A 1D phase macroscopic phase field model for dislocations anda second order Gamma-limit.

We study the asymptotic behaviour in terms of $\Gamma$-convergence of the following one dimensional energy $$ F_\e(u)= \mu_\e\int_I\int_I\frac{|u(x)-u(y)|^2}{|x-y|^2}\,dx\,dy +\eta_\e\int_I W\left(\frac{u(x)}{\e}\right)dx $$ 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 $\mu_\e$ and $\eta_\e$ 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. 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.