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

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
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