TABLE OF CONTENTS In the previous chapter we presented the basic equations for problems in non-linear solid mechanics in which strains remain small. We showed that the equations can be presented in a strong form as a set of partial differential equations or alternatively in terms of a variational principle or weak form expressed as an integral over the domain of interest. In the present chapter we use the weak form to construct approximate solutions based on the finite element method. This results in a Galerkin method for which general properties are well known. 1-4
Although it is assumed that the reader is familiar with finite element methods for small deformation linear problems, we present a full summary of the basic steps to construct a solution for the transient problem. We emphasize the differences between linear and non-linear effects as well as the numerical procedures used to establish the final discrete form of the equations which is the form used in computer analysis. We also consider both irreducible and mixed forms of approximation. The mixed forms are introduced to overcome deficiencies arising in use of low order elements based on irreducible forms. In particular, in this chapter we consider a mixed form appropriate for use in problems in which near incompressible behaviour can occur. In the second part of this book, we consider forms for structural problems where so-called 'shear locking' can occur in bending of thin rods, plates and shells.
We conclude this chapter by applying the methods...
