Micromechanics of dislocations in solids: $J$-, $M$-, and $L$-integrals and their fundamental relations

The aim of the present work is the unification of incompatible elasticity theory of dislocations and Eshelbian mechanics leading naturally to Eshelbian dislocation mechanics. In such a unified framework, we explore the utility of the $J$-, $M$-, and $L$-integrals. We give the physical interpretation of the $M$-, and $L$-integrals for dislocations, connecting them with established quantities in dislocation theory such as the interaction energy and the $J$-integral of dislocations, which is equivalent to the well-known Peach-Koehler force. The $J$-, $M$-, and $L$-integrals for dislocations have been studied in the framework of three-dimensional, incompatible, linear elasticity. First of all, the general formulas of the $J$-, $M$-, and $L$-integrals for dislocations are given. Next, the examined integrals are specified for straight dislocations. Finally, the explicit formulas of the $J$-, $M$-, and $L$-integrals are calculated for straight (screw and edge) dislocations in isotropic materials. The obtained results reveal the physical interpretation and significance of the $M$-, and $L$-integrals for straight dislocations. The $M$-integral between two straight dislocations (per unit dislocation length) is equal to the half of the interaction energy of the two dislocations (per unit dislocation length) depending on the distance and on the angle, plus twice the corresponding pre-logarithmic energy factor. The $L_3$-integral between two straight dislocations is the $z$-component of the configurational vector moment or the rotational moment about the $z$-axis caused by the interaction between the two dislocations. Fundamental relations between the $J$-, $M$-, and $L_3$-integrals are derived showing the inherent connection between them. The relations connecting directly the $J$-, $M$-, and $L_3$-integrals with the interaction energy are obtained.