An extension of Key's principle to nonlinear elasticity

A variational principle for finite isothermal deformations of anisotropic c ompressible and nearly incompressible hyperelastic materials is presented. It is equivalent to the nonlinear elastic field (Lagrangian) equations expr essed in terms of the displacement field and a scalar function associated w ith the hydrostatic mean stress. The formulation for incompressible materia ls is recovered from the compressible one simply as a limit. The principle is particularly useful in the development of finite element analysis of nea rly incompressible and of incompressible materials and is general in the se nse that it uses a general form of constitutive equation. It can be conside red as an extension of Key's principle to nonlinear elasticity. Various num erical implementations are used to illustrate the efficiency of the propose d formulation and to show the convergence behaviour for different types of elements. These numerical tests suggest that the formulation gives results which change smoothly as the material varies from compressible to incompres sible.