This monograph presents the geometric foundations of continuum mechanics. An emphasis is placed on increasing the generality and elegance of the theory by scrutinizing the relationship between the physical aspects and the mathematical notions used in its formulation. The theory of uniform fluxes in affine spaces is covered first, followed by the smooth theory on differentiable manifolds, and ends with the non-smooth global theory. Because continuum mechanics provides the theoretical foundations for disciplines like fluid dynamics and stress analysis, the author’s extension of the theory will enable researchers to better describe the mechanics of modern materials and biological tissues. The global approach to continuum mechanics also enables the formulation and solutions of practical optimization problems.
Foundations of Geometric Continuum Mechanics will be an invaluable resource for researchers in the area, particularly mathematicians, physicists, and engineers interested in the foundational notions of continuum mechanics.
Author(s): Reuven Segev
Series: Advances in Continuum Mechanics 49
Edition: 1
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 411
City: Cham
Tags: Continuum mechanics, Classical field theories, Differential geometry, Fluxes, Banach spaces, Manifolds, Bundles, Configuration space
Preface
Contents
1 Introduction
2 Prelude: Finite-Dimensional Systems
2.1 The Framework for the Problem of Statics
2.2 On the Solutions of the Problem of Statics
2.2.1 Existence of Solutions
2.2.2 Static Indeterminacy and Optimal Solutions
2.2.3 Worst-Case Loading and Load Capacity
Part I Algebraic Theory: Uniform Fluxes
3 Simplices in Affine Spaces and Their Boundaries
3.1 Affine Spaces: Notation
3.2 Simplices
3.3 Cubes, Prisms, and Simplices
3.4 Orientation
3.5 Simplices on the Boundaries and Their Orientations
3.6 Subdivisions
4 Uniform Fluxes in Affine Spaces
4.1 Basic Assumptions
4.2 Balance and Linearity
4.3 Immediate Implications of Skew Symmetry and Multi-Linearity
4.4 The Algebraic Cauchy Theorem
5 From Uniform Fluxes to Exterior Algebra
5.1 Polyhedral Chains and Cochains
5.2 Component Representation of Fluxes
5.3 Multivectors
5.4 Component Representation of Multivectors
5.5 Alternation
5.6 Exterior Products
5.7 The Spaces of Multivectors and Multi-Covectors
5.8 Contraction
5.9 Pullback of Alternating Tensors
5.10 Abstract Algebraic Cauchy Formula
Part II Smooth Theory
6 Smooth Analysis on Manifolds: A Short Review
6.1 Manifolds and Bundles
6.1.1 Manifolds
6.1.2 Tangent Vectors and the Tangent Bundle
6.1.3 Fiber Bundles
6.1.4 Vector Bundles
6.1.5 Tangent Mappings
6.1.6 The Tangent Bundle of a Fiber Bundle and Its Vertical Subbundle
6.1.7 Jet Bundles
6.1.8 The First Jet of a Vector Bundle
6.1.9 The Pullback of a Fiber Bundle
6.1.10 Dual Vector Bundles and the Cotangent Bundle
6.2 Tensor Bundles and Differential Forms
6.2.1 Tensor Bundles and Their Sections
6.2.2 Differential Forms
6.2.3 Contraction and Related Mappings
6.2.4 Vector-Valued Forms
6.2.5 Density-Dual Spaces
6.3 Differentiation and Integration
6.3.1 Integral Curves and the Flow of a Vector Field
6.3.2 Exterior Derivatives
6.3.3 Partitions of Unity
6.3.4 Orientation on Manifolds
6.3.5 Integration on Oriented Manifolds
6.3.6 Stokes's Theorem
6.3.7 Integration Over Chains on Manifolds
6.4 Manifolds with Corners
7 Interlude: Smooth Distributions of Defects
7.1 Introduction
7.2 Forms and Hypersurfaces
7.3 Layering Forms, Defect Forms
7.4 Smooth Distributions of Dislocations
7.5 Inclinations and Disclinations, the Smooth Case
7.6 Frank's Rules for Smooth Distributions of Defects
8 Smooth Fluxes
8.1 Balance Principles and Fluxes
8.1.1 Densities of Extensive Properties
8.1.2 Flux Forms and Cauchy's Formula
8.1.3 Extensive Properties and Cauchy Formula—Local Representation
8.1.4 The Cauchy Flux Theorem
8.1.4.1 Assumptions
8.1.4.2 Notation
8.1.4.3 Construction
8.1.5 The Differential Balance Law
8.2 Properties of Fluxes
8.2.1 Flux Densities and Orientation
8.2.2 Kinetic Fluxes and Kinematic Fluxes
8.2.3 The Flux Bundle
8.2.4 Flow Potentials and Stream Functions
9 Frames, Body Points, and Spacetime Structure
9.1 Frames, Balance, and Fluxes in Spacetime
9.2 Worldlines
9.3 Material Points, the Material Universe, and Material Frames
10 Stresses
10.1 Force Fields on Manifolds
10.2 Traction Stresses and Cauchy's Formula on Manifolds
10.3 The Power in Terms of the Traction Stresses
10.4 Forces and Stresses for Kinematic Fluxes
10.5 Force Fields and Traction Stresses for Kinetic Fluxes
10.6 Cauchy's Theorem for Traction Stresses
10.7 Variational Stresses
10.8 The Divergence of a Variational Stress Field
10.9 The Case Where a Connection Is Given
10.9.1 The Tangent Bundle of a Vector Bundle
10.9.2 Linear Connections on a Vector Bundle and Covariant Derivatives
10.9.3 The Covariant Divergence of the Stress
11 Smooth Electromagnetism on Manifolds
11.1 Electromagnetism in a Lorentz Frame
11.1.1 The Metric-Independent Maxwell Equations in Four-Dimensional Spacetime
11.1.2 The Lorentz Force
11.1.3 Metric-Invariant Maxwell Stress Tensor
11.2 Metric-Independent p-Form Electrodynamics
11.2.1 The Maxwell Equations for p-Form Electrodynamics
11.2.2 The Lorentz Force and Maxwell Stress for p-Form Electrodynamics
12 The Elasticity Problem
12.1 Kinematics
12.2 Statics
12.2.1 Preliminaries on Vector Bundle Morphisms and Pullbacks
12.2.2 Force Fields
12.2.3 Traction Stresses
12.2.4 Variational Stresses and Constitutive Relations
12.2.5 The Problem of Elasticity
12.3 Eulerian Fields
Appendix: Notation Used in This Chapter
13 Symmetry and Dynamics
13.1 Symmetry Group Action: Totals and Invariance
13.2 Dynamics
13.2.1 Dynamics of a Particle in a Proto-Galilean Spacetime
13.2.2 Dynamics of a Body in a Proto-Galilean Spacetime
Part III Non-smooth, Global Theories
14 Banachable Spaces of Sections of Vector Bundles over Compact Manifolds
14.1 The Cr-Topology on Cr(π)
14.2 Iterated (Nonholonomic) Jets and Iterated Jet Extensions
14.2.1 Iterated Jets
14.2.2 Local Representation of Iterated Jets
14.2.3 The Iterated Jet Extension Mapping for a Vector Bundle
15 Manifolds of Sections and Embeddings
15.1 The Construction of Charts for the Manifold of Sections
15.2 The Cr-Topology on the Space of Sections of a Fiber Bundle
15.2.1 Finite Local Representation of a Section
15.2.2 Neighborhoods for Cr(ξ) and the Cr-Topology
15.2.3 Open Neighborhoods for Cr(ξ) Using Vector Bundle Neighborhoods
15.3 The Manifold of Embeddings
15.3.1 The Case of a Trivial Fiber Bundle: Manifolds of Mappings
15.3.2 The Space of Immersions
15.3.3 Open Neighborhoods of Local Embeddings
15.3.4 Open Neighborhoods of Embeddings
16 The General Framework for Global Analytic Stress Theory
17 Dual Spaces Corresponding to Spaces of Differentiable Sections of a Vector Bundle: Localization of Sections and Functionals
17.1 Spaces of Differentiable Sections Over a Manifold Without Boundary and Linear Functionals
17.2 Localization for Manifolds Without Boundaries
17.3 Localization for Compact Manifolds with Corners
17.4 Supported Sections, Static Indeterminacy, and Body Forces
17.5 Supported Functionals
17.6 Generalized Sections and Distributions
18 de Rham Currents
18.1 Basic Operations with Currents
18.2 Local Representation of Currents
18.2.1 Representation by 0-Currents
18.2.2 Representation by n-Currents
18.2.3 Representation of the Boundary of a Current
19 Interlude: Singular Distributions of Defects in Bodies
19.1 Layering Currents and Defect Currents
19.2 Dislocations
19.2.1 Edge Dislocations
19.2.2 Screw Dislocations
19.2.3 Interfaces
19.3 Singular Distributions of Inclinations and Disclinations
20 Vector-Valued Currents
20.1 Vector-Valued Currents
20.2 Various Operations for Vector-Valued Currents and Local Representation
20.2.1 The Inner Product of a Vector-Valued Current and a Vector Field
20.2.2 The Tensor Product of a Current and a Co-vector Field
20.2.3 Representation by 0-Currents
20.2.4 Representation by n-Currents
20.2.5 The Exterior Product of a Vector-Valued Current and a Multi-Vector Field
20.2.6 The Contraction of a Vector-Valued Current and a Form
20.2.7 Alternative Representation by n-Currents
21 The Representation of Forces by Stresses and Hyperstresses
21.1 Representation of Forces by Hyper-Stresses
21.2 The Representation of Forces by Nonholonomic Stresses
21.3 Smooth Stresses
21.4 Stress Measures
21.5 Force System Induced by Stresses
21.6 On the Kinematic Mapping
21.7 Global Elastic Constitutive Equations and the Problem of Elasticity
22 Simple Forces and Stresses
22.1 Simple Variational Stresses
22.2 The Vertical Projection
22.3 Traction Stresses
22.4 Smooth Traction Stresses
22.5 Further Aspects of Stress Representation
22.6 Example: Non-smooth p-Form Electrodynamics on Manifolds
22.7 Flat Forces
23 Whitney's Geometric Integration Theory and Non-smooth Bodies
23.1 The Flat Norm: Motivation
23.2 Flat Chains
23.3 Federer's Definition of Flat Chains
23.4 Sharp Chains
23.5 Cochains
23.5.1 Flat Cochains
23.5.2 Sharp Cochains
23.5.3 The Cauchy Mapping
23.5.4 The Representation of Sharp Cochains by Forms
23.5.5 The Representation of Flat Cochains by Forms
23.6 Coboundaries and the Differential Balance Equations
24 Optimal Fields and Load Capacity of Bodies
24.1 Introduction
24.2 Balance Equations
24.2.1 Sobolev Spaces of Sections of a Trivial Vector Bundle
24.2.2 Wkp-Forces, (p,k)-Stresses, and Stress Field Optimization
24.2.3 Traces and Loading Distributions
24.2.4 Loading Distributions and Stresses
24.2.5 The Junction Problem for Fluxes
24.3 Preliminaries on Rigid Velocity Fields
24.3.1 The Subspace of Rigid Velocities
24.3.2 Approximation by Rigid Velocities
24.4 Other Differential Operators
24.4.1 General Structure
24.4.2 Example: The Space LD(Ω)
24.5 Quotient Spaces
24.5.1 Distortions
24.5.2 Example: The Space of LD-Distortions
24.5.3 Total Forces and Equilibrated Forces
24.5.4 Stresses for Unsupported Bodies Under Equilibrated Loadings
24.6 Subspaces
24.6.1 Supported Bodies and the Space LD(Ω)c
24.6.2 Stress Analysis for Supported Bodies
24.7 Product Structures
24.7.1 Product Structures on Subbundles of JkW
24.7.2 Stress Analysis for Elastic Plastic Bodies
References
Index
