Elasticity equations for incompressible and nearly incompressible materials by a variational theorem.

A variational theorem is presented which is equivalent to the elastic field equations expressed in terms of the displacements and a function of the mean pressure. The variational theorem is particularly useful in the development of approximate structural analysis techniques for incompressible and nearly incompressible materials. An approximate analysis method derived by utilizing the Ritz technique in conjunction with the variational principle is illustrated. The results obtained from the application of this procedure to a sample problem are contrasted with those obtained by a similar technique that employs the theorem of minimum potential energy. The comparison illustrates that the results obtained from the analysis based upon the theorem of minimum potential energy becomes quite inaccurate for nearly incompressible materials whereas the solution technique based upon the new variational theorem yields accurate results.