Uniform distribution of dislocations in Peierls–Nabarro models for semi-coherent interfaces

Abstract

In this paper we introduce Peierls-Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy-converges to a limit functional comprised of two contributions: one is given by a constant c ∞ > 0 gauging the minimal energy induced by dislocations at the interface, and corresponding to a uniform distribution of edge dislocations; the other one accounts for the far field elastic energy induced by the presence of further, possibly not uniformly distributed, dislocations. After assuming periodic boundary conditions and formally considering the limit from semi-coherent to coherent interfaces, we show that c ∞ is reached when dislocations are evenly-spaced on the one dimensional circle.

A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization

“…The fundamentals of our approach rest on recent results concerning sparsity for variational inverse problems: it has been empirically observed that the presence of a regularizer promotes the existence of sparse solutions, that is, minimizers that can be represented as a finite linear combination of simpler atoms. This effect is evident in reconstruction problems [34,40,63,64,65], as well as in variational problems in other applications, such as materials science [37,38,46,53]. Existence of sparse solutions has been recently proven for a class of general functionals comprised of a fidelity term, mapping to a finite dimensional space, and a regularizer: in this case atoms correspond to the extremal points of the unit ball of the regularizer [11,14].…”

A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation

“…We point out that, at least under some geometric assumptions on Ω and prescribing the set S u accordingly 2 , one can relate the elastic energy to the H 1 2 -norm of the jump of u, whence leading to 1d models which have been already investigated (see [28]; related models for different lattice mismatches can be found in [26,27] and references therein). Related to this setting is [31,19] where a similar analysis is done in 2d.…”

Derivation of a Line-Tension Model for Dislocations from a Nonlinear Three-Dimensional Energy: The Case of Quadratic Growth

“…Previous results were indeed confined to special geometries where the models could be treated in a two-dimensional framework and studied by means of Γ-convergence in different relevant energetic regimes. In these reduced models dislocations can be seen either as points in the cross section of a cylindrical domain (see [11,17,19,23,35]) with a strong similarity to the case of filaments of currents in superconductors [42,43,30] or as lines confined to a single slip plane and their energy described by nonlocal phase field models (generalizing the Peierls-Nabarro model [31,32,24,10,12,13,14,20,15]).…”