Still states of bistable lattices, compatibility, and phase transition

TL;DR

This paper characterizes the effective energy landscape of a two-dimensional bistable lattice, revealing a flat energy bottom due to numerous force-free deformed states, and explores compatibility conditions for such states.

Contribution

It provides a complete characterization of the flat bottom set of the effective energy in 2D bistable lattices, extending previous 1D analyses by including compatibility conditions.

Findings

The effective energy density is zero within a specific strain set.

A family of still states densely fills the flat bottom set.

Compatibility conditions for deformations are derived and analyzed.

Abstract

A two-dimensional bistable lattice is a periodic triangular network of non-linear bi-stable rods. The energy of each rod is piecewise quadratic and has two minima. Consequently, a rod undergoes a reversible phase transition when its elongation reaches a critical value. We study an energy minimization problem for such lattices. The objective is to characterize the effective energy of the system when the number of nodes in the network approaches infinity. The most important feature of the effective energy is its "flat bottom". This means that the effective energy density is zero for all strains inside a certain three-dimensional set in the strain space. The flat bottom occurs because the microscopic discrete model has a large number of deformed states that carry no forces. We call such deformations still states. In the paper, we present a complete characterization of the "flat bottom" set in terms of the parameters of the network. This is done by constructing a family of still states whose average strains densely fill the set in question. The two-dimensional case is more difficult than the previously studied case of one-dimensional chains, because the elongations in two-dimensional networks must satisfy certain compatibility conditions that do not arise in the one-dimensional case. We derive these conditions for small and arbitrary deformations. For small deformations a complete analysis is provided.