Lagrange multiplier and variational equations in mechanics

The equilibrium of a structure is characterized by either Euler’s equations completed with boundary and some internal conditions, or by variational equations of a stationary point of the total potential energy defined in a set of admissible functions. Generally this set is defined as a reciprocal image of a constraint function defined between two Banach spaces E and F; and has a manifold structure. Test functions of the variational formulation belong to the Banach tangent space of this set at the stationary point. Though variational equations are suitable for numerical methods through finite elements, the restriction of test functions only in the tangent space is source of some difficulties during numerical implementation. Lagrange multipliers, when they exist, offer the best way to bypass these obstacles. In this paper we present some conditions that guarantee the existence of Lagrange multipliers and establish the links between the new variational equations obtained and the initial variational formulation. We show how it has been applied in incompressible fluid or incompressible elastic solid mechanics. The Lagrange multipliers appear as the hydrostatic pressure which modifies their constitutive laws. We also show the efficiency of the Lagrange multipliers in the limit analyses of problems encountered in the homogenization process and particularly on junction of multistructures. In recent works on junction of elastic multi-dimensional structures, the limit final coupled equations are obtained studiously after some complex calculations. The Lagrange multiplier approach on junction of multistructures herein, which is the main result of this paper, substantially simplifies the analysis, without using any ad-hoc assumption as in previous work and paves the way to treat nonlinear junction equations.