THERMODYNAMIC RESTRICTIONS ON CONSTITUTIVE-EQUATIONS FOR 2ND-DEFORMATION-GRADIENT INELASTIC BEHAVIOR

The aim of this paper is to examine the possibility of a model of non-local inelasticity on thermodynamic grounds in the absence of higher-order stresses or additional kinematic entities. On considering a material that cannot support higher-order stresses, it is found that thermodynamics does not allow an explicit nonlocal dependence in the free-energy and hence the stress and entropy response, but allows it in the strength and internal variable rate response. Rate-independent plasticity presents its own peculiarities in the proof of this result, which are also inherent in deriving the constitutive restrictions for the local material. An account of the basic thermodynamic framework and the reasoning used in these proofs is provided in listing the result for the conventional rare-independent solid. This procedure represents the building blocks for the derivation of the constitutive restrictions of the non-local theory in which second-deformation-gradient-dependent constitutive postulates for inelastic materials are introduced. Such postulates involve a material length scale due to the gradient dependence and achieve uniqueness in constitutive response trivially. The constitutive restrictions are shown not to preclude the existence of a class of second-deformation-gradient-dependent inelastic constitutive postulates for which the corresponding bvp of incremental equilibrium requires no extra boundary conditions above the conventional specification. The general considerations mentioned above are utilized to illustrate the constitutive structure of a rate-independent metal single crystal deforming by multiple slip, whereby contact with earlier work in this area is established. The thermodynamic results are also used to derive conditions for the existence of a gradient-dependent potential for the inelastic strain rate for inelastic rate-dependent materials.