Fields of bounded deformation for mesoscopic dislocations

Abstract

In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties of the auxiliary model fields such as displacement and displacement gradient which are obtained directly from the main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without the introduction of gauge fields or, equivalently, without cuts in the Riemann foliation associated with the dislocated crystal. In a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and discuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. The elastic-plastic decomposition of the strain within this approach is also given a precise meaning.

A compatible‐incompatible decomposition of symmetric tensors in Lp with application to elasticity

“…In a second stage, the first main result has been applied to suggest another proof of certain Korn inequalities in L p . From an application viewpoint, it should be stressed that the structure theorem is useful in dislocation models because it can be proven (e.g., ) that in the presence of dislocations, inc e is related to the curl of the dislocation density κ , and hence, the field F is a dislocation‐induced tensor satisfying incinc F = f (Curl κ ), whereas u is related to the mechanical equilibrium equations and f a certain material‐dependent function.…”

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.…”

Incompatibility-governed singularities in linear elasticity with dislocations

“…5 A weak solution also exists in this case, since by Remark 1, one has w |∂Ω ∈ W 1/p ,p (∂Ω, R 3 ) for 1 ≤ p < 2. Now, by classical lifting theorems, the non-homogeneous problem is recast into a homogeneous problem with a right-hand side in W −1,p (Ω, R 3 ) for which a solution v ∈ W 1,p (Ω, R 3 ) exists as shown in [20].…”

“…It is indeed well known that these fields are not square integrable at this scale. Proving a Kröner identity at the mesoscale, such as incε = Curl κ L , was carried on by the author in a series of works [12][13][14] for some simple families of lines. Although, a proof of such relations for general lines was still missing.…”