A second order constitutive theory for hyperelastic material

The second order constitutive equation for a hyperelastic material with arbitrary symmetry is derived[ In developing a second order theory\ it is necessary to be discriminating in the choice of measures of defor! mation[ Here the derivation is done in terms of the Biot strain\ which has a direct physical interpretation in that its eigenvalues are the principal extensions of the deformation[ The constitutive equation is specialized for the cases of isotropy and transverse isotropy[ The isotropic equation derived here is compared with equations obtained by other authors in terms of the displacement gradient and the Green strain