In this paper we derive a line tension model for dislocations in 3D starting from a geometrically nonlinear elastic energy with quadratic growth. In the asymptotic analysis, as the amplitude of the Burgers vectors (proportional to the lattice spacing) tends to zero, we show that the elastic energy linearizes and the line tension energy density, up to an overall constant rotation, is identified by the linearized cell problem formula given in [S.

We study the limit behaviour of singularly-perturbed elliptic functionals of the formwhere u is a vector-valued Sobolev function, v ∈ [0, 1] a phase-field variable, and ε k > 0 a singular-perturbation parameter; i.e., ε k → 0, as k → +∞. Under mild assumptions on the integrands f k and g k , we show that if f k grows superlinearly in the gradient-variable, then the functionals F k Γ-converge (up to subsequences) to a brittle energy-functional; i.e., to a free-discontinuity functional whose surface integrand does not depend on the jump-amplitude of u. This result is achieved by providing explicit asymptotic formulas for the bulk and surface integrands which show, in particular, that volume and surface term in F k decouple in the limit.The abstract Γ-convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.

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We study variational models for dislocations in three dimensions in the line-tension scaling. We present a unified approach which allows to treat energies with subquadratic growth at infinity and other regularizations of the singularity near the dislocation lines. We show that the asymptotics via Gamma convergence is independent of the specific choice of the energy and of the regularization procedure.

In this paper we study the asymptotic behaviour of phase-field functionals of Ambrosio and Tortorelli type allowing for small-scale oscillations both in the volume and in the diffuse surface term. The functionals under examination can be interpreted as an instance of a static gradient damage model for heterogeneous materials. Depending on the mutual vanishing rate of the approximation and of the oscillation parameters, the effective behaviour of the model is fully characterised by means of Γ-convergence.