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Spatial and Material Forces in Nonlinear Continuum Mechanics: A Dissipation-Consistent Approach (Solid Mechanics and Its Applications, 272)

✍ Scribed by Paul Steinmann


Publisher
Springer
Year
2022
Tongue
English
Leaves
418
Category
Library

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✦ Synopsis



This monograph details spatial and material vistas on non-linear continuum mechanics in a dissipation-consistent approach. Thereby, the spatial vista renders the common approach to nonlinear continuum mechanics and corresponding spatial forces, whereas the material vista elaborates on configurational mechanics and corresponding material or rather configurational forces. Fundamental to configurational mechanics is the concept of force. In analytical mechanics, force is a derived object that is power conjugate to changes of generalised coordinates. For a continuum body, these are typically the spatial positions of its continuum points. However, if in agreement with the second law, continuum points, e.g. on the boundary, may also change their material positions. Configurational forces are then power conjugate to these configurational changes. A paradigm is a crack tip, i.e. a singular part of the boundary changing its position during crack propagation, with the relatedconfigurational force, typically the J-integral, driving its evolution, thereby consuming power, typically expressed as the energy release rate. Taken together, configurational mechanics is an unconventional branch of continuum physics rationalising and unifying the tendency of a continuum body to change its material configuration. It is thus the ideal formulation to tackle sophisticated problems in continuum defect mechanics. Configurational mechanics is entirely free of restrictions regarding geometrical and constitutive nonlinearities and offers an accompanying versatile computational approach to continuum defect mechanics. In this monograph, I present a detailed summary account of my approach towards configurational mechanics, thereby fostering my view that configurational forces are indeed dissipation-consistent to configurational changes.

✦ Table of Contents


Preface
Contents
Notation and Terminology
1 Introduction
1.1 The Need for Continuum Mechanics
1.2 Spatial Versus Material Forces
1.3 The Need for Material Forces
1.4 Vistas on Material Forces
1.5 Computational Material Forces
1.6 The Nature of This Treatise
References
2 Kinematics in Bulk Volumes
2.1 Configurations
2.1.1 Reference Configuration
2.1.2 Material Configuration
2.1.3 Spatial Configuration
2.2 Deformation in Bulk Volumes
2.2.1 Nonlinear Deformation Maps
2.2.2 Tangent Maps: Deformation Gradients
2.2.3 Compatibility Conditions for Tangent Maps
2.2.4 Cotangent Maps: Cofactors
2.2.5 Compatibility Conditions for Cotangent Maps
2.2.6 Measure Maps: Determinants
2.2.7 Strain Measures
2.2.8 Velocities
2.2.9 Gradients of Velocities
References
3 Kinematics on Dimensionally Reduced Smooth Manifolds
3.1 Configurations
3.1.1 Material Configuration
3.1.2 Spatial Configuration
3.2 Deformation on Boundary Surfaces
3.2.1 Nonlinear Deformation Maps
3.2.2 Tangent Maps: Deformation Gradients
3.2.3 Compatibility Conditions for Tangent Maps
3.2.4 Cotangent Maps: Cofactors
3.2.5 Compatibility Conditions for Cotangent Maps
3.2.6 Measure Maps: Determinants
3.2.7 Velocities
3.3 Deformation on Boundary Curves
3.3.1 Nonlinear Deformation Maps
3.3.2 Tangent Maps: Deformation Gradients
3.3.3 Compatibility Conditions for Tangent Maps
3.3.4 Cotangent Maps: Cofactors
3.3.5 Compatibility Conditions for Cotangent Maps
3.3.6 Measure Maps: Determinants
3.3.7 Velocities
Reference
4 Kinematics at Singular Sets
4.1 Configurations
4.1.1 Material Configuration
4.1.2 Spatial Configuration
4.2 Coherent Singular Surfaces
4.2.1 Jump in Nonlinear Deformation Maps
4.2.2 Jump in Tangent Maps
4.2.3 Jump in Cotangent Maps
4.2.4 Jump in Measure Maps
4.2.5 Jump in Velocities
4.2.6 Summary of Coherence Conditions
4.3 Coherent Singular Curves

4.3.1 Jump in Nonlinear Deformation Maps
4.3.2 Jump in Tangent Maps
4.3.3 Jump in Cotangent Maps
4.3.4 Jump in Measure Maps
4.3.5 Jump in Velocities
4.3.6 Summary of Coherence Conditions
4.4 Coherent Singular Points
4.4.1 Jump in Nonlinear Deformation Maps
4.4.2 Jump in Tangent Maps
4.4.3 Jump in Cotangent Maps
4.4.4 Jump in Measure Maps
4.4.5 Jump in Velocities
4.4.6 Summary of Coherence Conditions
References
5 Generic Balances
5.1 Generic Volume Extensive Quantity
5.1.1 Preliminaries
5.1.2 Global Format (Material Control Volume)
5.1.3 Local Format (Material Control Volume)
5.1.4 Global Format (Spatial Control Volume)
5.1.5 Local Format (Spatial Control Volume)
5.1.6 Balance Tetragon
5.2 Generic Surface Extensive Quantity

5.2.1 Preliminaries
5.2.2 Global Format (Material Control Surface)
5.2.3 Local Format (Material Control Surface)
5.2.4 Global Format (Spatial Control Surface)
5.2.5 Local Format (Spatial Control Surface)
5.2.6 Balance Tetragon
5.3 Generic Curve Extensive Quantity
5.3.1 Preliminaries
5.3.2 Global Format (Material Control Curve)
5.3.3 Local Format (Material Control Curve)
5.3.4 Global Format (Spatial Control Curve)
5.3.5 Local Format (Spatial Control Curve)
5.3.6 Balance Tetragon
Reference
6 Kinematical Balances'* 6.1 Spatial Tangent Map 6.1.1 Global Format (Material Control Volume) 6.1.2 Local Format (Material Control Volume) 6.1.3 Global Format (Spatial Control Volume) 6.1.4 Local Format (Spatial Control Volume) 6.2 Spatial Cotangent Map 6.2.1 Global Formulation (Material Control Volume) 6.2.2 Local Format (Material Control Volume) 6.2.3 Global Format (Spatial Control Volume) 6.2.4 Local Format (Spatial Control Volume) 6.3 Spatial Measure Map 6.3.1 Global Format (Material Control Volume) 6.3.2 Local Format (Material Control Volume) 6.3.3 Global Format (Spatial Control Volume) 6.3.4 Local Format (Spatial Control Volume) 6.4 Summary:Balances' of Spatial Maps
6.5 Material Tangent Map
6.5.1 Global Format (Spatial Control Volume)
6.5.2 Local Format (Spatial Control Volume)
6.5.3 Global Format (Material Control Volume)
6.5.4 Local Format (Material Control Volume)
6.6 Material Cotangent Map
6.6.1 Global Format (Spatial Control Volume)
6.6.2 Local Format (Spatial Control Volume)
6.6.3 Global Format (Material Control Volume)
6.6.4 Local Format (Material Control Volume)
6.7 Material Measure Map
6.7.1 Global Format (Spatial Control Volume)
6.7.2 Local Format (Spatial Control Volume)
6.7.3 Global Format (Material Control Volume)
6.7.4 Local Format (Material Control Volume)
6.8 Summary: `Balances' of Material Maps
References
7 Mechanical Balances
7.1 Mass
7.1.1 Global Format (Material Control Volume)
7.1.2 Local Format (Material Control Volume)
7.1.3 Global Format (Spatial Control Volume)
7.1.4 Local Format (Spatial Control Volume)
7.1.5 Balance Tetragon
7.2 Spatial Momentum
7.2.1 Global Format (Material Control Volume)
7.2.2 Local Format (Material Control Volume)
7.2.3 Global Format (Spatial Control Volume)
7.2.4 Local Format (Spatial Control Volume)
7.2.5 Spatial Stress Measures
7.2.6 Balance Tetragon
7.3 Vector Moment of Spatial Momentum
7.3.1 Global Format (Material Control Volume)
7.3.2 Local Format (Material Control Volume)
7.3.3 Global Format (Spatial Control Volume)
7.3.4 Local Format (Spatial Control Volume)
7.3.5 Balance Tetragon
7.4 Scalar Moment of Spatial Momentum

7.4.1 Global Format (Material Control Volume)
7.4.2 Local Format (Material Control Volume)
7.4.3 Global Format (Spatial Control Volume)
7.4.4 Local Format (Spatial Control Volume)
7.4.5 Balance Tetragon
References
8 Consequences of Mechanical Balances
8.1 Kinetic Energy
8.1.1 Local Format (Material Control Volume)
8.1.2 Global Format (Material Control Volume)
8.1.3 Local Format (Spatial Control Volume)
8.1.4 Global Format (Spatial Control Volume)
8.1.5 Conjugated Spatial Stress and Strain Measures
8.1.6 Balance Tetragon
8.2 Material Momentum
8.2.1 Local Format (Material Control Volume)
8.2.2 Global Format (Material Control Volume)
8.2.3 Local Format (Spatial Control Volume)
8.2.4 Global Format (Spatial Control Volume)
8.2.5 Material Stress Measures
8.2.6 Balance Tetragon
8.3 Vector Moment of Material Momentum
8.3.1 Local Format (Material Control Volume)
8.3.2 Global Format (Material Control Volume)
8.3.3 Local Format (Spatial Control Volume)
8.3.4 Global Format (Spatial Control Volume)
8.3.5 Balance Tetragon
8.4 Scalar Moment of Material Momentum

8.4.1 Local Format (Material Control Volume)
8.4.2 Global Format (Material Control Volume)
8.4.3 Local Format (Spatial Control Volume)
8.4.4 Global Format (Spatial Control Volume)
8.4.5 Balance Tetragon
References
9 Virtual Work
9.1 Referential Perspective
9.1.1 Kinematics in Bulk Volumes
9.1.2 Kinematics on Boundary Surfaces
9.1.3 Kinematics on Boundary Curves

9.1.4 Preliminaries
9.2 Virtual Displacements
9.2.1 Spatial Virtual Displacements
9.2.2 Material Virtual Displacements
9.2.3 Total Variations of Kinematic Quantities
9.3 Spatial Virtual Work Principle
9.3.1 Integrands in Material Configuration
9.3.2 Integrands in Spatial Configuration
9.3.3 Integrands in Reference Configuration
9.3.4 Balance Tetragons
9.4 Material Virtual Work Principle
9.4.1 Integrands in Material Configuration
9.4.2 Integrands in Spatial Configuration
9.4.3 Integrands in Reference Configuration
9.4.4 Balance Tetragons
10 Variational Setting
10.1 Extended Hamilton Principle
10.1.1 Variational Statement
10.1.2 Euler–Lagrange Equations: Summary
10.1.3 Euler–Lagrange Equations: Derivation
10.1.4 Euler–Lagrange Equations: Analysis
10.1.5 Euler–Lagrange Equations: PB/PF Operations
10.2 Extended Dirichlet Principle
10.2.1 Variational Statement
10.2.2 Euler–Lagrange Equations: Summary
10.2.3 Euler–Lagrange Equations: Derivation
10.2.4 Euler–Lagrange Equations: Analysis
10.2.5 Euler–Lagrange Equations: PB/PF Operations
11 Thermodynamical Balances
11.1 Interior Total Energy
11.1.1 Global Format (Material Control Volume)
11.1.2 Local Format (Material Control Volume)
11.1.3 Global Format (Spatial Control Volume)
11.1.4 Local Format (Spatial Control Volume)
11.1.5 Balance Tetragons
11.2 Exterior Total Energy
11.2.1 Global Format (Material Control Volume)
11.2.2 Local Format (Material Control Volume)
11.2.3 Global Format (Spatial Control Volume)
11.2.4 Local Format (Spatial Control Volume)
11.2.5 Balance Tetragon
11.3 Entropy
11.3.1 Global Format (Material Control Volume)
11.3.2 Local Format (Material Control Volume)
11.3.3 Global Format (Spatial Control Volume)
11.3.4 Local Format (Spatial Control Volume)
11.3.5 Balance Tetragon
12 Consequences of Thermodynamical Balances
12.1 Clausius–Duhem Assumption
12.2 Dissipation Power Inequality (DPI) I, II
12.2.1 Dissipation of Interior Total Energy
12.2.2 Dissipation of Exterior Total Energy
12.3 Exploitation of DPI in the Domain
12.4 Dissipation Power Inequality (DPI) III
12.4.1 Dissipation of Interior Total Energy
12.4.2 Dissipation of Exterior Total Energy
12.5 Exploitation of DPI on the Boundary
12.6 Exploitation of DPI at Singular Surfaces
12.7 Duality of Spatial and Material Stress
12.8 Four-Dimensional Formalism
References
13 Computational Setting
13.1 Continuous Spatial Virtual Work
13.2 Discretized Spatial Virtual Work
13.2.1 Finite Element Discretization
13.2.2 Finite Element Algebraization
13.2.3 Finite Element Linearization
13.3 Continuous Material Virtual Work
13.4 Discretized Material Virtual Work
13.4.1 Finite Element Discretization
13.4.2 Finite Element Algebraization
13.5 Computational Examples
13.5.1 Homogeneous Pacman-Shaped Domain
13.5.2 Heterogeneous Pacman-Shaped Domain
References


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