Currents and dislocations at the continuum scale

Abstract

A striking geometric property of elastic bodies with dislocations is that the deformation tensor cannot be written as the gradient of a one-to-one immersion, its curl being nonzero but equal to the density of the dislocations, a measure concentrated on the dislocation lines. In this work, we discuss the mathematical properties of such constrained deformations and study a variational problem in finite-strain elasticity, where Cartesian maps allow us to consider deformations in L p with 1 ≤ p < 2, as required for dislocation-induced strain singularities. Firstly we address the problem of mathematical modeling of dislocations. It is a key purpose of the paper to build a framework where dislocations are described in terms of integral 1-currents and to extract from this theoretical setting a series of notions having a mechanical meaning in the theory of dislocations. In particular, the paper aims at classifying integral 1-currents, with modeling purposes. In the second part of the paper, two variational problems are solved for two classes of dislocations, at the mesoscopic, and at the continuum scale. By continuum it is here meant that a countable family of dislocations is considered, allowing for branching and cluster formation, with possible complex geometric patterns. Therefore, modeling assumptions of the defect part of the energy must also be provided, and discussed.

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

“…In the presence of line-like defects such as Volterra dislocations [35], we are typically faced to the following issue: we have an elastic body Ω with a dislocation loop L and we assume that we have the means for determining at any point of Ω \ L the stretch and rotation, in other words we are given a metric tensor C. It turns out that we are only able to construct such a deformation F as in (2.1) and (2.2) in Ω \ Π L where Π L is a surface containing L and dividing Ω into two subdomains Ω + and Ω − [59]. Let S L ⊂ Π L be the surface enclosed by the loop, i.e.…”

“…However in L this representation fails, and hence the aforementioned approach holds piecewise. Nonetheless, something can be said at L, namely by use of Stokes theorem [59], one finds…”

Analytic and geometric properties of dislocation singularities

“…All these phenomena should be understood as mathematical objects which must be described in a geometrically unified way. To achieve this goal, the theory of currents has shown as extremely appropriate, as described in [29]. It should be emphasized that integral currents and Cartesian currents also allow one to handle singular problems in Continuum mechanics by means of the notions of distributional Jacobian and cofactors of the deformation tensor.…”

“…In the language of currents, the measure (1.2) is denoted as b ⊗ L, with L standing for an integral 1-current, as described in [29]. In the language of Physics, it is simply the (transpose of) the dislocation density Λ L := τ ⊗ bH 1 L , which should be considered as the problem data.…”

Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity

“…The analysis in the three-dimensional case is substantially more subtle. A formulation of dislocations in terms of currents was considered also in [19]. The aim of this paper is to study the lower semicontinuity and relaxation of functionals of the type (1.1).…”