Incompatibility-governed singularities in linear elasticity with dislocations

Abstract

The purpose of this paper is to prove the relation incε = Curl κ relating the elastic strain ε and the contortion tensor κ, related to the density tensor of mesoscopic dislocations. Here, the dislocations are given by a finite family of closed Lipschitz curves in Ω ⊂ R 3. Moreover the fields are singular at the dislocations, and in particular the strain is non square integrable. Moreover, the displacement fields shows a constant jump around each isolated dislocation loop. This relation is called after E. Kröner who first derived the same formula for smooth fields at the macroscale.

Analytic and geometric properties of dislocation singularities

“…Let us set V k := Curl G k . Now − Curl V k = Λ T L k , and Theorem 6.3 provides functions 33) for all ω ∈ D 3 (Ω × T 3 ), and…”

Direct Expression of Incompatibility in Curvilinear Systems

“…However, div κ t 0 except in particular cases, for instance, if one considers pure edge dislocation loops in 3D (that is, satisfying tr Λ = 0) and, therefore, the knowledge of the right-hand side of (1.1) is, in general, not sufficient to uniquely determine the field κ. Note that, in this case, the Frank tensor curl t ε and the dislocation density are unequivocally related, since equation (1.1) reduces to curl t ε = κ in Ω with ε × N = 0 on ∂Ω, by virtue of (1.4) and a uniqueness result, as proved by Scala and Van Goethem [14].…”

“…Specifically, Kröner's relation reads curl κ = inc ε, (1.1) where the contortion tensor κ is related to the tensor-valued dislocation density Λ by κ = Λ − (1/2) tr ΛI 2 . At the mesoscopic scale, the dislocation density reads Λ = Λ L = τ ⊗ bH 1 L , where H 1 L stands for the one-dimensional Hausdorff measure concentrated in the dislocation loop L. At the mesoscale, Kröner's relation also holds, as proved by the author [15,16]. At the macro (or continuous) scale, (which is the scale considered in the present work), Λ is a smooth tensor obtained from its mesoscopic counterpart by some regularization.…”

Direct expression of incompatibility in curvilinear systems

“…Specifically, Kröner's relation reads Curl κ = inc , (1.1) where the contortion tensor κ is related to the tensor-valued dislocation density Λ by κ = Λ − 1 2 tr ΛI 2 . At the mesoscopic scale the dislocation density reads Λ = Λ L = τ ⊗ bH 1 L , where H 1 L stands for the one-dimensional Hausdorff measure concentrated in the dislocation loop L. At the mesoscale, Kröner's relation also holds, as proved in [10,11]. At the macro (or continuous) scale (which is the scale considered in the present work), Λ is a smooth tensor obtained from its mesoscopic counterpart by some regularization.…”