Covariant hypo-elasticity

The theory of constitutive behavior, with explicit reference to hypo-elastic materials, is addressed with a geometric approach which, following physical arguments, leads to a covariant formulation. The essential role played by a careful distinction between spatial vectors, material-based spatial vectors and material vectors is emphasized. Definite answers to debated issues, such as the proper definition of stress rate, the formulation of integrability conditions, the fulfilment of material frame-indifference, and the task of evaluating the stress state evolution, are given. Simple shear and extension of a specimen of a hypo-elastic material are investigated as applications of the theory. Improper statements and unsound physical responses of hypo-elastic materials are overcome by the covariant theory, thus restoring the proper role to this constitutive model, for both theoretical and computational purposes.