The Line-Tension Approximation as the Dilute Limit of Linear-Elastic Dislocations

Abstract

We prove that the classical line-tension approximation for dislocations in crystals, that is, the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the Γ -limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length ψ(b, t), which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length ψ 0 (b, t) obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation of the classical energy down to its H 1 -elliptic envelope.

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

“…For a bi-Lipschitz map f : R n → R n , f T is the current 5) where τ is the tangent to f (γ) with the same orientation as τ , τ (f (x)) = D τ f (x)/|D τ f (x)|. As above, D τ f (x) denotes the tangential derivative of f along γ, which exists H 1 -almost everywhere on γ since f is Lipschitz on…”

“…For a specific problem of physical interest, namely, dislocations in a cubic crystal, we give in Section 4 an algebraic lower bound and an explicit expression for the H 1 -elliptic envelope in the case of small Burgers vector. An application of the tools derived here to the study of dislocations in a three-dimensional discrete model of crystals, which has partly motivated the present work, will be discussed separately [5].…”

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

“…This reduces the problem to a two-dimensional domain Ω ⊂ R 2 . So far most rigorous results have been restricted to this setting, but in recent work of Conti, Garroni & Ortiz [CGO15] the line tension limit has been derived in a full three-dimensional setting for linear elasticity and in the dilute limit (see also [CGM14], [SvG14]). Thus an extension of the results below to a more general three-dimensional setting might be possible, but is far from obvious.…”

“…In crystal plasticity one considers only those fields γ p which correspond to a superposition of slips across a given set of slip planes and corresponding Burgers vectors, see, e.g., [AD15] or [CGO15]. Similarly one can consider multiple dislocations and their corresponding slip lines.…”

Homogenization of vector-valued partition problems and dislocation cell structures in the plane

“…This has been done in several situations, ranging from point singularities in two dimensions [6,36,40] to a general three-dimensional setting [16]. Most of these results require the additional assumption that the admissible dislocation densities correspond to well separated dislocations in order to obtain compactness and to guarantee that sequences with bounded regularized elastic energy concentrate on dislocation lines with finite length and multiplicity, i.e., μ has finite total variation.…”