Weak solutions for one-dimensional non-convex elastodynamics

TL;DR

This paper investigates the existence and properties of weak solutions in one-dimensional elastodynamics with non-convex energy, revealing microstructure formation and solution regularity for phase transition models.

Contribution

It demonstrates local existence of weak solutions for all smooth initial data and characterizes microstructure formation in elastodynamics with non-convex energy functions.

Findings

Microstructures form instantaneously for certain initial strains.

Existence of local weak solutions for all smooth initial data.

Solutions can be smooth initially and develop microstructures later.

Abstract

We explore local existence and properties of classical weak solutions to the initial-boundary value problem of a one-dimensional quasilinear equation of elastodynamics with non-convex stored-energy function, a model of phase transitions in elastic bars proposed by Ericksen [19]. The instantaneous formation of microstructures of local weak solutions is observed for all smooth initial data with initial strain having its range overlapping with the phase transition zone of the Piola-Kirchhoff stress. As byproducts, it is shown that such a problem admits a local weak solution for all smooth initial data and local weak solutions that are smooth for a short period of time and exhibit microstructures thereafter for some smooth initial data. In a parallel way, we also include some results concerning one-dimensional quasilinear hyperbolic-elliptic equations.