Geometry and thermomechanics of structural rearrangements: Ekkehart Kröner's legacy Plenary lecture presented at the 80th Annual GAMM Conference, Augsburg, 25‐28 March 2002

Starting with the influential ideas of E. Kröner on the Non‐Riemannian geometrical theory of defects and a brief review of his works, we show how this legacy has developed into a true mechanics of materials on the material manifold. This synthesis, that essentially combines Kröner's ideas and those of his contemporary, J. D. Eshelby, yields a powerful unification of some of the most fruitful developments of continuum thermomechanics at the end of the XXth century and the beginning of the XXIst. In particular, a general theory of material inhomogeneities emerges which includes the study of the progress, under the action of well defined driving forces, of many microscopic and macroscopic defects or field singularities. In particular, we emphasize the relationship between the notion of material Eshelby stress and the notion of local structural rearrangements that are responsible for many of the macroscopically observed irreversible behaviors of solid‐like matter (e.g., anelasticity, damage, phase transitions, growth). All these phenomena take place directly on the material manifold. That is, their arena and that of the associated driving forces, also called configurational forces, or material forces, require a general setting that emphasizes this material peculiarity. This is the object of the paper in which all types of applications are also mentioned.

[1]  Gérard A. Maugin,et al.  Material Inhomogeneities in Elasticity , 2020 .

[2]  G. Maugin,et al.  An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures , 2010 .

[3]  R. Mueller,et al.  On material forces and finite element discretizations , 2002 .

[4]  Dietmar Gross,et al.  On configurational forces in the context of the finite element method , 2002 .

[5]  Marcelo Epstein,et al.  Thermomechanics of volumetric growth in uniform bodies , 2000 .

[6]  G. Maugin,et al.  Eshelby's stress tensors in finite elastoplasticity , 2000 .

[7]  G. Maugin.  On the universality of the thermomechanics of forces driving singular sets , 2000 .

[8]  G. Maugin.  On shock waves and phase-transition fronts in continua , 1998 .

[9]  M. Epstein,et al.  On the geometrical material structure of anelasticity , 1996 .

[10]  Gérard A. Maugin,et al.  Material Forces: Concepts and Applications , 1995 .

[11]  E. Kröner.  Configurational and Material Forces in the Theory of Defects in Ordered Structures , 1993 .

[12]  G. Maugin.  The Thermomechanics of Plasticity and Fracture , 1992 .

[13]  M. Epstein,et al.  The energy-momentum tensor and material uniformity in finite elasticity , 1990 .

[14]  F. Hehl,et al.  General Relativity with Spin and Torsion: Foundations and Prospects , 1976 .

[15]  P. M. Naghdi,et al.  SOME REMARKS ON ELASTIC-PLASTIC DEFORMATION AT FINITE STRAIN , 1971 .

[16]  En-Jui Lee.  Elastic-Plastic Deformation at Finite Strains , 1969 .

[17]  E. Kröner,et al.  Elastic moduli of perfectly disordered composite materials , 1967 .

[18]  E. Kröner,et al.  Elasticity theory of materials with long range cohesive forces , 1967 .

[19]  E. Kröner.  Dislocation: A New Concept in the Continuum Theory of Plasticity , 1963 .

[20]  J. D. Eshelby,et al.  The force on an elastic singularity , 1951, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[21]  M. Peach,et al.  THE FORCES EXERTED ON DISLOCATIONS AND THE STRESS FIELDS PRODUCED BY THEM , 1950 .

[22]  C. Christov,et al.  Nonlinear Duality Between Elastic Waves and Quasi-particles , 2002 .

[23]  George Herrmann,et al.  Mechanics in material space , 2000 .

[24]  G. Maugin.  Thermomechanics of forces driving singular point sets , 1998 .

[25]  Cristian Dascalu,et al.  Forces matérielles et taux de restitution de l'énergie dans les corps élastiques homogènes avec défauts , 1993 .

[26]  A. G. Herrmann.  On conservation laws of continuum mechanics , 1981 .

[27]  G. Maugin.  Magnetized deformable media in general relativity , 1971 .

[28]  K. Anthony.  Die theorie der disklinationen , 1970 .

[29]  C. Wang,et al.  On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations , 1967 .

[30]  Walter Noll,et al.  Materially uniform simple bodies with inhomogeneities , 1967 .

[31]  E. Kröner,et al.  Nicht-lineare Elastizitätstheorie der Versetzungen und Eigenspannungen , 1959 .

[32]  E. Kröner,et al.  Kontinuumstheorie der Versetzungen und Eigenspannungen , 1958 .