Crystal plasticity: The Hamilton-Eshelby stress in terms of the metric in the intermediate configuration

Abstract

The Hamilton-Eshelby stress is a basic ingredient in the description of the evolution of point, lines and bulk defects in solids. The link between the Hamilton-Eshelby stress and the derivative of the free energy with respect to the material metric in the plasticized intermediate configuration, in large strain regime, is shown here. The result is a modified version of Rosenfeld-Belinfante theorem in classical field theories. The origin of the appearance of the Hamilton-Eshelby stress (the non-inertial part of the energy-momentum tensor) in dissipative setting is also discussed by means of the concept of relative power

Covariance in plasticity

“…[8]). 17 -In absence of plastic phenomena, both (5.9) and (5.10) have counterparts in the Doyle-Ericksen formula [30,31] and Rosenfeld-Belinfante theorem [27]. In other words, the stress P is determined by energetic variations induced by variations in the metric in the ambient space (pull-back a part), whereas P is determined by analogous variations in the material metric, so by mutations in the structure of the matter.…”

“…The matter is standard and I do not reproduce it here. -Relation (5.10) has been obtained using a different procedure in [27] under the assumption that the intermediate configuration be isotropic. Here, I remove that assumption and show the validity of the formula in a more general setting.…”