Configurational Forces : Thermomechanics, Physics, Mathematics, and Numerics book cover
1st Edition

Configurational Forces
Thermomechanics, Physics, Mathematics, and Numerics

ISBN 9781439846124
Published December 1, 2010 by Chapman & Hall
562 Pages 19 Color & 58 B/W Illustrations

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Book Description

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, Noether’s 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 forces—one of the latest and most fruitful advances in macroscopic field theories. Through many applications, it shows the depth and efficiency of this theory.

Table of Contents

Continuum Mechanics in the Twentieth Century
The Objective of This Book
The Contents of This Book
Historical Note

Standard Continuum Mechanics
Theory of Motion and Deformation
Basic Thermomechanics of Continua
Examples of Thermomechanical Behaviors

Eshelbian Mechanics for Elastic Bodies
The Notion of Eshelby Material Stress
Eshelby Stress in Small Strains in Elasticity
Classical Introduction of the Eshelby Stress by Eshelby’s Original Reasoning
Another Example Due to Eshelby: Material Force on an Elastic Inhomogeneity
Gradient Elastic Materials
Interface in a Composite
The Case of a Dislocation Line (Peach–Koehler Force)
Four Formulations of the Balance of Linear Momentum
Variational Formulations in Elasticity
More Material Balance Laws
Eshelby Stress and Kröner’s Theory of Incompatibility

Field Theory
Elements of Field Theory: Variational Formulation
Application to Elasticity
Conclusive Remarks

Canonical Thermomechanics of Complex Continua
Canonical Balance Laws of Momentum and Energy
Examples without Body Force
Variable α as an Additional Degree of Freedom
Comparison with the Diffusive Internal-Variable Theory
Example: Homogeneous Dissipative Solid Material Described by Means of a Scalar Diffusive Internal Variable
Conclusion and Comments

Local Structural Rearrangements of Matter and Eshelby Stress
Changes in the Reference Configuration
Material Force of Inhomogeneity
Some Geometric Considerations
Continuous Distributions of Dislocations
Pseudo-Inhomogeneity and Pseudo-Plastic Effects
A Variational Principle in Nonlinear Dislocation Theory
Eshelby Stress as a Resolved Shear Stress
Second-Gradient Theory
Continuous Distributions of Disclinations

Discontinuities and Eshelby Stresses
General Jump Conditions at a Moving Discontinuity Surface
Thermomechanical Shock Waves
Thermal Conditions at Interfaces in Thermoelastic Composites
Propagation of Phase-Transformation Fronts
On Internal and Free Energies
The Case of Complex Media
Applications to Problems of Materials Science (Metallurgy)

Singularities and Eshelby Stresses
The Notion of Singularity Set
The Basic Problem of Fracture and Its Singularity
Global Dissipation Analysis of Brittle Fracture
The Analytical Theory of Brittle Fracture
Singularities and Generalized Functions
Variational Inequality: Fracture Criterion
Dual I-Integral of Fracture
Other Material Balance Laws and Path-Independent Integrals
Generalization to Inhomogeneous Bodies
Generalization to Dissipative Bodies
A Curiosity: "Nondissipative" Heat Conductors

Generalized Continua
Field Equations of Polar Elasticity
Small-Strain and Small-Microrotation Approximation
Discontinuity Surfaces in Polar Materials
Fracture of Solid Polar Materials
Other Microstructure Modelings

Systems with Mass Changes and/or Diffusion
Volumetric Growth
First-Order Constitutive Theory of Growth
Application: Anisotropic Growth and Self-Adaptation
Illustrations: Finite-Element Implementation
Intervention of Nutriments
Eshelbian Approach to Solid–Fluid Mixtures
Single-Phase Transforming Crystal and Diffusion

Electromagnetic Materials
Maxwell Could Not Know Noether’s Theorem but…
Electromagnetic Fields in Deformable Continuous Matter
Variational Principle Based on the Direct Motion
Variational Principle Based on the Inverse Motion
Geometrical Aspects and Material Uniformity
Remark on Electromagnetic Momenta
Balance of Canonical Momentum and Material Forces
Electroelastic Bodies and Fracture
Transition Fronts in Thermoelectroelastic Crystals
The Case of Magnetized Elastic Materials

Application to Nonlinear Waves
Wave Momentum in Crystal Mechanics
Conservation Laws in Soliton Theory
Examples of Solitonic Systems and Associated Quasiparticles
Sine Gordon Equation and Associated Equations
Nonlinear Schrödinger Equation and Allied Systems
Driving Forces Acting on Solitons
A Basic Problem of Materials Science: Phase-Transition Front Propagation

Numerical Applications
Finite-Difference Method
Finite-Volume Method—Continuous Cellular Automata
Finite-Element Method
Conclusive Remarks

More on Eshelby-Like Problems and Solutions
Analogy: Path-Independent Integrals in Heat and Electricity Conductions
The Eshelbian Nature of Aerodynamic Forces
The World of Configurational Forces



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Gérard A. Maugin is a distinguished professor and research director of the Institut Jean Le Rond d’Alembert 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.


"… 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.