Geometric continuum mechanics

Geometric Continuum Mechanics
Contents
Part I Kinematics, Forces and Stress Theory
	Manifolds of Mappings for Continuum Mechanics
		Contents
		1 Introduction
		2 A Short Review of Convenient Calculus in Infinite Dimensions
			2.1 The c∞-Topology
			2.2 Convenient Vector Spaces
			2.3 Smooth Mappings
			2.4 Main Properties of Smooth Calculus
			2.6 Recognizing Smooth Curves
			2.7 Frölicher Spaces
		3 Manifolds with Corners
			3.1 Manifolds with Corners
			3.3 Differential Forms on Manifolds with Corners
			3.4 Towards the Long Exact Sequence of the Pair (M,∂M)
			3.6 Riemannian Manifolds with Bounded Geometry
			3.7 Riemannian Manifolds with Bounded Geometry Allowing Corners
		4 Whitney Manifold Germs
			4.1 Compact Whitney Subsets
			4.3 Examples and Counterexamples of Whitney Pairs
			4.5 Our Use of Whitney Pairs
			4.6 Other Approaches
			4.7 Tangent Vectors and Vector Fields on Whitney Manifold Germs
			4.8 Mappings, Bundles, and Sections
			4.9 Is Stokes\' Theorem Valid for Whitney Manifold Germs?
		5 Manifolds of Mappings
			5.3 The Manifold Structure on C∞(M,N) and Ck(M,N)
			5.9 Sprays Respecting Fibers of Submersions
		6 Regular Lie Groups
			6.1 Regular Lie Groups
			6.3 The Diffeomorphism Group of a Whitney Manifold Germ
			6.5 The Connected Component of Diff(M) for a Whitney Manifold Germ M
			6.6 Remark
			6.7 Regular (Right) Half Lie Groups
			6.9 Groups of Smooth Diffeomorphisms on Rn
			6.12 Trouvé Groups
		7 Spaces of Embeddings or Immersions, and Shape Spaces
			7.1 The Principal Bundle of Embeddings
			7.2 The Space of Immersions and the Space of Embeddings of a Compact Whitney Manifold Germ
			7.3 The Orbifold Bundle of Immersions
		8 Weak Riemannian Manifolds
			8.1 Manifolds, Vector Fields, Differential Forms
			8.2 Weak Riemannian Manifolds
			8.3 Weak Riemannian Metrics on Spaces of Immersions
			8.5 Analysis Tools on Regular Lie Groups and on Diff(M) for a Whitney Manifold Germ
			8.6 Right Invariant Weak Riemannian Metrics on Regular Lie Groups and on Diff(M) for a Whitney Manifold Germ
			8.8 Examples of Weak Right Invariant Riemannian Metrics on Diffeomorphism Groups
			8.12 Trouvé Groups for Reproducing Kernel Hilbert Spaces
		9 Robust Weak Riemannian Manifolds and Riemannian Submersions
			9.1 Robust Weak Riemannian Manifolds
			9.3 Covariant Curvature and O\'Neill\'s Formula
			9.5 Semilocal Version of Mario\'s Formula, Force, and Stress
			9.6 Landmark Space as Homogeneous Space of Solitons
			9.7 Shape Spaces of Submanifolds as Homogeneous Spaces for the Diffeomorphism Group
		References
	Notes on Global Stress and Hyper-Stress Theories
		Contents
		1 Introduction
		2 Notation and Preliminaries
			2.1 General Notation
			2.2 Manifolds with Corners
			2.3 Bundles, Jets, and Iterated Jets
				2.3.1 Jets
				2.3.2 Iterated (Non-holonomic) Jets
				2.3.3 Local Representation of Iterated Jets
			2.4 Contraction
		3 Banachable Spaces of Sections of Vector Bundles over Compact Manifolds
			3.1 Precompact Atlases
			3.2 The Cr-Topology on Cr(π)
			3.3 The Jet Extension Mapping
			3.4 The Iterated Jet Extension Mapping
		4 The Construction of Charts for the Manifold of Sections
		5 The Cr-Topology on the Space of Sections of a Fiber Bundle
			5.1 Local Representatives of Sections
			5.2 Neighborhoods for Cr(ξ) and the Cr-Topology
			5.3 Open Neighborhoods for Cr(ξ) Using Vector Bundle Neighborhoods
		6 The Space of Embeddings
			6.1 The Case of a Trivial Fiber Bundle: Manifolds of Mappings
			6.2 The Space of Immersions
			6.3 Open Neighborhoods of Local Embeddings
			6.4 Open Neighborhoods of Embeddings
		7 The General Framework for Global Analytic Stress Theory
		8 Duals to Spaces of Differentiable Sections of a Vector Bundle: Localization of Sections and Functionals
			8.1 Spaces of Differentiable Sections over a Manifold Without Boundary and Linear Functionals
			8.2 Localization of Sections and Linear Functionals for Manifolds Without Boundaries
			8.3 Localization of Sections and Linear Functionals for Manifolds with Corners
			8.4 Supported Sections, Static Indeterminacy and Body Forces
			8.5 Supported Functionals
			8.6 Density Dual and Smooth Functionals
			8.7 Generalized Sections and Distributions
		9 de Rham Currents
			9.1 Basic Operations with Currents
			9.2 Local Representation of Currents
				9.2.1 Representation by 0-Currents
				9.2.2 Representation by n-Currents
		10 Vector Valued Currents
			10.1 Vector Valued Forms
			10.2 Vector Valued Currents
			10.3 Local Representation of Vector Valued Currents
				10.3.1 The Inner Product of a Vector Valued Current and a Vector Field
				10.3.2 The Tensor Product of a Current and a Co-vector Field
				10.3.3 Representation by 0-Currents
				10.3.4 The Exterior Product of a Vector Valued Current and a Multi-Vector Field
				10.3.5 The Contraction of a Vector Valued Current and a Form
				10.3.6 Representation by n-Currents
		11 The Representation of Forces by Hyper-Stresses and Non-holonomic Stresses
			11.1 Stresses and Non-holonomic Stresses
			11.2 Smooth Stresses
			11.3 Stress Measures
			11.4 Force System Induced by Stresses
		12 Simple Forces and Stresses
			12.1 Simple Stresses
			12.2 The Vertical Projection
			12.3 Traction Stresses
			12.4 Smooth Traction Stresses
			12.5 The Generalized Divergence of the Stress
			12.6 The Divergence for the Smooth Case
			12.7 The Invariance of the Divergence
			12.8 The Balance Equation
			12.9 Application to Non-holonomic Stresses
		13 Concluding Remarks
		References
	Applications of Algebraic Topology in Elasticity
		Contents
		1 Introduction
		2 Differential Geometry
			2.1 Exterior Calculus
		3 Algebraic Topology
			3.1 Homology and Cohomology Groups
				3.1.1 Group Theory
				3.1.2 Combinatorial Group Theory
				3.1.3 Chain Complexes and Homology Groups
				3.1.4 Cohomology Groups
				3.1.5 Relative Homology Groups
				3.1.6 Duality Theorems in Algebraic Topology
			3.2 Homotopy and the Fundamental Group
			3.3 Classification of Compact 2-Manifolds with Boundary
			3.4 Curves on Oriented Surfaces
			3.5 Theory of Knots
			3.6 Topology of 3-Manifolds
		4 Kinematics of Nonlinear Elasticity
		5 Compatibility Equations in Nonlinear Elasticity
			5.1 Compatibility Equations for the Deformation Gradient F
			5.2 Examples of Non-simply-connected Bodies and Their F-Compatibility Equations
				5.2.1 2D Elasticity on a Torus and a Punctured Torus
				5.2.2 2D Elasticity on Arbitrary Compact Orientable 2-Manifolds
				5.2.3 3D Elastic Bodies with Holes
			5.3 F-Compatibility Equations in the Presence of Dirichlet Boundary Conditions
			5.4 Compatibility Equations for the Right Cauchy–Green Strain C
			5.5 Compatibility Equations in Linearized Elasticity
		6 Differential Complexes in Nonlinear Elasticity
		References
	De Donder Construction for Higher Jets
		Contents
		1 Introduction
		2 Spaces of Smooth Sections
		3 Boundary Forms
		4 Application to Variational Problems
			4.1 Critical Points of Action Functionals
			4.2 Euler–Lagrange Equations
			4.3 De Donder Equations
			4.4 Symmetries and Conservation Laws
		5 Example
			5.1 Cauchy Problem
			5.2 De Donder Forms
			5.3 Symmetries
		Appendix
			Jets
			Prolongations
		References
Part II Defects, Uniformity and Homogeneity
	Regular and Singular Dislocations
		Contents
		1 Introduction
		2 Regular Lattices
			2.1 Frame Fields
			2.2 Material Parallelism
			2.3 The Dual View
			2.4 Integral Perspective
		3 Singular Lattices
			3.1 De Rham Currents
			3.2 Singular Layerings
			3.3 A Screw Dislocation
				3.3.1 Two-Dimensional Prelude
				3.3.2 The Screw
			3.4 A Conservation Law
			3.5 Branching
			3.6 Interfaces
			3.7 A Volterra Disclination
				3.7.1 Wedge Disclination as Interface
				3.7.2 Wedge Disclination as Superposition of Edge Dislocations
			3.8 Disengagements or Distriations
				3.8.1 Affine Subspaces, and Decomposable Multivectors and Multicovectors
				3.8.2 The Smooth Case
				3.8.3 The Singular Case
		4 From Discrete to Continuous Dislocations
		5 The Movement of Dislocations
			5.1 Introduction
			5.2 Frame Bundle Automorphisms
			5.3 Material Convection and Material Evolution
			5.4 Evolution Laws
		References
	Homogenization of Edge-Dislocations as a Weak Limit of de-Rham Currents
		Contents
		1 Introduction
		2 De-Rham Currents
		3 Layering Form for an Edge-Dislocation
		4 Homogenization of Distributed Edge-Dislocations
		5 Singular Torsion and Its Homogenization
		6 Homogenization for General Surfaces
		Appendix: Gluing Constructions
		References
	A Kinematics of Defects in Solid Crystals
		Contents
		1 Introduction
		2 Crystal States and Scalar Invariants
			2.1 Crystal States
			2.2 Scalar Invariants
			2.3 Classifying Manifold
		3 Burgers Vectors, Invariant Integrals, and Neutrally Related States
			3.1 Burgers Vectors and Invariant Integrals
			3.2 Neutral Related States
		4 Lie Groups
			4.1 Constant Dislocation Density Tensor
			4.2 Isomorphic Lie Groups and Algebras
			4.3 Campbell–Baker–Hausdorff Formula, Canonical Group J
			4.4 Higher Dimensional Lie Groups
			4.5 Homogeneous Spaces
		5 Discrete Groups
			5.1 Construction of Discrete Group from Given Crystal State
			5.2 Analogue of Crystallographic Restriction
			5.3 Mal\'cev\'s Results
			5.4 Canonical Basis of Discrete Groups
			5.5 Lattice Structure of Discrete Nilpotent Groups
			5.6 Symmetries of Discrete Nilpotent Groups D
		6 Discrete Structures
			6.1 Structures Obtained by Discrete Flow Along the Two Lattice Vector Fields
		7 Geometrical Setting
			7.1 Canonical Lattice Connection
			7.2 Examples
		8 Conclusion
		References
	Limits of Distributed Dislocations in Geometric and Constitutive Paradigms
		Contents
		1 Introduction
			1.1 Geometric and Constitutive Paradigms
			1.2 Description of the Main Results
			1.3 Structure of This Paper
		2 The Constitutive Paradigm of Noll and Wang
			2.1 Relation Between Geometric and Constitutive Paradigms
		3 Homogenization of Dislocations: Geometric Paradigm
			3.1 Sketch of Proof of Theorem 2
		4 Homogenization of Dislocations: Constitutive Paradigm
			4.1 Proof of Theorem 3
		5 The Role of Torsion in the Equilibrium Equations
		References
	On the Homogeneity of Non-uniform Material Bodies
		Contents
		1 Introduction
		2 Groupoids
		3 Lie Algebroids
			3.1 Introduction
			3.2 Definition
			3.3 The Lie Algebroid of a Lie Groupoid
				3.3.1 The β-Bundle
				3.3.2 Left-Invariant Vector Fields on a Lie Groupoid
				3.3.3 The Associated Lie Algebroid
		4 Characteristic Distribution
		5 Material Groupoid and Material Distribution
		6 Homogeneity
		7 Example
		References