Virtual Power Principle: A Lie Covariant Approach. Applications to Non-Linear Elasticity, Turbulence, Visco-elasticity

A covariant formulation of the virtual power principle based on Lie derivatives is proposed. The Lie covariant approach does not require an inner product and the Cauchy deformation tensor to start, but, at first order in a Galilean Euclidean setting, gives the usual linear results classically obtained with the Cauchy deformation tensor. The Lie approach may also enable to dfferentiate a fluid from a solid from an analytical point of view, and leads to propose a model for hysteresis. In the non-linear first order case we get covariant models for visco-elasticity, non linear fluids and non linear elasticity which dffer from usual models. In the second order case, enriched modelizations are obtained.