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Eshelbian dislocation mechanics: $J$-, $M$-, and $L$-integrals of straight dislocations

In this work, using the framework of (three-dimensional) Eshelbian dislocation mechanics, we derive the $J$-, $M$-, and $L$-integrals of a single (edge and screw) dislocation in isotropic elasticity as a limit of the $J$-, $M$-, and $L$-integrals between two straight dislocations as they have recently been derived by Agiasofitou and Lazar [Int. J. Eng. Sci. 114 (2017) 16-40]. Special attention is focused on the $M$-integral. The $M$-integral of a single dislocation in anisotropic elasticity is also derived. The obtained results reveal the physical interpretation of the $M$-integral (per unit length) of a single dislocation as the total energy of the dislocation which is the sum of the self-energy (per unit length) of the dislocation and the dislocation core energy (per unit length). The latter can be identified with the work produced by the Peach-Koehler force. It is shown that the dislocation core energy (per unit length) is twice the corresponding pre-logarithmic energy factor. This result is valid in isotropic as well as in anisotropic elasticity. The only difference lies on the pre-logarithmic energy factor which is more complex in anisotropic elasticity due to the anisotropic energy coefficient tensor which captures the anisotropy of the material.