Covariance in plasticity

Abstract

Covariance is imposed to the second law of thermodynamics and consequences are shown for elasticperfectly plastic bodies. In this setting, pointwise balances of standard and configurational actions, constitutive restrictions on the first Piola-Kirchhoff stress and the Eshelby one, and the structure of the dissipation are all derived from a unique source.

Mechanics of Material Mutations

“…The approach has been coupled with deformations in Coleman and Gurtin (1967) and Halphen and Nguyon (1975), with a subsequent rich literature, in the majority of cases related to plasticity and/or damage (see, e.g., Krajcinovic, 1996). The balance of microstructural actions can be reduced to the evolution equation that 42 Consequences of the covariance of the balance of energy of elastic simple bodies are discussed in Marsden and Hughes (1983), while for the covariance of the second law of thermodynamics, in the case of elastic-plastic materials, the first theorem published is in Mariano (2013). appears in internal variable schemes in the absence of external body actions (including even possible rotational microstructural inertia), microstress, and when the self-action is the sum of conservative and dissipative components (see Mariano, 2002 for details). However, the relation is just formal: the difference in the use of the notion of an observer continues to distinguish the two approaches.…”

Trends and challenges in the mechanics of complex materials: a view

“…What should we imagine for phenomena involving irreversible strain (as in plasticity) or dissipative stress components (as in viscoelasticity)? For finitestrain plasticity, a theorem proven in 2013 furnishes the answer [42]. Versions of it apt for traditional viscoelasticity and the multi-field setting for complex materials are possible.…”

“…In particular, it is convenient to express in terms of time derivative of ψ the covariance condition. To this aim, we can adapt to the setting discussed here the pertinent expression introduced in the special case of finite-strain plasticity discussed in [42]. (Incidentally, as regards plasticity, should we identify ν with the slip rate in single crystals or the plastic factor in the multiplicative decomposition of F, or else a function of elastic invariants, we could recover existing models of straingradient plasticity as special prominent offspring of the model-building framework discussed here.)…”

“…Here I sketch just the essential steps of the procedure. Applications to finite-strain plasticity based on the multiplicative decomposition of the deformation gradient can be found in [42].…”

A three-dimensional study of coupled grain boundary motion with junctions

“…nonlinear elasticity), these balance laws, which are independent of the standard balance laws, are derived from the horizontal variation of the energy functional [20]. In nonconservative setting such balances are usually postulated (rather than derived) as independent relations [21,22]; an exception is the recent work by Mariano [23,24] where the balance of configurational forces is derived from the invariance of the so-called relative power under roto-translating changes in observers. In our work (see also [2,6]), the configurational balance laws appear in the form of kinetic relations (see §4) which are motivated from the local dissipation inequalities obtained from (3.9).…”