Finite Element Implementation
The state-of-the-art methodology for modeling nonlinear geometrical and material response in the field of solid mechanics is the finite element method. Embedded in any finite element computational model is a material model. The solution of a set of nonlinear equilibrium equations in the finite element analysis of finite deformations is typically achieved through the employment of a Newton-Raphson iteration procedure. This requires the linearization of the equilibrium equations, which necessitates an understanding of the directional derivative. A relatively simple numerical technique for solving nonlinear equations in computational finite elasticity consists of employing the so-called incremental/iterative solution technique of Newton’s type. It is an efficient method with the desirable feature of a quadratic convergence rate near the solution point. It requires a consistent linearization of all of the quantities associated with the nonlinear problem, generating efficient recurrence update expressions. The nonlinear problem is then replaced by a sequence of easily solved linear equations at each iteration. The element stiffness formulation, based upon the “first elasticity tensor,” is carried out.