Eshelby force and power for uniform bodies

Abstract

Inspired by the seminal works of Eshelby (Philos Trans R Soc A 244A:87–112, 1951, J Elast 5:321–335, 1975) on configurational forces and of Noll (Arch Ration Mech Anal 27:1–32, 1967) on material uniformity, we study a thermoelastic continuum undergoing volumetric growth and in a dynamical setting, in which we call the divergence of the Eshelby stress the Eshelby force. In the classical statical case, the Eshelby force coincides with the negative of the configurational force. We obtain a differential identity for the modified Eshelby stress, involving the torsion of the connection induced by the material isomorphism of a uniform body, which includes, as a particular case, that found by Epstein and Maugin (Acta Mech 83:127–133, 1990). In this identity, the divergence of the modified Eshelby stress with respect to this connection of the material isomorphism takes the name of modified Eshelby force. Moreover, we show that Eshelby’s variational approach (1975) can be used to formulate not only the balance of material momentum, but also the balance of energy. In this case, we find that what we call Eshelby power is the temporal analogue of the Eshelby force, and we obtain a differential identity for the modified Eshelby power. This leads to concluding that the driving force for the process of growth–remodelling is the Mandel stress. Eventually, we find that the relation between the differential identities for the modified Eshelby force and modified Eshelby power represents the mechanical power expended in a uniform body to make the inhomogeneities evolve.