A Phenomenological Constitutive Model of Non-Conventional Elastic Response

Abstract

In this paper, a general model of elastic (non-dissipative) behavior is developed. This model belongs to a class of models, developed for the description of complex bodies, in which the local state is assumed to be determined not only by the deformation, but also by a family of additional material parameters. The latter, unlike some additional structures used in the mechanics of complex bodies (e.g., directors, order parameters, internal degrees of freedom), are not considered as interactions of microscopic nature; rather they are considered as variables of macroscopic nature that describe the internal structure of the material, while their rates describe the evolution of the internal structure in the course of deformation. Accordingly, these variables are assumed to evolve continuously with time in a manner that guaranties the reversibility of the applied dynamical process. A covariant theory for the continuum in question is derived by means of invariance properties of the global form of the spatial energy balance equation, under the superposition of arbitrary spatial diffeomorphisms. In particular, it is shown that the assumption of spatial covariance of the equation of balance of energy yields the standard conservation and balance laws of classical mechanics but it does not yield the standard Doyle-Ericksen formula. In fact, the "Doyle-Ericksen formula" derived in this work, has some extra terms in it, which are related directly to the internal structure of the material, as the latter is controlled by the additional parameters. In a similar manner, by assuming the absolute temperature as an additional state variable and by employing the invariance properties of the local form of the spatial balance of energy under superimposed spatial diffeomorphisms, which also include a temperature rescaling, a nonisothermal covariant constitutive theory is naturally obtained. A formal comparison of the proposed elastic material with the standard hyperelastic (Green elastic) solid is also presented.

On the quasi‐yield surface concept in plasticity theory

“…We note that Eq. is invariant under a replacement of t by and accordingly enforces reversible response (see for further details). On the other hand, plastic behavior is an irreversible one , a fact which calls for an appropriate modification of the rate equation .…”

On Spatial Covariance, Second Law of Thermodynamics and Configurational Forces in Continua

“…Accordingly, the proposed scheme resembles the standard internal variables approaches, in the sense that the material metric c in the spatial configuration does not enter explicitly the balance of energy equation. In order to exploit the invariance properties of the balance of energy equation we evaluate Equation (50) at 0 t t  (Yavari et al [10], Panoskaltsis and Soldatos [14]). Then, by applying the standard procedure (see Section 3), the left hand side of Equation (50) reads in view of Equation (49)…”

“…v c A further discussion on this subtle point may be found in Panoskaltsis and Soldatos [14], where an equation like (61) is considered as a first-order principle.…”

“…The integrability conditions can be found by means of Frobenius theorem (see, e.g., Bishop and Goldberg ([4], pp. 151-153) and they have been analyzed in a material setting in Panoskaltsis and Soldatos [14].…”

“…Moreover, he derived the standard splitting of the total energy into internal and kinetic energies and the transformation law for external forces. Since then a lot of effort has been placed and a series of papers have appeared in the literature dealing with the concept of covariant energy balance in both a conservative (see, e.g., Yavari et al [10]; Kanso et al [11]; Yavari and Ozakin [12]; Yavari and Marsden [13]; Panoskaltsis and Soldatos [14]) as well as a dissipative (see, e.g., Yavari [15]; Panoskaltsis et al [16,17]) setting.…”