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E-grāmata: Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics

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(Universite Pierre et Marie Curie, Paris, France)
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Exploring recent developments in continuum mechanics, Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics presents the general framework for configurational forces. It also covers a range of applications in engineering and condensed matter physics.

The author presents the fundamentals of accepted standard continuum mechanics, before introducing Eshelby material stress, field theory, variational formulations, Noethers theorem, and the resulting conservation laws. In the chapter on complex continua, he compares the classical perspective of B.D. Coleman and W. Noll with the viewpoint linked to abstract field theory. He then describes the important notion of local structural rearrangement and its relationship to Eshelby stress. After looking at the relevance of Eshelby stress in the thermodynamic description of singular interfaces, the text focuses on fracture problems, microstructured media, systems with mass exchanges, and electromagnetic deformable media. The concluding chapters discuss the exploitation of the canonical conservation law of momentum in nonlinear wave propagation, the application of canonical-momentum conservation law and material force in numerical schemes, and similarities of fluid mechanics and aerodynamics.

Written by a long-time researcher in mechanical engineering, this book provides a detailed treatment of the theory of configurational forcesone of the latest and most fruitful advances in macroscopic field theories. Through many applications, it shows the depth and efficiency of this theory.

Recenzijas

" an excellent introduction into this wide branch of mechanics, and, at the same time, it provides scientists already involved in the field extended references to specific aspects of Eshelbian mechanics. On an equal level, the thermomechanics, physics, mathematics and numerics of configurational forces are covered. Starting with elastic bodies, the theory is extended step-by-step to complex and generalized continua. Discontinuities of various kinds, fracture, moving interfaces, wave motion, etc., are treated, and elastic, elastoplastic, elastomagnetic and thermoelastic materials are discussed. The huge amount of material is arranged in a clear and rigorous manner. This is a book of a master in his field." Reinhold Kienzler (Bremen), Zentralblatt MATH



The book is indeed written by a true master of his field and is thus great fun to read and to study. It is of interest not only for specialists in configurational forces but for all those who are concerned with the broad field of continuum modeling. It is for example amazing to see how apparently dissimilar fields such as e1ectro-magneto-mechanics and biological growth or nonlinear waves are connected by the underlying and thus unifying concept of configurational forces. The book is extremely rich in detail and depth; at the same time it will be helpful for the beginner and the expert alike. In summary I assess that this is one of the few books that should be on the bookshelf of any researcher in mechanics and/or applied mathematics. MATHEMATICAL REVIEWS

1 Introduction
1(10)
1.1 Continuum Mechanics in the Twentieth Century
1(2)
1.2 The Object of This Book
3(3)
1.3 The Contents of This Book
6(2)
1.4 Historical Note
8(3)
2 Standard Continuum Mechanics
11(36)
2.1 Theory of Motion and Deformation
11(12)
2.1.1 Material Body
11(1)
2.1.2 Motion (or Deformation Mapping)
11(3)
2.1.3 Some Deformation Measures
14(1)
2.1.4 Displacement Field
15(2)
2.1.5 Convection, Pull Back, and Push Forward
17(1)
2.1.6 Time Derivatives and Rates
18(1)
2.1.7 Rate of Deformation
19(1)
2.1.8 Rigid-Body Motions
20(3)
2.2 Basic Thermomechanics of Continua
23(11)
2.2.1 Balance Laws
23(3)
2.2.2 Cauchy Format of the Local Balance Laws of Thermomechanics
26(1)
2.2.3 Piola-Kirchhoff Format of the Local Balance Laws of Thermomechanics
27(2)
2.2.4 Thermodynamic Hypotheses
29(1)
2.2.5 General Thermomechanical Theorems
30(4)
2.3 Examples of Thermomechanical Behaviors
34(13)
2.3.1 Thermoelastic Conductors
34(6)
2.3.2 A Sufficiently Large Class of Dissipative Solids
40(2)
2.3.3 Example: Elastoplasticity in Finite Strains
42(4)
2.3.4 Small Strain Rate Independent Elastoplasticity
46(1)
3 Eshelbian Mechanics for Elastic Bodies
47(30)
3.1 The Notion of Eshelby Material Stress
47(4)
3.1.1 Quasistatic Eshelby Stress
47(2)
3.1.2 Dynamic Generalization
49(1)
3.1.3 Weak Form of the Balance of Material Momentum
50(1)
3.2 Eshelby Stress in Small Strains in Elasticity
51(1)
3.2.1 Reduced Form of the Eshelby Stress and Field Momentum
51(1)
3.3 Classical Introduction of the Eshelby Stress by Eshelby's Original Reasoning
52(3)
3.4 Another Example Due to Eshelby: Material Force on an Elastic Inhomogeneity
55(2)
3.5 Gradient Elastic Materials
57(1)
3.6 Interface in a Composite
58(1)
3.7 The Case of a Dislocation Line (Peach-Koehler Force)
59(4)
3.8 Four Formulations of the Balance of Linear Momentum
63(2)
3.9 Variational Formulations in Elasticity
65(4)
3.9.1 Variation of the Direct Motion
65(1)
3.9.2 A Two-Fold Classical Variation: The Complementary Energy
66(2)
3.9.3 Inverse Motion
68(1)
3.10 More Material Balance Laws
69(4)
3.11 Eshelby Stress and Kroner's Theory of Incompatibility
73(4)
4 Field Theory
77(34)
4.1 Introduction
77(1)
4.2 Elements of Field Theory: Variational Formulation
78(12)
4.2.1 Noether's Theorem
78(5)
4.2.2 Example 1: Space-Time Translation
83(2)
4.2.3 Relationship to Lie-Group Theory
85(2)
4.2.4 Example 2: Rotations
87(3)
4.2.5 Example 3: Change of Scale
90(1)
4.3 Application to Elasticity
90(17)
4.3.1 Direct-Motion Formulation for Classical Elasticity
90(7)
4.3.2 Inverse-Motion Formulation for Classical Elasticity
97(2)
4.3.3 Second-Gradient Elasticity
99(4)
4.3.4 Hamilton's Equations
103(1)
4.3.5 Lie-Poisson Brackets
104(3)
4.4 Conclusive Remarks
107(1)
Appendix A4.1 Field Theory of the Peach-Koehler Force in Matter with Continuously Distributed Dislocations
108(3)
5 Canonical Thermomechanics of Complex Continua
111(28)
5.1 Introduction
111(2)
5.2 Reminder
113(1)
5.3 Canonical Balance Laws of Momentum and Energy
114(3)
5.3.1 A Canonical Form of the Energy Conservation
114(1)
5.3.2 Canonical (Material) Momentum Conservation
115(2)
5.4 Examples without Body Force
117(3)
5.4.1 Pure Homogeneous Elasticity
117(1)
5.4.2 Inhomogeneous Thermoelasticity of Conductors
117(1)
5.4.3 Homogeneous Dissipative Solid Material Described by Means of a Diffusive Internal Variable
118(2)
5.5 Variable α as an Additional Degree of Freedom
120(7)
5.5.1 General Formulation
120(4)
5.5.2 The Only Dissipative Process Is Heat Conduction
124(1)
5.5.3 The Coleman-Noll Continuum Thermodynamics Viewpoint
125(1)
5.5.4 The Field-Theoretic Viewpoint
126(1)
5.6 Comparison with the Diffusive Internal-Variable Theory
127(2)
5.7 Example: Homogeneous Dissipative Solid Material Described by Means of a Scalar Diffusive Internal Variable
129(3)
5.8 Conclusion and Comments
132(2)
5.8.1 Two Viewpoints on the Equation of Material Momentum
133(1)
Appendix A5.1 Four-Dimensional Formulation
134(2)
Appendix A5.2 Another Viewpoint
136(3)
6 Local Structural Rearrangements of Matter and Eshelby Stress
139(28)
6.1 Changes in the Reference Configuration
139(2)
6.2 Material Force of Inhomogeneity
141(1)
6.3 Some Geometric Considerations
142(6)
6.4 Continuous Distributions of Dislocations
148(1)
6.5 Pseudo-Inhomogeneity and Pseudo-Plastic Effects
149(5)
6.5.1 Materially Homogeneous Thermoelasticity
150(1)
6.5.2 Elastoplasticity in Finite Strains
151(1)
6.5.3 Other Cases
152(2)
6.6 A Variational Principle in Nonlinear Dislocation Theory
154(2)
6.7 Eshelby Stress as a Resolved Shear Stress
156(2)
6.8 Second-Gradient Theory
158(3)
6.9 Continuous Distributions of Disclinations
161(2)
Appendix A6.1 Unification of Three Lines of Research
163(4)
7 Discontinuities and Eshelby Stress
167(42)
7.1 Introduction
167(4)
7.2 General Jump Conditions at a Moving Discontinuity Surface
171(4)
7.2.1 Equations in the Cauchy Format in the Actual Configuration
171(1)
7.2.2 Equations in the Piola-Kirchhoff Format
172(3)
7.3 Thermomechanical Shock Waves
175(5)
7.4 Thermal Conditions at Interfaces in Thermoelastic Composites
180(3)
7.5 Propagation of Phase-Transformation Fronts
183(7)
7.5.1 Definition
183(1)
7.5.2 Driving Force and Kinetic Relation
184(4)
7.5.3 Nondissipative Phase Transition---Maxwell Rule
188(2)
7.6 On Internal and Free Energies
190(4)
7.7 The Case of Complex Media
194(6)
7.8 Applications to Problems of Materials Science (Metallurgy)
200(9)
7.8.1 Equilibrium Shape of Precipitates
200(3)
7.8.2 Inelastic Discontinuities and Martensitic Phase Transitions
203(4)
7.8.3 Remark on a Mechano-Biological Problem
207(2)
8 Singularities and Eshelby Stress
209(42)
8.1 The Notion of Singularity Set
209(1)
8.2 The Basic Problem of Fracture and Its Singularity
210(2)
8.3 Global Dissipation Analysis of Brittle Fracture
212(2)
8.4 The Analytical Theory of Brittle Fracture
214(6)
8.5 Singularities and Generalized Functions
220(6)
8.5.1 The Notion of Generalized Function or Distribution
220(1)
8.5.2 Basic Equations
221(1)
8.5.3 Case of Cracks
222(4)
8.6 Variational Inequality: Fracture Criterion
226(3)
8.7 Dual I-Integral of Fracture
229(3)
8.8 Other Material Balance Laws and Path-Independent Integrals
232(6)
8.8.1 Notion of L- and M-Integrals
232(2)
8.8.2 Distributional Approach: Energy of Cavities and Inclusions
234(2)
8.8.3 Distributional Approach: Dislocations
236(2)
8.9 Generalization to Inhomogeneous Bodies
238(2)
8.10 Generalization to Dissipative Bodies
240(5)
8.10.1 The Problem of Thermal Brittle Fracture
240(3)
8.10.2 Fracture in Anelasticity
243(2)
8.11 A Curiosity: "Nondissipative" Heat Conductors
245(6)
8.11.1 Conservation Equations
245(2)
8.11.2 The Problem of Thermoelastic Fracture
247(1)
8.11.3 Recovery of Classical Thermoelasticity
248(3)
9 Generalized Continua
251(34)
9.1 Introduction
251(1)
9.2 Field Equations of Polar Elasticity
252(10)
9.2.1 Elements of Kinematics
252(1)
9.2.2 Lagrange Equations of Motion
253(2)
9.2.3 Canonical Balance Laws
255(2)
9.2.4 Accounting for Inertia
257(2)
9.2.5 Dissipative Case
259(3)
9.3 Small-Strain and Small-Microrotation Approximation
262(2)
9.4 Discontinuity Surfaces in Polar Materials
264(3)
9.5 Fracture of Solid Polar Materials
267(2)
9.6 Other Microstructure Modelings
269(7)
9.6.1 Micromorphic Compatible Case
269(2)
9.6.2 Micromorphic Incompatible Case
271(2)
9.6.3 Special Cases
273(3)
Appendix A9.1 Liquid Crystals
276(7)
Appendix A9.2 Material Force Acting on a Center of Dilatation
283(2)
10 Systems with Mass Changes and/or Diffusion
285(44)
10.1 Introduction
285(2)
10.2 Volumetric Growth
287(5)
10.3 First-Order Constitutive Theory of Growth
292(6)
10.4 Application: Anisotropic Growth and Self-Adaptation
298(2)
10.5 Illustrations: Finite-Element Implementation
300(7)
10.5.1 Monotonic Growth and Adaptation
301(1)
10.5.2 Anisotropic Growth in the Float Model
302(2)
10.5.3 Growth Behavior under Cyclic Loading
304(3)
10.6 Intervention of Nutriments
307(3)
10.7 Eshelbian Approach to Solid-Fluid Mixtures
310(11)
10.7.1 Kinematics
310(3)
10.7.2 Balance of Mass
313(2)
10.7.3 Stress Power, Kinetic Energy, and Acceleration
315(1)
10.7.4 The Principle of Virtual Power
316(2)
10.7.5 Energy Equation in the Absence of Dissipation
318(1)
10.7.6 Constitutive Equations for Unconstrained Solid-Fluid Mixtures
319(1)
10.7.7 Constitutive Equations for Saturated Porolastic Media
320(1)
10.8 Single-Phase Transforming Crystal and Diffusion
321(8)
10.8.1 Introductory Remark
321(1)
10.8.2 Eigenstrains and Molar Concentrations
321(2)
10.8.3 Thermodynamic Equations
323(2)
10.8.4 Chemical Potential and Eshelby Stress
325(4)
11 Electromagnetic Materials
329(54)
11.1 Maxwell Could Not Know Noether's Theorem but
329(3)
11.1.1 The Notions of Maxwell Stress and Electromagnetic Momentum
329(2)
11.1.2 Lorentz Force on a Point Charge
331(1)
11.2 Electromagnetic Fields in Deformable Continuous Matter
332(7)
11.2.1 Maxwell's Equations in General Matter in RL
333(1)
11.2.2 Maxwell's Equations in Terms of Comoving Field in RC(x, t)
333(1)
11.2.3 Maxwell's Equations in the Material Framework
334(1)
11.2.4 Ponderomotive Force and Electromagnetic Stresses and Momentum
335(2)
11.2.5 Equations of Motion in Cauchy Format
337(1)
11.2.6 Equations of Motion in Piola-Kirchhoff Format
338(1)
11.3 Variational Principle Based on the Direct Motion
339(3)
11.3.1 Prerequisite
339(2)
11.3.2 The Variational Principle Per Se
341(1)
11.4 Variational Principle Based on the Inverse Motion
342(2)
11.5 Geometric Aspects and Material Uniformity
344(2)
11.6 Remark on Electromagnetic Momenta
346(2)
11.7 Balance of Canonical Momentum and Material Forces
348(1)
11.8 Electroelastic Bodies and Fracture
349(10)
11.8.1 General Equations
349(3)
11.8.2 Evaluation of the Energy-Release Rate in Electroelastic Fracture
352(1)
11.8.3 Electroelastic Path-Independent Integrals
353(2)
11.8.4 An Application: Antiplane Crack in a Dielectric with Induced Piezoelectricity
355(3)
11.8.5 Note on Linear Piezoelectricity
358(1)
11.9 Transition Fronts in Thermoelectroelastic Crystals
359(9)
11.9.1 General Equations
359(2)
11.9.2 Constitutive Relations
361(1)
11.9.3 Canonical Balance Laws
362(1)
11.9.4 Jump Relations at a Front
363(5)
11.10 The Case of Magnetized Elastic Materials
368(10)
11.10.1 Introductory Remark
368(2)
11.10.2 The Equations Governing Elastic Ferromagnets
370(2)
11.10.3 Canonical Balance of Material Momentum
372(1)
11.10.4 Phase-Transition Fronts in Thermoelastic Ferromagnets
373(3)
11.10.5 Domain Walls in Ferromagnets
376(2)
Appendix A11.1 Proof of Theorem 11.1 and Corollary 11.2
378(5)
12 Application to Nonlinear Waves
383(42)
12.1 Wave Momentum in Crystal Mechanics
383(2)
12.1.1 Definition
383(1)
12.2.2 Beware of One-Dimensional Systems!
384(1)
12.2 Conservation Laws in Soliton Theory
385(4)
12.3 Examples of Solitonic Systems and Associated Quasiparticles
389(7)
12.3.1 Korteweg-de Vries Equation
389(2)
12.3.2 "Good" Boussinesq Equation
391(1)
12.3.3 Generalized Boussinesq Equation
392(2)
12.3.4 Mechanical System with Two Degrees of Freedom
394(2)
12.4 Sine Gordon Equation and Associated Equations
396(4)
12.4.1 Standard Sine Gordon Equation
396(2)
12.4.2 Sine Gordon-d'Alembert Systems
398(2)
12.5 Nonlinear Schrodinger Equation and Allied Systems
400(3)
12.5.1 The Standard Cubic Nonlinear Schrodinger Equation
400(1)
12.5.2 Zakharov and Generalized Zakharov Systems
401(2)
12.6 Driving Forces Acting on Solitons
403(3)
12.7 A Basic Problem of Materials Science: Phase-Transition Front Propagation
406(7)
12.7.1 Some General Words
406(1)
12.7.2 Microscopic Condensed Matter-Physics Approach: Solitonics
407(2)
12.7.3 Macroscopic Engineering Thermodynamic Approach
409(2)
12.7.4 Mesoscopic Applied-Mathematics Approach: Structured Front
411(1)
12.7.5 Theoretical Physics Approach: Quasiparticle and Transient Motion
411(1)
12.7.6 Thermodynamically Based Continuous Automaton
412(1)
Appendix A12.1 Eshelbian Kinematic-Wave Mechanics
413(12)
A12.1.1 Some Words of Introduction
413(1)
A12.1.2 Kinematic-Wave Theory and Elasticity
414(4)
A12.1.3 Application to Nonlinear Dispersive Waves in Elastic Crystals
418(4)
A12.1.4 Return to the Notion of Eshelby Stress
422(3)
13 Numerical Applications
425(38)
13.1 Introduction
425(1)
13.2 Finite-Difference Method
426(2)
13.3 Finite-Volume Method---Continuous Cellular Automata
428(9)
13.4 Finite-Element Method
437(22)
13.4.1 General Principle
437(5)
13.4.2 Implementation
442(1)
13.4.3 First Example: Homogeneous Block under Pressure
443(3)
13.4.4 Second Example: Clamped Block under Tension
446(3)
13.4.5 Third Example: Inhomogeneous Cantilever Beam
449(2)
13.4.6 Fourth Example: Crack Propagation Using Material Forces
451(5)
13.4.7 Fifth Example: Layer on a Base Material
456(1)
13.4.8 Sixth Example: Misfitting Inhomogeneity in Two-Phase Materials
456(2)
13.4.9 Note on Topological Optimization
458(1)
13.5 Conclusive Remarks
459(4)
14 More on Eshelby-Like Problems and Solutions
463(12)
14.1 Introduction
463(1)
14.2 Analogy: Path-Independent Integrals in Heat and Electricity Conductions
463(4)
14.3 The Eshelbian Nature of Aerodynamic Forces
467(4)
14.4 The World of Configurational Forces
471(4)
Bibliography 475(40)
Index 515
Gérard A. Maugin is a distinguished professor and research director of the Institut Jean Le Rond dAlembert at the Université Pierre et Marie Curie and CNRS. He has taught at numerous universities around the world and has been involved in research projects with organizations such as the French Ministry of National Defense, US National Science Foundation, US Army Research Office, US Office of Naval Research, National Research Council of Canada, NATO, the European Community, and I.A.E.A-UNESCO. A member of many scientific societies, Dr. Maugin has received several awards throughout his career, including the Max Planck Research Award for Engineering Sciences given by the Max Planck Society and the Alexander von Humboldt Foundation.
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