  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.
