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

Description

Representative Volume Elements and Unit Cells: Concepts, Theory, Applications and Implementation clearly explains how representative volume elements and unit cells should to be accurately constructed in terms of finite element analysis, the imposition of boundary conditions, prescription of loads, meshing and post-processing. The theoretical background for using FEA in materials characterization is given in full, hence readers will gain a clear and detailed understanding of the underlying principles that they can use in the development of their own models. Full derivations for some commonly used unit cells are presented so that users can directly apply them in their own developments. Finally, a piece of code designed as an add-on to Abaqus software, UnitCells©, is provided on a companion site for the material characterization of some of the common types of composites. in addition, a user manual and relevant templates are included. 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 V W Y Back Cover