Abstract
Let $\Omega \subset \Bbb{R}^2$ denote a bounded domain whose boundary $\partial \Omega$ is Lipschitz and contains a segment $\Gamma_0$ representing the austenite-twinned martensite interface. We prove
$$\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla u(x,y))dxdy=0}$$
for any elastic energy density $\varphi : \Bbb{R}^2 \rightarrow [0,\infty)$ such that $\varphi(0,\pm 1)=0$. Here ${\cal W}(\Omega)$ consists of all Lipschitz functions $u$ with $u=0$ on $\Gamma_0$ and $|u_y|=1$ a.e. Apart from the trivial case $\Gamma_0 \subset \reel \times \{a\},~a\in \Bbb{R}$, this result is obtained through the construction of suitable minimizing sequences which differ substantially for vertical and non-vertical segments.
Similar content being viewed by others
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Elfanni, A., Fuchs, M. The behaviour of microstructures with small shears of the austenite-martensite interface in martensitic phase transformations . Z. angew. Math. Phys. 54, 937–953 (2003). https://doi.org/10.1007/s00033-003-1102-3
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s00033-003-1102-3