A Rigidity Result for a Reduced Model of a Cubic-to-Orthorhombic Phase Transition in the Geometrically Linear Theory of Elasticity

Abstract

We study a simplified two-dimensional model for a cubic-to-orthorhombic phase transition occuring in certain shape-memory-alloys. In the low temperature regime the linear theory of elasticity predicts various possible patterns of martensite arrangements: Apart from the well known laminates this phase transition displays additional structures involving four martensitic variants -- so called crossing twins. Introducing a variational model including surface energy, we show that these structures are rigid under small energy perturbations. Combined with an upper bound construction this gives the optimal scaling behavior of incompatible microstructures. These results are related to papers by Capella and Otto as well as to a paper by Dolzmann and Müller.

References in corpus (1)

• Energy scaling and branched microstructures in a model for shape-memory alloys with SO(2) invariance

Cited by in corpus (9)

• On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square
• Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations
• On Scaling Laws for Multi-Well Nucleation Problems without Gauge Invariances
• On the Energy Scaling Behaviour of Singular Perturbation Models with Prescribed Dirichlet Data Involving Higher Order Laminates
• On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_3$ Structure
• Minimal Energy for Geometrically Nonlinear Elastic Inclusions in Two Dimensions
• On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators
• On a probabilistic model for martensitic avalanches incorporating mechanical compatibility
• On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions