Continuum Mechanics and Lagrange equations with generalised coordinates

The main aim of the actual problem is to obtain Lagrange equations when the chosen parameters do not respect material rigidity, so inducing strains (and Continuum Mechanics). The proposed method consist of two principal parts: first the definition of a family of generalised displacements involving strains and second the elimination of the Cauchy stress tensor in the Virtual Work Principle valuable in Continuum Mechanics. As a final statement the rigidity law is introduced on the parameters to complete the obtained equations. On a friction problem, it is highlighted the necessity to really distinguish between these mathematical compatibility conditions taking account of the nature of the material and other relations expressing some experimental boundary conditions like friction laws.