The relative power and its invariance

Abstract

Abstract. -The relative power of actions in Cauchy bodies su¤ering mutations due to defect evolution is introduced. It is shown that its invariance under the action of the Euclidean group over the ambient space and the material space allows one to obtain (i) the balance of standard and configurational actions and (ii) the identification of configurational ingredients from a unique source.

Mechanics of Material Mutations

“…Along this path, configurational forces, couples, and stresses are postulated a priori and are identified later (at least some of them) in terms of energy and standard stresses, by using a procedure based on a requirement of invariance with respect to reparameterization of the boundary pertaining to the region in B occupied by what we are considering to be a defect (see details in Gurtin, 1995; see also Maugin, 1995 for other approaches). Alternatively, I use the velocity field previously mentioned to write what I call relative power (see Mariano, 2009 for its first definition in a nonconservative setting, with improvements in Mariano, 2012a), which is the power of canonical external actions on a generic part of the body augmented by what I call the power of disarrangements, which is a functional involving energy fluxes determined by the rearrangement of matter and configurational forces and couples due to breaking of material bonds and mutationinduced anisotropy. Canonical balances of standard and microstructural actions and the ones of configurational actions follow directly from a 10 The one used by Eshelby (1975) in his seminal article for determining the action on a volumetric defect in an elastic body undergoing large strain.…”

A certain counterpart in dissipative setting of the Noether theorem with no dissipation pseudo-potentials

“…As a consequence, we derive balances of standard and configurational actions from a unique source. The approach refines the proposal in references [12,13] and does not make use of the inverse mapping, which would be questionable because the growth of a defect destroys at least locally the commonly assumed one-to-one instantaneous correspondence between reference and current configurations. Also, our approach avoids a procedure adopted in references [14,15]: the introduction of a configurational stress and other a priori unknown actions that require a subsequent identification in terms of standard actions.…”

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

“…where ∂ x ψ is the explicit derivative of ψ with respect to x. The definition of P dis b (w) differs from what I have proposed previously in [3]. This new version seems to me more satisfactory from the viewpoint of a physical interpretation of the entities defined.…”

“…In dissipative setting there are various ways to arrive at an analogous interpretation. The one proposed in [3] seems to require less assumptions and structure than others (see references and comparisons in [3]). Here I vary in a specific aspect what is proposed in [3] because I think that in this way the physical evidence of that proposal can be clarified better.…”

“…In presence of inertia, there is an inertia contribution in the bulk force b because it can be decomposed additively into inertial and non-inertial components, and a tensor contribution which is given by the kinetic energy time the unit second-rank tensor -the resulting tensor added to P generates the standard energy-momentum tensor. Comparisons with other procedures producing the balances of configurational actions and explaining their role can be found in [3]. Reasons of physical significance, evidence, eventually elegance can allow one to discriminate and prefer one line of reasoning to the other.…”