Analytic and geometric properties of dislocation singularities

Abstract

This paper deals with the analysis of the singularities arising from the solutions of the problem ${-}\,{\rm Curl\ } F=\mu $, where F is a 3 × 3 matrix-valued Lp-function ($1\les p<2$) and μ a 3 × 3 matrix-valued Radon measure concentrated in a closed loop in Ω ⊂ ℝ3, or in a network of such loops (as, for instance, dislocation clusters as observed in single crystals). In particular, we study the topological nature of such dislocation singularities. It is shown that $F=\nabla u$, the absolutely continuous part of the distributional gradient Du of a vector-valued function u of special bounded variation. Furthermore, u can also be seen as a multi-valued field, that is, can be redefined with values in the three-dimensional flat torus 𝕋3 and hence is Sobolev-regular away from the singular loops. We then analyse the graphs of such maps represented as currents in Ω × 𝕋3 and show that their boundaries can be written in term of the measure μ. Readapting some well-known results for Cartesian currents, we recover closure and compactness properties of the class of maps with bounded curl concentrated on dislocation networks. In the spirit of previous work, we finally give some examples of variational problems where such results provide existence of solutions.

Variational Evolution of Dislocations in Single Crystals

“…[7]). This dynamics indeed is a variational evolution in the sense of Mielke and coauthors [22] together with the existence results established by the authors for the statics problem [31][32][33].…”

“…To the knowledge of the authors, no such energetic evolution was ever considered for three-dimensional dislocation clusters. It is the first purpose of this paper to recall and expose how the ideas developed in [30][31][32][33] meet in a very natural way the lines of Mielke and coauthors well-established existence theory for the quasi-static evolution of rateindependent systems. The basic ingredients are as follows: (i) a variational model at the statics level, and (ii) an appropriate notion of dissipation distance.…”

“…We consider a quasi-static evolution in the presence of a nonlinear stored-elastic energy following the framework proposed in [31][32][33], which relies on a polyconvex elastic energy, plus two higher order terms depending on the first derivatives of F , and in particular on the dislocations density. We supplement the system with a time-depending external bulk force f and a traction g acting on the Neumann part of the boundary, which drives the evolution.…”

“…The particular feature of this process is that the crystal must be considered on a large range of temperatures and hence the motion of dislocations can not be assumed as restricted to some pre-established glide planes, as for a crystal at ambient temperature. This is the motivation of our series of works [30][31][32][33] whose main feature is to consider dislocations in their three-dimensional generality (based on the pioneering contribution [28]). The second specificity of our approach is to consider finite-strain elasticity, since a dislocation does by essence induce large deformations near its singularity.…”

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

“…We refer to [82] for a recent contribution to the topic. The mathematical nature of the displacement field has been clarified in [60,61]: given the set of dislocations L, an integral current (i.e., the generalization of a closed Lipschitz loop) and a deformation tensor F satisfying (2.3), there exists u ∈ W 1,p (Ω,…”