Invariants of the stretch tensors and their application to finite elasticity theory

The principal invariants of the stretch tensors in the polar decomposition of the deformation gradient stand in one-to-one relation to the principal invariants of their squares, the Cauchy-Green deformation tensors. These relations are used to obtain the derivatives of the first set of invariants with respect to the deformation gradient. The latter generate expressions for the stretch and rotation factors in the polar decomposition as explicit functions of the deformation gradient, and lead to new formulae for the stress and elastic moduli for isotropic elastic materials.