# Low volume-fraction microstructures in martensites and crystal plasticity

@article{Conti2015LowVM, title={Low volume-fraction microstructures in martensites and crystal plasticity}, author={Sergio Conti and Barbara Zwicknagl}, journal={arXiv: Analysis of PDEs}, year={2015} }

We study microstructure formation in two nonconvex singularly-perturbed variational problems from materials science, one modeling austenite-martensite interfaces in shape-memory alloys, the other one slip structures in the plastic deformation of crystals. For both functionals we determine the scaling of the optimal energy in terms of the parameters of the problem, leading to a characterization of the mesoscopic phase diagram. Our results identify the presence of a new phase, which is… Expand

#### 9 Citations

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

- Physics, Mathematics
- 2015

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… Expand

Geometry of martensite needles in shape memory alloys

- Computer Science, Physics
- ArXiv
- 2019

This work presents a two-dimensional shape optimization model based on finite elasticity and discusses its numerical solution, indicating that the tapering profile of the needles can be understood within infinite elasticity, but not with linearized elasticity. Expand

Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys

- Mathematics, Physics
- 2020

We consider a singularly-perturbed two-well problem in the context of planar geometrically linear elasticity to model a rectangular martensitic nucleus in an austenitic matrix. We derive the scaling… Expand

Piecewise affine stress-free martensitic inclusions in planar nonlinear elasticity

- Mathematics, Medicine
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2017

We consider a partial differential inclusion problem which models stress-free martensitic inclusions in an austenitic matrix, based on the standard geometrically nonlinear elasticity theory. We show… Expand

On the Energy Scaling Behaviour of Singular Perturbation Models Involving Higher Order Laminates

- Mathematics
- 2021

Motivated by complex microstructures in the modelling of shape-memory alloys and by rigidity and flexibility considerations for the associated differential inclusions, in this article we study the… Expand

Energy Bounds for a Compressed Elastic Film on a Substrate

- Physics, Computer Science
- J. Nonlinear Sci.
- 2017

The fracture term is included, transforming the elastic minimisation into a free boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination in a compressed elastic film which delaminates from a substrate. Expand

A branched transport limit of the Ginzburg-Landau functional

- Physics, Mathematics
- 2017

We study the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a… Expand

On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

- Mathematics
- 2021

In this article we derive an (almost) optimal scaling law for a singular perturbation problem associated with the Tartar square. As in [Win97, Chi99], our upper bound quantifies the well-known… Expand

Branching of twins in shape memory alloys revisited

- Physics, Materials Science
- 2019

We study the branching of twins appearing in shape memory alloys at the interface between austenite and martensite. In the framework of three-dimensional non-linear elasticity theory, we propose an… Expand

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