom faces (z¼ bz and z¼ -bz) are related through translational symmetries. ujz¼bz  ujz¼bz ¼ 0;

vjðx;y;bz Þ  vjðx;y;bz Þ

¼0

and

wjðx;y;bz Þ  wjðx;y;bz Þ ¼ 2bz ε0z :

(8.240) These boundary conditions have been obtained entirely from the symmetry considerations without any artificial interference. They are sufficient for the UC under this loading condition, bearing in mind that the rigid body translation in the z-direction has been constrained according to (8.33) as a part of the implementation of the reflectional symmetries. The boundary conditions for vertices and edges are waived here, but they can be found in (Li et al., 2011) and are provided on the designated website associated with this monograph in digital form as a template for interested users to adapt. 8.4.2.2 Boundary conditions under s0yz On face x ¼ 0: ujð0;y;zÞ ¼ 0:

(8.241)

On face x¼ bx/2: uj

  uj

vj

 þ vj

1 b ;y;z 2 x

1 b ;y;z 2 x

wj

¼0

1 b ;y;z 2 x

1 b ;y;z 2 x

  wj

1 b ;y;z 2 x

 ¼ 2v11  ¼ 0.

1 b ;y;z 2 x

ðz > 0Þ

(8.242)

301

Further symmetries within a UC

On face y ¼ 0: ujðx;0;zÞ ¼ wjðx;0;zÞ ¼ 0:

(8.243)

On face y¼ by/2: uj

  uj

¼0

 þ vj

 ¼ 2v11

x; y¼12by ;z

vj

x; y¼12by ;z

x; y¼12by ;z

wj

x;

y¼12by ;z

ðz > 0Þ.

x; y¼12by ;z

 þ wj

(8.244)

 ¼ by gyz 0

y¼12by ;z

x;

On the top and bottom faces (z¼ bz and z¼ -bz) ujðx;y;bz Þ  ujðx;y;bz Þ ¼ 0; ¼0

vjðx;y;bz Þ  vjðx;y;bz Þ and

wjðx;y;bz Þ  wjðx;y;bz Þ ¼ 0: (8.245)

8.4.2.3 Boundary conditions under s0xz On face x ¼ 0: vjð0;y;zÞ ¼ wjð0;y;zÞ ¼ 0:

(8.246)

On face x¼ bx/2: uj

 þ uj

1 b ;y;z 2 x

  vj

vj

1 b ;y;z 2 x

wj

 ¼ 2u11

1 b ;y;z 2 x

¼0

1 b ;y;z 2 x

 þ wj

1 b ;y;z 2 x

ðz > 0Þ.

(8.247)

 ¼ bx gxz 0

1 b ;y;z 2 x

On face y ¼ 0: vjðx;0;zÞ ¼ 0: On faces y¼ by/2:

(8.248)

302

Representative Volume Elements and Unit Cells

uj

 þ uj

vj

  vj

x;12by z

x;12by z

x;12by z

 ¼ 2u11

¼0

x;12by z

  wj

wj

x;12by z

x;12by z

ðz > 0Þ.

(8.249)

¼0

On the top and bottom faces (z¼ bz and z¼ -bz): ujðx;y;bz Þ  ujðx;y;bz Þ ¼ 0; ¼0

vjðx;y;bz Þ  vjðx;y;bz Þ and

wjðx;y;bz Þ  wjðx;y;bz Þ ¼ 0: (8.250)

8.4.2.4 Boundary conditions under s0xy On face x ¼ 0: vjð0;y;zÞ ¼ wjð0;y;zÞ ¼ 0:

(8.251)

On face x¼ bx/2: uj

  uj

vj

 þ vj

1 b ;y;z 2 x

1 b ;y;z 2 x

¼0

1 b ;y;z 2 x

 ¼ bx gxy 0

bigð12bx ;y;z

  wj

wj

1 b ;y;z 2 x

bigð12bx ;y;z

ðz > 0Þ.

(8.252)

¼0

On face y ¼ 0: ujðx;0;zÞ ¼ wjðx;0;zÞ ¼ 0:

(8.253)

On face y¼ by/2: uj

 þ uj

¼0

vj

  vj

¼0

x;12by ;z

x;12by ;z

wj

x;12by ;z

x;12by ;z

x;12by ;z

  wj

¼0

x;12by ;z



 z>0 .

(8.254)

303

Further symmetries within a UC

On the top and bottom faces (z¼ bz and z¼ -bz) ujðx;y;bz Þ  ujðx;y;bz Þ ¼ 0; ¼0

vjðx;y;bz Þ  vjðx;y;bz Þ and

wjðx;y;bz Þ  wjðx;y;bz Þ ¼ 0: (8.255)

8.5 Centrally reflectional symmetry Whenever a reflectional or rotational symmetry has been used, one has to accept the consequence, i.e. that the boundary conditions would vary from loading case to loading case, as the price to pay for the reduced size in the UC. This is because reflectional and rotational symmetries only preserve the senses of some stress and strain components, typically those of direct stresses, but not others, typically of some shear stresses and strains, for which the concept of antisymmetry has to be resorted to. Involvement of any of these symmetries splits the applied loads, as well as the stresses and the strains, into two mutually exclusive categories, symmetric and antisymmetric. As a result, different boundary conditions will have to be prescribed under different loading conditions bearing different nature of symmetry, since they cannot coexist in a single analysis. This therefore prohibits any combination of loads of different natures. This has been the downside of reflectional and rotational symmetries. UC users have been left in a dilemma whether to use a full sized UC under a single set of boundary conditions for all loading conditions, or a UC of reduced size, which has to be analyzed using separate sets of boundary conditions under different loading conditions, stripping its capability of dealing with combined loading conditions. However, there could be one compromise when a special symmetry is present, central reflection (Li and Zou, 2011). A centrally reflectional symmetry is often observed in many objects. However, patterns and shapes possessing this symmetry often show other symmetries, such as reflections and rotations, making the recognition of central reflection either redundant or unobvious. A simple but distinctive shape of central reflection is a triclinic crystal (Nye, 1985), Fig. 8.12, which is a hexahedron of three parallel, unnecessarily orthogonal, pairs of faces and three independent side lengths. In this geometry, central reflection about its center is the only symmetry available. The analytical description of the central reflection, denoted as CR hereafter in the section, can be given as a mapping

304

Representative Volume Elements and Unit Cells

c

β

P' O

b α

P a

γ

a≠b≠c α ≠ β ≠ γ≠ 90º

Fig. 8.12 A triclinic crystal.

CR :

P

/

P0:

(8.256)

where P is an arbitrary point in the triclinic body as the original and P0 as its image under the mapping. Assuming the coordinates of P and P0 are (x, y, z) and (x0 , y0 , z0 ), respectively, one has ðx0  x0 ; y0  y0 ; z0  z0 Þ ¼  ðx x0 ; y  y0 ; z  z0 Þ or ðx0 ; y0 ; z0 Þ ¼ ð2x0  x; 2y0  y; 2z0  zÞ;

(8.257)

where (x0, y0, z0) are the coordinates of the center O for the central reflection. Using the notations for symmetries as introduced in Chapter 2, a 2 central reflection is then a combination of SO A and CA where A is an arbitrary axis passing through the center O. Central reflection is therefore not independent. However, a special feature of this is that the sense of the stresses and strains, whether in the upper length scale or in the lower length scale, remain unchanged under this particular symmetry transformation. If such a symmetry is available, by making use of it, the size of the UC can be halved whilst all loading cases can be analyzed using a single set of boundary conditions. It is important to observe that all stresses and strains preserve their senses under a central reflection. Symbolically, this can be expressed as a mapping of field strain and macroscopic strains under the central reflection     CR : εx ; εy ; εz ; gyz ; gxz ; gxy P / εx ; εy ; εz ; gyz ; gxz ; gxy P 0 (8.258)

305

Further symmetries within a UC

  ε0x ; ε0y ; ε0z ; g0yz ; g0xz ; g0xy

CR :



/

  ε0x ; ε0y ; ε0z ; g0yz ; g0xz ; g0xy (8.259)



where εx ; εy ; εz ; gyz ; gxz ; gxy are the field strains in the UC whilst   ε0x ; ε0y ; ε0z ; g0yz ; g0xz ; g0xy are the macroscopic strains which are the volume averages of the field strains, whereas their counterparts for reflectional and rotational symmetries do not behave like this, e.g.     0 0 0 0 0 0 0 0 0 0 0 0 : ε ; ε ; ε ; g ; g ; g ; ε ; ε ; g ;  g ;  g SO / ε x x y z yz xz xy x y z yz xz xy   ε0x ; ε0y ; ε0z ; g0yz ; g0xz ; g0xy

Cx2 :

/

(8.260)   ε0x ; ε0y ; ε0z ; g0yz ;  g0xz ;  g0xy :

(8.261) are applied one after another, the changes of the sense of the two involved shear components cancel each other, resulting in (8.259). The same applies to their stress counterparts, as illustrated in Fig. 8.13, where a reflectional transformation about the center of the cube would not cause any change and the same applies to the outward normals to the cube, also shown in Fig. 8.13. With the outward normals, surface traction can be obtained from the stresses. Traction boundary conditions can be derived from the symmetry considerations although they are not required in conventional FE analyses (Li, 2012). Antisymmetry does not get involved When SO A

and CA2

Vz

nz Wyz

W xz

z

ny

Vy

y

W xy

x

nx Vx

Fig. 8.13 Stresses showing perfect symmetry under centrally reflectional symmetry transformation.

306

Representative Volume Elements and Unit Cells

at all as far as stresses/traction and strains are concerned. This breaks the aforementioned dilemma straightaway. Apparently, this is not what a reflectional symmetry or a rotational symmetry alone could offer. The size of the UC having such symmetry can now be halved without any penalty, except for the fact that the boundary conditions for the new UC become more complicated, and the new UC of reduced size will come with a single set of boundary conditions which are applicable to all loading conditions and their combinations. The complication introduced is by no means trivial, and deserves a full account which will be given in the present section. One must pay attention to the mapping for displacements CR :

ðu  u0 ; v  v0 ; w  w0 Þ

/

 ðu0  u0 ; v 0  v0 ; w 0  w0 Þ; (8.262)

where (u, v, w) and (u0 , v0 , w0 ) are the displacements at P and P0 , respectively, and (u0, v0, w0) are those at O. These should be dealt with properly and used correctly in order to obtain boundary conditions rationally, since boundary conditions for UCs have to be described in terms of displacements. Consider the application of a centrally reflectional symmetry in structural analyses, which is relatively simpler than its applications in the formulation of UCs. The triclinic crystal as shown in Fig. 8.12 has been taken as a symbolic representation of a centrally reflectionally symmetric structure loaded in a symmetric manner under the same symmetry, Fig. 8.14(a). Applying the symmetry, the structure can be analyzed with only half of it, Fig. 8.14(b), if appropriate boundary conditions are imposed on the shaded section plane. Since relative displacements (to point O) are all centrally reflected to opposite directions according to the symmetry transformation (8.5), one has

(a) F

(b)

c

O P’

F

b

P F

b/2

a Fig. 8.14 Application of reflectional symmetry: (a) a centrally symmetric structure; (b) the symmetric half of the structure.

307

Further symmetries within a UC

u  u0 ¼ ðu0  u0 Þ v  v0 ¼ ðv 0  v0 Þ w  w0 ¼ ðw 0  w0 Þ

u þ u0 ¼ 2u0 or

v þ v0 ¼ 2v0

(8.263)

w þ w 0 ¼ 2w0

where (u, v, w) and (u0 , v0 , w0 ) are displacements at P and P0 on the section surface as original and image under the symmetry transformation. Eq. (8.263) deliver the desirable boundary conditions for the section surface, when P takes positions of all points on any half of the section plane, e.g. the half on the left hand side of the dash-dot chain in Fig. 8.14(b). Eq. (8.263) are based on the displacements relative to point O. This leaves the displacements at O completely free as far as the symmetry is concerned, which accommodate the rigid body translations of the structure. To facilitate an FE analysis, they need to be constrained. To achieve this, they can be prescribed to any fixed values. Without loss of generality, one can prescribe uO ¼ 0 vO ¼ 0

(8.264)

wO ¼ 0: A proper FE analysis will also require the rigid body rotations constrained, which can be done in a conventional manner. It should be pointed out that the selection of the section plane is not unique. It can be any surface, flat or curved, as long as it is centrally reflectionally symmetric about center O. Necessarily, the section plane or surface must pass through the center O. The boundary conditions as obtained in (8.263) and (8.264) are generally applicable whichever section plane or surface is chosen. They provide the boundary conditions if the structure is to be analyzed with a half sized model. Considerations will be now made to those UCs as formulated in Chapter 6 using translational symmetries alone which also possess a further centrally reflectional symmetry. In general, for the full size UCs formulated using translational symmetries alone, the boundary conditions take a form as given by Equation (6.16) of Chapter 6. In presence of a central reflection within these UCs, their size can be halved and they can still be analyzed with a single set of boundary conditions for all loading cases. In 2D problems, central reflection is identical to 180 rotation, the boundary conditions for which can be obtained from those given in Section 8.3.1. Because of that, application of central reflectional symmetry

308

Representative Volume Elements and Unit Cells

Side surface: Type III

(a) P

Back surface: Type III

P'

P

Top surface: Type II P'

Front surface: Type III

P''

Side surface: Type III

O

Bottom surface: Type I

(b)

(c) y

y

x

x

O

Fig. 8.15 Examples of UCs halved in size using central reflection: (a) square UC for UD composite; (b) hexagonal UC for UD composite; (c) plain weave textile.

to 2D problems will not be further discussed. Attention will be paid to 3D UCs. In many existing 3D UCs, whether they were originally proposed for particulate, UD fiber reinforced or textile reinforced composites, the

Further symmetries within a UC

309

existence of a further central reflection is very common. Some examples are shown in Fig. 8.15. In terms of size reduction of the UCs shown in Fig. 8.15, using a plane reflectional symmetry would achieve the same effects. However, the UCs of reduced size obtained using the plane reflection will have different sets of boundary conditions under different loading conditions. In order to derive the boundary conditions for the faces in the half sized UC obtained via the use of central reflection, the faces or parts of the faces present on the new cell are classified into three mutually exclusive types. Type I: The newly created face which partitions the full size cell into two-halves. An example is the bottom surface as shown in Fig. 8.15(a) in the case of a square UC. Central reflection maps a half of this face to the other half. This leads to the required boundary conditions for this type of face, as given in Eqs. (8.263) and (8.264). Type II: Faces whose originals under both translational and centrally reflectional symmetry transformations are on the other half. An example is the top surface as shown in Fig. 8.15(a) in the case of a square UC. It can be established that half of such a face is the image of the other half of the same face under the symmetry transformation as a combination of a translation and the central reflection. The required boundary conditions for this type of face will result from such relationship between the two-halves of the face, viz. (8.263) and (6.16), given conditions (8.264). u þ u0 ¼ u  u00 ¼ ðx  x00 Þε0x v þ v 0 ¼ v  v 00 ¼ ðx  x00 Þg0xy þ ðy  y00 Þε0y w þ w 0 ¼ w  w 00 ¼ ðx  x00 Þg0xz þ ðy  y00 Þg0yz þ ðz  z00 Þε0z ; (8.265) where (u,v,w), (u0 ,v0 ,w0 ) and (u00 , v00 , w00 ) are displacements at points P(x,y,z), P0 (x0 , y0 , z0 ) and P00 (x00 , y00 , z00 ), respectively. P00 is on the opposite side of the full size UC as the original of P under the translational symmetry transformation and the original of P0 under the centrally reflectional symmetry transformation, as depicted in Fig. 8.15(a). Points P0 and P collapse to the same point C at the center of the face, where the boundary conditions are obtained as

310

Representative Volume Elements and Unit Cells

1 uC ¼ ðx  x00 Þε0x 2 1 1 vC ¼ ðx  x00 Þg0xy þ ðy  y00 Þε0y 2 2

(8.266)

1 1 1 wC ¼ ðx  x00 Þg0xz þ ðy  y00 Þg0yz þ ðz  z00 Þε0z 2 2 2 Type III: Faces whose originals under the central reflection are on the other half whilst the originals under translation remain in the same half. Examples are the left and right side surfaces and the front and back surfaces as shown in Fig. 8.15(a) in the case of a square UC. As the central reflection maps the face to the half outside of the new UC under consideration, the displacements on these faces are not affected by the central reflection. The required boundary conditions remain the same as those derived from pure translational considerations in the full size cell, as given in (6.66), in general. Since Type III faces require no extra efforts, an optimised partition of the full size UC into two halves should have as many faces of this type as possible. For instance, in the case of a multisided prism, one would choose the partition plane perpendicular to the axis of the prism, as all faces on the cylindrical surface will be of Type III. Relatively, Type II faces are the most complicated to treat and each of them has to be split into two to start with. Their numbers should be minimised when possible. The partition in Fig. 8.15(a) has two pairs of faces in Type III (front and back, left and right) and this is as many as one could get. There is one face in Type II (top) and this is as few as one can get. If the full-sized cell is partitioned diagonally, the number of faces in Type III will reduce to one pair (front and back) whilst those in Type II increase to two (both sides), resulting in a worse choice. Similar argument can be made to the partition in Fig. 8.15(b). Take a rectangular prismatic UC generated from straight translations in three orthogonal directions for example, in which both cases as in Fig. 8.15(a) and (c) fall. Suppose it is partitioned by a plane perpendicular to the y-axis, without loss of generality, as shown in Fig. 8.16. Face y01 on the left hand side of the bottom surface of the prism is in the partition plane and it is hence Type I. The image of Face y01 under centrally reflectional symmetry is Face y02, i.e. the right half of the bottom surface. Under translational symmetry in the y-direction, the original of the left hand side half of the top surface, Face yþ 1 , is on the missing half of the full size UC and that under central reflection is also on the missing

311

Further symmetries within a UC

(a)

2bz

by

9

16

Face y1+

1 0

C

P'

1 1 Face z+

z

I

6 5

XVII

II

XI III

V O

VI Face y20

IV

XV I XII

x

XVIII

C

XV

7

O 4

XI V

Face x+

8

XI X

XX XII I

14

13

1

(b) 15

P

Face y2+

12

2 3

Face z-

y

Face x-

VII

X IX

VIII

2bx Face y10

Fig. 8.16 A rectangular prismatic UC after using central reflection about point O: (a) faces and vertices; (b) edges.

half. Face yþ 1 is therefore of Type II. Its image under combined y-translation and central reflection is Face yþ 2 , i.e. the right hand side half of the top surface, where points P and P0 as involved in Eq. (8.266) are also depicted. The original of Face xþ under the translational symmetry is Face x on the same half of the UC. The original of Face xþ under the centrally reflectional symmetry is on the missing half of the UC. This pair of faces, Face xþ and Face x, are therefore Type III, so are Face zþ and Face z. A complete (both necessary and sufficient) set of the boundary conditions for such a UC are presented as follows. Between corresponding points on faces (excluding edges): 0

uxþ  ux ¼ 2bx ε0x vxþ  vx ¼ 2bx g0xy

abbreviated as Uxþ  Ux ¼ Fx

wxþ  wx ¼ 2bx g0xz

Type III faces

1

B C B 0 C B x  x ¼ 2bx C @ A y 0  y ¼ z0  z ¼ 0 (8.267) 0

uyþ1 þ uyþ2 ¼ 0 vyþ1 þ vyþ2 ¼ 2by ε0y wyþ1 þ wyþ2 ¼ 2by g0yz

abbreviated as Uyþ1 þ Uyþ2 ¼ Fy

Type II faces

1

B C B C B y  y00 ¼ 2by C @ A x  x00 ¼ z  z00 ¼ 0 (8.268)

312

Representative Volume Elements and Unit Cells

þ where Face yþ 1 includes the z < 0 part of the intersection between y1 and þ yþ 2 on one side of C (excluding C) and Face y2 includes the z > 0 part of the intersection on the other side of C, as indicated by the dotted frame in Fig. 8.16(a),

0

uC ¼ 0 1 abbreviated as UC ¼ Fy 2

vC ¼ by ε0y wC ¼ by g0yz

centre of Type II faces

B B B y  y00 ¼ 2by B @ x  x00 ¼ z  z00 ¼ 0

1 C C C C A

(8.269) uy01 þ uy02 ¼ 0 vy01 þ vy02 ¼ 0

abbreviated as Uy01 þ Uy02 ¼ 0

ðType I facesÞ

wy01 þ wy02 ¼ 0 (8.270) where Face y01 includes the z < 0 part of the intersection between y01 and y02 on one side of O (excluding O) and Face y02 includes the z > 0 part of the intersection on the other side of O, as indicated by the dotted frame in Fig. 8.16(a), uO ¼ 0 vO ¼ 0

abbreviated as UO ¼ 0

ðcentre of Type I facesÞ

(8.271)

wO ¼ 0 0

u zþ  u z ¼ 0 v zþ  v z ¼ 0 wzþ  wz ¼ 2bz ε0z

abbreviated as Uzþ  Uz ¼ Fz

Type III faces

1

B C B C B z  z00 ¼ 2by C: @ A 00 00 xx ¼yy ¼0 (8.272)

The nodes on corresponding faces should be ordered in such a way that the correspondences follow the underlying symmetry transformations. If the relationships between faces are applied to the edges I to IV (excluding vertices), one could have at least the following four sets of

313

Further symmetries within a UC

conditions between corresponding points, using the abbreviations as introduced above ! ! ! ! ! ! ! ! U II  U I ¼ Fz U III  U II ¼ Fx U III  U IV ¼ Fz U IV  U I ¼ Fx : (8.273) where the arrows on top of U indicate the relative directions in the ordering the nodes along these edges. When the two arrows point to the same direction, the nodes on the two edges are ordered in the same direction. Otherwise, they are opposite to each other. It is obvious that not all conditions in (8.273) are independent, as a simple linear combination of the first three will reproduce the fourth. If a set of conditions is considered to be used to eliminate the displacements along one edge as unknowns from the sys! ! tem, e.g. the first set in (8.273) is to eliminate U II using U I , then independent sets of conditions can be obtained as follows. ! ! ! ! ! ! U II  U I ¼ Fz U III  U I ¼ Fx þ Fz U IV  U I ¼ Fx : (8.274) In other words, displacements on edges II, III and IV can all be eliminated using those on edge I. Similarly, complete but also independent sets of conditions for all other edges can be obtained as follows. ! ! ! ! U VI þ U V ¼ Fx U IX þ U V ¼ 0 U X  U V ¼ Fx (8.275) ! ! U VII  U XII ¼ Fz

! U VIII þ U XII ¼ 0

! U XI þ U XII ¼ Fz (8.276)

! U XIV þ U XIII ¼ Fx þ Fy

! ! U XV  U XX ¼ Fz

! U XVII þ U XIII ¼ Fy

! U XVI þ U XX ¼ Fy

! ! U XVIII  U XIII ¼ Fx (8.277)

! U XIX þ U XX ¼ Fy  Fz :

(8.278) Similar argument can be applied to the vertices for which complete and independent sets of conditions are given as follows. 1 1 U1 ¼  Fx  Fz 2 2

1 1 U3 ¼  Fx þ Fz 2 2

314

Representative Volume Elements and Unit Cells

1 1 U5 ¼ Fx þ Fz 2 2 1 U2 ¼  Fx 2 1 U4 ¼ Fz 2

1 1 U7 ¼ Fx  Fz 2 2

(8.279)

1 U6 ¼ Fx 2

1 U8 ¼  Fz 2

1 1 1 U9 ¼  Fx þ Fy  Fz 2 2 2 1 1 1 U13 ¼ Fx þ Fy þ Fz 2 2 2 1 1 U10 ¼  Fx þ Fy 2 2 1 1 U12 ¼ Fy þ Fz 2 2

(8.280) 1 1 1 U11 ¼  Fx þ Fy þ Fz 2 2 2 1 1 1 U15 ¼ Fx þ Fy  Fz 2 2 2

(8.281)

1 1 U14 ¼ Fx þ Fy 2 2

1 1 U16 ¼ Fy  Fz : 2 2

(8.280)

8.6 Guidance to the sequence of exploiting existing symmetries As a general guideline, a logical sequence in exploiting existing symmetries is as follows. (1) Translational symmetries; (2) Central reflection when available, if the user wishes to have a single set of boundary conditions for all loading cases and also to allow mixed loading cases; (3) Reflectional symmetries when available, if the user is allowed to deal with individual loading cases separately without any need to analyze mixed loading cases. Reflectional symmetries should be given priority if it co-exists with rotational symmetries because the former tend to result in relatively simpler forms of boundary conditions. (4) Rotational symmetries when available. Similar to the reflectional symmetries, the use of rotational symmetries will force individual loading cases to be analyzed separately. The employment of them will also mean that the user is prepared to handle more complications in boundary conditions. The translational symmetries in the architecture at the lower length scale deliver the homogeneity at the upper length scale as elaborated in Chapter 3.

Further symmetries within a UC

315

In presence of such symmetries, a free body of finite extents in all directions can be identified which covers the space the material occupies through translational symmetries. The free body can be employed as a UC after boundary conditions have been prescribed to it according to the symmetries involved, although it might not be of the smallest size. As a UC, it can be analyzed under a common set of boundary conditions for all loading conditions. Using these symmetries alone will not impose any restriction to the material to be categorised and characterised in terms of its effective properties, except the homogeneity of the material at the upper length scale, and it can still be as anisotropic as it could be. Central reflection is not always available. Even if it is, presenting it may require careful identification of the free body as described above. The advantageous feature of this symmetry is that it halves the size of the UC to be analyzed without placing any restriction on the anisotropy. The halfsized UC can still be analyzed under a single set of boundary conditions for all loading cases, although the boundary conditions can be slightly more complicated. Advantages can be taken of reflectional and rotational symmetries if they are present to reduce the size of the UC to be analyzed. Any use of them, however, will impose a restriction to the analysis, as well as the categorisation of the material. With any of these symmetries, a material’s principal plane or axis will be identified. The presence of a single principal axis will put the material into the monoclinic category. If there exists a second principal plane or axis orthogonal to the first, the material will be orthotropic. Since under these symmetries some loading cases in terms of individual average stress components are symmetric, whilst others are antisymmetric, different boundary conditions may have to be employed under different loading conditions as fully elaborated in this chapter. Given the functionality of these types of symmetries, translational symmetries have to be employed first. Otherwise, using other symmetries in attempt to reduce the domain from infinite to finite will be flawed as pointed out in Chapter 5.

8.7 Concluding statement As established in Chapter 6, translational symmetries alone are sufficient to construct UCs resulting in UCs defined through relative displacement boundary conditions. In practice, however, additional symmetries are often available in UCs established using translational symmetries alone.

316

Representative Volume Elements and Unit Cells

How to take advantage of these additional symmetries, viz. reflections and rotations, has been the subject of the present chapter. Whilst each use of such an additional symmetry reduces the UC to be analyzed to half of its original size, it does not come without penalties. One has to be prepared for more complicated boundary conditions for the UC to be analyzed. One may also have to deal with different effective stress components as loading cases under different boundary conditions. Additional symmetries can be present in combined form between reflections and rotations and they can come in multiplicity. Careful selection of tessellations in imposing the translational symmetries to generate the original UCs is essential in order to preserve various additional symmetries in those UCs generated using translational symmetries alone. A particular type of symmetry, central reflection, though it is not an independent type of symmetry geometrically, offers substantial savings in computational cost as a single set of boundary conditions is applicable to all loading conditions, unlike in case of reflections or rotations. The only price to pay is that the boundary conditions become more complicated to a degree, but it certainly is a price worthwhile to pay, especially if the associated UCs are meant to be used extensively, e.g. in virtual testing for material characterisation. As the boundary conditions get more and more sophisticated, the significance of ‘sanity checks’ simply cannot be overstated when implementing those boundary conditions. In many ways, the present chapter reinforces the point advocated in the Chapter 6 that construction of a UC is about establishing the boundary conditions the UC will be subjected to. It is worth noting that the Kdofs involved in formulations of UCs with additional symmetries are to be treated in exactly the same way as in problems using translational symmetries alone. This applies both to defining the nodal displacements/concentrated forces at the Kdofs as input and processing the output. It can be proven, by adapting accordingly the derivations presented in Chapter 7, that in UCs involving further symmetries the relative displacement and traction boundary conditions at Kdofs remain the essential and natural boundary conditions, respectively. In most cases, the boundary conditions presented in this chapter appear to be tedious. However, they are systematic and hence suitable for programming. Readers are reminded that the finite element method was unthinkable to apply manually, but, once coded appropriately, it has become a universally applicable tool, without which modern engineering can hardly

Further symmetries within a UC

317

sustain itself. The claimed systematic nature has been demonstrated through a code, UnitCells©, as a secondary development of Abaqus/CAE (Li et al., 2015). It will be described in the final chapter of this book.

References Abaqus Analysis User’s Guide, 2016. Abaqus 2016 HTML Documentation. Li, S., 1999. On the unit cell for micromechanical analysis of fibre-reinforced composites. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455, 815e838. Li, S., 2008. Boundary conditions for unit cells from periodic microstructures and their implications. Composites Science and Technology 68, 1962e1974. Li, S., 2012. On the nature of periodic traction boundary conditions in micromechanical FE analyses of unit cells. IMA Journal of Applied Mathematics 77, 441e450. Li, S., Reid, S.R., 1992. On the symmetry conditions for laminated fibre-reinforced composite structures. International Journal of Solids and Structures 29, 2867e2880. Li, S., Zhou, C., Yu, H., Li, L., 2011. Formulation of a unit cell of a reduced size for plain weave textile composites. Computational Materials Science 50, 1770e1780. Li, S., Zou, Z., 2011. The use of central reflection in the formulation of unit cells for micromechanical FEA. Mechanics of Materials 43, 824e834. Li, S., Jeanmeure, L.F.C., Pan, Q., 2015. A composite materials characterisation tool: UnitCells. Journal of Engineering Mathematics 95, 279e293. Nye, J.F., 1985. Physical Properties of Crystals. Clarendon Press, Oxford.

CHAPTER 9

RVE for media with randomly distributed inclusions 9.1 Introduction Modern materials often exhibit different internal architectures at different length scales or in different perspectives. Some of those architectures might be of completely irregular physical and geometric characteristics, such as that in particulate reinforced composites, in which reinforcing particulates are usually dispersed at random, or the random fibre distribution in the transverse cross-section of UD composites. Micromechanical modelling of such problems requires the use of representative volume elements (RVEs). The definition of an RVE and its associated considerations have been presented in Chapter 4, followed by a critical review of commonly observed misperceptions and misuse of RVEs in the literature in Chapter 5. Having reached the level of understanding thus far, the need is apparent for RVE formulation and analysis techniques that are logically derived based on the correct concepts and deliver results that are mathematically accurate. The task of this chapter is to offer such an approach to address the problem and elaborate on the steps taken in order to deliver the subject without ambiguity. Exact solution would be obtained if the problem could be analysed over an infinite domain, or the correct boundary conditions had been prescribed if a finite domain is employed as an RVE. However, none of these options is practical in general. The proposed approach is based on the principle of Saint-Venant. In the context of the present application, it states that the results obtained over the regions within the RVE sufficiently distant from the boundary, where inaccurate boundary conditions are prescribed, should be close enough to the exact solution, provided that the forces associated with such incorrect boundary conditions are statically equivalent to those appearing in the infinite domain or those in the exact boundary conditions. However, the principle of Saint-Venant itself does not provide any estimate of how large the distance from the boundary should be. In fact, it varies from problem to problem in general and the characteristic measure of this Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00009-8

© 2020 Elsevier Ltd. All rights reserved.

319

j

320

Representative Volume Elements and Unit Cells

distance, or decay length, as defined in Chapter 4, will have to be determined in an ad hoc manner for a given type of problems.

9.2 Displacement boundary conditions and traction boundary conditions for an RVE The description of uniform displacement and uniform traction boundary conditions as referred to in Chapter 4 is in fact rather simplistic in a sense that it does not provide instructions of how such boundary conditions are to be implemented. The elaborated description of these boundary conditions will therefore be provided here to benefit potential users. Consider a rectangular area, 0  ya and 0  z  b, as the volume element in the yz-plane in a 2D space for the ease of presentation, which can be easily extended to a cuboid in a 3D space. Assuming a uniaxial stress s0y is to be prescribed in the y-direction in order to determine the effective Young’s modulus and Poisson’s ratio in this direction, the traction boundary conditions can be given as follows.
py
y¼0 ¼ s0y and pz jy¼0 ¼ 0;
and pz jy¼a ¼ 0; py
y¼a ¼ s0y (9.1)
and pz jz¼0 ¼ 0; py
z¼0 ¼ 0
and pz jz¼b ¼ 0; py
z¼b ¼ 0 where py and pz are the traction components. If one is interested in the determining the effective shear modulus, the traction boundary conditions for an RVE under a pure shear stress s0yz in the yz-plane are to be applied as
and pz jy¼0 ¼ s0yz py
y¼0 ¼ 0
py
y¼a ¼ 0 and pz jy¼a ¼ s0yz (9.2)
py
z¼0 ¼ s0yz and pz jz¼0 ¼ 0
and pz jz¼b ¼ 0: py
z¼b ¼ s0yz For FEM applications, these conditions will have to be topped up with appropriate constraints to rule out rigid body motions, in particular, rigid body rotations. The correct method of constraining rigid body rotations when the domain deviates from orthogonal regularity is elaborated in

321

RVE for media with randomly distributed inclusions

Section 5.5 of Chapter 5. Regarding the application of tractions, an important point is to be made about the vanishing traction components as involved in (9.1) and (9.2). Since traction boundary conditions are natural boundary conditions, they are prescribed when loads are applied. Loads of zero magnitude imply the absence of loads and therefore they do not need to be applied explicitly in an FE analysis. In other words, vanishing traction boundary conditions should be left unattended and this is strictly correct as far as FE modelling is concerned, whilst they will be satisfied approximately through energy minimisation in the same way as equilibrium conditions are satisfied, as well as the nontrivial boundary conditions. In (9.1) and (9.2), what needs to be applied is therefore reduced to the following.

py
y¼0 ¼  s0y and py
y¼a ¼ s0y for uniaxial tension; and (9.3) pz jy¼0 ¼ s0yz
py
z¼0 ¼ s0yz

and and

pz jy¼a ¼ s0yz
py
z¼b ¼ s0yz

for pure shear.

(9.4)

Note that if one wishes to apply uniaxial stress s0z in the z-direction, boundary conditions for such case will be similar to those given by (9.3) but will have to be applied on surfaces z¼0 and z¼b. The so-called uniform displacement boundary conditions are not usually prescribed purely in terms of displacements. The boundary conditions for the same loading case as (9.1) and (9.2) are actually defined in a mixed form as follows. vjy¼0 ¼ 0

and

pz jy¼0 ¼ 0

and pz jy¼a ¼ 0 vjx¼a ¼ aε0y
py
z¼0 ¼ 0 and pz jz¼0 ¼ 0
py
z¼b ¼ 0 and pz jz¼b ¼ 0
py
y¼0 ¼ 0 and wjy¼0 ¼ 0
py
y¼a ¼ 0

and

vjz¼0 ¼ 0

and pz jz¼0 ¼ 0

vjz¼b ¼ bg0yz

and

wjy¼a ¼ 0 pz jz¼b ¼ 0

for uniaxial tension; and

for pure shear.

(9.5)

(9.6)

322

Representative Volume Elements and Unit Cells

These boundary conditions should be topped up with the constraints to eliminate rigid body motions remaining in the system, viz. translation in the z-direction for (9.5) and rigid body rotation for (9.6). Ignoring the vanishing natural boundary conditions, one is left with the following, respectively. vjy¼0 ¼ 0 (

and

vjy¼a ¼ aε0y

for uniaxial tension; and

wjy¼0 ¼ wjy¼a ¼ 0 vjz¼0 ¼ 0

and

vjz¼b ¼ bg0yz

for pure shear

(9.7) (9.8)

One can apply uniaxial strain ε0z in a similar manner as ε0y is prescribed in (9.7). Note that the vanishing traction boundary conditions in (9.5) and (9.6) should not be replaced by vanishing displacements. Otherwise, they will correspond to a completely different loading conditions. It should also be pointed out that although (9.1) and (9.2) or (9.3) and (9.4) are unique forms of prescribing the relevant traction boundary conditions, the way of constraining rigid body motions as a necessary part of the imposition of boundary conditions can take different forms, and rigid body motion constraints have not been included in (9.1) and (9.2) or (9.3) and (9.4). On the other hand, (9.5) and (9.6) or (9.7) and (9.8) are not unique forms of prescribing displacement boundary conditions for the relevant effective stress states. This is due to the fact that any combination of rigid body motions can be superimposed on top of them to present a different appearance without changing the stress state, as has been explained in Section 6.2 of Chapter 6. For instance, one can have 1 vjy¼0 ¼  aε0y 2

and

1 vjy¼a ¼ aε0y 2

for uniaxial tension; and (9.9)

wjy¼0 ¼ 0

and

1 wjy¼a ¼ ag0yz 2

for pure shear (9.10) 1 0 vjz¼0 ¼ 0 and vjz¼b ¼ bgyz 2 They will be to exactly the same effects as (9.7) and (9.8). Any lack of uniqueness is a source of potential confusions and it is the responsibility of the users to apply the displacement boundary conditions correctly.

RVE for media with randomly distributed inclusions

323

In much the same way as in UCs, when (9.7) and (9.8) are employed, average strains ε0y and g0yz serve as Kdofs at which loads can be applied. The concept of Kdofs and their applications have been introduced in Section 6.6 of Chapter 6. Whilst displacement boundary conditions as provided above could be perfectly correct for UCs, as obtained from the symmetry conditions due to the regularity of the architecture at the lower length scale, the traction boundary conditions are usually incorrect unless the material is homogeneous even at its lower length scale. In absence of regularity as is usually the case for RVEs, even the displacement boundary conditions cannot be prescribed correctly, in general, let alone the traction boundary conditions. Any application of such boundary conditions will have to be considered as some kind of approximation. The extent of such an approximation determines the validity of the RVE introduced, as will be the subject of this chapter. Hill (1963) suggested that the volume element was representative if one could obtain sufficiently close results under both the uniform traction and the uniform displacement boundary conditions. As commented in Chapter 4, this is unnecessary, as will be elaborated fully in the next a few sections.

9.3 Decay length for boundary effects Without perfect symmetry, neither uniform tractions nor uniform displacements will give exact boundary conditions for the volume element concerned. However, either of them will represent a statically equivalent uniaxial loading condition if homogeneity is present at the upper length scale. According to the principle of Saint-Venant, either of them should result in perfectly correct stress and strain distributions an appropriate distance into the volume element from its boundary. This distance varies from point to point along the boundary. The maximum value of this distance for all relevant loading conditions is the decay length, that has already been introduced in Chapter 4. The inner part of the volume element a decay length away from its boundary in all directions forms a subdomain of the volume element. As long as the original volume element is large enough to allow such a subdomain to be also representative in terms of volume fractions of constituents, the subdomain will be an RVE, a perfect one. The effective properties of the material should not be extracted directly from the original RVE itself, but from the subdomain where the stress and strain distributions are free from the effects of incorrect boundary conditions, in

324

Representative Volume Elements and Unit Cells

terms of either traction or displacement. The question that remains is what should be the decay length. The principle of Saint-Venant does not provide any clear guideline for it, and in fact, the decay length varies from material to material and from one effective property to another, hence it will have to be determined by the user on a case-by-case basis. In order to estimate the decay length for UD composites, one with regularly packed fibres in its transverse cross-section is considered as an example. In a composite so chosen, symmetry is present and hence correct boundary conditions can be obtained if the border of volume element is selected along symmetry lines as a stack of repeating cells. In order to examine the effect of inaccurate boundary conditions, one side of the stack of such cells is deliberately truncated away from the symmetry line as shown in Fig. 9.1, such that no correct boundary conditions can be prescribed on it. Two analyses were conducted with each stack of cells, one involving uniform traction boundary conditions, as defined by (9.3), whilst another was carried out with uniform displacement (9.7) being prescribed. Both types of boundary conditions are statically equivalent to uniaxial tension. Comparing the stress distributions from the two analyses in the truncated cells, one can observe how far the disturbance introduced by the incorrect boundary conditions propagates as a way of evaluating the decay length. The results shown in Figs. 9.2 and Fig. 9.3 are von Mises stress contours plotted over a squarely packed array and a hexagonally packed array, respectively, where Fig. 9.2(A) and Fig. 9.3(A) were obtained with uniform displacement being applied, whilst in results shown in Fig. 9.2(B) and Fig. 9.3(B) were obtained with uniform traction being prescribed. The von Mises stress is not usually relevant for composites in general. However, in this case, both the fibre and matrix are assumed to be isotropic and the von Mises stress should be meaningful in the context of the present discussion. Only complete cells have been shown in Figs. 9.2 and 9.3, since the stress distributions in the truncated ones are deemed to be wrong. In Figs. 9.2 and 9.3, the columns with title ‘Case-0’ show stress contour over the stacks that did not involve the truncated cells hence had correct boundary conditions applied to them. Such stacks were analysed in order to compare the results with those obtained for truncated stacks. The remaining columns were obtained under incorrect boundary conditions, as each of those stacks had on the top an incomplete cell truncated at different locations as shown in Fig. 9.1. The pattern of contour plots in the column of Case-0 in Fig. 9.2(A) is identical for every cell in the stack, because Case-0 represents a correct solution to the problem. Comparing the patterns in Case-1, Case-2 and Case-

RVE for media with randomly distributed inclusions

325

Fig. 9.1 Arrays of periodic cell of (A) square packing (B) hexagonal packing.

3 with that in Case-0, it can be observed that the further away the cells are from the top border where incorrect boundary conditions are prescribed, i.e. moving from cell No. 1 to cell No. 5, the closer the comparison of the stress contours with that in Case-0. In fact, there is no noticeable difference

326

Representative Volume Elements and Unit Cells

(a) L1 1→

(b) Case-0 No.1

Case-1

Casse-2

Case-3

Case-0 C

L1 →

Case-1

Case-2

Case-3

No.1

L2 2→

No.2 2

L2 →

No.2

L3 →

No.3 3

L3 →

No.3

L4 L →

No.4 4

L4 →

No.4

L5 5→

No.5 5

L5 → No.5

Fig. 9.2 Von Mises stress contours over complete periodic cells from the square packed arrays (A) under prescribed uniform displacement and (B) under prescribed uniform traction.

between stress contours from cell No.2 onwards. Comparing respective cases in Fig. 9.2(A) and (b), it becomes obvious that relatively, the boundary effects decay over a longer distance if a uniform traction is prescribed, although disturbances in the latter case settle down eventually by cell No. 4 or so. The same trends are repeated in Fig. 9.3 for hexagonal packing where the plots have been arranged in exactly the same order as those in Fig. 9.2. In other words, under uniform displacement the decay length is approximately equal to the length of one cell, whilst under uniform traction it can be several cells. Apparently, the boundary conditions corresponding to prescribed uniform displacement represent a better approximation to the exact boundary conditions than those of prescribed uniform traction. Therefore, it is advisable that the displacement boundary condition should be employed. In this case, the typical decay length will be a couple of characteristic lengths. More justifications can be found in (Wongsto and Li, 2005). For regular patterns, the characteristic length is the cell size. In the case of

327

RVE for media with randomly distributed inclusions

(a)

(b) Case-0

L1 o

Case-11

Casse-2

Case-3

L1 o

Case-0

Case-1

C ase-2

Case-3

No o.1

No.1

L2 o

L2 o

No o.2

No.2

L3 o

L3 o

No o.3

No.3

L4 o

L4 o

No o.4

No.4

L5 o

L5 o

No o.5

No.5

Fig. 9.3 Von Mises stress contours over complete periodic cells from the hexagonally packed arrays (A) under prescribed uniform displacement and (B) under prescribed uniform traction. Generating the microstructures of randomly distributed physical and geometric features.

irregular patterns, it can be expected to be the average distance between the centres of two adjacent fibres. This will be further verified in Section 9.5.

9.4 Generation of random patterns In order to facilitate theoretical construction of RVEs, a means of generating a randomly distributed features is desirable. A UD composite having random fibre distribution over its transverse cross-section will be taken as a special case in a 2D space. The process of reproducing the geometry of cross-section is equivalent to generating randomly spaced, nonoverlapping circular discs in a plane within a domain. There could be different approaches to achieve this goal. Digitising microscopic photographs such as that shown in Fig. 4.1 is an obvious one, but the procedure could be computationally costly and time-consuming. Employing a suitable numerical algorithm would be more effective in this respect. One possibility is to adopt a ‘coin-dropping’, or random insertion process, as was

328

Representative Volume Elements and Unit Cells

implemented by Bulsara et al. (1999). Whilst the random nature of the obtained distribution is obvious, it may be difficult to achieve a specified disc (constituent) volume fraction, especially at high volume fractions. Constituent volume fraction is often a dominant parameter in determining the performance of the composite and a reasonable control of it is essential. A different process was employed by Gusev et al. (2000) using a Monte Carlo method. An alternative scheme was proposed in Wongsto and Li (2005), which is capable of reproducing any practical volume fraction and yet maintaining the random nature in distribution. The outline of the procedure is as follows. A set of regularly packed discs of radius R in the yz-plane is generated. An example of such set with hexagonally packed discs is shown in Fig. 9.4. The spacing between discs, 2b, can be calculated making use of the constituent volume fraction, Vf, as 2 pR ffiffi . Next, two frames are introduced and fixed in the plane, Frame b2 ¼ 2p 3V f

1 and Frame 2, with the latter being placed inside the former. The distance from the border of Frame 2 to that of Frame 1 should be equal to a decay length, which, according to the conclusion reached in the previous section, is a few characteristic lengths, b. As in Fig. 9.4, it is approximately 4b. These frames do not have to be perfect squares, but they both have to contain

Frame-1

Frame-2

Fig. 9.4 Hexagonally packed discs at Vf ¼ 65% involving 105 discs.

RVE for media with randomly distributed inclusions

329

sufficient number of discs. They have been chosen to be square, i.e. b¼a, as shown in Fig. 9.4 for the sake of argument. As a next step, each disc is moved in a random manner as illustrated in Fig. 9.5. A random angle q between 0 and 360 is generated first to determine the direction for the disc to shift. Along this direction, a maximum distance of shift can be found, which is the smaller of the distance to the border of Frame 1 and that to the point when the disc comes into contact with another, marked as r in Fig. 9.5. Then a second random number, k, is generated between 0 and 1 and the actual distance of shift is given as kr. This process is performed for every disc to complete one iteration. One iteration is effectively one shake of the layout. A sufficiently large number of iterations will have to be made, e.g. 250, before the original regularity disappears. After that, a check on the volume fractions within both frames will be conducted after each iteration. The iterations terminate when the volume fractions in both frames are close enough to the original one, allowing a small discrepancy, e.g. 2% error, as the criterion for terminating iterations. The procedure can be easily programmed and, given parameters of R, Vf and the number of discs to be involved, a set of coordinates can be obtained as the centres of the randomly distributed discs. As examples of the applications of the scheme outlined above, three cases have been generated at a volume fraction of 65% as shown in Fig. 9.6(AeC). They are of the similar volume fraction to that shown in Figure 4.1 and share all the features present in Figure 4.1. A further case was generated at Vf ¼ 60% as shown in Fig. 9.6(D), for which some experimental data are available (Soden et al., 1998). Computationally, this procedure is very efficient, and any of the cases can be generated literally in a matter of seconds on an ordinary PC. Cases as shown in Fig. 9.6 were meshed and analysed with the results being presented in the next section.

U kU

T

z y

Fig. 9.5 Schematic diagram for perturbation process.

330

Representative Volume Elements and Unit Cells

(a) Vf1 = 65.69%, and Vf2 =64.17%

(b) Vf1 = 65.84% and Vf2 = 64.26%

(c) Vf1 = 66.38% and Vf2 = 64.59%

(d) Vf1 = 61.00% and Vf2 = 59.40%

Fig. 9.6 Random fibre distribution obtained for a case with original volume fraction Vf ¼ 65% after (A) 265 iterations; (B) 286 iterations and (C) 382 iterations. Fibre distribution for a case with Vf ¼ 60% after 262 iterations e plot (D). Volume fractions within Frame 1 and Frame 2 are denoted as V1f and V2f , respectively.

9.5 Strain and stress fields in the RVE and the subdomain The cases as generated in the previous section represent transverse cross-sections of UD composites when randomly distributed discs are viewed as fibres and the surrounding area as the matrix. To proceed with the subsequent micromechanical FE analysis, area delimited by Frame 1 will be treated as the RVE. For the purpose of extracting and processing

RVE for media with randomly distributed inclusions

331

the results afterwards, the areas inside and outside Frame 2 are meshed individually, whilst the compatibility of the mesh will be maintained across the border of Frame 2 so that the part of the mesh inside Frame 2 can be extracted as a subdomain. Examples of these meshes are shown in Figs. 9.7 and 9.8, where respective RVEs correspond to random distributions presented in Fig. 9.6(A) and (D). In Fig. 9.7, a further frame was introduced within Frame 2. This does not compromise the validity of the mesh except that it had an additional consideration during meshing. The purpose of it will be clear when the results are discussed. Any gap between two neighbouring fibres will be meshed into at least two elements across the gap for representativeness of the mesh. In the unlikely event of two touching fibres, compromises will have to be made using triangular elements of very sharp corners. The analysis is carried out based on the generalised plane strain idealisation as has been described in Chapter 6 in the context of UC applications. The quadratic triangular and quadrilateral generalised plane strain elements CPEG6 and CPEG8 of Abaqus (2016) were used for the longitudinal/transverse tension and transverse shear deformation whilst heat conduction elements DC2D6 and DC2D8 were adapted for the anticlastic problem of longitudinal shear based on an analogy as described in Section 6.4.1.1 of Chapter 6.

Fig. 9.7 The mesh for the RVE with fibres distributed at random for the case of Vf¼65%.

332

Representative Volume Elements and Unit Cells

Fig. 9.8 The meshes for one of the RVEs with fibres distributed at random at Vf¼60%.

It has been established in Section 9.3 that imposition of a uniform normal displacement is a more advantageous method of prescribing the loading to the RVE. Therefore, to achieve a macroscopically uniaxial stress state in the y-direction, displacements as defined in (9.7) were applied along the relevant sides of the boundary. This can be conveniently achieved by making use of the Kdof ε0y . As has been established in Chapter 7, a uniform displacement boundary condition can be prescribed by applying a concentrated ‘force’ As0y , where A¼a2 is the planar area of the RVE, to the Kdof instead of a prescribed nodal ‘displacement’ without changing the nature of boundary conditions, provided that displacements on the boundary have been associated with the Kdofs. This ensures a uniform displacement imposed on the boundary whilst leaving its magnitude for the subsequent analysis to determine as the ‘nodal displacement’ at the Kdof concerned. To facilitate the analysis, for all the cases with 65% fibre volume fraction, both the fibre and matrix are assumed to be isotropic and homogeneous with elastic properties Em ¼ 1 GPa, nm ¼ 0.3 and Ef ¼ 10 GPa and nf ¼ 0.2 for matrix and fibre, respectively. For the case with 60% fibre volume fraction, material properties of both fibre and matrix are listed in Table 9.1 (Soden et al., 1998).

333

RVE for media with randomly distributed inclusions

Table 9.1 Constituents properties of the Silenka Eglass 1200 tex and epoxy UD composite.

Matrix Young’s modulus Matrix Poisson’s ratio Fibre Young’s modulus Fibre Poisson’s ratio Fibre volume fraction

3.35 (GPa) 0.35 74 (GPa) 0.2 60%

The analyses were carried out on the domain Frame 1. To obtain the pattern of deformation and stress distribution in the RVE, a load as the ‘concentrated force’ equivalent to 1 MPa average uniaxial stress in the y-direction was applied to it. The results extracted from the subdomain Frame 2 were considered to represent the correct responses of the material, free from the effects of incorrectly prescribed boundary conditions along the border of Frame 1. The analysis and data extraction process have been applied to all the four cases shown in Fig. 9.6. The results are presented and discussed below. The von Mises stress contour plot over the subdomain Frame 2 corresponding to the case with Vf¼65% as shown in Fig. 9.6(A) is displayed in Fig. 9.9. The contour plot is shown in a deformed configuration with the magnitude of deformation being amplified by a factor of 300. The areas with the highest values of stress are always found at interfaces, especially

Fig. 9.9 Contours of von Mises stress in Frame 2 as a part of RVE for Case-1 under macroscopically uniaxial transverse tension.

334

Representative Volume Elements and Unit Cells

in areas where fibres are close to each other in the direction of loading (Zou and Li, 2002). The distortion of the shape of fibre cross-section is small relative to overall deformation because of fibre stiffness being higher than that of the matrix. Therefore, the deformation is mostly taken by the matrix although stresses in the fibres tend to show higher levels than those in the matrix. This explains why observed stress concentrations are so located. As a result of deformation, the edges of Frame 2 do not remain straight. This provides an indication how wrong it would be if Frame 2 was analysed alone and straight edges had been assumed when prescribing displacement boundary conditions. To further emphasise this, a case was generated where the domain Frame 2 was analysed directly, with uniform displacements prescribed along the straight edges of this domain. The stress contour for this case is shown in Fig. 9.10 with the same deformation magnification factor of 300. Comparing this case with results obtained from the more accurate analysis conducted on Frame 1 from which Frame 2 was extracted, as shown in Fig. 9.9, significant difference can be found. The maximum von Mises stress in the incorrect analysis were found to be 9.879 MPa in the fibres and 4.501 MPa in the matrix whilst their counterparts obtained in the more accurate analysis were 14.371 MPa and 10.997 MPa, respectively. A potential but significant risk of employing incorrect analysis is that if one uses the von Mises criterion with a given yield stress for the matrix, the load for the onset of plastic deformation in the matrix will be miscalculated.

Fig. 9.10 Von Mises stress contour over the Frame 2 alone analysed with inaccurate boundary conditions.

RVE for media with randomly distributed inclusions

335

In this particular case, for instance, the respective load was underestimated by approximately a factor of 2.5. Given the erroneous analysis as shown in Fig. 9.10, reasonable accuracy can still be achieved in the subdomain within Frame 2 in Fig. 9.7 as mentioned before. To demonstrate this, the von Mises stress contour plots for this same subdomain but from different analyses, one on Frame 1 and the other on Frame 2, are shown in Fig. 9.11. Both plots share many common features, although the distance to the borders of Frame 2 is not quite equal to a decay length and the innermost frame may not be large enough to be representative. Some of the differences in the patterns of the contour

Fig. 9.11 Frame 3 extracted from (a) Fig. 9.10 and (b) Fig. 9.11, both with a deformation magnification factor of 300.

336

Representative Volume Elements and Unit Cells

plots are due to the difference in the scales of the stress levels as shown in the legends. The consistency in their trends reinforces the justification made to the proposed methodology based on the fact that under approximate boundary conditions, correct solutions can be obtained from an inner subdomain at least one decay length away from the boundary where the approximate boundary conditions are prescribed. The underlying mechanical principle is that of Saint-Venant.

9.6 Post-processing for average stresses, strains and effective properties As was mentioned in Chapter 5, a proper description of the procedure for post-processing the results from the analyses of RVEs and UCs has been a topic often avoided the literature. In the case of RVEs, if stresses and strains are to be extracted from a subdomain, one cannot make use of Kdofs when post-processing the results as the Kdofs are for the complete RVE, not the subdomain. The stresses and strains will have to be averaged one way or another, which involves the integration of stresses and strains over the subdomain. Since implementing the integration procedure of the stresses and strains as a computer code is a demanding job for most users, the following simplifications, which keep the mathematical rigour, can be helpful. One will find Green’s formula    Z  I I ny vfy vfz ds (9.11) þ dA ¼  fz dy þ fy dz ¼ ½ fy fz  vy vz nz A

vA

vA

and the Gauss theorem (divergence theorem) 8 9 >  ZZZ  = < nx > vfx vfy vfz þ þ dU ¼ % ½ fx fy fz  ny dS > vx vy vz ; : > vU nz U

(9.12)

useful, where ½ fx fy  and ½ fx fy fz  are continuously differentiable vector fields in 2D and 3D, A and U are the domains of interest in 2D and 3D spaces, respectively, with vA and vU being their boundaries, ½ ny nz  and ½ nx ny nz  are the unit outward normals to vA and vU, s is the arc length of vA and S the surface area of vU. In particular, if one introduces ½ fx fy  ¼ ½ gx hx gy hy  for 2D cases and ½ fx fy fz  ¼ ½ gx hx gy hy gz hz  for 3D cases, respectively, where g and h are two separate continuously

337

RVE for media with randomly distributed inclusions

differentiable vector fields like f, Green’s formula and the Gauss theorem give the following integration by parts, respectively     Z  I Z  ny vgy vhy vgz vhz hy þ hz dA ¼ ½ fy fz  þ gz ds  gy dA vy vz vy vz nz A

A

vA

(9.13) ZZZ  U

 vgy vgx vgz hx þ hy þ hz dU ¼ vx vy vz

¼ % ½ gx hx

8 9 nx > > > > >  ZZZ  = < > vhy vhx vhz gz hz  ny dS  þ gy þ gz gx dU > > vx vy vz > > > > : ; U nz (9.14)

gy hy

vU

With these mathematical formulae, the average strains and stresses in 2D case can be derived. Specifically, making use of Green’s formula (9.13) and referring to Fig. 9.12 for symbols involved, average strains are determined as  Z Z Z  I 1 1 vv 1 vv v0 1 0 εy ¼ εy dA ¼ 0dy þ vdz dA ¼  dA ¼ A A vy A vy vz A A

¼ ¼

1 A 1 A

A

I vdz ¼

1 A

A

ZC3 vdz þ C2

vA Zz2

vjy¼y2 dz 

Zz2

1 A

z1

1 A

ZC1 vdz ¼ C4

1 A

vA ZC4

ZC3 vdz  C2

(9.15)

z1

z z2 z1

C4

C1

y1

C3

C2

y2

vdz C1

vjy¼y1 dz;

b

1 A

y

a

Fig. 9.12 An RVE and its subdomain.

338

Representative Volume Elements and Unit Cells

1 ε0z ¼ A

Zy2 y1

g0yz ¼

1 A

Z gyz dA ¼ A

1B @ A

Z 

ZC2

1B @ A 0

1 A

A

(9.16)

ZC3

ZC4 wdz 

C2

Zy2

ZC1 vdy þ

C3

Zy2 vjz¼z2 dy 

y1

 I vw vv 1 vdy þ wdz ¼ þ dA ¼ vy vz A vA

vdy þ C1

¼

wjz¼z1 dy; y1

0 ¼

Zy2

1 wjz¼z2 dy  A

C wdzA ¼

C4

Zz2 vjz¼z1 dy þ

y1

1

Zz2 wjy¼y2 dz 

z1

1 C wjy¼y1 dzA;

z1

(9.17) where notation A is used both as the subdomain for post-processing and the area of it for averaging without confusion. To apply (9.15)e(9.17), it is essential that the sides of the subdomain are straight and parallel to the coordinate axes. To evaluate the integrations along the sides of the subdomain in the expressions above consistently, appropriate numerical integration rules should be employed for all elements involved along the side. If the elements are linear, i.e. 3- or 4-noded, the trapezium rule should be applied. For quadratic elements, one should use the Simpson’s rule. These are to reflect the fact that the contributions from different nodes are of different weights depending on the location of the node. It is clear that the coordinates of the nodes on all sides must be available and used appropriately as a part of the post-processing. Similar arguments can be made for average strains in 3D cases, for which the Gauss theorem should be used instead, resulting in the following expressions for average strains 8 9 >  ZZZ  = < nx > 1 vu v0 v0 1 0 εx ¼ þ þ dU ¼ % ½ u 0 0  ny dS > U vx vy vz U vU ; : > nz U 1 0 ZZ ZZ 1B C (9.18) udS  udS A; ¼ @ U Sx2

Sx1

339

RVE for media with randomly distributed inclusions

ε0y ¼

1 U

ZZZ  U

 v0 vv v0 1 þ þ dU ¼ % ½ 0 v vx vy vz U vU

1 0 ZZ ZZ 1B C vdS  vdSA; ¼ @ U Sy2

ε0z ¼

1 U

(9.19)

8 9 nx > > > = < > w  ny dS > > > ; : > nz

(9.20)

Sy1

ZZZ  U

8 9 nx > > > = < > 0  ny dS > > > ; : > nz



v0 v0 vw 1 þ þ dU ¼ % ½ 0 vx vy vz U vU

0

1 0 ZZ ZZ 1B C wdS  wdS A; ¼ @ U Sz2

g0yz

1 ¼ U

Sz1

ZZZ  U

 v0 vw vv 1 þ þ dU ¼ % ½ 0 vx vy vz U vU

w

8 9 nx > > > > > > > < > = v  ny dS > > > > > > > : > ; nz 1

0 ZZ ZZ ZZ ZZ 1B C wdS  wdS þ vdS  vdSA; ¼ @ U Sy2

Sy1

Sz2

Sz1

(9.21)

g0xz

1 ¼ U

ZZZ  U

 vw v0 vu 1 þ þ dU ¼ % ½ w vx vy vz U vU

0

8 9 nx > > > > > > > < > = u  ny dS > > > > > > > : > ; nz 1

0 ZZ ZZ ZZ ZZ 1B C wdS  wdS þ udS  udS A; ¼ @ U Sx2

Sx1

Sz2

Sz1

(9.22)

340

g0xy

Representative Volume Elements and Unit Cells

1 ¼ U

ZZZ  U

 vv vu v0 1 þ þ dU ¼ % ½ v vx vy vz U vU

u

8 9 nx > > > > > > > = < > 0  ny dS > > > > > > > ; : > nz 1

0 ZZ ZZ ZZ ZZ 1B C vdS  vdS þ udS  udSA; ¼ @ U Sx2

Sx1

Sy2

Sy1

(9.23) where U has been used both as the subdomain and its volume and Sx1 , Sx2 , Sy1 , Sy2 , Sz1 and Sz2 are the six faces of the subdomain, which are flat and parallel to the coordinate planes as indicated in the subscripts. The numerical integrations on the faces of the subdomain will have to be evaluated consistently using a 2D numerical integration according to the tessellations on the faces and the order of the elements involved. It should be noted that users tend to base the calculations of the average stresses and strains on their intuition to obtain some formulae seemingly similar but not exactly the same as rigorously derived above. The point to make through the above derivation is to demonstrate that mathematical rigour is often readily available without introducing additional numerical demand. To convert the volume integration to surface integration for average stresses, similar process can be followed, but the outcomes tend to be slightly more complicated than what one would expect intuitively. Using integration by parts again along with necessary mathematical manipulations, one has  ZZZ ZZZ  1 1 vx vx vx 0 sx ¼ sx dU ¼ sx þ sxy þ sxz dU U U vx vy vz U 8 U9 >  ZZZ  = < nx > 1 1 vsx vsxy vsxz ¼ % x½ sx sxy sxz  ny dS  þ þ xdU > U vU U vx vy vz ; : > nz U 0 0 1 ZZ ZZ ZZ 1B C 1B ¼ @x2 sx dS  x1 sx dS A þ @ xsxy dS U U Sx2

Sx1

Sy2

0 1 ZZ ZZ ZZ C 1B C  xsxy dS A þ @ xsxz dS  xsxz dS A U Sy1

1

Sz2

Sz1

(9.24)

341

RVE for media with randomly distributed inclusions

vs

vsxz xy x where equilibrium equation vs vx þ vy þ vz ¼ 0 has been made use of. Similarly, 1 0 ZZZ ZZ ZZ 1 1B C s0y ¼ sy dU ¼ @ ysxy dS  ysxy dS A U U U

Sx2

0 þ

1B @y2 U

ZZ

ZZ sy dS  y1

Sy2

Sy1

1

Sx1

1 0 ZZ ZZ C 1B C sy dS A þ @ ysxz dS  ysxz dS A U Sz2

Sz1

(9.25) 1 0 ZZ ZZ 1 1B C s0z ¼ sz dU ¼ @ zsxz dS  zsxz dS A U U Sx2 Sx1 1 1 0U 0 ZZ ZZ ZZ ZZ 1B C 1B C þ @ zsyz dS  zsyz dS A þ @z2 sz dS  z1 sz dS A U U ZZZ

Sy2

Sy1

Sz2

Sz1

(9.26) s0yz

1 ¼ U

ZZZ U

1 syz dU ¼ U

1 ¼ % y½ sxz U vU

syz

ZZZ  U

 vy vy vy sxz þ syz þ sz dU vx vy vz

8 9 > > > nx > > > = > < sz  ny dS > > > > > > > : nz ;

1  U

ZZZ  U

 vsxz vsyz vsz þ þ ydU vx vy vz

1 1 0 0 ZZ ZZ ZZ ZZ 1B C 1B C ¼ @ ysxz dS  ysxz dS A þ @y2 syz dS  y1 syz dS A U U Sx2

Sx1

Sy2

Sy1

0 1 ZZ ZZ 1B C þ @ ysz dS  ysz dS A U Sz2

Sz1

(9.27)

342

Representative Volume Elements and Unit Cells

s0xz ¼

1 U

1 0 ZZ ZZ 1B C sxz dU ¼ @ zsx dS  zsx dS A U

ZZZ U

Sx2

Sx1

1 1 0 0 ZZ ZZ ZZ ZZ 1B C 1B C þ @ zsxy dS  zsxy dS A þ @z2 sxz dS  z1 sxz dS A U U Sy2

Sy1

Sz2

Sz1

(9.28)

s0xy ¼

1 U

0

ZZZ sxy dU ¼ U

1B @x2 U

ZZ

ZZ sxy dS  x1

Sx2

1 C sxy dS A

Sx1

1 1 0 0 ZZ ZZ ZZ ZZ 1B C 1B C xsy dS  xsy dS A þ @ xsyz dS  xsyz dS A þ @ U U Sy2

Sy1

Sz2

Sz1

(9.29) Apparently, the shear components of the average stress tensor can be derived differently resulting in different expressions as follows that should nevertheless produce the same numerical outcomes. 1 1 0 0 ZZ ZZ ZZ ZZ 1B C 1B C s0yz ¼ @ zsxy dS  zsxy dS A þ @ zsy dS  zsy dS A U U 0 1B þ @z2 U

Sx2

Szx

ZZ

ZZ syz dS  z1

Sz2

1

Sy2

Sy1

C syz dS A

Sz1

(9.27a)

343

RVE for media with randomly distributed inclusions

0 s0xz ¼

1B 2 @x U

ZZ

ZZ sxz dS  x1

Sx2

Sx1

1

1 0 ZZ ZZ C 1B C sxz dS A þ @ xsyz dS  xsyz dS A U Sy2

1 0 ZZ ZZ 1B C þ @ xsz dS  xsz dS A U Sz2

Sy1

Sz1

(9.28a) 1 1 0 0 ZZ ZZ ZZ ZZ 1B C 1B C s0xy ¼ @ ysx dS  ysx dS A þ @y2 sxy dS  y1 sxy dS A U U Sx2

Sx1

Sy2

1 0 ZZ ZZ 1B C þ @ ysxz dS  ysxz dS A U Sz2

Sy1

Sz1

(9.29a) One can easily reduce the above from their 3D form to their 2D counterparts. 0 1 ZZ Zy2 Zy2 1 1B C s0x ¼ sx dA ¼ @x2 sx jx¼x2 dy  x1 sx jx¼x1 dyA A U y1 y1 A (9.30) 0 x 1 Z2 Zx2 1 þ @ xsxy jy¼y2 dx  ysxy jy¼y1 dxA U x1

s0y ¼

1 A

x1

0

ZZ sy dA ¼ A

0 þ

1@ y2 U

1B @ U

Zy2

Zy2 ysxy jx¼x2 dy 

y1

Zx2

y1

Zx2 sy jy¼y2 dx  y1

x1

x1

1

sy jy¼y1 dxA

1 C ysxy jx¼x1 dyA (9.31)

344

Representative Volume Elements and Unit Cells

s0xy ¼

1 A

0

ZZ sxy dA ¼

þ

1@ y2 A

Zy2 ysx jx¼x2 dy 

y1

A

0

1B @ A

Zy2

y1

Zx2

Zx2 sxy jy¼y2 dx  y1

x1

1 C ysx jx¼x1 dyA

1

(9.32)

sxy jy¼y1 dxA

x1

Equally, one can also have 1 0 Zy2 Zy2 1B C s0xy ¼ @x2 sxy jx¼x2 dy  x1 sxy jx¼x1 dyA A 0 þ

1@ A

y1

y1

Zx2

Zx2 xsy jy¼y2 dx 

x1

1

(9.32a)

xsy jy¼y1 dxA

x1

Relatively, the expressions of average stresses as surface integrations derived above are less likely to be obtained intuitively. For instance, intuition easily leads to the following two expressions as the average direct stress on each of the two surfaces perpendicular to the y-axis. ZZ ZZ y2  y1 y2  y1 0 0 sy ¼ sy dS or sy ¼ sy dS (9.33) U U Sy2

Sy1

If one has achieved complete convergence in terms of RVE size and absolute accuracy, there should not be any difference between either of (9.33) and mathematically derived expression (9.25) in this particular case. However, given the approximate nature of FEM, the accuracy of the derived expressions will be completely consistent with the FEM approximation whilst the intuitive ones lack such consistency. A point to be made here is that although the post-processing is not really exciting, it does not mean that it should be tampered casually, especially when consistent results can be obtained which are not too difficult to evaluate. FEM does not aim at exact solution. However, its accuracy erodes if it is not followed in a consistent manner. The consistency is underlined by the rationality, i.e. the strict definition of physical properties, mathematical rules and logic. For the integrity of science and engineering, it is worth one’s while to search for the rational solutions. Rational ones might demand a bit more skills and patience

RVE for media with randomly distributed inclusions

345

to obtain, but they are not necessarily more complicated to implement. The beauty of truth is that rational ones are often simpler. A good example on the same subject is the post-processing for UCs through the Kdofs as established in Chapter 7. Calling intuitive methods devised for obtaining the desired outcomes the ‘engineering approaches’ could sometimes be an insult to the name. Proper engineering approaches are taken by competent engineers to achieve approximate solutions in absence of rigorous theoretical solutions, whilst keeping necessary control of the errors introduced by the extra assumptions to deliver the approximations. Use of casual approaches that add unnecessary errors to the solutions where theoretically accurate solutions can be obtained can never be justified, especially if the casual approaches are not any simpler than their rigorous counterparts.

9.7 Conclusions It is the subject of this chapter how RVEs should be constructed and analysed and how the results should be extracted through appropriate postprocessing. Practices in this subject as found in the literature tend to be loosely presented without much mathematical rigour. However, through the elaborations and derivations, the coherence in the subject has been firmly established. Together with the conclusion from Chapter 4, an RVE can be constructed from the original configuration without falsifying periodicity. Uniform displacement boundary conditions are prescribed to generate a macroscopically uniaxial stress state or pure shear stress state. A subdomain of a number of characteristic lengths into the RVE should be extracted for post-processing in order to obtain average stresses and strains, from which effective material properties can be evaluated. The evaluation of average stresses and strains can be based on mathematically derived formulae in consistence with the FEM. Intuition is neither necessary nor helpful in this particular case.

References Abaqus Analysis User’s Guide, 2016. Abaqus 2016 HTML Documentation. Bulsara, V.N., Talreja, R., Qu, J., 1999. Damage initiation under transverse loading of unidirectional composites with arbitrarily distributed fibers. Composites Science and Technology 59, 673e682. Gusev, A.A., Hine, P.J., Ward, I.M., 2000. Fiber packing and elastic properties of a transversely random unidirectional glass/epoxy composite. Composites Science and Technology 60, 535e541. Hill, R., 1963. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357e372.

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Representative Volume Elements and Unit Cells

Soden, P.D., Hinton, M.J., Kaddour, A.S., 1998. Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Composites Science and Technology 58, 1011e1022. Wongsto, A., Li, S., 2005. Micromechanical FE analysis of UD fibre-reinforced composites with fibres distributed at random over the transverse cross-section. Composites Part A: Applied Science and Manufacturing 36, 1246e1266. Zou, Z., Li, S., 2002. Stresses in an infinite medium with two similar circular cylindrical inclusions. Acta Mechanica 156, 93e108.

CHAPTER 10

The diffusion problem 10.1 Introduction There is a range of physical and engineering problems governed by a diffusion partial differential equation, such as heat conduction. Mathematically, it is a boundary value problem with a governing partial differential equation of the parabolic type in general. However, for material characterisation, it is sufficient to consider its steady state and the problem degenerates to an elliptic one which is perfectly suited for FEM. Diffusion is a type of problems rather different from stress analysis mathematically, and even more so physically. However, with appropriate arrangement, analogies can be drawn between these two types of problems in multiple ways. Without diverting the discussion to a comprehensive introduction of the subject of diffusion problems, it will rely mostly on the existing analogies to the problem of stress analysis in order to deliver the contents in this chapter.

10.2 Governing equation The diffusion problem is governed by a partial differential equation as follows. vqx vqy vqz vT þ þ ¼ a2 vt vx vy vz

(10.1)

where ½ qx qy qz T is the diffusion flux vector, a a diffusion-related material property, T the concentration field and t the time. In the context of heat conduction, ½ qx qy qz T is the heat flux vector, a2 ¼ cr, with c being the heat capacity and r the density of the material, and T is the temperature field. For the characterisation of the diffusion properties of the medium, one only needs to deal with a much simplified form of the equation in its steady state as follows.

Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00010-4

© 2020 Elsevier Ltd. All rights reserved.

347

j

348

Representative Volume Elements and Unit Cells

vqx vqy vqz þ þ ¼0 (10.2) vx vy vz Such a problem is easily solvable with most of the modern commercial FE codes, which have a readily available facility for solving the heat conduction problem. Even if the actual problem is not exactly a heat conduction problem, an analogy can be resorted to so that the heat conduction problem modelling capabilities in commercial FE codes can be employed to solve it. Modelling anticlastic deformation for the longitudinal shear of UD composites as addressed in Section 6.4.1.1 can serve as an example of application of such analogy. In order to employ a conventional FE solver to characterise the medium which is multi-phased at its lower length scale, the user must provide the diffusion coefficients of each constituent involved in the medium as the input. The diffusion coefficients are defined through the following constitutive relationship between the diffusion flux and the gradient of the concentration field as 9 8 > > vT > > > > > 8 9 > > 3> 2 vx > > > > q k k k > x= > > > 11 12 13 < < = 7 vT 6 qy ¼ 4 k21 k22 k23 5 ; (10.3) > > vy > : > ; > > > qz k31 k32 k33 > > > > > > > vT > > > > ; : vz where kij (i,j ¼ 1,2,3) are the diffusion coefficient matrix of a phase in the medium which is given in its completely anisotropic form. Substituting (10.3) into (10.1) or (10.2), it can be seen that the governing equation for the diffusion problem is a second order partial differential equation about the concentration field. This applies to each phase involved in the medium at the lower length scale. Between the phases, basic continuity requirements have to be met, such as the continuity in the concentration field, T, and the diffusion influx normal to the border between the phases. In addition, the problem will be assigned the appropriate boundary conditions as will be one of the major focuses later in this chapter. The objective of characterisation of the medium involved in a diffusion problem is to evaluate the effective constitutive relationship in the upper length scale between the average gradients of the concentration field and the average diffusion fluxes as follows.

349

The diffusion problem

8 9 2 0> > q > > x k011 > > < = 6 0 q0y ¼  6 4 k21 > > > > > : q0 > ; k031 z

k012 k022 k032

8 9 0> > vT > > > 3> > > > > 0 > > vx k13 > > < 0= 7 vT 0 7 ; k23 5 > vy > > > 0 > > > vT 0 > > k33 > > > > > : ; vz

(10.4)

where k0ij (i,j ¼ 1,2,3) are the effective diffusion coefficients. To evaluate them theoretically, one needs to solve the diffusion problem involving structural features at the lower length scale. Having solved the problem, one can obtain the average diffusion fluxes and average concentration gradients. The relationship between them identifies the effective diffusion coefficients. Use can be made of FEM to deliver such a solution if one employs either a UC, when regularity is present at the lower length scale of the medium, or an RVE in absence of regularity at the lower length scale of the medium that is homogeneous at the upper length scale. The diffusion coefficients for all phases involved should be provided as input. Appropriate formulation of the UC or RVE will come up with proper boundary conditions to be prescribed to the UC or RVE and this will be the subject of the next section. In order to extract the effective diffusion coefficients of the medium represented by the UC or RVE, a certain post-processing will be required, but the demands can be minimised if the model has been formulated properly as will be elaborated in due course in this chapter. As argued in Chapter 3, this should be preceded with appropriate material categorisation. If there exists a reflectional symmetry, for example, about the coordinate plane perpendicular to the x-axis, or a rotational symmetry about an axis, e.g. the x-axis, the medium is monoclinic and its diffusion coefficient matrix reduces to 3 2 0 k011 0 7  0 6 7 6 K ¼ 6 0 k022 k023 7: (10.5) 5 4 0 k032 k033 When there exists another reflectional symmetry about coordinate plane perpendicular to the y- or z-axis, or a rotational symmetry about the y or zaxis, the medium is orthotropic, hence

350

Representative Volume Elements and Unit Cells

2

k011

 0 6 K ¼6 4 0 0

0 k022 0

0

3

7 0 7 5; k033

whilst isotropy is achieved if k011 dium, giving 2 0 3 2 1 k 0 0  0 6 7 06 0 K ¼4 0 k 0 5 ¼ k 40 0 0 0 k0

(10.6)

¼ k022 ¼ k033 ¼ k0 in an orthotropic me0

0

3

1

7 05

0

1

(10.7)

In Eqs. (10.3)e(10.5), the diffusion coefficient matrix has been given in a form which does not have to be symmetric. In fact, some commercial FE codes, e.g. ANSYS/Fluent, allow it to be asymmetric, where there is a special way to accommodate the fact that the diffusion coefficient matrix appears as a full matrix. There has hardly been much discussion further into the subject to define the special categories of the material, such as monoclinic and orthotropic. Similarly, the matter of the symmetry, or the lack of it, of the diffusion coefficient matrix has hardly been addressed either, again, as lack of categorisation, in general. Without pretending to be specialists on the subject of diffusion, a sober question should be raised here. The diffusion problem, as most other physical processes, should be governed by the laws of thermodynamics. The behaviour can therefore be described in terms of the so-called internal energy, with the diffusion fluxes and the concentration gradients as the internal state variables. The relationship between these internal state variables, i.e. the constitutive equations, results from a constraint to the internal energy from the second law of thermodynamics, which remains the thermodynamics consideration underlying most constitutive relationships for most physical processes, if not all. Because of the energy consideration, the symmetry of the diffusion coefficient matrix can be argued in the same way as for the elastic stiffness matrix in Section 3.2.2. In this respect, one could surely present experimental data to demonstrate the lack of symmetry in the diffusion coefficient matrix. Equally, such evidence is available against the symmetry of the stiffness matrix. However, it has been well-established in elasticity that any lack of symmetry in the stiffness matrix as measured from experiments is deemed to be due to the experimental error. The authors believe that the same

The diffusion problem

351

statement can and should be made in relation to the diffusion coefficient matrix, in general. Otherwise, the existence of the internal energy would be brought into question and consequently the laws of thermodynamics would be challenged. The same as in elasticity, asymmetric diffusion coefficient matrix can be allowed in the incremental constitutive relationship for nonlinear problems, as this will not compromise any physical rules. It can be further argued along the same line of elasticity that the diffusion coefficient matrix has to be positive definite, like the stiffness matrix in the theory of elasticity. Otherwise, the first law of thermodynamics would be compromised. Given the argument as presented above, the symmetry of the 33 diffusion coefficient matrix can be assumed. If so, an important conclusion can be drawn as follows. Mathematically, for any nn real symmetric positivedefinite matrix, there always exist n real positive eigenvalues, repeated or not, and n eigenvectors. In the case of the 33 diffusion coefficient matrix, the 3 eigenvectors will define three principal axes perpendicular to each other. In a coordinate system formed using the principal axes as the coordinate axes, the medium, irrespective to the presence of reflectional or rotational symmetries in the medium, will be orthotropic. The three eigenvalues will be the principal diffusion coefficients. In other words, the highest degree of anisotropy for the diffusion problem is orthotropy, in general, no matter how complicated the architecture of the medium is at its lower length scale, as long as the behaviour of the medium remains linear, i.e. all diffusion coefficients are constant whilst the concentration and its gradients vary. The identification of the principal axes can be relatively straightforward in presence of symmetries, reflectional and rotational. In absence of them, it will be a matter of a mathematical eigenvalue problem to solve for the 33 diffusion matrix.

10.3 Relative concentration field The objective of this section is to derive the appropriate boundary conditions for UCs when the medium shows regularity in its structure at the lower length scale. For the purpose of characterisation of such medium, constant gradients of the concentration field at the upper length scale should be prescribed, which are in fact the average concentration gradients at the lower length scale

352

Representative Volume Elements and Unit Cells

9 8 0> > vT > > > > > > > > > > vx > > > > > > = < 0  0 vT VT ¼ : (10.8) > > vy > > > > > > > > > 0> > > vT > > > > ; : vz Assuming the availability of translational symmetries, which is the precondition for the applicability of UCs, the relative concentration between the those at the corresponding points P0 and P in different cells is vT 0 vT 0 vT 0 Dx þ Dy þ Dz: (10.9) vx vy vz With the average concentration gradients as the Kdofs, this equation gives the relative concentration boundary conditions if P is placed at the one part of the boundary whilst P0 at the other part of the boundary of the same UC and ½ Dx Dy Dz T define the translation associated with the symmetry for these two parts of the boundary of the same UC. As in the mechanical problem, different parts of the boundary of the UC may correspond to different translational symmetries and hence different values of ½ Dx Dy Dz T . One has to make sure that such relative concentration boundary conditions have been imposed on every part of the boundary of the UC. Similar to constraining the rigid body displacements for mechanical UCs, one of the nodes in a UC for the diffusion problem should be assigned a reference concentration. For the characterisation of a linear material, the magnitude of the reference concentration will make no difference and one can set it to zero without loss of generality. These will all be illustrated through an example of a cuboidal UC in the next section. Similarly, it is the users’ responsibility to eliminate redundant boundary conditions at the edges and vertices of the UC. The procedure is the same as for its mechanical counterpart. At the Kdofs, the user has the choice of prescribing average concentration gradients or concentrated diffusion flux. If one prescribes unit average concentration gradients in each of the three coordinate directions one at a time, the concentrated diffusion fluxes as obtained from the solution at the these three Kdofs will define a column of the effective diffusion coefficient matrix after being divided by the volume of the UC. T0  T ¼

The diffusion problem

353

One could prescribe concentrated diffusion fluxes at the Kdofs instead, if the properties of interest are the effective diffusion resistances rather than diffusion coefficients, where former is the inverse of the latter.

10.4 An example of a cuboidal unit cell Consider a medium of orthogonal translational symmetries in the x-, y- and z-directions by distances of 2a, 2b and 2c, respectively. A cuboidal UC of side lengths of 2a, 2b and 2c can be defined, as shown in Figure 6.9 of Chapter 6. The available translations can be defined as TP'  TP ¼ 2iaTx0 þ 2jbTy0 þ 2kcTz0 ;

(10.10)

where i, j and k are the number of periods translated in each of the directions from point P to P0 , and Tx0 ; Ty0 and Tz0 are the average concentration gradients in the three coordinate directions. For faces x ¼ -a and x ¼ a (i ¼ 1, j ¼ 0 & k ¼ 0) excluding edges and vertices T jða;y;zÞ  T jða;y;zÞ ¼ 2aTx0 :

(10.11)

For faces y ¼ -b and y ¼ b (i ¼ 0, j ¼ 1 & k ¼ 0) excluding edges and vertices T jðx;b;zÞ  Tðx;b;zÞ ¼ 2bTy0 :

(10.12)

For faces z ¼ -c and y ¼ c (i ¼ 0, j ¼ 0 & k ¼ 1) excluding edges and vertices T jðx;y;cÞ  T jðx;y;cÞ ¼ 2cTz0 :

(10.13)

For edges parallel to the x-axis (excluding vertices) T jðx;b;cÞ  T jðx;b;cÞ ¼ 2bTy0 T jðx;b;cÞ  T jðx;b;cÞ ¼ 2bTy0 þ 2cTz0

(10.14)

T jðx;b;cÞ  T jðx;b;cÞ ¼ 2cTz0 For edges parallel to the y-axis (excluding vertices), after eliminating any redundant conditions T jða;y;cÞ  T jða;y;cÞ ¼ 2aTx0 T jða;y;cÞ  T jða;y;cÞ ¼ 2aTx0 þ 2cTz0 T jða;y;cÞ  T jða;y;cÞ ¼ 2cTz0

(10.15)

354

Representative Volume Elements and Unit Cells

For edges parallel to the z-axis (excluding vertices), after eliminating any redundant conditions T jða;b;zÞ  T jða;b;zÞ ¼ 2aTx0 T jða;b;zÞ  T jða;b;zÞ ¼ 2aTx0 þ 2bTy0

(10.16)

T jða;b;zÞ  T jða;b;zÞ ¼ 2bTy0 For the vertices, after eliminating any redundant conditions, one has T jða;b;cÞ  T jða;b;cÞ ¼ 2cTz0 T jða;b;cÞ  T jða;b;cÞ ¼ 2aTx0 T jða;b;cÞ  T jða;b;cÞ ¼ 2aTx0 þ 2bTy0 T jða;b;cÞ  T jða;b;cÞ ¼ 2bTy0

(10.17)

T jða;b;cÞ  T jða;b;cÞ ¼ 2aTx0 þ 2cTz0 T jða;b;cÞ  T jða;b;cÞ ¼ 2bTy0 þ 2cTz0 T jða;b;cÞ  T jða;b;cÞ ¼ 2aTx0 þ 2bTy0 þ 2cTz0 The reference temperature can be set at the vertex (-a,-b,-c) as T jða;b;cÞ ¼ 0

(10.18)

Through the above boundary conditions, the average concentration gradients Tx0 ; Ty0 and Tz0 are introduced as the Kdofs. If they are prescribed unit values as follows 8 9 0> 8 9 8 9 8 9 > T > > > < x> = > 0 0 1 > > : T0 ; z as three loading cases, then out of the analysis, one obtains the concentrated diffusion fluxes 8 9ð3Þ 8 9ð1Þ 8 9ð2Þ 0> 0> > > > q q0 > q > > > > > > > > > = = = < x> < x> < x> q0y q0y ; and q0y : (10.20) > > > > > > > > > > > > > > > > 0; > 0; > 0; : : : qz qz qz

The diffusion problem

355

Putting them into a 33 matrix and dividing the matrix by the volume of the UC, one obtains the diffusion coefficient matrix of the medium represented by the UC. Translational symmetries also result in periodic boundary conditions in diffusion flux, but they are natural boundary conditions and should not be imposed for FE analysis, as the periodic traction boundary conditions in the mechanical counterpart. It can be concluded that the procedure follows that for mechanical counterpart. As a result, all UCs as presented in Chapter 6, as well as the considerations on Kdofs as discussed in Chapter 7 and additional symmetries as presented in Chapter 8 can be reproduced for the diffusion problem in order to evaluate the diffusion characteristics of the medium. Readers can refer to the references (Gou et al., 2018; Gou et al., 2017; Gou et al., 2017b; Gou et al., 2018b; Gou et al., 2018c; Gou et al., 2015; Li et al., 2011) for more examples of applications of UCs and other related aspects of the diffusion problem.

10.5 RVEs When the medium concerned does not shown any regularity in its structure at the lower length scale, an RVE will have to be employed instead of a UC. Most of the discussions made on RVEs for mechanical counterpart can be translated to the diffusion problem in a reasonably straightforward manner, if concentration is interpreted as displacement and diffusion flux as stress. Much of the discussion for RVEs as presented in Chapter 4 and Chapter 9 can be thus adapted for the diffusion problem here, as well the relevant concepts, such as the decay length. It is generally true that prescribing the concentration boundary conditions results in smaller decay length than prescribing diffusion flux. Relatively, the prescription of boundary conditions in the diffusion problem is slightly simpler and less confusing than that in the mechanical counterpart since one has only a single variable to worry about, either concentration as a scalar or the component of the diffusion flux vector normal to the boundary, whilst in the mechanical problem, their counterparts are both vectors of three components to be considered, namely, displacements or tractions or a logical combination of them as discussed in Section 9.2 of Chapter 9. Consider a cuboid within 0  xa, 0  y  b and 0  z  c as the RVE in a 3D space. In order to determine the effective diffusion coefficients of the

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medium, average concentration gradients Tx0 , Ty0 and Tz0 can be prescribed through the following concentration boundary conditions T jð0;y;zÞ ¼ 0;

T jða;y;zÞ ¼ aTx0

T jðx;0;zÞ ¼ 0;

T jðx;b;zÞ ¼ bTy0

T jðx;y;0Þ ¼ 0;

T jðx;y;cÞ ¼ cTz0 :

(10.21)

Using FEM, one can evaluate the average diffusion flux through each face of the RVE. Through boundary conditions (10.21), average concentration gradients Tx0 , Ty0 and Tz0 are introduced and they can serve the same role as the Kdofs as for UCs. One can then use the argument around Eqs. (10.19) and (10.20) for UCs to deliver the effective diffusion coefficient matrix. A noticeable difference from UCs is that, whilst the analysis is carried out with a complete RVE, the post-processing is conducted for a subdomain of the RVE which is the decay length distance away from its boundary. As stated in Chapter 9, it is the user’s responsibility to ensure that the subdomain remains representative.

10.6 Post-processing for average concentration gradients and diffusion fluxes To obtain effective concentration and diffusion flux, mathematical manipulations similar to those presented in Chapter 9 can be performed. The averages should be taken from an appropriate subdomain of the RVE a decay length distance inside the RVE, i.e. x1x  x2, y1y  y2 and z1z  z2. Following similar arguments as were made as for average stresses and strains in Chapter 9, the expression for average concentrations are obtained as follows:  ZZZ ZZZ  1 vT 1 vT v0 v0 0 Tx ¼ dU ¼ þ þ dU U vx U vx vy vz U U 8 9 1 0 nx > > > > ZZ ZZ = < 1 1B C ¼ % ½ T 0 0  ny dS ¼ @ TdS  TdSA > > U vU U > ; : > Sx2 Sx1 nz (10.22a)

357

The diffusion problem

 ZZZ  vT 1 v0 vT v0 dU ¼ þ þ dU vy U vx vy vz U U 8 9 1 0 nx > > > > ZZ ZZ < = 1 1B C TdS  TdS A ¼ % ½ 0 T 0  ny dS ¼ @ > > U vU U > : > ; Sy2 Sy1 nz

Ty0 ¼

1 U

ZZZ

(10.22b)  ZZZ  vT 1 v0 v0 vT dU ¼ þ þ dU vz U vx vy vz U U 8 9 1 0 n > > x> > ZZ ZZ = < 1 1B C ¼ % ½ 0 0 T  ny dS ¼ @ TdS  TdS A > > U vU U > ; : > Sz2 Sz1 nz

Tz0 ¼

1 U

ZZZ

(10.22c) where U has been used both as the subdomain for post-processing and the its volume for averaging, and Sx1 , Sx2 , Sy1 , Sy2 , Sz1 and Sz2 are the six faces of the subdomain, which are parallel to the coordinate planes as indicated by the subscripts. The numerical integrations on the faces of the subdomain will have to be evaluated consistently using a 2D numerical integration according to the tessellations on the faces and the order of the elements involved. Their 2D forms can be given as Tx0 ¼

1 A

Zy2

1 T jx¼x2 dy  A

y1

Ty0 ¼

1 A

Zx2 x1

Zy2 T jx¼x1 dy

(10.23a)

T jy¼y1 dx

(10.23b)

y1

1 T jy¼y2 dx  A

Zx2 x1

where A denotes both the 2D subdomain for post-processing and the its area for averaging. It should not be difficult to obtain the above expressions intuitively without necessary mathematical derivations in the case of the diffusion problem.

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If unit average temperature gradients have been prescribed to the RVE the way specified in (10.19), the average concentration gradients as obtained in (10.22) and (10.23) should reproduce the prescribed values approximately. The degree of approximation should offer a good assessment of the representativeness of the RVE and the measure of the decay length taken. To reduce the volume integrations for diffusion fluxes to surface integrations, one has  ZZZ ZZZ  1 1 vx vx vx 0 qx ¼ qx dU ¼ qx þ qy þ qz dU U U vx vy vz U

¼

1 % x½ qx U vU 0

¼

U

1B @x2 U

qy

8 9 nx > > > > > > > > >  ZZZ  < > = 1 vqx vqy vqz qz  ny dS  þ þ x dU > > U vx vy vz > > > > > > U > : > ; nz

ZZ

ZZ qx dS  x1

Sx2

Sx1

1

1 0 ZZ ZZ C 1B C qx dS A þ @ xqy dS  xqy dS A U Sy2

Sy1

1 0 ZZ ZZ 1B C xqz dS  xqz dS A þ @ U Sz2

Sz1

(10.24) vq

vqz y x where diffusion governing equation vq vx þ vy þ vz ¼ 0 has been used. Similarly, 1 0 ZZZ ZZ ZZ 1 1B C q0y ¼ qy dU ¼ @ yqx dS  yqx dS A U U U

Sx2

Sx1

0 þ

1B @y2 U

ZZ

ZZ qy dS  y1

Sy2

1 C qy dS A

Sy1

1 0 ZZ ZZ 1B C þ @ yqz dS  yqz dS A U Sz2

Syz

(10.25)

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The diffusion problem

q0z ¼

1 U

ZZZ U

1 0 ZZ ZZ 1B C qz dU ¼ @ zqx dS  zqx dS A U Sx2

Sx1

1 0 ZZ ZZ 1B C þ @ zqy dS  zqy dS A U Sy2

0 þ

1B @z2 U

Sy1

ZZ

ZZ qz dS  z1

Sz2

1 C qz dS A

Sz1

The above can be easily reduced from their 3D form to their 2D counterparts. 1 0 ZZ Zy2 Zy2 1 1B C q0x ¼ qx dA ¼ @x2 qx jx¼x2 dy  x1 qx jx¼x1 dyA A A y1

A

0 þ

1@ A

y1

Zx2

Zx2

 xqy 

y¼y2

dx 

x1

q0y ¼

1 A

0

ZZ qy dA ¼ A

1B @ A

Zy2 yqx jx¼x2 dy 

y1

1 þ @y2 A

1

y¼y1

dxA

(10.26)

x1

Zy2 0

 xqy 

1 C yqx jx¼x1 dyA

y1

Zx2

 qy 

y¼y2

x1

Zx2 dx  y1

1  qy y¼y dxA 1

x1

It is obvious that the above are unlikely to be obtained from one’s intuition. Following the same procedure presented with Eqs. (10.19) and (10.20) for UCs, the effective diffusion coefficients can be determined. In general, the obtained effective diffusion coefficients give rise to a full 33 matrix, unless the medium happens to show monoclinic, orthotropic or isotropic characteristics in the coordinate system employed. However, as argued in Section 10.2, the matrix can always be diagonalised so that the medium can be presented as orthotropic within its principal coordinate system.

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10.7 Conclusions Application of UCs and RVEs to the diffusion problem on which a range of practical engineering problems are based, including heat conduction, fluid flow through porous medium, etc., has been addressed. In many ways, the approaches taken are analogous to their mechanical counterparts as introduced in previous chapters. Commercial FE codes are available offering solvers to this class of problems, sparing users from the task of devising the solver. However, it is squarely users’ responsibility to impose the correct boundary conditions and to post-process the results sensibly. Only then, numerical solutions will become relevant. This applies to all fields of physics and engineering. Carrying out an exercise on material categorisation as advocated in Chapter 3 of this book, it has been claimed that the diffusion coefficient matrix should be symmetric and positive definite as required by the laws of thermodynamics. As a result, any medium, no matter how complicated its architecture is at the lower length scale, can only be as anisotropic as being orthotropic.

References Gou, J.-J., Gong, C.-L., Gu, L.-X., Li, S., Tao, W.-Q., 2018a. The unit cell method in predictions of thermal expansion properties of textile reinforced composites. Composite Structures 19, 99e117. Gou, J.-J., Gong, C.-L., Gu, L.-X., Li, S., Tao, W.-Q., 2017a. Unit cells of composites with symmetric structures for the study of effective thermal properties. Applied Thermal Engineering 126, 602e619. Gou, J.-J., Fang, W.-Z., Dai, Y.-J., Li, S., Tao, W.-Q., 2017b. Multi-size unit cells to predict effective thermal conductivities of 3D four-directional braided composites. Composite Structures 163, 152e167. Gou, J.-J., Ren, X.-J., Dai, Y.-J., Li, S., Tao, W.-Q., 2018b. Study of thermal contact resistance of rough surfaces based on the practical topography. Computers & Fluids 164, 2e11. Gou, J.-J., Ren, X.-J., Fang, W.-Z., Li, S., Tao, W.-Q., 2018c. Two small unit cell models for prediction of thermal properties of 8-harness satin woven pierced composites. Composites Part B 135, 218e231. Gou, J.-J., Zhang, H., Dai, Y.-J., Li, S., Tao, W.-Q., 2015. Numerical prediction of effective thermal conductivities of 3D four-directional, braided composites. Composite Structures 125, 499e508. Li, H., Li, S., Wang, Y., 2011. Prediction of effective thermal conductivities of woven fabric composites using unit cells at multiple length scales. Journal of Materials Research 26 (3), 384e394.

CHAPTER 11

Boundaries of applicability of representative volume elements and unit cells 11.1 Introduction Representative volume elements (RVEs) and unit cells (UCs) are basic concepts that have found wide use in material science and engineering and turned into helpful tools and techniques for scientists and engineers. Since they are so basic, they tend to be taken for granted and employed casually. In Chapter 5, examples have been cited how misperceptions found their way into the applications of these concepts and techniques leading to confusion. The fact that the subject, basic as it is, has led to this monograph of a volume so far suggests at least that these concepts and techniques can and should be established on a firm basis and as a result there are set rules to follow. Having elaborated on the erroneous treatments associated with the subject in Chapter 5 and proper treatments of various aspects of the subject in each of the previous chapters, the objective of the present chapter is to set the boundary for the applicability of these concepts and techniques beyond which any further application would constitute an abuse. Without appropriate precaution warnings, such unfounded extensions could be easily made to the existing RVEs or UCs.

11.2 Predictions of elastic properties and strengths Effective elastic properties represent some kind of average behaviour over the volume of the material involved in an RVE or UC and they tend to be relatively insensitive to some of the subtle details, e.g. slight deviation from perfect regularity in the structure at the lower length scale, to justify the use of a UC. Within a given UC, if one slightly alters the internal configuration, such as a particular geometric feature, it should not alter significantly the values of the predicted effective elastic properties, provided that the alterations do not change substantially the main features such as volume fraction, orientation of fibers/tows, etc. For this reason, one can usually Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00011-6

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have relatively high level of confidence on the effective elastic properties predicted using RVEs and UCs. On the other hand, the predictions of the strength characteristics of a material represented by an RVE or UC cannot be made with an equal level of confidence. Strength is usually dictated by the stresses whilst stress distributions are sensitive to the detailed geometric and material variations. Any irregularity in the structure at the lower length scale is likely to have significant effects on the predicted stresses. The similar consideration resulted in the concept of stress concentration as a well-known subject. As a result, strengths predicted based on RVEs, assuming their representativeness, or UCs, assuming perfect regularity, could show significant disparity from realistic values, such as those from testing, due to any mismatch between the assumed position and physical reality as present in tested specimens. Consequently, the level of confidence on the obtained strength predictions will be compromised relative to the predicted effective elastic properties. For this reason, realistic allowance in terms of accuracy should always be given to strength predictions. In other words, one should always be sceptical about the genuineness of any claims of a perfect accuracy in predicted strengths using RVEs or UCs. Alternatively, a prediction might be perfectly accurate for the specific case as analysed, but this particular case might not be representative enough to allow the results to be applied to the practical problem the analysed case was meant to represent. Since stress distributions, especially stress concentration, are sensitive to the local geometric details, as stated above, the representativeness of an RVE should be evaluated with this in mind. An RVE will be representative only if it contains all characteristic features in right degree of concentration. Consequently, in order to capture a representative and critical stress state, an RVE of a substantially larger size may have to be employed than that used for predicting effective elastic properties. For problems represented by UCs, predicted stresses can be affected by any irregularity in the patterns as opposed to the idealized regularity at the lower length scale. Perfect regularity is the underlying assumption justifying the use of UCs and hence the formulation of UCs lacks the representation of any deviation from perfect regularity. The effects of this on the predicted strengths should be appropriately evaluated before basic level of confidence can be established, for instance, by employing an array of UCs with some of them being dislocated as designated. The internal geometric features within the UC are another significant consideration as stress distributions tend to be sensitive to them. One must strike a right balance in this respect. On one

Boundaries of applicability of representative volume elements and unit cells

363

hand, appropriate idealization and simplification are necessary. On the other hand, one has to ensure the idealization and simplification do not distort the stress distributions too much hence the UC remains sufficiently representative. To justify this, the simplest thing one can do is to conduct systematic parametric studies to avoid missing any blind spot behind the idealization and simplification introduced. Obtaining accurate stresses is crucial for strength prediction. However, stresses alone are not sufficient to predict failure. One needs an appropriate failure criterion to associate a given stress state with the prediction of failure. There are many failure criteria in the literature, especially for composites. They do not seem to agree with each other. More depressingly, many of them do not even agree with common sense or simple logic in one respect or another (Li and Sitnikova, 2018). As an effort to bring common sense or simple logic back into such failure criteria, the Tsai-Wu criterion for UD composites was scrutinized and rationalized as an example in (Li et al., 2017), though the authors acknowledge that this criterion is by no means universally applicable, nor most satisfactory in accuracy. The point advocated through the exercises as referred to above is that for any failure criterion to be of practical relevance, it should at least show a basic level of consistency, and no assumption should be resorted to without appropriate justifications. Unfortunately, this is a position not quite achieved in the state-of-the-art of the failure theories as far as composites are concerned, as was clearly demonstrated through the study known as the World-Wide Failure Exercise (Hinton et al., 2004; Hinton and Kaddour, 2012, 2013). Before a well-formulated failure criterion is established, an appropriate reservation on the predicted strength will always have to be kept and the predicted results should not be relied upon blindly before they are appropriately validated by other independent means, e.g. through experiments.

11.3 Representative volume elements The whole idea about RVEs is their representativeness. In Chapters 4 and 9, the focus was mainly on their representativeness when forming an RVE, in terms of homogeneity at the upper length scale, volume fractions of all constituents and other relevant features. Homogeneity also implies infinite extent in the domain of interest at the lower length scale. This is usually justified in a multiscale sense, i.e. any geometric feature, such as a finite element at the upper length scale, is infinitely large at the lower length scale. Therefore, the complete domain from which the RVE is selected should not

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Representative Volume Elements and Unit Cells

show any limit at the lower length scale. In other word, any hard limit on the extent of an RVE which is defined at the lower length scale, such as a boundary as present in the domain of the problem at the upper length scale, should be considered as a breach of the assumption of RVEs as this compromises the representativeness of the RVE. As was stated in Chapter 9, in multiscale material characterization the complete domain to be analysed at the lower length scale is usually assumed infinite according to the precise definition of the problem. The validity of reducing the domain from infinite to a finite extent rests on the representativeness of the domain selected, i.e. the RVE. The representativeness of an RVE can always be ensured if it is of a sufficiently large size. Once one reaches a sufficiently large sized RVE, any further increase in size will make no difference on the predicted properties from the RVE. For the efficiency in modelling, one is always interested in the smallest size of such an RVE, any further reduction of which would lead to noticeable variations in the outcomes. Given the post-processing scheme as established in Chapter 9 employing a subdomain of a decay length inside the RVE analysed, it is essential that the subdomain remains representative. As also mentioned previously, the measure of representativeness is subjective, as it varies depending on the effective properties it is meant to characterise. One RVE representative enough in one respect of physical study may not be so in another respect. Having selected an RVE representative in the perspective concerned, its applicability then relies on the uniformity of the deformation for mechanical problems in the upper length scale. In terms of the diffusion problem, it is the uniformity of the gradients of the concentration field. The effective properties obtained through RVEs would lose their representativeness in problems at the upper length scale where excessive gradients are present. A quantitative measure of the excessiveness can be established considering the variation of the field at the upper length scale over the characteristic length of the RVE and the variation of the field within the RVE at the lower length scale. The former should be much smaller than the latter. However, when the two become comparable, the gradients in the upper length scale can be considered excessive and uniformity can no longer be taken for granted. For the mechanical problems, extension of the applications to the realm of finite deformation is not forbidden, as will be addressed in Chapter 13, although a great deal of complications arises as a consequence of the deformation being finite and due care will have to be exercised. An obvious issue is the fact that the definitions of stresses and strains are no longer unique by then and the user must have a clear understanding which of them he/she is interested in.

Boundaries of applicability of representative volume elements and unit cells

365

Their definitions depend on the analysis formulations employed in the FE codes, e.g. total Lagrangian formulation and updated Lagrangian formulation. Such difference needs to be duly reflected in the post-processing of the results from the analysis of an RVE. Mixing one with another is a typical source of error. Applying an analysis involving finite deformation without such knowledge is considered as a blind extension which is a dangerous move as will become clear in Chapter 13. As will be established in Chapter 13, incorporating finite deformation in the analysis of UCs is far more complicated than simply applying deformation larger than that in linear problems and switching the finite deformation functionality on. What marks the boundary of the applicability of an RVE in terms of deformation is localization, i.e. when global softening takes place whilst deformation starts to localize at a certain area which will soon lead to an imminent mode of failure. A typical example of such localization is necking as observed in the uniaxial tensile tests of specimens of ductile materials. The global softening is indicated by the load drops in the load-displacement curves. Any localized behaviour violates the representativeness of the RVE under consideration, since the localized pattern of deformation present in the RVE analysed is no longer expected in any of the surrounding area of the RVE (otherwise, it would not be localized), which negates the presumption of uniformity in deformation. Finite deformation is one source of nonlinearity. The other source is material nonlinearity when material behaves differently under different stress levels or experiences different deformation histories. The RVE should still be free from localization to remain applicable. The measure of representativeness in terms of the size of the RVE can also be affected by the involvement of the nonlinearity of any of the constituent materials in the RVE. Since nonlinearity introduces additional variability in the behaviour of the RVE, for an RVE to remain representative, its size may have to be increased in such cases. The material nonlinearity in the diffusion problem, i.e. when the diffusion coefficients varies with the value of the concentration field, could put a categorical restriction on the applicability of RVEs straightaway since the uniform gradients of the concentration field imply variation in the concentration field which undermines the uniformity of the material. As a consequence, the diffusion fluxes at the upper length scale will no longer be uniform in general corresponding to uniform gradients of the concentration field at the upper length scale.

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Representative Volume Elements and Unit Cells

11.4 Unit cells Unit cells are introduced through the regularity of patterns in the structures at the lower length scale, both physically and geometrically, giving rise to properties which can be described as translationally symmetric or periodic. The assumption behind the introduction of multiple length scales is that features and the regular patterns at the lower length scale are infinitesimally small when observed in the upper length scale. As a result, a UC at the lower length scale can be considered as a material point at the upper length scale in order to define the effective properties of the material at the upper length scale. In other words, any finite volume of material at the upper length scale, such as a finite element, should contain a sufficiently large number of UCs. This sets one boundary of applicability of UCs and the outcomes of material characterization in terms of applications at the upper length scale. Closely associated with consideration above, and given the fact that a UC is always an RVE, the applicability of UCs is also subject to the condition that the deformation at the upper length scale is uniform. Here the uniformity means that the strain fields and stress fields are constant, not the displacement fields. For the diffusion problem, the concentration gradients and diffusion fluxes should be constant at the upper length scale but not the concentration field itself. In this respect, the discussion made to the RVEs in the previous section applies equally to UCs, for instance, the initiation of localization defines the limit of applicability. Similarly to the discussion on RVEs, finite deformation alone does not prevent the use of UCs although the actual application can be greatly complicated by the different definitions of stresses and strains as will be elaborated in Chapter 13. The boundary of the applicability is the emergence of localized deformation. With regard to the material nonlinearity for UCs, given the regularity in the architecture at the lower length scale, the applicability of UCs is not affected by the material nonlinearity for mechanical applications, provided that no localized deformation takes place, but should not be assumed for the diffusion problem.

11.5 Conclusions Having derived necessary formulations associated with RVEs and UCs in the previous chapters, it is the purpose of this chapter to issue a necessary warning message about the applicability of RVEs and UCs as a timely reflection, since none of them is meant to be universally applicable. Meaningful

Boundaries of applicability of representative volume elements and unit cells

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tools can be offered by these idealizations when they are applied within the boundaries of their applicability. Beyond such boundaries, blind extensions are recipes for confusion. The boundaries of their applicability are drawn by the assumptions based on which the concepts of RVEs and UCs are introduced and their formulations are established and hence should be respected by the RVE and UC users.

References Hinton, M.J., Kaddour, A.S., 2012. Evaluation of theories for predicting failure in polymer composite laminates under 3-D states of stress. (WWFE-II, Part A comparison between theories & B comparison with experiments). Journal of Composites Materials 46 (19-20), 47(6e7), March 2013. Hinton, M.J., Kaddour, A.S., Soden, P.D., 2004. Failure Criteria in Fibre-Reinforced-Polymer Composites e the World-wide Failure Exercise. Elsevier. Li, S., Sitnikova, E., 2018. A critical review on the rationality of popular failure criteria for composites. Composites Communications 8, 7e13. Li, S., Sitnikova, E., Liang, Y., Kaddour, A.-S., 2017. The Tsai-Wu failure criterion rationalised in the context of UD composites. Composites Part A: Applied Science and Manufacturing 102, 207e217.

CHAPTER 12

Applications to textile composites 12.1 Introduction 12.1.1 Background To appreciate the advantageous characteristics of textile composites, it is helpful to review the position of conventional use of fibre reinforced composites. Laminates made of unidirectional (UD) fibre reinforced plies account for most of structural applications of composites as UD composites allow their stiffness and strength benefits along the fibre direction to be exploited fully. Placing laminae at different fibre orientations helps to compensate the low performance of individual laminae in the direction transverse to fibres. However, the layered construction only allows such compensation to be realised in the plane of the laminate, whilst in the direction transverse to the plane of the laminate, there has been a genuine and generic weakness for such laminated composites hindering their applications in areas where integrity is crucial, for instance, in components susceptible to impact or fatigue. Various means have been explored by researchers in order to reinforce the transverse performances, such as stitches, z-pins, etc. However, none of them seems to have won prevailing dominance as an established approach for alleviating weak transverse properties, because none delivers the required performances. One of the reasons behind the disappointment lies in the lack of understanding of the mechanical properties. Improvement of inferior transverse properties was the motivation behind these developments, where the word ‘transverse’ probably has been sometimes understood too literally. As a result, the reinforcements introduced were in the transverse direction, e.g. z-pins and stitches. They did indeed reinforce the material resisting deformation and failure resulting from direct stress in the transverse direction, e.g. against mode I delamination. However, stress is a second rank tensor and each component involves two directions. When these two directions coincide with each other, the stress component is direct stress and, when they differ, the component is shear. Sufficient reinforcements against direct stresses do not necessarily deliver the desired resistance to shear. As a close resemblance, a case can be cited in terms of in-plane behaviour since it Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00012-8

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is a relatively more familiar aspect of the composites. As is well-known, 0 UD plies are weak in the 90 direction. Plies at 90 can be introduced to reinforce performance in the in-plane transverse direction. In fact, the 0 and 90 plies reinforce each other in their respective transverse directions, forming the so-called cross-ply laminates. Such laminates can be made strong and stiff enough against direct stresses in these two directions. However, the introduction of the 90 plies helps little with the in-plane shear performance of the 0 plies, and vice versa. The resistance against in-plane shear of a cross-ply laminate is hardly any better than that of the composite having 0 or 90 plies alone. In practical applications of conventional laminates, this weakness is overcome by introducing off-axis plies, typically, 45 plies. When the number of plies in each of these four directions become equal, the so-called quasi-isotropic laminates are produced. They offer shear resistance in the same way as isotropic materials do. In the out-of-plane transverse direction, shear is often more problematic than the direct stress as many typical failure mechanisms, such as those due to lateral impacts, are dominated by transverse shear stresses associated with mode II fracture as was the predominant consideration behind the delamination problem for laminated composites. It is true that the resistance of the interlaminar interfaces to the mode I fracture is usually significantly lower than that against mode II, e.g. in terms of critical energy release rates, as conventionally denoted by GIc and GIIc. However, the mode I fracture driving force is usually much lower than that of mode II as expressed in terms of energy release rates, conventionally denoted by GI and GII. Often, the energy release rate in mode I vanishes since the direct stress over the fracture plane is compressive, typically in the case of lateral impact. Therefore, mode II is often more critical than mode I. Proper solutions will have to include reinforcements in direction inclined relative to the transverse direction, preferably at 45 , to be most effective, in order to resist the transverse shear stresses. As a by-product, these inclined reinforcements improve the resistance of a component to the transverse direct stress as well. In other words, appropriate amount of such inclined transverse reinforcements can deliver sufficient resistance to the mode I fracture in loading scenarios resulting in this mode of fracture. Modern developments of textile industry have allowed many sophisticated 3D textile preforms to be fabricated, which, once appropriately impregnated with binding matrix and processed into composites, offer exactly what conventional laminated composites are short of, i.e. the integrity, in particular, in the thickness direction, often with fibre tows naturally

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inclined relative to the plane of the fabric due to the interlocking undulations. Like with other means of transverse reinforcements, the improved transverse performances are achieved at the price of the in-plane performances as a trade-off. Designers have to come to their assessments where the material and structure can afford such a compromise before committing to one or another. There are structural parts for which the in-plane properties are substantially higher than required whilst transverse strengths are insufficient. The start-of-the-art developments in high performance fibres are also creating further headroom for in-plane properties. Under such circumstances, 3D textile composites could find their niche by having some of their in-plane properties reduced to gain those in the transverse direction. Textile preforms, especially 3D ones, are demanding to fabricate. Reasonably accurate assessment of their performances is essential for their acceptance in engineering as serious candidates for structural materials. The availability of appropriate means of assessing the effective properties of such composites is a pre-condition if one intends to achieve any degree of acceptance by designers. Without a viable approach to predict them theoretically, one would have to rely on experimental means. However, for 3D textile composites, the experimental approach is simply impractical. Small changes in parameters of some of the textile preforms could sometimes mean rather different fabrication process, dismissing experimental ‘trial and error’ approach as a viable route. The lack of analysis and design tools is undoubtedly the bottleneck to the wide acceptance of textile composites in their systematic structural applications. Designers always prefer simple approaches, if such approaches are available. On the other hand, over-simplified theories, such as the rule of mixtures, provide very limited scope, and can sometimes be even misleading. In applications of laminated composites, an ideal compromise is struck by the classic laminate theory (CLT). However, its applicability is restricted to laminated composites where the domain of interest is layered, rendering its natural suitability for layer-wise defined models. Such convenience is unfortunately unavailable in textile composites. In order to take advantage of such materials, one will have to be prepared for more sophisticated models as a compromise. Fortunately, modern computing capability and development of commercial FE codes have progressed to such an extent that advantages can be taken of them to ease the pressure from the sophistications of theoretical models, provided that the models are formulated appropriately to represent the reality. Here once again, the key is the representativeness

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of the models to be formulated. Casually created models are not necessarily wrong, especially those models dealt with using FEM. As long as a case runs, it offers a correct solution of some kind, but only correct to the problem the model defines mathematically. The problem meant to be represented by a model is not necessarily the problem defined by the model. Therefore, the subtle difference in wording between ‘the problem defined by the model’ and ‘the problem meant to be represented by a model’ could mean outcomes worlds apart. Throughout the elaborations in this monograph, it has always been the focus that the models, UCs and RVEs, should be formulated to represent what they meant to represent. It has been demonstrated that this is doable as far as UCs and RVEs are concerned. As an example of the formulations delivered in the monograph, most common types of UCs and some of the RVEs for typical applications to composites have been implemented to form a computer code, called UnitCells©, developed by the authors and their coworkers over the past decade or so at the University of Nottingham. It has the involved operations, in particular, the imposition of boundary conditions, automated so that the responsibility left for the users is mostly to define the geometric parameters and constituent material properties before the effective properties of the composites could be evaluated. Chapter 14 is designated to offer a suitable introduction to this development. With such automated codes, design of textile composites through computational tools becomes feasible. The definition of geometric parameters as mentioned above can be relatively straightforward for UD and particulate reinforced composites after certain appropriate idealisations. In the case of a laminate, they are no more than the definition of the layup. This is why laminated composites are so appealing in this respect. For a textile preform, given its attractive regularity in its architecture, an appropriate idealisation is far from being straightforward, especially for 3D textile preforms. Idealisation is necessary in the presentation of the geometry. An appropriate idealisation should aim at delivering a full mathematical description of the geometry as realistically as possible. This process is referred to as parameterisation of the architecture at the meso-scale so that it can be fully reconstructed using a limited set of topological and geometric parameters. Whilst maintaining the representativeness, the fewer these parameters the better and the easier for the approach to be accepted by engineering practitioners. The focus of the present chapter will be placed on the applications of UCs to textile composites. Given that textile preforms are usually of regular

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meso-structures, defining a UC is not excessively difficult, as will be demonstrated, but still by no means trivial. Having read Chapter 6 and 8, reader would have concluded that the word ‘tedious’ offers the most relevant description to the exercise of derivation of boundary conditions. The actual derivation of the boundary conditions will therefore be avoided but they could be easily re-established by referring to relevant previous chapters and sections. The attention will be placed on the identifications of various symmetries present in the architectures involved. In most cases, the cuboidal UC as formulated in Subsection 6.4.2.2 could be blindly taken as a UC once the periodicity in the architecture has been identified. It is almost certainly not the optimum one. If one adopts the UC approach as a one-off casual exercise, it might not be a bad option. However, the potential of the UCs as a methodology is meant to offer a tool for systematic material characterisation and it is likely to be used repeatedly. In fact, the experience of the authors has been to run tens of thousands numerical testing cases as a batch job to offer some kind of database for purposeful and systematic material characterisation. In this case, the efficiency of the UCs formulation makes huge differences, sometimes between possible and impossible. Complications of UCs for textile composites are associated with identification of the geometric patterns in the textile preforms in terms of symmetries present in the architecture, the employment of the right symmetries to minimise the size of the UC and the parameterisation of the architecture. Whilst there are some general rules to follow as established in previous chapters, users will have to exercise their own judgement whilst following the basic rules to optimise their effects. It is extremely helpful if one identifies all available symmetries in the architecture before attempting to select the most beneficial ones to be employed to construct a UC. In the context of the present chapter, it will serve as valuable exercises to digest the ethos of this monograph, even if some of the available symmetries may not be useful in terms of the construction of the UC and some might not even be independent. Symmetries can also help with the identification of the smallest building blocks which can then be used to construct the UC concerned, which reduces the demand on the creation of the geometric model. If the genuinely smallest building block can be identified for an architecture, achieving a geometric description of the architecture paves the way for parameterisation. This will be illustrated throughout this chapter case by case. Although possible combinations of available symmetries in reality cannot be

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exhausted, practices with at least some of them will improve users’ grasp of the applications of symmetries in general. There are three general types of textile preforms, woven, braided and knitted. Knitted fabrics are usually too flexible to be useful candidates for structural applications, although high flexibility might be attractive for special functional applications. On balance of significance, they will not be included in this study, although there is no reason why the considerations given in this chapter cannot be adapted for the analysis of composites made of knitted preforms and the extension should be the most straightforward. Emphasis in this chapter will be placed on the former two types. Fabrics are usually moulded through processes such as resin transfer moulding (RTM), resin infusion, etc. A composite forms as the resin cures, often at an elevated temperature. More sophisticated processes are employed in order to obtain metal or ceramic matrix composites which are favoured in high temperature applications. Whilst these forming processes vary widely, the composites produced can be analysed in much the same way except for their different constituent properties.

12.1.2 Composites made of woven preforms Weaves typically involve warp and weft tows interlocked appropriately to form a piece of integral fabric. Warp and weft tows are orthogonally orientated in the plane of the fabric. Warp tows extend in longitudinal direction whilst the weft tows go in the transverse direction over a limited width of the fabric in the plane of the weave. As a weave is fabricated, it usually keeps its width whilst building up in length. Weaves can be 2D or 3D. The former involves two arrays of warp tows placed in a certain order interlacing a single array of weft tows, whilst the latter involves multiple arrays of each type with appropriate depth of undulation in the warp tows to interlock the weft tows which keep mostly straight. Typical 2D weaves are plain, twill and satin weaves, as shown in Fig. 12.1. In textile industry, different weave patterns are mostly to offer different surface textures. For composites applications, there are a number of basic considerations to account for in terms of the effects of reinforcement. To form these weaves, fibre tows have to undulate to different extents. The more pronounced are the undulations, the more compromised is the stiffness of the composites in the plane of the fabric. As satin weaves have relatively long segments of straight tows, they offer relatively high stiffness. For this reason, satin weaves of high harnesses tend to perform better in

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Applications to textile composites

(a)

(b)

(c)

Fig. 12.1 Typical 2D weaves: (a) plain weave, (b) twill weave and (c) satin weave (4 harness). The images were produced in Texgen software (Long and Brown, 2011).

structural applications. Plain weaves are the opposite with twill weaves standing in between. The next consideration is their formability or drapability, i.e. the flexibility to deform and to fit to moulds of curved surfaces without creases. In this respect, these weaves tend to follow the same order as for the stiffness consideration, i.e. satin being the most formable whilst plain weave the least with twill weaves as an intermediate option. In many modern applications, woven fabrics are often used as a sacrificial layer as an anti-scratching measure where these weaves are employed as surface protective layers. To this particular effect, these weaves tend to follow the opposite order, i.e. plain weave delivers the best performance whilst satin weave the worst. In practical applications, weaves should be selected according to the purpose. Often they are employed for multiple purposes. Designers will have to strike a balance between various considerations. 3D woven composites are usually of structural functionalities. Some of the preforms are genuine 3D weaves as evolved from fabrics involved in the traditional textile industry, whilst others are effectively stacks of dry UD plies that are secured using some binding tows through the thickness of the stack. A typical example of the latter is the so-called non-crimp fabrics (NCF) in which there are usually two types of fibre tows in the longitudinal direction, one straight as the genuine reinforcements and one undulated as the binders, in addition to appropriate amount of straight weft fibre tows. The major consideration behind the application of NCF is to maintain the stiffness and strength benefits of fibres whilst they are laid straight. The drawback is that the effects of reinforcement of the binders through-the-thickness are limited because, as was stated earlier, the binders

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Representative Volume Elements and Unit Cells

tend to be too literally ‘through-the-thickness’. The modelling of such NCF composites is similar to that of genuine weaves. In fact, it is even simpler because of simpler geometries of straight fibre tows. Therefore, the focus of the present discussion will be on the composites made of genuine weaves. 3D weaves are similar to their 2D counterparts in the sense that both involve warp and weft tows in an orthogonal manner. The difference is however that these tows are stacked with all or some of the warp tows interlocking the weft tows in a certain designated pattern. An example of 3D weave is shown in Fig. 12.2. Varying the spacing, harness and depth, infinite number of different interlocking patterns can be produced. With these in mind, the patterns can usually be parameterised to identify the most practical weaves that can be fabricated. Elaborate description of such a parametrisation will be presented in Subsection 12.4.1. A genuine and also critical weakness of composites based on woven preforms is their lack of in-plane shear stiffness and strength due to the orthogonal arrangement of the fibre tows. In-plane shear is a crucial consideration in many structural parts. In aero-structures, for instance, shear is one of the major loading cases for skins of wings and fuselages, where stiffness is often the priority. For most of the moving components, such as ailerons, flaps, rudder, elevators, and in particular engine fan blades, flutter is a critical design boundary which is largely controlled by the torsional rigidity of the component. This relies on the in-plane shear stiffness. For laminated composites made of stacked 2D weaves, the weakness can be improved by laying a number of plies at 45 in a designated manner. Similar measures can be taken in NCF composites by introducing off-axis layers of fibre tows. However, the manufacturing process of such NCFs will become significantly more sophisticated, especially in the mechanisms of introducing the off-axis fibre tows and designing and accommodating the binder paths through-the-thickness.

Fig. 12.2 An example of 3D weave.

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12.1.3 Composites made of braided preforms Braiding is a completely different from weaving as a textile fabrication process. In a traditional braid, there are two off-axis arrays of fibre tows interlacing each other as the fabric builds up in length along its major axis. The direction of each tow is called an axis and the basic braid is therefore called a 2D 2-axial braid, as shown in Fig. 12.3(a). For structural application, to obtain desired properties in the principal axis, additional fibre tows can be incorporated in the principal directions, making such braids 2D 3- or 4-axial ones. The number of axes is one of the characteristic descriptors of the braid. The angle between the off-axis tows to the longitudinal direction is called braiding angle which is another important descriptor. The topology of a 2D 2-axial braids is identical to that of the plain weave, but tows in the former are usually non-orthogonal, and the manufacturing process is completely different from weaving. Like plain weaves which do not have much resistance to in-plane shear, 2D braids are usually rather stretchy in directions of its principal axes, as it is a some sort of mechanism rather than a structure and it is therefore shapeunstable. However, it can have reasonable shear resistance. This position is greatly improved after introducing another axis to form a 2D 3-axial braids, as shown in Fig. 12.3(b) and (c). The 2D 3-axial braids are shape stable. For such braids, it is fairly easy to achieve a balance properties in directions all round, e.g. to show an effectively isotropic characteristics, if isotropy is desirable. Depending on the way how the third axis interlaces with the existing two, the textures can be rather different. In that as shown in Fig. 12.3(b), the third axis is usually along the length direction of the braid and it keeps straight. Therefore, the composite is expected to offer higher performances in this direction as compared to the other directions and its potential of structural applications is obvious (Roberts et al., 2002, 2009a, 2009b). (a)

(b)

(c)

Fig. 12.3 Typical 2D braids (a) 2-axial, (b) and (c) 3-axial but different interlacing.

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Representative Volume Elements and Unit Cells

The architecture in Fig. 12.3(c) offers effective in-plane isotropy apparently. However, it is very porous. Unless for a specific niche, the potential of its structural applications is limited. A 3D braid has some or all axes running at angles out of a plane. The most basic 3D braid is the so-called 3D 4-axial braid, as shown in Fig. 12.4(a) with (b) as a zoomed-in view which also forms a unit cell. The fabrication process of such braid is often referred to as 4-step braiding which can be highly automated. Like 2D 2-axial braids, it is flexible in the length direction as well as sideways. However, practically, it is reasonably straightforward to introduce more axes of fibre tows into the braid to make it structurally stable whilst offering higher performances in designated directions. Fig. 12.4(c) and (d) show two examples of this kind in their idealistic (a)

(b)

(c)

(d)

Fig. 12.4 3D braided composites: (a) 3D 4-axial braid, (b) a unit cell for 3D 4-axial braid, (c) 3D 5-axial braid(fifth tows in the fifth axis) and (d) 3D 5-axial braid (fifth tows in the sixth axis).

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forms. The former has the additional axis in the longitudinal direction which is usually described as the fifth axis whilst the latter has the sixth axis with the fifth being void. There is no reason why one cannot have the seventh axis and have all axes incorporated at the same time to form a 3D 7-axial braid.

12.2 Use of symmetries when defining an effective UC In what to follow after this section, comprehensive observations will be made to various typical textile preforms where plenty of symmetries are available which can be classified in groups as elaborated in Chapter 2. The objective of the analysis of UCs is always the characterisation of the material represented by the UC. In order to achieve this, the UC will have to be subjected to loading cases consisting of six average stresses or strains and a change in temperature. The use of the symmetries has implications on the physical fields involved in the problem, which will be interpreted as boundary conditions for the UC concerned. These symmetries are listed below as a timely revision of the concepts of conventional symmetries as introduced in Chapter 2 and the centrally reflectional symmetry as elaborated in Section 8.5 and, in particular, in close association with their applications. (1) Translations (2) Reflections (3) Rotations (4) Central reflection. Some guidelines on the use of these symmetries have been provided in Section 8.6. They will be put in practice in the sections to follow. As a quick recapitulation, translational symmetries as fully elaborated in Section 6.4 should be exhausted first to define a UC of a minimum size that can be achieved using this type of symmetry. These symmetries will lead to a single set of boundary conditions applicable to all loading cases. The use of them does not come with any implication on the anisotropy of the material. It is usually straightforward to identify translations in orthogonal directions. It is advisable that attention also ought to be paid to available translational symmetries in non-orthogonal directions. The use of them can often help to significantly bring down the size of the UC to be analysed. The price to pay is slightly more sophisticated boundary conditions, but they are manageable. However, the gain in terms of reducing computational demands is high and, hence, such an exercise is certainly worth the effort. The central reflection as introduced in Section 8.5 and (Li and Zou, 2011) share two commonalities with the translational symmetries: (a) it leads

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Representative Volume Elements and Unit Cells

to a single set of boundary conditions applicable to all loading cases, and (b) it does not carry any implication on the degree of anisotropy. The level of complexity in the obtained boundary conditions is similar to those from rotational symmetries as introduced in Section 8.4 (Li and Reid, 1992) but higher than those from conventional reflectional symmetries as can be found in Section 8.2. In the case when a central reflection happens to coexist with a conventional reflection or a rotation, the choice should reflect the following considerations, based on a UC already obtained from translational symmetries alone. If a single set of boundary conditions for all loading cases is desirable, one should go for central reflection when available. Unless there is a further central reflection available in the UC of reduced size, the UC of such reduced size will be the final choice. When making use of a central reflectional symmetry to reduce the size of the UC, the partitioning is not unique. The choice of the partitioning plane should be based on (a) availability of another central reflectional symmetry in the UC of reduced size, and (b) convenience in the construction of the geometric model and subsequent meshing, where a general guideline is to avoid fragmenting geometric entities more than absolutely necessary and keep them as regular in shape as possible. If the objective is to minimise the size of the UC whilst putting up with multiple sets of boundary conditions, one should go for conventional reflections, followed by rotations and central reflection, as available. The boundary conditions derived from rotational symmetries are more complicated than those from reflectional symmetries and hence when rotational symmetries co-exist with reflectional symmetries, the latter should be given priority in general. When applying a rotational symmetry to reduce the size of a UC, the partitioning is not unique and therefore the considerations as given the central reflection regarding the partitioning apply equally here. There are two commonalities between a reflectional and a rotational symmetry: (a) Some loading cases are symmetric whilst others are antisymmetric and, as a result, different loading cases may have to be analysed under different boundary conditions. Usually, the direct average stresses, one of the average shear stresses and the temperature changes are symmetric whilst the remaining two shear stresses are antisymmetric. After a single application of a reflectional or a rotational symmetry, two separate analyses will be necessary to fulfil the task of materials characterisation.

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Usually, after application of another reflectional or rotational symmetry, the combinations of symmetric and antisymmetric conditions require four separate analyses before the material can be fully characterised, where three average direct stresses and the temperature change are prescribed as four loading cases in one analysis and each of the three average shear stresses as an individual analyses. For this reason, sometimes, the effects of reduced sizes in UCs may not always show anticipated reduction in computing time and readers should be aware of this. (b) Each of them defines a principal plane or axis of the material, reducing the degree of anisotropy of the material represented by the UC. As another recapitulation, the axis perpendicular to a principal plane is a principal axis and the plane perpendicular to a principal axis is a principal plane; and a material with two principal axes perpendicular to each other, or two principal planes perpendicular to each other is an orthotropic material. Another consideration to make when determining the sequence of applying available symmetries is the availability of further symmetries afterwards. This is largely dictated by the initial positioning of the full sized UC generated based on translational symmetries alone. Some of the symmetries may not be of mechanical significance, usually, when the planes or the axes associated with these symmetries are inclined, because none of the loading cases stresses shows these symmetries. However, it may still be of value to explore them to exhaust all available symmetries. Although the outcome will no longer offer a UC, it may be considered as a building block for the construction of geometric model and subsequent meshing. Experienced FEM users can tell how demanding these tasks are in 3D modelling, in general. Identification of such building blocks may be of great assistance in this respect.

12.3 Unit cells for two-dimensional textile composites 12.3.1 Idealisations in the thickness direction Whether the textile preform is described as 2D or 3D, it is always 3D in nature having both the in-plane and the out-of-plane dimensions, and is to be modelled as such. The variety of preform types is mostly in the inplane aspect which therefore deserves lengthy elaboration as will be pursued from the next subsection on. Out-of-plane, i.e. in the thickness direction, there is only a few major considerations that can be made regarding this

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aspect of the composite, depending on the nature of the applications. If a piece of textile preform is used in a laminate as a layer whilst the laminate is going to be analysed using the classic laminate theory (CLT), one would only need the in-plane properties and the composite can be considered to be under a plane-stress condition, the same as a lamina in the CLT. Effectively, the top and bottom surfaces will be considered as free from any constraint, as depicted in Fig. 12.5(a). In is worth noting that whilst this is relatively simpler to model, adopting such an approach makes it completely impossible to obtain the out-of-plane properties. When applications of such composites involve their properties in the thickness direction, the construction of the composite in this direction will have to be incorporated. Unless there is strong reason to treat the layers differently because of the physical conditions given, one can assume periodicity in the thickness direction. If the period is assumed to be thickness of a

(a) More layers Free surface Free surface More layers

(b) More layers

More layers

Relative displacement boundary conditions Relative displacement boundary conditions

(c) More layers

More layers

Relative displacement boundary conditions

Relative displacement boundary conditions

Constant normal displacement boundary conditions Symmetry boundary conditions

Fig. 12.5 Idealisations in the thickness direction: (a) free surfaces, (b) relative displacement boundary conditions on assumption of translational symmetry, and (c) translational symmetry with additional mid-plane reflectional symmetry.

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layer of the composite, as illustrated in Fig. 12.5(b), the formulation of the UC can be based on a translational symmetry in the thickness direction. Effectively, this implies that the composite has been assumed to be of infinite thickness at the lower length scale. Use of the translational symmetries alone will not bring any restrictions or implications to the effective properties of the material in terms of its anisotropy, whilst reflectional and/or rotational symmetries reduce the degree of anisotropy, as was elaborated in Subsection 3.2.2. One should be aware of that when generating a UC model of material that is meant to be genuinely anisotropic. The model would lose it representativeness if any undue measure has been taken in the model construction which prevented such anisotropy from being possible. Bearing in mind that the imposition of translational symmetry requires identical tessellation on the top and bottom surfaces, meshing can sometimes be challenging. A potential solution to this issue is as follows. If the period is assumed to be equal to double thickness of a single layer and if the two layers involved are further assumed to be reflectionally symmetric about the plane of the interface between the two layers, a UC that does not require identical tessellations on the top and bottom surfaces can be established, as shown in Fig. 12.5(c). However, as was argued in Section 3.2.2.1, use of a reflectional symmetry will automatically place the material into the monoclinic category in terms of its effective properties, hence it will no longer show general anisotropy if it happens to be a feature as present in the system. Apparently, each of three idealisations comes with its advantages and disadvantages and each has its own implications in terms of the category of the material and the characteristics of the effective material properties. Users are advised to make their choices according to the suitability of each of the options so that the option selected is the most representative for the physical problem.

12.3.2 Plain weave For a piece of composite made of plain weave preform, using translational symmetries in orthogonal directions along the fibre tows as described in Subsection 6.4.1.2, square UCs can be obtained as shown in Fig. 12.6(a). Each of the red squares can be employed as a UC. As a UC, the same boundary conditions are applicable to any of squares from A to D. However, if one wishes to reduce the size of the UC to be analysed, there are significant differences amongst them.

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(a)

Representative Volume Elements and Unit Cells

(b)

(c)

Fig. 12.6 UCs for a plain weave: (a) square from orthogonal translations, (b) rectangular from non-orthogonal translations, and (c) the smallest UC.

UC A and UC B are effectively the same, except for a shift by half of a period in either vertical or horizontal direction. One can follow the procedure as presented in Section 8.4.2 to reduce the size of the UC to be analysed to 1/16 of the original one and hence the derivations will not be repeated here. According to the recommendations given in Section 8.6, after exhausting the translational symmetries (orthogonal ones for the time being), it would be central reflection as introduced in Section 8.5 to look for if one wishes to use a single set of boundary conditions for all loading cases. None of A and B has such symmetry. If one shifts either by a quarter of the period in both the horizontal and vertical directions, it will produce C. Apparently, C has centrally reflectional symmetry, and the size of the UC to be analysed can be reduced to half of it as indicated by the green shade as shown in Fig. 12.6(a). Detailed boundary conditions resulting from the centrally reflectional symmetry have been derived in Section 8.5 with broader elaborations in (Li and Zou, 2011) and will therefore not be repeated here. For clarity, the green shaded rectangle within UC C in Fig. 12.6(a) is enlarged and re-plotted Fig. 12.6(b). Within it, one can still observe more symmetries, including rotation about the central horizontal and vertical axes, respectively. They can reduce the size of the UC to a quarter of the green rectangle. Taking the bottom left quarter, there is a further reflectional symmetry in it. Once it is used, it will reproduce the smallest UC, i.e. 1/16 of the original one, the same size as that obtained in Subsection 8.4.2. However, this is not the optimum sequence of using the symmetries. If one is to employ either reflectional or rotation symmetries after central reflection was used, the central reflectional symmetry would not show any more benefit

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than reflectional or rotational ones. Using it early on makes the derivation for boundary conditions more complicated. It will be better to start with reflections, after translations, of course, as dealt with in Subsection 8.4.2. Within UC D, there is no any further symmetry. One can see the differences in selecting the position of the UC. If one is not interested in further symmetries, this will be just as good as any others. Even so, the positioning of the UC should still not be done casually. In a plain weave, the geometric pattern exhibits identical topological characteristics in both the vertical and horizontal directions. It is good to preserve this feature within the UC selected, as is the case for A, B and C. The considerations behind this are as follows. (1) It is easier to generate the geometric model in terms of its internal partitions between different phases of constituents as will be elaborated toward the end of this subsection where another rotational symmetry will be used. (2) When meshing, the same tessellation plan can be used in both directions, meaning that less efforts are required. (3) It is easier to achieve equal properties in both directions, although it can always be achieved with or without this feature, provided that the mesh is sufficiently fine. However, without this geometric feature being preserved, one might need to have more refined mesh in order to achieve equal properties in both directions. If one is prepared to deal with slightly more complicated boundary conditions as provided in Subsection 6.4.1.3 for unit cells resulting from translational symmetries along two non-orthogonal directions, a UC can be obtained as any of the black rectangles as shown in Fig. 12.6(b). The two directions of translational symmetries are the horizontal and the inclined one, respectively, through which the neighbouring cells as shown in dashed lines to the UC can be considered as the images of the UC under translational symmetry transformations. Whilst using the boundary conditions obtained under these translations symmetries, the UC can be taken from any position. If such selected UC is considered as a full-sized one, then the position of the UC will have implications on the availability of further symmetries in the UC. Apparently, within the black rectangle, there is no further symmetry. On the other hand, UC shown in yellow has a central reflection. Therefore, the size of the UC can be reduced to half as shown in yellow shade in Fig. 12.6(b) without compromising the position that a single set of boundary conditions can be employed for all loading conditions. Relative to its

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Representative Volume Elements and Unit Cells

counterpart from a central reflection after orthogonal translations as shown in Fig. 12.6(a), it is half of that size. If one wishes to employ a single set of boundary conditions for all loading cases, the smallest UC that can be obtained is this square yellow shade as shown in Fig. 12.6(b). There are in fact more symmetries within the yellow shade, namely, one reflection about each of the horizontal and vertical axes. The final size of the smallest UC will be identical to that obtained previously. However, one will lose the benefit of a single set of boundary conditions for all loading cases. Similarly, if one selects the purple rectangle in Fig. 12.6(b) as a UC, there are reflections about the vertical and horizontal axes. Employing these symmetries, the size of the UC can be reduced to that marked by the purple shade, within which there is a further rotational symmetry about the vertical axis. One way or another, after exploring these symmetries, the smallest UC can always be produced, shown as the white squares in Fig. 12.6(b). It is worth noting that in terms of appearance, it is identical to that obtained from the square full-sized UC in Fig. 12.6(a). In fact, if the smallest size of the UC is the objective, the most convenient route in terms of the derivation of boundary conditions is through the square full-sized UC shown in Fig. 12.6(a). The procedure is to employ two rotational symmetries after two reflectional symmetries has been fully elaborated in Subsection 8.4.2. As has been repeatedly emphasised throughout the monograph, the lack of uniqueness is a generic feature of UCs. However, the user should be in control what he/she wants ultimately. The choices are often around the ease of derivation and application, the computational cost and whether all loading cases should be analysed with a single set of boundary conditions. To make the right choice, the key is the awareness of all symmetries available in the problem and of the implications of each type of symmetries employed. There is a further geometric symmetry present in the smallest UC as sketched in Fig. 12.6(c). It is a 180 rotations about an axis lying diagonally in the mid plane as shown there. This symmetry does not offer any advantages in terms of formulation and analysis of such UC because relevant loading cases will not exhibit this symmetry. However, it can still be useful in terms of the construction of the geometric model of the UC as well as the meshing. Since the smallest UC incorporates fragments of only two tows, one can introduce a partitioning surface, which is likely to be curved, between these two tows. In order to take advantage of this symmetry, the partitioning surface has to pass through the axis of rotation. In fact, one only needs to define half of it on one side of the axis, whilst the other half can always be generated from the existing one through the rotational symmetry.

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Applications to textile composites

With such a partitioning surface in place, one will have much desired freedom in defining the shape of the fibre tows. Again, because of the symmetry, one only needs to introduce one of the two and the other can be generated by a symmetry transformation. As long as the tow is kept on one side of the partitioning surface, interpenetration of the two tows will never occur, as is often the difficulty, especially when dealing with composites of relatively high fibre volume fractions. To conclude this subsection, as a recapitulation of the topic of material categorisation, the reflectional and rotational symmetries present in this particular meso-structure also guarantee the orthotropy of the composite represented by the UCs, in fact, squarely symmetric, as defined in Subsection 3.2.2, provided that the orthogonally arranged fibre tows are identical in both directions. In reality, most people would come to this conclusion following their intuition that happens to be perfect in this case but is apparently not always reliable, as will be reflected in the next subsection.

12.3.3 Twill weave For a twill weave, using the orthogonal translations in its plane, the UC can be obtained as marked by red square in Fig. 12.7(a). As in the plain weave case, the square can be placed anywhere without affecting the boundary conditions for the unit cell derived from the translational symmetries alone. However, if one is prepared to employ non-orthogonal translations as established in Subsection 6.4.1.3, a UC of a quarter of the size of the square one can be obtained which is marked by a black rectangle in Fig. 12.7(b) with its image drawn in dashed lines from the translational symmetry transformation (a)

(b)

Fig. 12.7 Twill weave (a) UC from orthogonal translations, and (b) UCs based on nonorthogonal translations before exploiting further symmetries.

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Representative Volume Elements and Unit Cells

along the inclined axis. Again, its location can be arbitrary if one does not intend to reduce the size of UC by employing further symmetries available. Otherwise, the location should be selected carefully. In particular, a central reflection can be identified in a UC shown as the green rectangle in Fig. 12.7(b). With it, the size of the UC can be further halved to that marked by the green shade, for which a single set of boundary conditions will be applicable to all loading cases. If one is prepared to use multiple sets of boundary conditions for different loading conditions, another rotation, about the out-of-plane axis shown as a black blob in Fig. 12.7(b), can be employed. This reduces the UC to a small square marked by the purple shade. This will be the smallest UC one can obtain for the analysis of twill weaves. So far, no use has been made of reflectional or rotational symmetries and consequently no indication has been obtained yet on the degree of anisotropy of the composite. As far as geometry is concerned, however, there is a further rotational symmetry about the axis at 45 within the smallest UC as indicated in Fig. 12.7(b). It will not have much mechanical significance. One can nevertheless use it to define the geometry and to help with meshing in the same way as elaborated for respective UC representing plain weave composite in Subsection 12.3.2. The presence of two rotational symmetries about axes perpendicular to each other, one out of the plane indicated by the blob and one inclined at 45 indicated by the dash-dot line, places the composites made of twill weaves into the category of orthotropy. However, as was elaborated in Subsection 3.2.2.2, the principal axes for this type of weave are not along in the directions of the fibre tows, drawing a limit to one’s intuition.

12.3.4 Satin weaves Satin weaves are architecturally more variable than the plain and twill weaves. First of all, it needs a harness number to describe one of the topological characteristics. The ones shown in Fig. 12.8 are both 4 harness ones but of slightly different topologies. These are by no means the only possible options, not even necessarily the most typical ones, either. They are however employed to help with the elaboration to follow. With two orthogonal translations, a UC can be established as marked by a red square in Fig. 12.8(a). Given the translational nature of the symmetries employed, the square can be arbitrarily placed without changing the boundary conditions for the UC. However, for this particular architecture, given its regularity within the UC, one can identify three rotational symmetries,

391

Applications to textile composites

(a)

(b)

Fig. 12.8 Satin weaves of 4 harness (a) regular offset, and (b) irregular offset.

two of them are about the diagonal axes as shown. As a result, this UC can be reduced in size to that marked by the red shaded triangle. Despite offering a substantial reduction in size of UC, use of these symmetries incurs different sets of boundary conditions, hence this is not the optimal choice of the UC, especially since there is a better alternative available. The presence of a third rotational symmetry about the axis out of the plane, e.g. through the intersection of the two diagonal axes, does not offer any new information. The two rotational symmetries identified above have a very important implication on the categorisation of this particular architecture as shown in Fig. 12.8(a). It is apparently orthotropic according to Subsection 3.2.2, but the principle axes are again not coincident with the fibre tow directions. This conclusion should not be generalised for satin weaves of other arrangements, as it is the consequence of the regularity as present in this particular architecture. Given the regularity in offset pattern in the satin weave as shown in Fig. 12.8(a), non-orthogonal translations are available. Using them, the size of the UC can be significantly reduced to the black rectangle as also shown in the same figure. Again, this can be arbitrarily placed and analysed using a single set of boundary conditions for all loading conditions. This is the smallest size of UC one can obtain for the satin weave as shown in Fig. 12.8(a). Fig. 12.8(b) shows another 4 harness satin weave. The same red square defines a perfect UC there. However, due to the irregularity of the offset pattern within the UC, no more symmetries are available. The size of the UC cannot be reduced no matter how carefully the square UC is placed

392

Representative Volume Elements and Unit Cells

and the composite made of it has to be categorised into the family of generally anisotropic materials. Therefore, even though the two weaves as shown in Fig. 12.8 fall into the same textile category of 4 harness satin weaves, their mechanical characteristics are fundamentally different.

12.3.5 2D 2-axial braid A characteristic difference in the architecture of braids from that of weaves is that there is lack of reflectional symmetry in general, given the presence of off-axis fibre tows. However, translational symmetries are always present which offer the basis for the constructions of UCs. Usually, there are also plenty of central reflectional and rotational symmetries. For a 2D 2-axial braid, with orthogonal translations only, a rectangular UC can be defined as marked in Fig. 12.9 by a red rectangle. If the user is not interested in any reduction in size of the UC by employing other symmetries available, the red rectangle can be arbitrarily placed within the plane the braid. The boundary conditions will not be affected by the location of UC and one set of them will be applicable to all loading cases. However, if one wishes to take advantage of available symmetries in the structure, the location should be selected carefully. In particular, in order to employ central reflection, benefitting from a single set of boundary conditions for all loading cases, the UC should be chosen as the green rectangle in Fig. 12.9. After applying this symmetry, UC will be halved in size into the one marked by the green shade within the green rectangle. However, no further symmetries are available in this UC hence its size cannot be reduced any further.

Fig. 12.9 The basic 2D 2-axial braid.

Applications to textile composites

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If achieving the minimum size of the UC is the objective, one can employ symmetries available in the UC specified by the red rectangle in Fig. 12.9. Two rotational symmetries can be easily identified about the vertical and horizontal axes. The size of the UC can be reduced to a quarter as indicated by the red shade within the red rectangle. Given the presence of two rotational symmetries, the composite is categorised into orthotropic family. The axes of rotations are the principal axes as was elaborated in Section 3.2.2. Apparently, none of them is aligned with the in-plane directions of fibre tows. To exhaust the symmetries available in a 2D 2-axial braid for the purpose of UC definition, a central reflectional symmetry can be identified within the red shade as shown in Fig. 12.9, which can be used to halve the size if this UC. Since the rotational symmetries have been employed, the UC will have to be analysed under different boundary conditions for different loading cases. The further use of the central reflection will reduce the size of the UC but will not make the boundary conditions any simpler. Since the symmetry this time is central reflection, the way of partitioning the red shade is not unique. On balance, considering the subsequent meshing convenience, the black line is an optimum solution under the present consideration. The resulting triangles on either side of the black partition can be employed as a UC and this is the smallest UC one can have for this type of braid. Following the procedure elaborated fully in Chapter 8, the relative displacement boundary conditions can be derived to have the UC fully formulated. If one is prepared to employ non-orthogonal translations, a UC can be obtained as any of diamond shapes in Fig. 12.9, which are half of the size of the red rectangle. Same as the rectangular counterpart, the diamond UC can be arbitrarily placed as far as the prescription of the relative displacement boundary conditions is concerned. However, its location should be carefully selected if one wishes to take advantage of further available symmetries. Specifically, if one wants to take advantage of centrally reflectional symmetry, the UC should be selected as marked by the green diamond in Fig. 12.9. The UC can be reduced to that marked by the green shade within the green diamond shape. This has been so partitioned that the geometric features involved would to be most regular and hence convenient for subsequent meshing. As explained in Section 12.2, the central reflection is usually meant to finalise the UC definition, unless there is another central reflection in the UC. Otherwise, the benefit of analysing all loading cases

394

Representative Volume Elements and Unit Cells

under a single set of boundary conditions will be lost. Once the green shade has been chosen, there is no further symmetry within it. Although this is twice the size of the smallest UC as obtained previously, it has its own advantages as stated. On balance, it is likely to be the optimum choice as the UC for this architecture. If one chooses the black or the purple diamond in order to minimise the size of the UC, there are rotational symmetries about the vertical and horizontal axes, respectively. The UC can be reduced to a quarter of the diamond as marked by the black and purple shades, respectively. They are of the same size as the red shade and of the same effects in terms of boundary conditions. A subtle difference between the black and purple shaded UCs is that the black one involves two fragments of fibre tows whilst the purple one involves three. This will make some difference in the geometric construction of the UC and the subsequent meshing. In general, the fewer the fragments, the better. The difference is subtle. However, if a user starts to make sense out of subtleties at this level, concepts of symmetry, boundary conditions, loading conditions, computational efficiency and meshing convenience will no longer be any barrier for him/her to take full advantage of the resources available. As a general rule in the definitions of UCs, one can end up with identical UCs following different routes. Also true, one could follow identical procedure leading to UCs of different appearances but still to the same effects, provided that they have all been formulated correctly. This is the case because one can locate UCs differently when using translational symmetries to start with and employ different ways of partitioning when making use of rotational symmetries and central reflections. Once the procedure has been established and followed correctly, the lack of uniqueness should therefore not constitute any confusion. What a serious user ought to achieve is an optimum choice on balance of considerations about the computational cost, the ease of deriving the boundary conditions, whether all loading cases should be analysed with a single set of boundary conditions, and whether the obtained geometry of the UC is suitable for subsequent meshing. To summarise, the choices for users are as follows: (1) The rectangular UC (the frame not the shade) with a single set of boundary conditions whilst the boundary conditions will be in their simplest form. (2) The diamond-shaped UC with a single set of boundary conditions, which is smaller in size than that in the previous option, but the boundary conditions will be slightly more complicated.

395

Applications to textile composites

(3) The triangular UC of the smallest size but under different boundary conditions for different loading cases, bearing in mind of the subtle differences as discussed. (4) The parallelogram UC (green shade) with a single set of boundary conditions for all loading conditions, which is probably the optimum choice on balance of all considerations. Any intermediate positions, e.g. the red rectangular shade or the green isosceles triangle, represent a waste of resources one way or another. The authors wish to reflect an interesting philosophical quote from (Ny, 1966): ‘Gaining less than attainable and consuming more than necessary are both wasting’ (authors’ translation). It should also be a guideline for construction of UCs using symmetries available with respect to the computational cost, convenience in implementation and human efforts in derivation and meshing.

12.3.6 2D 3-axial braids As stated before, the topology for 2D 3-axial braids is no longer unique. Fig. 12.10 shows example of one of them (Roberts et al., 2002; Roberts et al., 2009a; Roberts et al., 2009b; Xu et al., 2019a). Using orthogonal translations alone, the UC one can obtain as marked by a red rectangle in Fig. 12.10(a), which can be placed at the arbitrary location within the braid. However, if it is positioned as shown, there is a rotational symmetry about the horizontal axis through the middle of the UC. As is often the case, the UC obtained based on orthogonal translations does not usually represent the optimum choice in terms of further symmetries available for UC size (a)

(b)

Fig. 12.10 A 2D 3axial braid (a) unit cells from different translational symmetries, and (b) UCs of reduced sizes.

396

Representative Volume Elements and Unit Cells

reduction and the number and shapes of fragments involved for the subsequent meshing. An alternative is to employ non-orthogonal translations, in which case the shape of the UC becomes as marked by the green parallelogram. It is of the same size as the rectangular UC. Similarly to the previous case, its location can be arbitrary, yet some choices are better than others in terms of availability of more symmetries. Specifically, there are plenty of symmetries to take advantage of within the UC shown in Fig. 12.10. In particular, its size can be halved by employing the available centrally reflectional symmetry, which halves the size of the UC, reducing it into that marked by the green shade whilst keeping the analysis under a single set of boundary conditions for all loading cases. There are more symmetries within this half sized UC which are worth exploring. To assist with such exploration, the green shade in Fig. 12.10(a) has been redrawn into a parallelogram frame in Fig. 12.10(b). One of the available symmetries is rotation about the axis out of the plane passing through its centre as indicated by a blob in Fig. 12.10(b). If the partitioning plane is chosen to be parallel to shorter sides of the UC, the UC reduces to that marked by the yellow shade in Fig. 12.10(b). Within it, another rotational symmetry about the horizontal axis is available. Applying it reduces the size of the UC for this type of the braid to the very minimum, with smallest UC being marked by a black triangle. This is produced if the partitioning plane containing the axis of rotation is perpendicular to the plane of the fabric. Again, it is by no means the only option. In fact, a surface (not shown but its projection to the plane of the fabric will be identical to the yellow shade) through the middle of the vertical tow and passing the interface between the two inclined tows will be an optimum choice for the subsequent meshing as it involves only two parts of fibre tows and their shapes are kept as regular as they could be. Given the two rotational symmetries as utilised above, the composite made of such braided preform is orthotropic with the horizontal and vertical axes as two of the principal axes. The other form of 2D 3-axial braid as shown in Fig. 12.11 can be examined with a view to define an appropriate UC to represent it. Orthogonal translations lead to the red rectangle as a UC as shown in Fig. 12.11. Two rotational symmetries about the vertical and horizontal axes, respectively, are available, use of which reduces the size of this UC to that of the red shade. There is a further rotational symmetry about the out-of-plane axis through the centre of the red shade, which can halve the size again. The

Applications to textile composites

397

Fig. 12.11 2D 3-axial braid of different geometry.

process here is very similar to that as employed for the hexagonally packed UD composite as presented in Subsection 8.4.1. The difference is that rotational symmetries have been used to reduce to the red shade, instead of reflectional symmetries that were employed to reduce the size of hexagonally packed UD composite. As is in the case of the UC for hexagonally packed UD composite, the choice of the partitioning plane in the red shade to implement the final rotational symmetry is not unique. Forming a triangle as shown in the green shade seems to be the optimum in this case as it involves only three fragments of fibre tows, all of reasonable shapes for subsequent meshing. This UC is also the smallest one can achieve for this type of braid. The use of rotational symmetries signifies the orthotropy of the material. As will be shown below, this is yet an incomplete categorisation but it is what the considerations so far are capable of delivering. Non-orthogonal translations can result in two different UCs, one being diamond shaped and another hexagonal as shown in black in Fig. 12.11. They are both of smaller than that marked by the red rectangle. Both of them can be analysed under a single set of boundary conditions for all loading cases. The diamond-shaped UC has two pairs of sides whilst the hexagonal has three, but the latter is three-quarters of the size of the former. The balance is between computational cost and human efforts. Within the diamond-shaped UC, there are rotational symmetries about the two diagonals, which are of no mechanical significance but could help to define a building block for the construction of geometric model. The final symmetry to explore is rotation about the axis out of the page passing through the

398

Representative Volume Elements and Unit Cells

centre of the diamond-shaped UC. This will produce a UC twice as large as the smallest obtained previously without offering any significant benefit in any respect and hence will not be of any interest. The presence of the two rotational symmetries about the two diagonals suggests the orthotropy of the material with these two axes being the principal axes. Given the existence of principal axes in the same plane but in different directions, the isotopy in the plane of the fabric can be deduced. There is yet a more straightforward way to observe the isotropy in the plane of the fabric. Consider the hexagonal UC as shown in Fig. 12.11. There are more symmetries in it than in the diamond-shaped UC. Use of one of them, namely, the rotation about the vertical axis, halves the size of the hexagon, but the resultant UC is still 1/16 larger than that marked by a green triangle, hence it is not very effective in terms of reducing the size of the UC. However, there is also 60 rotational symmetry about the out-of-plane axis, i.e. C3. It does not help with reducing the size of the UC because it is not met by the loading conditions, but has significant implications on material categorisation. In presence of such symmetry, the material is effectively isotropic in the plane perpendicular to the axis of rotation, i.e. in the plane of the fabric. This is a characteristic difference from the 2D 3-axial braid discussed earlier. Throughout this section, symmetries of different types have been used repeatedly. The derivations of boundary conditions have been waived as one can follow the procedure as established in Chapter 8. Although every single case has its own characteristics, the common rule is the use of relative displacement field. The cases as presented here can serve as useful exercises for those interested in gaining competence in application of symmetries when formulating effective UCs.

12.4 Unit cells for three-dimensional textile composites 12.4.1 3D weaves Formation of a practical weave patterns, both 2D and 3D, in fabrication involves the relative movement of the warp and weft tows. It is the job of textile designers to regulate such movement to achieve the desired design. However, direct imitation of the fabrication process does not seem to be the most straightforward way of constructing the weave patterns generated. An alternative approach (Xu et al., 2019b) is to parameterise the topology, thus

Applications to textile composites

399

offering the capability of numerically generating a wide range of weave patterns. As was outlined in Section 12.1.2, 3D woven composites involve two arrays of orthogonally placed fibre tows. The weft tows are regularly arranged into aligned rows in the horizontal direction and are stacked in the vertical direction to form a matrix. Consider an assembly of tows where all warp tows undulate regularly with a given depth in an identical form and pace. They cross a number of columns of weft tows before turning as the warp tows undulate. This scenario does not involve any interlocking and the two arrays of the tows still stay loose, as shown in Fig. 12.12(a), where a slightly slanted view of this pattern is given, showing perfect (a)

(b)

(c)

Fig. 12.12 Schematic view of the interlocking (a) two aligned slices, (b) slices after shifting, and (c) four slices illustrating full interlocking as a result of shifting slices.

400

Representative Volume Elements and Unit Cells

alignment of weft tows in depth. Only two sets of warp tows, one highlighted in blue and another in pink, are shown in the figure for the clarity of the view. This assembly can be considered as a stack of two slices along the weft direction, with each containing just one set of warp tows. Interlocking takes place if one horizontally shifts the slice at the back (with pink warp tows) relative to that in the front (with blue warp tows) by a full spacing between the weft tows and the weft tows are re-aligned and re-joined, as shown in Fig. 12.12(b). With multiple slices involved, the interlocking can be viewed more clearly and realistically as illustrated in Fig. 12.12(c). The mechanism as demonstrated through Fig. 12.12 reveals an important observation. A single slice can be representative enough, which offers great convenience in defining an appropriate UC for any architecture of the weave. With the periodicity in a slice in both vertical and horizontal directions, a UC can be defined as a complete period of the slice in each of the horizontal and vertical directions. Mapping from the UC to any of its images within the slice corresponds to a translation as a combination of these two orthogonal translations. Mapping from a UC in one slice to that in another slice requires another translation, along the axis inclined to the weft direction in the plane of the row of weft tows, which is non-orthogonal to the former two. Together, these three translations form a complete set of three-dimensional non-orthogonal translations through which the entire space of the 3D weave can be covered by the images of the UC under one set of the translational symmetry transformations. The UC is therefore very similar to those in the previous section established using nonorthogonal translations, when the woven preform as presented in Fig. 12.12(c) is viewed from the top. If the discussions on the 2D textile preforms in Subsections 12.3.2 and 12.3.3 are supplemented with the consideration that the periodicity is present in the thickness direction, as outlined in Subsection 12.2.1, the definition of the UC and its relationship with its surrounding cells can be readily inherited. A close scenario could be the satin weave as presented in Subsection 12.2.4. As the symmetries employed so far are of translational nature, they do not carry any implication on the degree of anisotropy of the material. The material represented by the UC will have to be considered and characterised as a completely anisotropic material in general. However, depending on the internal architecture within the UC and hence the material, there might be more symmetries. With appropriate identification of the UC, translational symmetries should have been exhausted. One could look for any

Applications to textile composites

401

centrally reflectional, conventionally reflectional and rotational symmetries within the UC. This is an extremely useful step to take. First of all, any further reflectional or rotational symmetry helps to identify a principal plane or axis of the material and therefore reduces the degree of anisotropy. Readers are reminded that there is no industrial standard available to support the characterisation of materials with degree of anisotropy beyond orthotropy. Therefore, there has been no established means to validate theoretical results beyond orthotropic materials practically. Secondly, these additional symmetries can help to reduce the size of the UC to be analysed. Even if one does not wish to tackle the sophistication in boundary conditions coming with the symmetries, these symmetries, if any, can help to identify the smallest building block which can greatly simplify the subsequent construction of the geometric model and mesh generation. It is worth noting that in 3D modelling in general, mesh generation always requires a significant effort and is usually labour-intensive and time-consuming. The same as in the 2D weaves, one should concentrate on the translational symmetries first to define a most efficient UC. As stated in Section 8.1, the domain of a UC, once defined through translational symmetries, can be offset by any distance in any direction. Often by doing so, one might identify a more efficient selection of the UC for the architecture concerned as has been demonstrated in previous sections. When there are sufficient number of arrays of warp tows stacked in the thickness direction, periodicity can be assumed in this direction, allowing a single period to be taken to form a UC. One might wish to incorporate the effects of surface layers by conducting special analysis specifically for the surface layers. The fall-back position is that one has to analyse the complete thickness. In general, considerations for employing additional symmetries in 3D weaves are similar to those elaborated for 2D textiles in the previous section. The problem specific to 3D weaves is the geometric realisation of 3D woven architecture, the solution of which is offered below. The external geometry of UC for 3D weave is simple. Given the translational symmetries available, the simplest shape for the UC is apparently a cuboid but out of a staggered pattern in the plane of the fabric. In order to define its topology, a number of integer parameters will be introduced to help with the parameterisation of the weave pattern and the complete determination of the UC ultimately. With respect to the shifting mechanism, a parameter nstep will be employed to specify resulting dislocation in terms of the number of

402

Representative Volume Elements and Unit Cells

columns of the weft tows as a topological descriptor. To make sense of it, 0 means no shift, as is the case shown in Fig. 12.12(a), 1 means shift by one column of weft tows as is the case shown in Fig. 12.12(b) and (c). Other numbers will become relevant when more sophisticated architectures are addressed. The range for this parameter is therefore greater than 0 but less than the number of columns of weft tows in a full period in the direction of shifting. In the illustration as provided in Fig. 12.12, the weft tows are packed in straight columns and rows. To allow variations in the weave patterns that can be accommodated in a unified approach, every other column can be given an offset vertically to produce a pattern as shown in Fig. 12.13(a) and, alternatively, one can offset rows in a similar fashion but horizontally, as shown in Fig. 12.13(b). This is another mechanism involved in describing the internal topology of the UC related to the weave pattern. A topological parameter noffset is introduced as a switch, taking its value amongst -1 for column offset in the vertical direction; 1 for row offset horizontally and 0 for the lack of offset. This parameter will not take any other value.

(a)

(b)

Fig. 12.13 Offset of (a) the columns in the vertical direction and (b) rows of the weft tows in the vertical and horizontal direction.

Applications to textile composites

403

With the layout of the weft tows being specified, the geometric definition of the warp tows is addressed next. A warp tow turns its direction up and down as it travels through the weft tows on its way to generate the desired undulation. When it turns, it will have to be arrested by some weft tows. A warp tow can turn immediately once it passes the weft tow, or it can skip a few of them before making the turn. A topological parameter nskip is therefore introduced to define the number of weft tows it skips before turning. nskip ¼ 1 means an immediate turn, and so on, as illustrated in Fig. 12.14(a). The warp tow does the same as the default when it turns back upwards, but it can be defined separately like in a satin weave if one wishes to add the level of sophistication. The next parameter describing the undulation is ndeep, which specifies the number of rows of the weft tows a warp tow moves past through the thickness before making a turn, as is illustrated in Fig. 12.14(b). The further it goes, the deeper the undulation. A warp tow would remain flat if

(a)

(b)

(c)

Fig. 12.14 Illustration of the topological parameters specifying the undulation of the warp tows (a) nskip, (b) ndeep and (c) nsteep.

404

Representative Volume Elements and Unit Cells

ndeep ¼ 0 and it would form a 2D fabric if ndeep ¼ 1. To form a proper 3D weave, ndeep must be greater than 1. Whilst parameter ndeep defines the depth of the undulation, it does not describe how quickly it reaches its bottom before making its turn. The next parameter nsteep is designated for this purpose. It specifies how many rows of the weft tows a warp tow passes by before crossing a column of the weft tows, as illustrated in Fig. 12.14(c). The higher the number, the steeper the undulation. Again, for a fabric to be genuinely 3D, nsteep will have to be greater than 0. Also, as it should be limited by the depth, hence should not be greater than ndeep. When it is equal to ndeep, the warp tow will travel to the other extreme in one go between two neighbouring columns of weft tows. With the five topological parameters as introduced above, the topology of a 3D weave can be defined with a reasonably wide coverage for most weave patterns of practical significance, whilst having the scope of being extended. The present discussion will be kept within this coverage. To demonstrate the use of these parameters in the definition of the topology of woven textile preforms, some typical weave patterns have been listed in Table 12.1 associated with the values or ranges of these parameters. In Table 12.1, the 2D weaves as discussed in the previous section have also been included to demonstrate the suitability of the suggested parametrisation scheme for simple problems before exploring its wider applications to more sophisticated problems. They are followed by two examples of typical 3D weaves. In each case, the topology of any particular weave can always be defined by a combination of five parameters introduced above. Table 12.1 Relationship between the five topological parameters and weaves defined. Topological parameters of the weave Architecture

ndeep & nsteep

nskip

nstep

2D weave

Plain Twill Satin

ndeep ¼ nsteep ¼ 1 ndeep ¼ nsteep ¼ 1 ndeep ¼ nsteep ¼ 1

nskip ¼ 1 nskip  2 nskip top  2 nskip bot ¼ 1

nstep ¼ 1 nstep ¼ 1 nstep ¼ 1

3D weave

Angle interlock (noffset ¼ 0) Offset interlock (noffset ¼ 1)

ndeep ndeep ndeep ndeep

nskip  1

nstep  1

nskip  1

nstep  1

2  nsteep  1 2  nsteep  1

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Applications to textile composites

Two 3D weaves, one from each category of the two included in Table 12.1, have been plotted in Fig. 12.15(a) and (b). They are a ply-toply interlock (angle interlock when ndeep ¼ 2) and an offset interlock weave pattern, respectively. In modelling of textile composites, in general, once the topology has been sorted out, a typical difficulty is to define the geometry of fibre tows so that the desired fibre volume fraction can be obtained. Achieving practical fibre volume fractions, let alone maximising them in textile composite models is by no means a straightforward task. Often, even if one assumed 100% of fibre volume fraction within tows, which was of course not realistic at all, the desirable fibre volume fraction could still be unattainable. In reality, fibre tows in textile composites tend to adapt their cross-sections from point to point along their length to leave room for other tows whilst filling up the vacant spaces, thus minimising the matrix contents. Simulating such cross-section variation in geometrical modelling is a demanding exercise, which is rarely attempted. Undulating and interlacing fibre tows have already kept solid modellers well occupied in most cases in order to avoid their interpenetration with the neighbouring fibre tows. Given this background, it is a good practice to adopt the tow geometry that would to

(a) a unit cell

((b)) a unit cell

Fig. 12.15 Typical 3D woven preform architectures as defined by the topological parameters (a) ply-to-ply interlock (noffset ¼ 0, nstep ¼ 1, nskip ¼ 1, ndeep ¼ 2, nsteep ¼ 2) and (b) offset interlock (noffset ¼ 1, nstep ¼ 2, nskip ¼ 1, ndeep ¼ 3, nsteep ¼ 3).

406

Representative Volume Elements and Unit Cells

minimise the voids between the fibre tows. If this is carried out carefully enough, the demand for use of variable cross-sections as means of increasing fibre volume fraction can probably be waived without compromising the outcomes of the simulation. An exercise of this kind will be conducted below to demonstrate the practice where a set of geometric parameters, some for the cross-sections of fibre tows and others for warp tow paths, will be introduced to complete the definition of UCs for this family of 3D woven composites. According to the microscopic observation of the cross-sections of fibre tows in 3D woven composites (Yu, 2016), the geometry of the crosssection varies mostly between an ellipse and a rectangle. Due to the interlocking mechanism, warp tows cross each other at a regular spacing. At the crossings, fibres in neighbouring tows are oriented differently, which prevents any interpenetration between them. Warp tows are therefore periodically constrained on their sides by neighbouring warp tows, whilst the presence of weft tows above and below keeps them in their positions. This biaxial compaction tends to shape the cross-section of warp tows so that is resembles a rectangle with rounded corners. Of course, in reality it varies to a degree along the length of each warp tow at different positions in relation to the configurations of neighbouring tows and can also be affected by the tightness of the packing of the tows in the fabric. For weft tows, the pressure on the top and bottom transmitted through the neighbouring layers above and below, or compaction applied by the tool during the forming process of the composite, tends to flatten the top and the bottom surfaces of their cross-sections. Toward the sides, weft tows tend to conform with the undulating warp tow paths. As a compromise of all these considerations, the tow cross-section is assumed to take a shape of an ellipse split in the middle and filled with a rectangle, as shown in Fig. 12.16. The elliptical profile on both sides is expected to offer the rounded corners for the warp tows as well as the curvature required for weft tows to conform to the surrounding warp tows. The ellipse can be described by the horizontal and vertical semi-axes, a and b, respectively, where a can be less b to allow the overall cross-section to look more rectangular than elliptical when appropriate. If the height of the rectangle is defined as H, which is one of geometric parameters of cross-section, then b ¼ H/2. At the given vertical semi-axis b, the horizontal semi-axis a determines the actual shape of the ellipse, or the roundness of the corners of the tow cross-section profile. To quantify this effect, another geometric parameter

Applications to textile composites

407

Fig. 12.16 The assumed fibre tow cross-section as an assembly of a rectangle and two semi-ellipses, one at each end of the rectangle.

of cross-section, g, is introduced as the ratio of the width of the elliptical part to the full width of cross-section, W, as follows: 2a (12.1) W For practical considerations, it is reasonable to restrict the aspect ratio of the tow cross-sections such that W  H. Given the construction of the cross-section as introduced in Fig. 12.16, the range of g has to be 0  g  1. Over this range, the cross-section varies from a rectangle to an ellipse. Additionally, g can also be prescribed a value of 2 outside its definition in Eq. (12.1), to specify a lenticular cross-section section which is rather popular in defined tow cross sections, in particular, for weft tows as they are wrapped by undulating warp tows. A number of special cases corresponding to different values of g are listed below associated with their specific geometric characteristics along with a graphic representation shown in Fig. 12.17. g¼

Fig. 12.17 Typical tow cross-section profiles as specified by the shape parameter g

408

Representative Volume Elements and Unit Cells

1) g ¼ 0: a rectangle (a square if W¼H); 2) H=W > g > 0: a rectangle with semi-elliptical ends (major axis being vertical and hence slightly rounded ends); 3) g ¼ H=W (implying a¼b): a rectangle with semi-circular ends or a circle if W¼H; 4) 1 > g > H=W : a rectangle with semi-elliptical ends (major axis being horizontal and hence large chamfers); 5) g ¼ 1: an ellipse (a circle if W¼H); 6) g ¼ 2: a lenticle. To summarise, the cross-sections of fibre tows are determined by three geometric parameters, height H, width W and a shape parameter g. With the tow cross-section parameterised, the next step is to define the warp tow path. It is assumed that a warp tow turns around the weft tow and therefore conforms with its profile. Before the turn, warp tow follows a horizontal straight line, although the length of this stretch of path could be zero if g ¼ 1 if nskip¼1. After the turn, it also returns back to a straight line, but inclined, to proceed to the next column of weft tows. A complete period of the warp tow path can be described as follows. It is divided into a number of symmetric segments. As a special case where ndeep¼nsteep, for the weave as shown in Fig. 12.18, there are four such segments. Taking the segment OSTR as the original, by a 180 rotation about point R, the segment O’S’T’R’ can be generated. Applying a reflectional symmetry about the xz-plane, the remaining two on the left can be obtained. Depending on the actual location of the warp tow within a UC, the order of the four segments within a complete period can be rearranged to reflect the phase difference between different warp tows. In cases where ndeep>nsteep, a period will involve more columns of weft tows and hence more warp tows. However, the warp tows can all be generated from a generic one by appropriate segmentation and relevant symmetry transformations.

Fig. 12.18 A complete period of the warp tow path.

Applications to textile composites

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Modelling of the warp tow as a solid object is straightforward as most solid modellers, including FE pre-processors, have a sweep function to generate a longitudinal solid by sweeping a cross-section along a given curve. Users are reminded that different codes might have different terminologies to describe the same function and, sometimes, the same terminology for different functions. Essentially, one needs to generate a longitudinal object as a stretch of a warp tow by moving a cross-section along a curve whilst keeping the cross-section perpendicular to the curve. To generate the warp tows, one only needs to have the segment OSTR determined. To allow for different shapes of cross-sections of warp and weft fibre tows, they are defined with two different sets of geometric parameters, denoted as Ha, Wa, ga and Hb, Wb and gb, respectively, corresponding to H, W and g as defined through Figs. 12.16 and 12.17. For the simplicity of the geometry and the convenience of computational generation of the geometric model, it is assumed that the warp tow stays in contact with the weft tow until it straightens as it leaves the weft tow. The top and bottom surfaces of the warp tow stay parallel throughout its entire length, including the stretch around the turn. The segment of the warp tow as shown in Fig. 12.18 can be divided into three sub-segments, OS, ST and TR. Sub-segment OS starts from the centre as a horizontal straight stretch resting on top of the row of weft tows. Its length is related to the topological parameter nskip and other geometric parameters as 1 1 LOS ¼ ðnskip  1ÞðWb þ Db Þ þ ð1  gb ÞWb (12.2) 2 2 Sub-segment OS is followed by a curved sub-segment ST which wraps around the elliptical weft tow. It then turns back to a straight subsegment inclined at an angle of q which is yet to be determined. A magnified and elaborated view of the OSTR segment is sketched in Fig. 12.19. To facilitate the definition of the warp tow path, parametric expressions for curves representing different segments have been derived as follows. Segment OS is a straight line defined as 1 z ¼ ðHa þ Hb Þ; y˛½0; LOS  (12.3) 2 Sub-segment ST is a parallel curve (Weisstein, 2019) to the elliptical weft tow profile. The semi-ellipse as the weft tow profile is expressed in its parametric form as

410

Representative Volume Elements and Unit Cells

Fig. 12.19 A typical segment of the warp tow path in relation to the interacting weft tow.

y ¼ a cosð4Þ þ LOS z ¼ b sinð4Þ

h p 4˛  ; 2

pi 2

(12.4)

where a ¼ gWb =2 is the horizontal semi-axis and b ¼ Hb/2 the vertical semi-axis. The curve parallel to the ellipse by a distance of Ha/2 outside of the ellipse can be obtained as (Weisstein, 2019)   bHa y¼ aþ cosð4Þ þ LOS ; 2r h pi 4 ; 4˛ (12.5) 0   2 aHa z¼ bþ sinð4Þ; 2r where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ a2 sin2 ð4Þ þ b2 cos2 ð4Þ.

(12.6)

It should be stressed that 4 is a parameter in the parametric form of definition of ellipses, not the angular polar coordinate at a point corresponding to 4 as defined by (12.4). Since the centre line of the warp tow over the segment ST is parallel to the elliptical profile of the weft tow, the tangential points T and T0, respectively, where the inner surface of the warp tow departs from the surface of the elliptical weft tow profile, correspond to the same value 40 of parameter 4. The slopes of their tangents at this point are identical, and equal to the slope of sub-segment TR, hence

411

Applications to textile composites

    dz  dz=d4  b tanðqÞ ¼  ¼ ¼ cotð40 Þ;   dy 40 dy=d4 40 a

(12.7)

where angle q is as specified in Fig. 12.19. Sub-segment TR is a straight line and hence its equation can be given in a straightforward manner as z  zR ¼ tanðqÞðy  yR Þ;

y˛½ yT ;

yR .

(12.8)

where (yR, zR) are coordinates of point R and the coordinates of point T are denoted as (yT, zT). The latter can be obtained from (12.5) at 4 ¼ 40 , where angle 40 is associated with q according to Eq. (12.7), and the way of determining the two will be outlined below. Coordinates of point R are defined differently at different combinations of topological parameters. As a special case to illustrate the relationship, assume ndeep ¼ nsteep ¼ 2. The variation of q is shown in Fig. 12.20 for three different offset scenarios. This is nevertheless a pure geometric relationship referring to the overall configuration in Fig. 12.18 and the definitions of relevant topological parameters and the coordinates of point R can be determined as follows. Without any offset, i.e. at noffset ¼ 0,

Fig. 12.20 Variation of q for weaves of different topological parameter noffset: (a) noffset ¼ 0 (b) noffset ¼ 1 with the distance of vertical offset being Doffset ¼ 12 ðHa þHb Þ; (c) noffset ¼ 1 with the distance of the horizontal offset being Doffset ¼ 12 ðWb þDb Þ.

412

Representative Volume Elements and Unit Cells

1 yR ¼ nskip ðWb þ Db Þ; 2 1 zR ¼  ðnsteep  1ÞðHa þ Hb Þ 2

(12.9)

With the vertical offset, yR ¼ nskip

Wb þ Db ; 2

Ha þ Hb ; zR ¼ ð2nsteep  1Þ 4

(12.10)

and with horizontal offset yR ¼ ðnskip þ nsteep  1Þ

Wb þ Db ; 2

(12.11) 1 zR ¼  ðnsteep  1ÞðHa þ Hb Þ 2 With coordinates of point R being defined, following relation can be established for determining 40 and q based on Eqs. (12.7) and (12.8)   b zj4¼40  zR ¼ tanðqÞðy  yR Þ ¼  cotð40 Þ yj4¼40  yR . a

(12.12)

Eq. (12.12) defines a transcendental equation for 40 . A closed-from solution is not usually attainable and one may have to resort to a numerical algorithm, e.g. the Newton’s iteration scheme, to obtain a numerical solution. Once it is obtained, Eq. (12.7) can be used to determine tan(q) in a straightforward manner. The position of point T can be uniquely determined from Eq. (12.5) with 4 ¼ 40 . This concludes the definition of segment OSTR. As stated above, the rest of the warp tow path can be obtained through respective symmetry transformations. As a summary, the geometry of a 3D weave can be completely determined by the following set of eight geometric parameters: (1) Warp tow thickness Wa; (2) Warp tow width Ha; (3) Warp tow shape parameter ga ; (4) Gap between neighbouring warp tows Da; (5) Weft tow thickness Wb; (6) Weft tow width Hb;

Applications to textile composites

413

(7) Weft tow shape parameter gb ; (8) Gap between neighbouring weft tows Db. These are of course in addition to the set of five topological parameters as defined earlier. The detailed elaboration as presented in this subsection has demonstrated that, as long as regularity is present, one can fully define a UC by making appropriate use of the symmetries as available. Without loss of crucial information, the architecture within the UC can be idealised in such a way that it can be parameterised as elaborated above. To achieve this, one will always need a number of parameters designated to the topology. It is imperative that the topology of the architecture can be represented completely and efficiently, i.e. by as few of them as possible. The remaining ones are for the geometry. An important consideration here is to minimise the volume of the voids which will have to be subsequently filled in by matrix alone. As a result, the volume fraction of the fibre tows will be maximised which delivers realistic fibre volume fractions in the composite represented by the UC. With the parameterisation as presented above for a wide enough range of 3D woven architectures, the tow volume fraction can easily reach about 70% in most cases. It would be reasonable to assume a fibre volume fraction of 70% in tows. This would result in an overall fibre volume fraction around 50%, which should be pretty much what 3D woven composites could realistically reach. With this, the account presented here should have offered a sufficiently practical way forward for materials scientists, engineers and designers contemplating serious applications of 3D woven composites.

12.4.2 3D braids The topology of some types of the 3D braids has been illustrated through idealised plots in Fig. 12.4. In reality, when they are practically fabricated with finite dimensions, the cells are not exactly orientated as in Fig. 12.4(a). Instead, they are more likely to be as sketched in Fig. 12.21 as a top view of layout as shown in Fig. 12.4(a), with Fig. 12.21(a) being a relatively more idealistic scenario and Fig. 12.21(b) as a more realistic position, especially after the braid has been subjected to compaction during the forming process into a composite. The shaded boundary indicates presence of a different braiding pattern in which fibre tows behave like light beams reflecting at the mirror before merging into the subsequent braiding to find their way back into the regular unit cells. With the topological construction remaining the same as before, the unit cells can be formed more realistically by assuming non-orthogonal

414

Representative Volume Elements and Unit Cells

(a) (b)

Fig. 12.21 More realistic packing the unit cells in 3D braids (a) relatively more idealistic, and (b) relatively more realistic.

translations in the plane as shown in Fig. 12.21. Readers should be wellfamiliar with them by now. Otherwise, a revisit to Subsection 6.4.1.3 would suffice. However, given the non-orthogonal nature of the translations, the derivation in Subsection 8.3.3 is no longer applicable as there is not as many symmetries as was in the orthogonal scenario. However, given the 3D 4-axial braiding mechanism, the presence of a rotational symmetry about the longitudinal axis can always be assumed. As a result, the size of the UC can be reduced to a half. This is unfortunately as far as one can go. Given the presence of a rotational symmetry, the material can categorised as monoclinic with the principal axis being in the axial direction. Without any further restriction in the geometric arrangements, it should not be assumed to be orthotropic in general. The difficulty for practical applications will be to achieve a reasonable tow volume fraction for this architecture of textile preform. Without committing to full details, a viable way forward should emerge if one broke the UC up and identified a number of smallest building blocks. Each building block should aim at involving minimum number of tow fragments so that the tow volume fraction can be relatively easily maximised within these building blocks before assembling them into a UC.

12.5 Conclusions Through systematic consideration of symmetries present in composites made of various typical textile preforms, unit cells for a wide range of such preforms can be identified to deliver the most efficient approach when applied in computational modelling. This facilitates material characterisation, design and optimisation of textile composites via computational means, breaking the barrier to the wide acceptance of textile composites

Applications to textile composites

415

in practical applications. Whilst delivering this capability, the exercises as performed in this chapter also serve as a timely recapitulation of the concepts and skills as some of the objectives of this monograph, which has covered a good number of largely neglected topics falling between boundaries of different disciplines, such as symmetry and material categorisation. A clear and positive outcome of this chapter, in a nutshell, can be a declaration that a design tool is ready for textile composites. It can be to the same effect as the classical laminate theory is to conventional laminated composites if one is assisted with a modest computing power and an FE code. To offer an indicative measure of its practicality, on a typical desktop PC, the definition of the architecture of the textile preform takes minutes. For many relatively simple architectures, actual running time can be minutes as well whilst sophisticated ones can be analysed in hours, if not quicker.

References

Li, S., Reid, S.R., 1992. On the symmetry conditions for laminated fibre-reinforced composite structures. Int. J. Solids & Structures 29, 2867e2880. Li, S., Zou, Z., 2011. The use of central reflection in the formulation of unit cells for micromechanical FEA. Mechanics of Materials 43, 824e834. Long, A.C., Brown, L.P., 2011. Modelling the geometry of textile reinforcements for composites: TexGen. In: Boisse, P. (Ed.), Composite Reinforcements for Optimum Performance. Woodhead Publishing. ISBN 9781845699659. Ny, T.-Z., 1966. The First and Second Laws of Thermodynamics. People’s Education Press, Beijing (in Chinese). Roberts, G.D., Goldberg, R.K., Binienda, W.K., Arnold, W.A., Littell, J.D., Kohlman, L.W., 2009a. Characterization of Triaxial Braided Composite Material Properties for Impact Simulation. NASA/TMd2009-215660; Sept. 2009. Roberts, G.D., Pereira, J.M., Braley, M.S., Arnold, W.A., Dorer, J.D., Watson, W.R., 2009b. Design and Testing of Braided Composite Fan Case Materials and Components. NASA/TMd2009-215811; Oct. 2009. Roberts, G.D., Revilock, D.,M., Binienda, W.K., Nie, W.Z., Mackenzie, S.B., Todd, K.B., 2002. Impact testing and analysis of composites for aircraft engine fan cases. Journal of Aerospace Engineering 15, 104e110. Weisstein, E.W., 2019. Parallel Curves, [Online]. MathWorld d A Wolfram Web Resource. Available:, 2019]. http://mathworld.wolfram.com/ParallelCurves.html. Xu, M., Sitnikova, E., Li, S., 2019a. Formulation of the size reduced unit cell for triaxial braided composites. In: 22nd International Conference on Composite Materials. Melbourne, Australia, 12-16 August 2019. Xu, M., Sitnikova, E., Li, S., Unification and Parameterisation of 3D Weaves and Formulation of Unit Cells for Woven Composites, 2019b. preparation for journal publication. Yu, T., 2016. Continuum Damage Mechanics Models and Their Applications to Composite Components of Aero-Engines. PhD thesis. The University of Nottingham.

CHAPTER 13

Application of unit cells to problems of finite deformation 13.1 Introduction The need to account for finite deformation arises when the small deformation assumption is no longer valid, that is, displacement and/or its first derivatives can no longer be considered small. Finite deformations represent a challenge, especially in modern applications of engineering materials and structures, including composites. Handling problems involving finite deformations, both numerically and experimentally, is by no means a trivial matter as compared to their small deformations counterparts. For example, in order to measure stress under assumption of the strains being small, all that is required in the simplest tensile coupon test is to divide the load applied by the cross-section of the specimen. The stress obtained will be conventionally called engineering stress, which has a unique definition, referring to the undeformed configuration of the specimen. Therefore, in order to apply a given amount of stress, the operator of the testing machine can follow precise instructions as to which load level the specimen should be loaded. When attempting the same but accounting for finite deformations, a range of difficulties arise. Specifically, the stress and strain definitions are no longer unique. They refer to different configurations, before and after deformation and hence any experimental measurements of stresses and strain in the test will have to be subject to interpretation. Suffice it to say, because stresses could be referring to a deformed configuration which is not known before load is applied, there is no way for the operator to tell the exact load level from the given stress level. Same considerations are equally true in numerical modelling of finite deformations. A closer examination of the state-of-the-art will reveal that if the users enter the problem without appropriate understanding of the formulation of the finite deformation problem, as well as its finite element representation, they might not even know what stresses and strains the set of numbers represents. Interpretation according to intuitive perception is no different Representative Volume Elements and Unit Cells ISBN: 978-0-08-102638-0 https://doi.org/10.1016/B978-0-08-102638-0.00013-X

© 2020 Elsevier Ltd. All rights reserved.

417

j

418

Representative Volume Elements and Unit Cells

from shot in dark. One needs a great deal of luck to hit something meaningful at all. Because of that, prior to attempting any nonlinear analyses of UCs, given the non-unique definition of stresses and strains under finite deformations, one must choose which nonlinear stress and strain measures are meant to be employed for the problem being analysed. Using inappropriate ones is almost as wrong as using a ruler to measure weight. Most of modern commercial FE codes employ a numerical algorithm suitable for solving nonlinear problems. The formulation of such algorithm is known as updated Lagrangian formulation (Washizu, 1975), developed specifically for this purpose. The documentation of FE codes usually gives little information regarding the finite deformation formulations they employ. In particular, from documentation of Abaqus (2016) one can only learn what type of stress and strain output that can be requested out of nonlinear analysis at individual integration points, with their choice being limited to few particular stress and strain types. Nevertheless, with the facility for nonlinear analysis, both geometric and material, being readily available, saving users the efforts of devising their own models and solution techniques, it is therefore rather tempting for users to rush into the exercise blindly. In the context of unit cell applications, users might take for granted that provided that the nonlinear analysis option has been switched on, it would be done automatically, with the UC formulation and post-processing techniques from linear analysis being directly applicable to problems involving finite deformations. Since such cases often involve material nonlinearity, the users put most of their efforts into implementing sophisticated constitutive behaviour of the material. The formulation of UCs themselves to be employed in such studies is in the meantime attended rather casually, hence is subject to the same misperceptions and erroneous treatments as reviewed in Chapter 5. Examples of this kind can be found in the literature, e.g. (Guo et al., 2007, deBotton et al., 2006). The typical analysis of UCs with nonlinearity aims to quantify the average stresses and strains under the applied loading, same as in linear analysis, hence one may expect that it may be possible to draw analogies and adapt existing linear problem formulations to solve nonlinear ones at least in this respect. This is indeed the case for UC formulation as will be presented in this chapter. Such extensions of linear formulations, however, should never be taken for granted, and they have to be made rationally (Sitnikova and Li, 2019). Due to scarcity of information available regarding the solution techniques employed in FE codes, semi-analytical approach was followed in

Application of unit cells to problems of finite deformation

419

this chapter to establish UC modelling at finite deformations. The particular aspects of UC linear formulations that can be readily adopted and extended to nonlinear formulation will be specified and logical assessment of the gaps in formulation will be conducted, based on which numerical procedures will be devised to recover the missing links.

13.2 Unit cell modelling at finite deformations 13.2.1 Boundary conditions When moving to the finite deformation problem, the formulation of UC as developed in Chapter 6 still remains applicable, in particular, in terms of application of translational symmetries; however, its handling requires interpretation. In particular, the deformation kinematics underlying the relative displacement field which leads to the boundary conditions for unit cells eventually 2 3 vU vU vU 6 7 9 8 9 8 9 6 vx vy vz 78   6 7> Dx > > > < =

=

= 6 vV 7 vV vV 7   6 v   v  ¼ VUDx ¼ 6 7 Dy > > > 6 vx vy vz 7> ; : > ; : > ; 6 7: w ðx0;y0;z0Þ w ðx;y:zÞ 6 vW vW vW 7 Dz 4 5 vx vy vz (13.1) according to the rule of differentiation, remains the same. Eq. (13.1) is identical to (6.1) and is given here for ease of referencing. In Eq. (13.1) as well as in the rest of the chapter, bold face characters denote vectors or tensors as conventionally adopted. The major difference from linear problems is the use of complete form of displacement gradient, 2 3 vU vU vU 6 7 6 vx vy vz 7 2 U 0 U 0 U 0 3 x y z 6 7 6 vV 7 6 vV vV 6 7 6 V0 V0 V0 7 VU ¼ 6 (13.2) 7¼ y z 5 6 vx vy vz 7 4 x 6 7 6 vW vW vW 7 Wx0 Wy0 Wz0 4 5 vx vy vz

420

Representative Volume Elements and Unit Cells

rather than its special form as in (6.9). The latter was employed to constrain the rigid body rotations of the UC in a special way which does not affect the strains and hence the stresses, as have been elaborated in Section 6.2. As was also argued there, for small deformation problem the rigid body rotations can be constrained in different ways. Provided that the complete displacement gradient matrix has been specified, the rigid body rotations will have been eliminated. This remains equally true in nonlinear problems. However, setting the off-diagonal components of the displacement gradient matrix to zero tends to over-constrain rigid body rotations in finite deformation problems. In other words, whilst constraining rigid body rotations, this tends to cause strains as well, as will be numerically demonstrated later. In the subsequent FE analysis, there will be no need to do that anymore. Therefore, rigid body rotations have been constrained when relative displacement field is defined in Eq. (13.2). The definition of various types of finite deformation tensors relies heavily on the deformation gradient, F, which is related to the displacement gradient as follows: F ¼ I þ VU;

(13.3)

where I is a unit tensor. The deformation gradient is then employed to define nonlinear strain measures, such as the logarithmic strain, ELOG, with its expression being as follows: pffiffiffiffiffiffiffiffiffi ELOG ¼ ln FFT ; (13.4) or Green strain, EG, defined as EG ¼

 1 T F F I 2

(13.5)

which is also commonly employed in nonlinear analysis, with the latter being more appropriate for the total Lagrangian formulation. In the finite deformation problem, the rotation has its own definition, not quite the same as in its small deformation sense. Namely, deformations gradient can be decomposed into a product of a rotation tensor and a stretch tensor, e.g. F ¼ UF ¼ JU

(13.6)

where F and J are the right and left stretch tensors which contain the information about deformation whilst U is the rotation tensor which is an orthogonal tensor and hence U1 ¼ UT . Note that these notations may be

Application of unit cells to problems of finite deformation

421

different from those conventionally adopted in textbooks; they have been employed here to avoid clashes with other notations used. Multiplying deformation gradient by its transpose as in expression for Green strain (Eq. 13.5) yields FT F ¼ ðUFÞT UF ¼ FT UT UF ¼ FT U1 UF ¼ FT F

(13.7a)

similarly, FFT ¼ JUðJUÞT ¼ JUUT JT ¼ JUU1 JT ¼ JJT

(13.7b)

where the rotation tensor has been eliminated. With such defined rotation, these strain measures are rotation-independent due to the way the deformation gradient is involved. The rotation eliminated here is of a more general sense than rigid body rotations but certainly includes rigid body rotations. With nonlinear strains being defined according to Eqs. (13.4) and (13.5), constraining any components of the displacement gradient, such as how it was done in linear case, will have an effect on the strain values calculated, which explains why the full form of displacement gradient in Eq. (13.1) will have to be used to avoid the rigid body rotations being overconstrained. Therefore, the UC model in nonlinear case will have nine Kdofs, each associated with a specific component of the displacement gradient. Since the components of the displacement gradient in UC with geometric nonlinearity no longer readily represent strains, they are meant to be average values of the components of displacement gradient for the entire unit cell at the upper length scale, according to its definition in (13.2). The average strains have to be derived from them, whichever formulation to follow.

13.2.2 Stress averaging As has already been stated, under finite deformation, none of the relationships is as simple as in the small deformation scenario anymore. The elements in the macroscopic displacement gradient can no longer be translated into strains linearly. The strains, as well as stresses, have lost their unique definition under finite deformation. There are multiple forms of them, each pertaining to the context of the specific formulation. Before they can be quantified, they have to be specified properly in the context of finite deformation. Consider these quantities at the macroscopic scale first in order to

422

Representative Volume Elements and Unit Cells

express the average stresses and strains. Examples of nonlinear strain measures that are commonly used in models with geometric nonlinearity are Green strain and logarithmic strain, given by Eqs. (13.5) and (13.4), respectively. As far as nonlinear stress characteristic are concerned, there is a variety of them as well. It is worth noting that there are well-established relations (Fung and Tong, 2001) between different nonlinear stress measures. Specifically, the first and second Piola-Kirchhoff stress tensor, denoted as Pand S, respectively, are related to Cauchy stress, s, as follows: P ¼ JsFT ;

(13.8)

and S ¼ JF1 sFT

(13.9)

where J ¼ detjFj is the Jacobian of F. Additional complication resulting from adapting finite deformations in modelling of UCs is the averaging of stresses and strains in order to obtain stress and strain state of the material in the upper length scale. Averaging these stresses and strains out of nonlinear analysis is still the responsibility of the user. This, as has been elaborated in Section 6.7, may pose a substantial challenge even in linear problems, whilst in nonlinear modelling the problem is further complicated by a wide selection of stress measures that can be employed. In Chapters 6 and 7, an efficient way of obtaining the average stresses and average strains for unit cells by making use of Kdofs has been fully and rigorously established in application to linear analysis. With average strains being obtained directly from nodal displacement output at Kdofs, the procedure for stresses averaging is also straightforward. It has been proven in that the reaction forces, Rij, obtained at Kdofs as an output from the analysis where the average strains were prescribed, are related to the average stresses as 1 Sij ¼ Rij V

(13.10)

where Sij is the average stress, and V is the volume of the unit cell before deformation. Eq. (13.10) is equivalent to Eq. (7.36), being written in more general terms. Since nodal displacements at Kdofs in nonlinear problems no longer represent the average strains, there is no reason why the reaction force output should be explicitly related to stresses. Even if it is related, it would not be clear whether it will be Cauchy, first or the second Piola-Kirchhoff stress. Furthermore, nine reactions will be obtained as outcome of the UC

Application of unit cells to problems of finite deformation

423

analysis in nonlinear case, which is surplus to number of components in Cauchy or the second Piola-Kirchhoff stress tensor, which are both symmetric. In fact, the calculus derivation as provided in Chapter 7 applies strictly to linear problem, except the discussion in Section 7.6 based on energy consideration. Namely, if the material remains linearly elastic, the strain energy should remain half of the product of stress and strain. This should be equal to the elastic work done by the applied “forces” to the respective “displacements”. In other words, the “forces” at the Kdofs and the “nodal displacements” at the same Kdofs should remain energy conjugates.

13.2.3 An assertion The logical assessment of formulation as presented in this and previous sections is as follows. By making use of the Kdofs in a unit cell, two sets of information are available out of an analysis: the displacement gradient VU as the prescribed “nodal displacements” imposed on the Kdofs and the reactions at Kdofs as the direct output from the finite element analysis. As argued above, nodal displacements and nodal forces are energy conjugates. It is well-established in the theory of finite deformations that the displacement gradient and the first Piola-Kirchhoff stresses are energy conjugates (Lm < Dx > = >

= Dy ¼ 0 > > : ; > : > ; Dz a 9 8 9 8 9 8 > = = >

= >

< Dx > Dy ¼ 0 þ a > > ; ; > : > ; > : > : 0 0 Dz 9 8 9 8 9 8 > = = >

= >

< Dx > Dy ¼ a þ 0 > > > > > > ; ; : ; : : a 0 Dz

wjH  wjD ¼ aWx0 ujA  ujD ¼ aUy0 vjA  vjD ¼ aVy0 wA  wjD ¼ aWy0 ujC  ujD ¼ aUz0 vjC  vjD ¼ aVz0 wjC  wjD ¼ aWz0 ujE  ujD ¼ aUx0 þ aUy0 vjE  vjD ¼ aVx0 þ aVy0 wjE  wjD ¼ aWx0 þ aWy0 ujB  ujD ¼ aUy0 þ aUz0 vjB  vjD ¼ aVy0 þ aVz0 wjB  wjD ¼ aWy0 þ aWz0 ujG  ujD ¼ aUx0 þ aUz0 vjG  vjD ¼ aVx0 þ aVz0 wjG  wjD ¼ aWx0 þ aWz0 ujG  ujD ¼ aUx0 þ aUy0 þ aUz0 vjG  vjD ¼ aVx0 þ aVy0 þ aVz0 wjG  wjD ¼ aWx0 þ aWy0 þ aWz0

427

8 9 8 9 8 9 > < Dx > = >

= >

= Dy ¼ 0 þ 0 > > : ; > : > ; > : > ; Dz 0 a 9 8 9 8 9 8 9 8 > = = >

= >

= >

< Dx > Dy ¼ 0 þ a þ 0 > > ; ; > : > ; > : > ; > : > : a 0 0 Dz

vjH  vjD ¼ aVx0

Application of unit cells to problems of finite deformation

AD

8 9 8 9 > < Dx > = >

= Dy ¼ 0 > > > > : ; : ; Dz 0 9 8 9 8 > =

= > < Dx > ¼ a Dy > > ; : > ; > : 0 Dz

428

Representative Volume Elements and Unit Cells

Table 13.2 Strain measures obtained in FE analysis.

logarithmic strains according to Eq. (13.4), or any other nonlinear strain measure as required. The next target is the stress. If the assertion is right, the stress obtained from Eqs. (13.10) and (13.12) will be the average Cauchy stress in the unit cell with its value being the same as predicted directly by Abaqus. It is indeed the case again as is shown in Table 13.3, namely, the Cauchy stresses from Abaqus output at the integration points are identical within the effective digits shown to those from the reactions at the Kdofs after being converted into Cauchy stress. The assertion has therefore been confirmed by the numerical results as shown. It is worth noting that there is one difference between the Abaqus output of Cauchy stresses and that obtained with (13.10) and (13.12). The latter one suggests that in cases given in the second and third line of Table 13.3 there are non-zero direct stress components of very small values as compared to that of other non-zero stresses. Typical stress contours in such calculations

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Table 13.3 Various stresses obtained from FE analysis.

Fig. 13.2 Contour stresses over the element with the displacement gradient corresponding to the second row of Table 13.2: (a) direct stress s11 ; (b) shear stress s12 .

are shown in Fig. 13.2. As can be seen, whilst the shear stress as shown in Fig. 13.2(b) is perfectly uniform, the contour plot for the direct stress in Fig. 13.2(a) is multi-coloured, suggesting some stress variation. However, the legend of the plot indicates that the stress is effectively zero. The value of this stress calculated from output at Kdofs is also of very small magnitude relative to the others, which suggests that it is simply numerical rounding error. Similar rounding errors can also be found in other examples below, and the same explanation applies to all such cases. Some more cases have been shown in Table 13.4 where the same displacement gradients are prescribed to a relatively small and a relatively

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Table 13.4 Stresses and strains corresponding to relatively small and very large deformations applied.

large value. It can be seen that at relatively small deformation, the differences between different types of strains and different types of stresses are small. In fact, if one kept reducing the magnitude of the deformation, the differences would eventually vanish to reproduce the small deformation idealisation. However, as the deformation increases, the disparity becomes significant, indicating the effects of finite deformation. 13.2.4.2 Applying concentrated forces at Kdofs In all the cases presented in the previous subsection, UC was loaded by applying nodal displacements at Kdofs, which are shown to be nothing else but the components of the displacement gradient tensor. In discussion on the input and output at Kdofs in a linear analysis in Sections 7.5e7.6, alternative method of load application was through prescribing the concentrated forces at Kdofs. The two methods could be used interchangeably depending on the nature of the problem being solved. To assess the applicability of this method of loading in nonlinear problems, a number of cases were devised where concentrated forces were applied at Kdofs in various combinations. Once such cases were attempted, it became clear that it often resulted in convergence difficulties where solver could not complete the case over the single increment. The authors have subsequently realised the cause and therefore present it here for the readers’

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reference. Whilst any combination of displacement gradients defines a physically permissible deformation and therefore strains and stresses can be obtained accordingly, arbitrary combinations of the components of the first Piola-Kirchhoff stress could correspond to an incompatible displacement gradient field and therefore does not represent any physically permissible displacement field. As a result, the solution may not exist in general, in which case, there would be nowhere for the iterations to converge to meaningfully. Nevertheless, if the calculations complete successfully, i.e. when the solution does exist, the results can be processed in exactly the same way as in cases where nodal displacements are applied, and the conclusion regarding the nature of output at Kdofs remains true. It can be therefore concluded that whilst prescribing concentrated forces at Kdofs is a valid method in the linear problem, for nonlinear analysis of UCs, is not recommended. The applicability of applied concentrated force at Kdofs in the linear problem is partially the sheer luck and partially the simplicity in the linear deformation kinematics from which strains could be integrated to give displacement as demonstrated in Chapter 7.

13.2.5 Procedure for post-processing The analysis of a finite deformation problem is expected to be more complicated than its small deformation counterpart. The elaboration presented above suggests that it is more complicated than one might have thought. As the geometry of the deformable body changes with the deformation, without appropriate attention being paid, the user may even not have a grip on the load applied in terms of average stresses or strains. The position is equally true that the output from the FE analysis may be subject to misinterpretation. With a unit cell model, the way of obtaining average stresses and strains has been a subject of confusion even for the small deformation problem. Fortunately, it has been established that they can be rigorously and yet simply obtained by making use of the Kdofs. The average stresses and average strains are directly related to the concentrated forces applied or reactions obtained at these Kdofs and the nodal displacements at these Kdofs either as the deformation in response to the loads applied or the imposed as boundary conditions. There is hardly any post-processing required in order to obtain average stresses and average strains in the unit cell concerned.

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It has now been established that similar relationships are available even in the finite deformation problem although they are based on principles that are more complicated. One will have all 9 displacement gradients activated in prescribing the relative displacement boundary condition for the unit cell and they are the Kdofs of the unit cell. The nodal displacements at these Kdofs give the displacement gradient directly. It needs to be converted into deformation gradient by adding a unit matrix to it. Both Green strain and logarithmic strain can then be obtained following their definitions in Eqs. (13.5) and (13.4), or any other nonlinear strain measure as desired, where a little of post-processing will be required, especially for the logarithmic strain, which should be simple enough with the help of Matlab for instance. Either provides an average strain for the unit cell, based on different formulations of the finite deformation problem. The concentrated forces obtained as reactions at these Kdofs, once divided by the original volume of the unit cell before deformation give the first Piola-Kirchhoff stress directly. This can be transformed into the second Piola-Kirchhoff stress or Cauchy stress after some simple 33 matrix manipulation according to Eqs. (13.11)e(13.12), respectively. Either provides an average stress for the unit cell, again, based on different formulations of the finite deformation problem. In terms of computational demands, the post-processing as described above vastly outperforms stress and strain averaging procedure involving integrating the stress and strain field within the unit cell. The former is meant to be at users’ fingertips whilst the latter cannot be properly performed without substantial programming based on full knowledge of finite elements theory and formulation. In addition to that, this complex approach offers less flexibility in terms of average strain and stress measures one can obtain, because one will have to apply averaging to Abaqus output, which is limited to Cauchy stresses and nominal strain or logarithmic strains only.

13.2.6 Rotations In small deformation problem, the rigid body rotations are defined by the antisymmetric components of the displacement gradient matrix, which has zero values for all diagonal components whilst the off-diagonal components are paired but of opposite sense within each pair. It can be easily proven that the displacement gradients given by such an antisymmetric form only results in vanishing strains and stresses. To illustrate the difference in terms of rigid body rotations between small and finite deformations, such displacement gradients are prescribed to the Kdofs in the finite deformation analyses.

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Table 13.5 Stress and strain measures obtained in loading cases that would result in rigid body rotation in linear analysis.

The results are provided in Table 13.5. It can be seen that at small values of such displacement gradients, the induced stresses are relatively low. Comparing the same magnitude of displacement gradient prescribed previously but unpaired, the stresses and strains are orders of magnitude higher. Neglecting such small stresses and strains returns the problem back to the small deformation regime. As their magnitude increases, the magnitude of stresses escalates disproportionally demonstrating fully the effects of geometric nonlinearity. These relatively trivial analyses confirm that the rigid body rotations in the sense of small deformation are no longer rigid body rotations in finite deformation problems. The antisymmetric components of displacement gradient contribute to strains as well as to the stresses. It was pointed out by Novozhilov (1953) that, whilst these components of displacement gradient were still associated in a way with rigid body rotations, they were so only in some kind of average sense and hence should not be understood literally. As was argued before, the stress components of small magnitude in the second Piola-Kirchhoff and Cauchy stress given in the second line of Table 13.5 are the consequence of rounding errors.

13.3 The uncertainties associated with material definition The first part of this chapter addressed the model formulation and post-processing of the results from the geometrically nonlinear analysis of unit cells. Although it sounded promising that the differences from the small deformation problem remain somewhat manageable, the whole study of

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finite deformation for anisotropic materials will be found to be overshadowed by an ill-defined aspect of the problem as will be revealed below, especially when Abaqus is employed as the solver, given the fact that Abaqus represents the state-of-the-art in the field of FEM nonlinear analysis in general. In the finite deformation problem of elastic isotropic materials, it has been conventionally assumed that the constitutive behaviour of the material remains unchanged during the deformation. This is acceptable provided that the strains are small enough to keep the material behaviour within its elastic limit whilst the finite deformation results mainly from finite rotations. However, the same assumption will no longer hold for anisotropic materials, in particular, fibre reinforced composites, since even if the strains remain small, finite rotations alone will alter the elastic behaviour by changing the fibre orientation. In practical structures, e.g. the wing of a modern passenger aircraft, it is not uncommon to allow a rotation about 20 at the wing tip. One could make an assumption to neglect such effects whilst accepting the consequence of numerical errors. Although this is not an ideal position to be in, it would at least be consistent. In Abaqus, the orientation of the material axes are adjusted by an amount of the “average rigid body rotations”.1 This is illustrated in the example as follows. To gain some insight into the matter, single element model was employed once again but this time it was a 2D plane stress CPS4 element of a square shape as shown in Fig. 13.2. The material was defined as perfectly elastic orthotropic material, which can be understood as a unidirectionally fibre reinforced composite to assist subsequent discussion. The material properties corresponded to those of T300 unidirectional lamina as cited from (Soden et al., 1998), with values of the longitudinal, transverse and in-plane shear moduli being E1 ¼ 138 GPa, E2 ¼ 11 GPa and G12 ¼ 5.5 GPa, respectively, and major Poisson ratio n12 ¼ 0.28. The local coordinate system for the material was so defined that its axis 1 lined up with the fibre direction, which was in the global x-axis. The axes of the local coordinate system were therefore aligned parallel to the sides of the element, as shown in Fig. 13.3(a). In order to have a revealing yet a simple enough deformation, it was constrained and loaded as shown schematically in Fig. 13.2. The shearing load was equally applied in terms of prescribed displacements at the two node on the right hand side (Fig. 13.4).

1

This information was obtained in a private communication with Abaqus support team.

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y

x

Fig. 13.3 Schematic of a shear loading of an FE model in a plane stress state.

Fig. 13.4 Change in direction of principal axes: (a) element with orientation of the axes before deformation; (b) deformed element and orientation after deformation using linear analysis, (c) deformed element and orientation after deformation from nonlinear analysis, and (d) the legends of contour plots for non-zero stresses and strains from the nonlinear analysis.

Deformed shapes of the element with the material orientation visualised are shown in Fig. 13.3(b) and (c), for small deformation (linear) and finite deformation (nonlinear), respectively. As is expected, in the former case, the material orientation is not altered, with the axes of local coordinate system remaining parallel to those of global coordinate system. With the geometric nonlinearity option activated in Abaqus, the axes of local coordinate system have re-oriented as a result of deformation. However, axis 1 does not

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line up with the horizontal sides of the element, which are supposed to be the direction the fibres lie in the case of a unidirectional composite, as one would expect. As can clearly be seen in Fig. 13.3(c), there is an angle between the expected fibre orientation, shown as solid black lines, and axis 1 of coordinate system updated after deformation, whilst the latter one is no longer aligned with x-axis of the global coordinates. As stated earlier, the actual amount of rotation in the axis is determined by the “average rigid body rotation”, which is a quantity of neither direct physical interpretation nor necessarily related to the rotation of fibres. It is not even documented anywhere in the open literature. Yet it is there in the software and it is incorporated in all such analyses without users’ notice or justification. Given the loose nature of the operation involved, the consequence is the uncertainties introduced by this operation. As this is not an aspect users always have the appetite to check carefully, the uncertainties are usually embedded in the results whilst users are kept completely innocent. The most worrying aspect is that there is no established theory in the literature incorporating the rotations of material axes in the formulation of the finite deformation problem, except for some casual treatment as incorporated in (Li et al., 2001) which was meant to facilitate a solution for a specific aspect of such a nonlinear problem. In other words, the nonlinearity purely due to the rotation of material’s principal axes as a result of the finite deformation has never been formally and consistently incorporated into in the elastic stress-strain relationship for anisotropic materials. Given this position, all finite deformation analyses up to date have been based on a completely hypothetical approach of updating the material’s principal directions which has neither mathematical nor physical justification whatsoever, and it has never been publicised in any form.

13.4 Concluding remarks The methodology for conducting FE modelling of UCs at finite deformations has been presented as an extension of UC formulation developed for small deformations. Semi-analytical approach was followed in order to convert stresses and strains between different formulations. Appropriate handling of these stresses and strains is a typical complication of such problem generically. The inputs and outputs of the analysis of a simple FE model were carefully monitored and interpreted to assist the identification of various quantities.

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It has been shown that after minimal processing average stresses and strains can be obtained from input and output at specific nodes defined in a FE UC model, referred to as key degrees of freedom in theoretical formulation of the model. The displacement inputs at key degrees of freedom are the components of the average displacement gradient, whilst the corresponding reaction forces at these nodes, normalised with respect to the volume of UC before deformation, are those of first Piola-Kirchhoff stress tensor. With them being known, alternative nonlinear stress and strain tensors can be reproduced, e.g. second Piola-Kirchhoff stress and Green strain, through appropriate conversion relations. This allows to quantify stressstrain response in UC analysis involving finite deformations. Whilst it has been established that the procedures of quantifying average stresses and strains remain relatively straightforward in UCs with geometric nonlinearity, a serious drawback of finite deformation formulation for anisotropic materials as implemented in Abaqus has been exposed. Based on a simple example, it has been demonstrated that the rotation of principal material axes due to deformation was not defined correctly, since the local coordinate system as output by the solver neither remained in its original orientation, nor was it aligned with the expected directions of the material. The implication of this is that the stresses and strains obtained from Abaqus are no longer given in the actual materials principle axes, but in some hypothetic directions Abaqus assumed on the users’ behalf not even with the users’ awareness. Until this uncertainty in material definition is resolved, applicability of finite deformation modelling to composites will remain questionable.

References Abaqus Analysis User’s Guide, 2016, HTML Documentation, Dassault Systemes, Rhode Island, USA Lm