Deformation concentration for martensitic microstructures in the limit of low volume fraction

Abstract

We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of Γ-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for SBV p functions whose jump sets have a prescribed orientation.

On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

“…Our result should be viewed in the context of scaling laws in the calculus of variations in general and more specifically in the modelling of shape-memory alloys and related phase transformation problems (see [37] and [45] for surveys on this). In the context of the modelling of shape-memory alloys, scaling laws, providing some insights on the possible behaviour of energy minimizers, have been deduced in various settings [3,5,6,8,9,14,15,20,[34][35][36][38][39][40][41][42]55]. For certain models, in subsequent steps, even finer properties (such as for instance almost periodicity results) have been derived [13].…”

A branched transport limit of the Ginzburg-Landau functional

“…From (1.2) and the discussion around (1.7), for type-I superconductors, the Ginzburg-Landau functional can be seen as a non-convex, non-local (in u) functional favoring oscillations, regularized by a surface term which selects the lengthscales of the microstructures. The appearance of branched structures for this type of problem is shared by many other functionals appearing in material sciences such as shape memory alloys [ CDZ17], we go one step further and prove that, after a suitable anisotropic rescaling, configurations of low energy converge to branched patterns. The two main difficulties in our model with respect to the one studied in [OV10] are the presence of an additional lengthscale (the penetration length) and its sharp limit counterpart, the Meissner condition ρB = 0 which gives a nonlinear coupling between u and B.…”

Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

“…A variant of I in which the Dirichlet boundary condition is replaced by the elastic energy of austenite outside R Lx ,Ly has been studied in [19], a vectorial extension in [23], resulting in a rich phase diagram. The limit ε ∼ θ 2 → 0 of a similar model was addressed in [21].…”

“…6 the relation between L 2 and L ∞ bounds changes, and requires a different treatment of the regions close to the top and bottom boundaries. While the explicit behaviour on θ is not necessary to obtain asymptotic self-similarity of minimizers (see Theorem 2), we expect that such bounds might be helpful for proving explicit self-similarity of minimizers in the limit of low volume fractions, which could be taken along the lines of [21]. For a slightly simpler model arising in the variational study of type-I-semiconductors, the explicit self-similar minimizer is completely charcterized in the limit of low-volume fractions in [28].…”