An implicit constitutive relation for describing the small strain response of porous elastic solids whose material moduli are dependent on the density

Abstract

In this short note, we develop a constitutive relation that is linear in both the Cauchy stress and the linearized strain, by linearizing implicit constitutive relations between the stress and the deformation gradient that have been put into place to describe the response of elastic bodies (Rajagopal, KR. On implicit constitutive theories. Applications of Mathematics 2003; 28: 279–319), by assuming that the displacement gradient is small. These implicit equations include the classical linearized elastic constitutive approximation as well as some classes of constitutive relations that imply limiting strain in tension, as special subclasses. Moreover, the constitutive relations that are developed allow the material moduli to depend on the density; thus, they can be used to describe the response of porous materials, such as porous metals, bone, rocks, and concrete undergoing small deformations.

On the response of anisotropic elastic bodies described by implicit constitutive relations

“…Moreover, materials such as rocks, bone, and so on are also anisotropic. Rajagopal [14] developed constitutive relations for isotropic elastic bodies undergoing small deformations, whose material moduli depend on the density. Here, we have extended that study to two specific classes of anisotropic bodies, transversely isotropic bodies, and bodies which have two preferred directions for response.…”

“…However, in porous elastic bodies wherein the pores are sufficiently small so that the body can yet be considered a continuum, the material moduli, even in small strain response is found to be density dependent, which in virtue of the balance of mass implies that they are dependent on the linearized strain (see discussion in Pauw [1], Nguyen et al, [2], Lydon and Balendran [3], Munro [4] for experiments on concrete, Zhang et al, [5], Luo and Stevens [6] for experiments on ceramics, Manoylov et al, [7], Kovác ȋk [8], Hirose et al, [9] for experiments in powder metallurgy, Helgason et al, [10], Vanleene et al, [11] experiments on bone, Cristescu [12], experiments on rocks). Recently, a constitutive relation to describe the response of porous elastic bodies undergoing small deformations has been put into place within the context of implicit constitutive relations for elastic bodies (see Rajagopal [13,14] and Rajagopal and Saccomandi [15]; also see Rajagopal and Wineman [16] in the case of linear viscoelastic bodies). These new constitutive relations are approximations of nonlinear implicit constitutive relations that describe the response of compressible elastic bodies developed earlier by Rajagopal [17,19,20] (see also Rajagopal and Srinivasa [21]).…”

Pure bending of an elastic prismatic beam made of a material with density-dependent material parameters

“…A special subclass of models for elastic response that are nonlinear in the small strain range are the models wherein the material moduli depend on the density (see, for example, Rajagopal [13] and Rajagopal and Saccomandi [14]). In particular, the literature survey in the work by Rajagopal and Saccomandi [14] reveals that the response of various solid materials can be modelled using models with ''density-dependent Young's modulus.''…”

“…A special subclass of models for elastic response that are nonlinear in the small strain range are the models wherein the material moduli depend on the density (see, for example, Rajagopal [13] and Rajagopal and Saccomandi [14]). In particular, the literature survey in the work by Rajagopal and Saccomandi [14] reveals that the response of various solid materials can be modelled using models with “density-dependent Young’s modulus.” If we abstain from the fact that the concept of Young’s modulus is well defined only in the fully linearised setting, and if the phrase “density-dependent Young’s modulus” is taken literally (see Rajagopal and Saccomandi [14] for further critical remarks in this respect), then the response of these materials is, for example, governed by the constitutive relation:…”

“…(see Rajagopal [13] or Murru and Rajagopal [15]), where Γ and E ref are constant material parameters. Note that the models of type (2a) can be shown to arise when one deals with non-linear elastic solids specified in terms of the Gibbs potential (see Průša et al [16] and Gokulnath et al [17] for details).…”