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

In this short note we derive constitutive relations for the response of elastic solids, which are transversely isotropic and those that have two preferred directions of symmetry wherein the material moduli of the body depend on its density, undergoing small deformations. Such constitutive relations have relevance to the response of rocks, concrete, ceramics, bones, and porous metals. The constitutive relations that we develop reduce appropriately to the classical linearized elastic constitutive relation, when appropriate material moduli are set to zero. The new constitutive relations, when restricted to those that are linear in both the stress and the linearized strain, have terms in addition to those for a classical linearized elastic body, and it would be interesting to investigate the consequence of these additional terms with regard to the response of the body.