A class of transversely isotropic non-linear elastic bodies that is not Green elastic

An implicit constitutive relation is proposed to study transversely isotropic bodies. The relation is obtained assuming the existence of a Gibbs potential that depends on the second Piola–Kirchhoff stress tensor, from which the Green Saint-Venant strain tensor is obtained as the derivative with respect to the stress. The responses of unconstrained as well as inextensible bodies are studied, and some boundary value problems are analysed. An inextensible body, where the constraint of inextensibility appears only in tension is also considered.