A 1D Macroscopic Phase Field Model for Dislocations and a Second Order $\Gamma$-Limit
Matteo Focardi, Adriana Garroni · Multiscale Modeling and Simulation · 2007
We study the asymptotic behavior in terms of $\Gamma$-convergence of the one dimensional energy $F_{\varepsilon}(u) = \mu_{\varepsilon} \int_I \int_I \frac{|u(x)-u(y)|^2}{|x-y|^2}\,dx\,dy + \eta_{\varepsilon} \int_I W(\frac{u(x)}{\varepsilon})dx,$ where I is a given interval, and W is a one-periodic potential that vanishes exactly on ${\bf Z}$. Different regimes for the asymptotic behavior of the parameters $\mu_{\varepsilon}$ 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. 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.