The energy scaling behaviour of singular perturbation models of staircase type in linearized elasticity for higher order laminates

TL;DR

This paper analyzes the energy scaling laws in singular perturbation models of staircase type within linearized elasticity, revealing how lamination order and rank-one directions influence bounds and demonstrating the bounds' sharpness.

Contribution

It introduces a detailed analysis of how lamination order and symmetrized rank-one directions affect energy scaling bounds in higher-order laminate models.

Findings

Lower scaling bounds depend on lamination order and rank-one directions.

Upper bounds are demonstrated for specific geometries and well configurations.

The bounds are shown to be sharp, emphasizing the importance of the two parameters.

Abstract

We investigate the scaling behaviour of a singular perturbation model within the geometrically linearized theory of elasticity involving data of higher lamination order. We study boundary data which are of staircase type and show rather general lower scaling bounds, both in the setting of prescribed Dirichlet data and for periodic configurations with a mean value constraint. In contrast to the setting without gauge invariances, these lower scaling bounds depend on \emph{two} parameters -- the order of lamination of the boundary data as well as the number of involved (non-)degenerate symmetrized rank-one directions. By discussing upper bounds in specific geometries and for a specific constellation of wells, we give evidence of the sharpness of these lower bound estimates. Hence, it is necessary to keep track of the outlined \emph{two} parameters in deducing scaling laws within the geometrically linearized theory of elasticity.