*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
$$
F_{\varepsilon}(u)=
\mu_{\varepsilon\int}_{I\int}_{I\frac{u}(x)-u(y)^{2}{x}-y^{2}\,dx\,dy
}
+\eta_{\varepsilon\int}_{I} W\left(\frac{u(x)}{\varepsilon}\right)dx
$$
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.

**Keywords:**
$\Gamma$-convergence, dislocations, Phase-transitions, Asymptotic development

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