Hierarchical pattern formation in elastic materials

Abstract

We study hierarchical structures such as branched twins in elastic materials based on a model of martensitic materials in which hierarchical twinning near the habit plane (austenite-martensite interface) is a new and crucial ingredient. The model includes (1) a triple-well potential (θ⁶ model) in local strain, (2) strain gradient terms up to second order in strain and fourth-order in gradient, and (3) all symmetry-allowed compositional fluctuation-induced strain gradient terms which favor hierarchical structures and enable communication between macroscopic (cm) and microscopic (Ao) regions essential for shape memory. Below the transition temperature (T₀) we obtain the conditions under which branching of twins is energetically favorable. This hierarchy of length scales provides a related hierarchy of time scales and thus the possibility of non-exponential decay. Results based on 2D simulations of the time-dependent Ginzburg-Landau (TDGL) equation are shown for twins, tweed and hierarchy formation. We also apply stability analysis to study the formation of modulated structures at early time and obtain an approximate phase diagram for the model.