On surface energies in scaling laws for singular perturbation problems for martensitic phase transitions

TL;DR

This paper compares various surface energies in singular perturbation problems for martensitic phase transitions, demonstrating that the derived scaling laws are robust across different energy types and anisotropies, with minimal dependence on detailed surface energy structures.

Contribution

It establishes that scaling laws in martensitic phase transformations are largely unaffected by the specific form of surface energies, including anisotropic and higher-order laminates.

Findings

Scaling laws are robust across diffuse, sharp, and interpolated surface energies.

Anisotropic surface energies yield the same scaling behavior as isotropic ones.

Only the innermost lamination direction significantly influences the scaling law.

Abstract

The objective of this article is to compare different surface energies for multi-well singular perturbation problems associated with martensitic phase transformations involving higher order laminates. We deduce scaling laws in the singular perturbation parameter which are robust in the choice of the surface energy (e.g., diffuse, sharp, an interpolation thereof or discrete). Furthermore, we show that these scaling laws do not require the presence of isotropic surface energies but that generically also highly anisotropic surface energies yield the same scaling results. More precisely, the presence of essentially generic partial directional derivatives in the regularization terms suffices to produce the same scaling behaviour as in the isotropic setting. The only sensitive directional dependences are directly linked to the lamination directions of the well structure -- and even for these only the ``inner-most'' lamination direction is of significance in determining the scaling law. In view of experimental applications, this shows that also for higher-order laminates, the precise structure of the surface energies -- which is often very difficult to determine experimentally -- does not have a crucial impact on the scaling behaviour of the investigated structures but only enters when considering finer properties.