On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions

Abstract

We provide (upper and lower) scaling bounds for a singular perturbation model for the cubic-to-tetragonal phase transformation with (partial) displacement boundary data. We illustrate that the order of lamination of the affine displacement data determines the complexity of the microstructure. As in \cite{RT21} we heavily exploit careful Fourier space localization methods in distinguishing between the different lamination orders in the data.

arXiv comment34 pages, 6 figures, comments welcome; added argument in Proposition 3.6, corrected typos

References in corpus (12)

• Energy scaling and branched microstructures in a model for shape-memory alloys with SO(2) invariance
• A Rigidity Result for a Reduced Model of a Cubic-to-Orthorhombic Phase Transition in the Geometrically Linear Theory of Elasticity
• Deformation concentration for martensitic microstructures in the limit of low volume fraction
• On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square
• Exact constructions in the (non-linear) planar theory of elasticity: From elastic crystals to nematic elastomers
• Quasiconvexity at the boundary and the nucleation of austenite
• On Scaling Laws for Multi-Well Nucleation Problems without Gauge Invariances
• On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_3$ Structure
• On the Energy Scaling Behaviour of Singular Perturbation Models with Prescribed Dirichlet Data Involving Higher Order Laminates
• A Compactness and Structure Result for a Discrete Multi-Well Problem with $SO(n)$ Symmetry in Arbitrary Dimension
• 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