# A $1$D phase field model for dislocations and a second order $\Gamma$limit

created by focardi on 29 Dec 2006
modified on 02 May 2008

[BibTeX]

Published Paper

Inserted: 29 dec 2006
Last Updated: 2 may 2008

Journal: SIAM Multiscale Model. Simul.
Volume: 6
Number: 4
Pages: 1098-1124
Year: 2007

Abstract:

We study the asymptotic behaviour in terms of $\Gamma$-convergence of the following one dimensional energy $$F\varepsilon(u)= \mu\varepsilon\intI\intI\frac{ u (x)-u(y) 2}{ x -y 2}\,dx\,dy +\eta\varepsilon\intI W\left(\frac{u(x)}{\varepsilon}\right)dx$$ where $I$ is a given interval, $W$ is a one-periodic potential that vanishes exactly on $*Z*$.

Different regimes for the asymptotic behaviour of the parameter $\mu_\varepsilon$ and $\eta_\varepsilon$ are considered. In a very diluted regime we get a limit defined on $BV(I)$ and proportional to the total variation of $u$. In this particular case we also consider the limit of a suitable boundary value problem for which we characterize the second order $\Gamma$-limit.

\noindent The study under consideration is motivated by the analysis of a variational model for a very important class of defects in crystals, the dislocations, and the derivation of macroscopic models for plasticity.