Abstract
This chapter presents approximations and numerical analysis of finite deformations of elastic solids. It focuses on Ritz–Galerkin approximations of weak solutions of a class of nonlinear elasticity problems using the finite-element method. It presents certain mathematical features of the approximation, such as, the establishment of a priori error estimates and convergence proofs for a class of finite-element approximations of quasi-static and dynamic problems infinite elasticity. The chapter presents the development of the theorems concerning Galerkin approximations assuming that certain ideal subspaces Sh(B) can be developed for arbitrary domains B. It also presents the numerical results obtained by applying the theory to a number of dynamic problems in finite elasticity in which shock and acceleration waves are developed. These include the longitudinal motion of thin incompressible rods, the bending of bars, and the stretching and inflation of thin elastic sheets.
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