The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner (1919-2000)

Goethem, Nicolas Van

doi:10.3934/jgm.2010.2.303

Cited by 16 publications

(20 citation statements)

References 32 publications

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“…It has been shown in [33] that it also gives a hint that if the first-grade defect variable is chosen to be Λ, the second-grade defect variable should be κ × ∇. Moreover, this choice of defect model variables closes 13 the model in the sense that η can also be written as (cf.…”

Section: Argument Of Structurementioning

confidence: 99%

“…In this work, the dislocation density is a second-order tensor. However, the chosen model defect variable will be the so-called contortion tensor (see, e.g., [33] for a discussion on its geometrical meaning) defined as follows.…”

Section: Definition 1 (Defect Set)mentioning

confidence: 99%

“…In the small strain assumption in 3D, let g := I − 2E B be the elastic metric. Then, it is not hard to see [21,33] that the above statement is equivalent to requiring that E B be compatible in our sense: η := −∇ × E B × ∇ = 0 with η = Ric − I 2 tr Ric. Therefore it immediately follows by identification with Eq.…”

Section: Argument Of Structurementioning

confidence: 99%

“…Surprisingly enough, the flow shows good properties directly from the incompatibility-contortion relation as proved in Theorem 4.1. Another reference should here be [33] where it is shown that Riemann curvature, Cartan torsion and torsion contortion in the crystal differentiable manifold have a physical meaning in terms of defect quantities. Let us recall that the notion of crystal manifold is a 60-years-old and rich idea due to Kröner [19] and Kondo [18].…”

Section: Applications Of Theorem 41 For Modeling Purposesmentioning

confidence: 99%

“…The geometric framework as introduced by Kröner has in particular attracted attention of the author [33] since the geometry of the defects (dislocations, disclinations and point defects) can be envisaged as a way to recover and/or discover relations between model variables which do not necessarily appear clear if one restricts to a purely mechanical analysis. Another way to clarify the role of the model variables is to prove formulas (some of which might be well-known and widely used), which were otherwise heuristically inferred, formally justified, let alone only conjectured.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A multiscale model for dislocations: From mesoscopic elasticity to macroscopic plasticity

Goethem

2012

Z Angew Math Mech

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A multiscale model for dislocations in single crystals is proposed. The aim of this paper is to provide homogenization of dislocations from the meso-to the macro-scale. In particular we prove a new formula relating macroscopic strain incompatibility and dislocation density. Moreover, it is shown that plasticity is recovered from homogenization of purely elastic mesoscopic crystals with dislocations, where the appropriate functional space of Special functions of Bounded Deformation appears as a natural choice. Nonstandard defect and deformation internal variables and their evolution in time appear while novel differential relations are sketched in the non-Riemannian crystal with a thermodynamical model as based on those results.

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Section: Argument Of Structurementioning

confidence: 99%

Section: Definition 1 (Defect Set)mentioning

confidence: 99%

Section: Argument Of Structurementioning

confidence: 99%

Section: Applications Of Theorem 41 For Modeling Purposesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A multiscale model for dislocations: From mesoscopic elasticity to macroscopic plasticity

Goethem

2012

Z Angew Math Mech

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The Incompatibility Operator: from Riemann’s Intrinsic View of Geometry to a New Model of Elasto-Plasticity

Amstutz

Goethem

2019

CIM Series in Mathematical Sciences

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The mathematical modelling in mechanics has a long-standing history as related to geometry, and significant progresses have often been achieved by the invention of new geometrical tools. Also, it happened that the elucidation of practical issues led to the invention of new scientific concepts, and possibly new paradigms, with potential impact far beyond. One such example is Riemann's intrinsic view in geometry, that offered a radically new insight in the Physics of the early 20th century. On the other hand, the rather recent intrinsic approaches in elasticity and elasto-plasticity also share this philosophical standpoint of looking from inside, i.e., from the "manifold" point of view. Of course, this approach requires smoothness, and is thus incomplete for an analyst. Nevertheless, its first aim is to highlight the concepts of metric, curvature and torsion; these notions are addressed in the first part of this survey paper. In a second part, they are given a precise functional meaning and their properties are studied systematically. Further, a novel approach to elasto-plasticity constructed upon a model of incompatible elasticity is designed, carrying this intrinsic spirit. The main mathematical object in this theory is the incompatibility operator, i.e., a linearized version of Riemann's curvature tensor. So far, this route not only has led the authors to a new model with a solid functional foundation and proof of existence results, but also to a framework with a minimal amount of ad-hoc assumptions, and complying with both the basic principles of thermodynamics and invariance principles of Physics. The questions arising from this novel approach are complex and intriguing, but we believe that the model is now sufficiently well posed to be studied simultaneously as a problem of mathematics and of mechanics. Most of the research programme remains to be done, and this survey paper is written to present our model, with a particular care to put this approach into a historical perspective.2010 Mathematics Subject Classification. 35J48,35J58,49S05,49K20,74C05,74G99,74A05,74A15, 80A17.

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Thermodynamic forces in single crystals with dislocations

Goethem

2013

Z. Angew. Math. Phys.

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The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner (1919-2000)

Cited by 16 publications

References 32 publications

A multiscale model for dislocations: From mesoscopic elasticity to macroscopic plasticity

A multiscale model for dislocations: From mesoscopic elasticity to macroscopic plasticity

The Incompatibility Operator: from Riemann’s Intrinsic View of Geometry to a New Model of Elasto-Plasticity

Thermodynamic forces in single crystals with dislocations

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