16 - Nonlinear Finite Element Method

Abstract

Engineers are interested in the response of structures, thereby in the solution of boundary value problems. Having established the equations that control elasto-plasfic, viscoplastic, and creep behavior, it is evident that these equations are so complex that exact analytical solutions of boundary value problems cannot, in general, be established. Instead, approximative solution methods must be looked for exact analytical solutions of boundary value problems. In this regard, today, the finite element (FE) method turns out to be the most powerful numerical means. For linear problems, use of the finite element method is straightforward. Elasto-plastic, viscoplastic, and creep problems, however, are nonlinear, and this gives rise to a number of questions that must be resolved before a reliable solution can be established. These new questions can be summarized into: formulation of the nonlinear finite element method, solution of the nonlinear global equations, and integration of the constitutive equations. This chapter first presents the formulation of the nonlinear finite element method. The chapter begins by formulating the nonlinear finite element method for general nonlinear solid mechanics. The formulation of the nonlinear FE method is very similar to that of the linear FE method; it is based on the weak formulation of the equations of motion—that is, on the principle of virtual work.