# A variational approach to single crystals with dislocations

created by scala on 04 Dec 2018
modified on 05 Dec 2018

[BibTeX]

Accepted Paper

Inserted: 4 dec 2018
Last Updated: 5 dec 2018

Journal: SIAM J. Math. Anal.
Year: 2018

Abstract:

We study the graphs of maps $u:\Omega\rightarrow\mathbb R^3$ whose curl is an integral $1$-current with coefficients in $\mathbb Z^3$. We characterize the graph boundary of such maps under suitable summability property. We apply these results to study a three-dimensional single crystal with dislocations forming general one-dimensional clusters in the framework of finite elasticity. By virtue of a variational approach, a free energy depending on the deformation field and its gradient is considered.

The problem we address is the joint minimization of the free energy with respect to the deformation field and the dislocation lines. We apply closedness results for graphs of torus-valued maps, seen as integral currents and, from the characterization of their graph boundaries we are able to prove existence of minimizers.