A mixed variational principle in nonlinear elasticity using Cartan’s moving frames and implementation with finite element exterior calculus

Abstract

This article is an attempt at offering a new perspective for the mechanics of solids using Cartan’s moving frames, specifically discussing a mixed variational principle in non-linear elasticity. We treat quantities defined on the co-tangent bundles of reference and deformed configurations as primary unknowns along with deformation. Such a treatment invites compatibility of the fields (defined on the co-tangent bundle) with the base-space (configurations of the solid body) so that the configuration can be realized as a subset of the Euclidean space. Using the moving frame, we rewrite the metric and connection through differential forms. These quantities are further utilized to write the deformation gradient and Cauchy–Green deformation tensor in terms of frame and co-frame fields. The geometric understanding of stress as a co-vector valued 2-form fits squarely within our overall program. We also show that, for a hyperelastic solid, an equation similar to the Doyle–Ericksen formula may be written for the co-vector part of the stress 2-form. Using this kinetic and kinematic understanding, we rewrite a mixed functional in terms of differential forms, whose extremum leads to the compatibility of deformation, constitutive relations, and equations of equilibrium. Finite element exterior calculus is then utilized to construct a finite-dimensional approximation for the differential forms appearing in the variational principle. These approximations are then used to construct a discrete functional which can be numerically extremized. The discretization leads to a mixed model as it involves independent approximations of differential forms related to stress and deformation gradient. The mixed variational principle is then specialized for the 2D case, whose discrete approximation is applied to problems in nonlinear elasticity. From the numerical study, it is found that the present discretization does not suffer from locking and related convergence issues.