On variational methods in finite and incremental elastic deformation problems with discontinuous fields

Abstract

With a view toward a numerical solution by means of the finite-element method, we give here a variational statement for large elastic deformations at finite strains which involves independent variation of the displacement, the (nonsymmetric first Piola-Kirchhoff) stress, and the deformation-gradient fields, and which includes both the boundary and the jump conditions. Then we present, for small deformations superimposed on the large, three variational statements, each involving three independent fields and each including both the boundary and the jump conditions. These statements are such that the first variation of the corresponding functional yields the field equations which characterize the equilibrium of the finitely-deformed state considered and also the field equations that pertain to the incremental deformations. Several specializations of these results are discussed. By way of illustration, finally, we present a finite-element formulation of the large deformation problem, using three independent fields, where each field is approximated by a piecewise-linear function within each element.