Linear finite-element equations for nonlinear deformation of slightly compressible elastic material

A finite-element method is developed for solving boundary-value problems associated with nonlinear static deformation of slightly compressible materials. A perturbation technique based on that developed by A. J. M. Spencer in Second Order Effects in Elasticity Plasticity and Fluid Dynamics (Pergamon 1964) is used to obtain the linear governing classical equations in general curvilinear coordinates. These equations are then formulated in weak forms for the finite-element method. However, solutions can only be developed when the corresponding solutions of the incompressible material are known. This method is attractive in the sense that only linear equations are to be solved for nonlinear boundary-value problems. A numerical example is reported.