Abstract

In the presence of dislocations, the elastic deformation tensor $F$ is not a gradient but satisfies the condition $CurlF={\Lambda }_{\mathcal{ℒ}}^T$ (with the dislocation density ${\Lambda }_{\mathcal{ℒ}}$ a tensor-valued measure concentrated in the dislocation $\mathcal{ℒ}$). Then $F\in L^p$ with $1\le p<2$. This peculiarity is at the origin of the mathematical difficulties encountered by dislocations at the mesoscopic scale, which are here modeled by integral $1$-currents free to form complex geometries in the bulk. In this paper, we first consider an energy-minimization problem among the couples $\left(F,\mathcal{ℒ}\right)$ of strains and dislocations, and then we exhibit a constraint reaction field arising at minimality due to the satisfaction of the condition on the deformation curl, hence providing explicit expressions of the Piola–Kirchhoff stress and Peach–Koehler force. Moreover, it is shown that the Peach–Koehler force is balanced by a defect-induced configurational force, a sort of line tension. The functional spaces needed to mathematically represent dislocations and strains are also analyzed and described in a preliminary part of the paper.

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