Search for: All records

We present an asymptotic framework that rigorously generates nonlinear constitutive relations between stress and linearized strain for elastic bodies. Each of these relations arises as the leading order relationship satisfied by a one-parameter family of nonlinear constitutive relations between stress and nonlinear strain. The asymptotic parameter limits the overall range of strains that satisfy the corresponding constitutive relation in the one parameter family while the stresses can remain large (relative to a fixed stress scale). This differs from classical linearized elasticity where a fixed constitutive relation is assumed, and the magnitude of the displacement gradient serves as the asymptotic parameter. Also unlike classical approaches, the constitutive relations in our framework are expressed as implicit relationships between stress and strain rather than requiring stress explicitly expressed as a function of strain, adding conceptual simplicity and versatility. We demonstrate that our framework rigorously justifies nonlinear constitutive relations between stress and linearized strain including those with density-dependent Young’s moduli or derived from strain energies beyond quadratic forms. 

There seems to be a basic misconception in several recent papers concerning the material symmetry of bodies in configurations that are pre-stressed. In this short paper we point to the source of the error and show that the material symmetry that is possible depends on the nature of the pre-stress. We also extend the results for material symmetry which have been well known within the context of simple elastic solids to the general class of simple materials. This generalization has relevance to the material symmetry of biological solids that are viscoelastic. 

Trabecular bone, a solid that has a heterogeneous porous structure, demonstrates nonlinear stress–strain relationship, even within the small strain region, when subject to stresses. It also exhibits different responses when subject to tension and compression. This study presents the development of an implicit constitutive relation between the stress and the linearized strain specifically tailored for trabecular bone-like materials. The structure of the constitutive relation requires the solution of the balance of linear momentum and the constitutive relations simultaneously, and in view of this, a two-field mixed finite element model capable of solving general boundary value problems governed by a system of coupled equations is proposed. We investigate the effects of nonlinearity and heterogeneity in a dogbone-shaped sample. Our study is able to capture the significant nonlinear characteristics of the response of the trabecular bone undergoing small strains in experiments, in both tension and compression, very well. 

Recently, Arumugam et al. (2023) developed a constitutive relation for the response of isotropic inhomogeneous compressible elastic solids in order to describe the response of the trabecular bone. Since porous solids such as bones, cement concrete, rocks, metallic alloys, etc., are anisotropic, in this short note we develop a constitutive relation for such bodies that exhibit transverse isotropy and also having two preferred directions of symmetry. Another characteristic of bones is that they exhibit different response characteristics in tension and compression, and hence any constitutive relation that is developed has to be capable of describing this. Also, the material moduli depend on both the density and the mean value of the stress (mechanical pressure), as is to be expected in a porous solid. In the constitutive relation that is developed in this paper, though the stress and the linearized strain appear linearly in the constitutive relation, the relationship is nonlinear. We also derive the response of such solids when undergoing uniaxial extension and compression, simple shear and torsion.