(BQ) Part 2 book "The finite element method (Volume 1: The basis)" has contents: Adaptive finite element refinement; the time dimension - semi-discretization of field and dynamic problems and analytical solution procedures; couple systems; computer procedures for finite element analysis,.and other contents. | 12 Incompressible materials, mixed methods and other procedures of solution Introduction We have noted earlier that the standard displacement formulation of elastic problems fails when Poisson's ratio becomes or when the material becomes incompressible. Indeed, problems arise even when the material is nearly incompressible with > 0:4 and the simple linear approximation with triangular elements gives highly oscillatory results in such cases. The application of a mixed formulation for such problems can avoid the di culties and is of great practical interest as nearly incompressible behaviour is encountered in a variety of real engineering problems ranging from soil mechanics to aerospace engineering. Identical problems also arise when the ¯ow of incompressible ¯uids is encountered. In this chapter we shall discuss fully the mixed approaches to incompressible problems, generally using a two-®eld manner where displacement (or ¯uid velocity) u and the pressure p are the variables. Such formulation will allow us to deal with full incompressibility as well as near incompressibility as it occurs. However, what we will ®nd is that the interpolations used will be very much limited by the stability conditions of the mixed patch test. For this reason much interest has been focused on the development of so-called stabilized procedures in which the violation of the mixed patch test (or BabusÏ ka±Brezzi conditions) is arti®cially compensated. A part of this chapter will be devoted to such stabilized methods. Deviatoric stress and strain, pressure and volume change The main problem in the application of a `standard' displacement formulation to incompressible or nearly incompressible problems lies in the determination of the mean stress or pressure which is related to the volumetric part of the strain (for isotropic materials). For this reason it is convenient to separate this from the total stress ®eld and treat it as an independent variable. Using the `vector'
