Separation of compressibility and shear deformation in the elastic energy density (L)

Abstract

A formulation of the elastic energy density for an isotropic medium is presented that permits separation of effects due to compressibility and shear deformation. The motivation is to obtain an expansion of the energy density for soft elastic media in which the elastic constants accounting for shear effects are of comparable order. The expansion is carried out to fourth order to ensure that nonlinear effects in shear waves are taken into account. The result is E≃E0(ρ)+μI2+13AI3+DI22, where ρ is density, I2 and I3 are the second- and third-order Lagrangian strain invariants used by Landau and Lifshitz, μ is the shear modulus, A is one of the third-order elastic constants introduced by Landau and Lifshitz, and D is a new fourth-order elastic constant. For processes involving mainly compressibility E≃E0(ρ), and for processes involving mainly shear deformation E≃μI2+13AI3+DI22.