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

TL;DR

This paper introduces a covariant formulation of the virtual power principle using Lie derivatives, providing new models for non-linear elasticity, fluids, and visco-elasticity that differ from traditional approaches.

Contribution

It presents a Lie covariant approach to the virtual power principle that does not rely on an inner product or Cauchy tensor, enabling new analytical models for complex materials.

Findings

Reproduces classical linear results at first order in a covariant framework

Differentiates fluids from solids analytically using Lie derivatives

Proposes covariant models for non-linear elasticity, fluids, and visco-elasticity

Abstract

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.