<p>A generalization of Ogden’s incompressible isotropic hyperelastic material model is proposed. This model has the following features in comparison with Ogden’s original model: (1) the expression for the Cauchy stress tensor for Ogden’s original model is written in the principal axes of the left stretch tensor, and the corresponding expression for this stress tensor for the generalized model is written in basis-free (eigenprojection-based) form; (2) the generalized model, unlike Ogden’s original model, includes as a special case the Hencky incompressible isotropic hyperelastic material model; (3) the generalized model, unlike Ogden’s original model, supports the form of Hooke’s law, i.e., in the case of infinitesimal strains, the expressions for both the specific potential strain energy and the Cauchy stress tensor for the generalized model become the corresponding expressions for Hooke’s law. In addition, for the new generalized Ogden’s model, a basis-free (eigenprojection-based) form of the fourth-order supersymmetric elasticity tensor was obtained.</p>