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

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

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

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

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

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

Preface xi
Part One Basics
1 Introduction --- background, objectives and basic concepts
3(8)
1.1 The concept of length scales and typical length scales in physics and engineering
3(1)
1.2 Multiscale modelling
4(1)
1.3 Representative volume element and unit cell
5(1)
1.4 Background of this monograph
6(1)
1.5 Objectives of this monograph
6(2)
1.6 The structure of this monograph
8(2)
References
10(1)
2 Symmetry, symmetry transformations and symmetry conditions
11(32)
2.1 Introduction
11(1)
2.2 Geometric transformations and the concept of symmetry
12(4)
2.3 Symmetry of physical fields
16(8)
2.4 Continuity and free body diagrams
24(4)
2.5 Symmetry conditions
28(13)
2.6 Concluding remarks
41(1)
References
42(1)
3 Material categorisation and material characterisation
43(24)
3.1 Background
43(2)
3.2 Material categorisation
45(15)
3.3 Material characterisation
60(4)
3.4 Concluding remarks
64(1)
References
64(3)
4 Representative volume elements and unit cells
67(12)
4.1 Introduction
67(1)
4.2 RVEs
68(3)
4.3 UCs
71(5)
4.4 Concluding remarks
76(1)
References
77(2)
5 Common erroneous treatments and their conceptual sources of errors
79(28)
5.1 Realistic or hypothetic background
79(3)
5.2 The construction of RVEs and their boundary
82(2)
5.3 The construction of UCs
84(12)
5.4 Post-processing
96(2)
5.5 Implementation issues
98(3)
5.6 Verification and the lack of `sanity checks'
101(1)
5.7 Concluding remarks
102(1)
References
103(4)
Part Two Consistent formulation of unit cells and representative volume elements
6 Formulation of unit cells
107(82)
6.1 Introduction
107(1)
6.2 Relative displacement field and rigid body rotations
108(6)
6.3 Relative displacement boundary conditions for unit cells
114(1)
6.4 Typical unit cells and their boundary conditions in terms of relative displacements
115(61)
6.5 Requirements on meshing
176(1)
6.6 Key degrees of freedom and average strains
177(2)
6.7 Average stresses and effective material properties
179(3)
6.8 Thermal expansion coefficients
182(1)
6.9 "Sanity checks" as basic verifications
183(2)
6.10 Concluding remarks
185(2)
References
187(2)
7 Periodic traction boundary conditions and the key degrees of freedom for unit cells
189(34)
7.1 Introduction
189(4)
7.2 Boundaries and boundary conditions for unit cells resulting from translational symmetries
193(4)
7.3 Total potential energy and variational principle for unit cells under prescribed average strains
197(1)
7.4 Periodic traction boundary conditions as the natural boundary conditions for unit cells
198(4)
7.5 The nature of the reactions at the prescribed key degrees of freedom
202(7)
7.6 Prescribed concentrated `forces' at the key degrees of freedom
209(2)
7.7 Examples
211(8)
7.8 Conclusions
219(1)
References
220(3)
8 Further symmetries within a UC
223(96)
8.1 Introduction
223(2)
8.2 Further reflectional symmetries to existing translational symmetries
225(26)
8.3 Further rotational symmetries to existing translational symmetries
251(40)
8.4 Examples of mixed reflectional and rotational symmetries
291(12)
8.5 Centrally reflectional symmetry
303(11)
8.6 Guidance to the sequence of exploiting existing symmetries
314(1)
8.7 Concluding statement
315(2)
References
317(2)
9 RVE for media with randomly distributed inclusions
319(28)
9.1 Introduction
319(1)
9.2 Displacement boundary conditions and traction boundary conditions for an RVE
320(3)
9.3 Decay length for boundary effects
323(4)
9.4 Generation of random patterns
327(3)
9.5 Strain and stress fields in the RVE and the sub-domain
330(6)
9.6 Post-processing for average stresses, strains and effective properties
336(9)
9.7 Conclusions
345(1)
References
345(2)
10 The diffusion problem
347(14)
10.1 Introduction
347(1)
10.2 Governing equation
347(4)
10.3 Relative concentration field
351(2)
10.4 An example of a cuboidal unit cell
353(2)
10.5 RVEs
355(1)
10.6 Post-processing for average concentration gradients and diffusion fluxes
356(4)
10.7 Conclusions
360(1)
References
360(1)
11 Boundaries of applicability of representative volume elements and unit cells
361(10)
11.1 Introduction
361(1)
11.2 Predictions of elastic properties and strengths
361(2)
11.3 Representative volume elements
363(3)
11.4 Unit cells
366(1)
11.5 Conclusions
366(1)
References
367(4)
Part Three Further developments
12 Applications to textile composites
371(46)
12.1 Introduction
371(10)
12.2 Use of symmetries when defining an effective UC
381(2)
12.3 Unit cells for two-dimensional textile composites
383(15)
12.4 Unit cells for three-dimensional textile composites
398(16)
12.5 Conclusions
414(1)
References
415(2)
13 Application of unit cells to problems of finite deformation
417(22)
13.1 Introduction
417(2)
13.2 Unit cell modelling at finite deformations
419(14)
13.3 The uncertainties associated with material definition
433(3)
13.4 Concluding remarks
436(1)
References
437(2)
14 Automated implementation: UnitCells© composites characterisation code
439(20)
14.1 Introduction
439(2)
14.2 Abaqus/CAE modelling practicality
441(9)
14.3 Verification and validation
450(6)
14.4 Concluding remarks
456(1)
References
456(3)
Index 459
Shuguang Li is a Professor of Aerospace Composites at the Institute for Aerospace Technology, Faculty of Engineering, University of Nottingham, UK. He obtained his PhD from the University of Manchester in 1993 and returned to his academic track as a lecturer at the University of Manchester in 1995 and was appointed to his present position in 2012. Professor Li has published well over 100 academic papers, most of them in highly reputable international journals. His main research interest is in the area of analysis of composite materials and structures, in particular, on modelling damage and failure, micromechanics and material characterisation. As an outcome of his research on micromechanical modelling of composites, a piece of software UnitCells© has emerged which offers material scientists and structural designers a useful tool for characterisation of composites in terms of effective elastic properties as well as strength properties. Elena Sitnikova is currently a Research Fellow at the Faculty of Engineering, the University of Nottingham. She received her MSc degree in Applied Mathematics, Mechanics from Saint Petersburg State University in 2004 and PhD in Engineering from the University of Aberdeen in 2010. Prior to joining the University of Nottingham in 2013, she took a position of Research Associate at the University of Liverpool. Dr Sitnikova authored a number of research papers in several fields of engineering. Her current interests are in the field of textile composites, with particular emphasis on problems of damage, material characterisation and high strain rate response of these materials.
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