Eshelby generalization for the dynamic [formula omitted] integrals

Abstract

The dynamic generalization—in the presence of inertia forces—of Eshelby's force on an elastic singularity is presented, where the total change of the energy of the system in two different defect motions, differing by an infinitesimal displacement throughout the history of the motion, is computed by considering the difference in the work of the tractions (with inertia forces considered as body forces) on a cut-out surface in Eshelby's thought ‘cut and re-insert’ experiment needed to realize the shift of the defect in the different motions. This expression, which coincides with a surface-independent obtained by Fletcher (1976) by applying Noether's theorem applied on the Lagrangian, is defined as the dynamic J integral. Changes of the energy of the system as computed by the changes in the work of the tractions (by the same thought experiment) needed to realize the rotation of the defect yield an expression that coincides with another expression obtained by Fletcher, and is defined as the dynamic L integral with meaning of a moment on an elastic singularity, while changes in the work of the tractions with respect to a self-similar scaling parameter coincide with another conserved expression in Fletcher, which is defined as the dynamic M integral. To cite this article: X. Markenscoff, C. R. Mecanique 334 (2006).