Seminar 30th October 2012 noon 54/8033
Phase transitions in solids: from atomistic waves to a continuum picture.
Johannes Zimmer
Bath
- Web page
- http://www.maths.bath.ac.uk/~jz203/people/
- Categories
- Complex Systems
- Submitter
- Luke Goater
Phase transitions in solids: from atomistic waves to a continuum picture The equations of elasticity in one space dimension, $u_{tt} = \sigma(u_x)_x$, become ill-posed if the potential energy density is nonconvex, or equivalently if $\sigma$ is non-monotone. This complication necessarily arises in the theory of so-called martensitic phase transitions, which are diffusionless solid-solid transformations where several stable phases can coexist.
Different regularisations of this ill-posed problem have been proposed; we will here focus on so-called kinetic relations, which relate the velocity of a moving interface to a driving force. Phenomenological kinetic relations have been proposed, but a natural question is whether they can in simple situations be derived from first principles, namely atomistic considerations.
To investigate this question, we study the simplest one-dimensional chain model of martensitic materials, where neighbouring atoms are coupled by a spring with bi-quadratic potential. We present existence results for travelling waves and discuss non-uniqueness of microscopic solutions. This non-uniqueness will be discussed in light of the macroscopic kinetic relation.