Numerical Formulation of Nonlinear Elasticity Problems

The development of nonlinear stiffness relations for three-dimensional finite elements of an elastic continuum is considered. On the basis of linear displacement approximations, consistent finite element representations are formulated for geometrically nonlinear problems involving large displacements and large strains. General nonlinear stiffness relations are derived for compressible materials and for incompressible materials of the Mooney-Rivlin and neo-Hookean type. Stiffness relations for triangular plates, membranes, and straight bars are also derived. It is shown that the connection of elements into the assembled system can be accomplished by a series of group transformations. A simple example is provided to demonstrate parts of the theory.