On the tangent modulus tensor in hyperelasticity

The tangent modulus tensor, denoted as D, plays a central role in nonlinear finite element simulation of elastomeric components such as seals. It is derived from the strain energy function w for isotropic elastomers. Using Kronecker product notation, a compact expression for D has been derived in Nicholson [1] and Nicholson and Lin [2] for invariant-based strain energy functions such as the Mooney-Rivlin model. In the current investigation, a corresponding expression is derived for stretch ratio-based strain energy functions such as the Ogden model. Compressible, incompressible and near-incompressible elastomers are addressed. The derived expressions are considerably more elaborate than their counterparts for invariant based models. As illustration, D is evaluated and presented for the torsion of a natural rubber shaft described by a three term Ogden model, using coefficients reported by Treloar.