Null Lagrangians, admissible tractions, and finite element methods

Abstract

The principle of virtual work in elastostatics is shown to imply the principle of minimum potential energy whenever the applied tractions obtain as natural Neumann data associated with a null Lagrangian. Tractions of this kind are referred to as variationally admissible, and are those for which finite element methods may be applied directly. The most general null Lagrangian for elastostatic problems is shown to be generated by 15 functions of the independent variables and the displacements (current configuration variables), and this representation remains valid both for linear and nonlinear elasticity. Explicit representations are given for all null Lagrangians and all variationally admissible tractions, the general form of a null Lagrangian being a third-degree polynomial in the configuration gradients. Explicit application to finite element formulations of elastostatic problems is given. A model problem of a Sturm-Liouville system with nonlinear Neumann data at one end is given in the appendix as an assist to the reader.