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A Well-posed Hypoelastic Model Derived From a Hyperelastic One

Hypoelastic models are widely used in industrial and military codes for numerical simulation of high strain dynamics of solids. This class of model is often mathematically inconsistent. More exactly, the second principle is not verified on the solutions of the model, and the initial state after a reversible cycle is not recovered. In the past decades, hyperelastic models, which are mathematically consistent, have been intensively studied. For their practical use, ones needs to entirely rewrite the commercial codes. Moreover, calibration of equation of states would be needed. In this paper two hypoelastic models for isotropic solids are derived from equivalent hyperelastic models. The hyperelastic models are hyperbolic for all possible deformations. It allows us to use robust Godunov's schemes for numerical resolution of these models. Two new objective derivatives corresponding to two different equations of state and defining the evolution of the deviatoric part of the stress tensor naturally appear. These derivatives are compatible with the reversibility property of the model : it conserves the specific entropy in a continuous motion. The most used hypoelastic model (Wilkins model) is recovered in the small deformation limit.