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197) a( z , z ) b( z , z ) A( z ), z Z. The current time is tn and the current iteration count is k + 1. 197) is obtained as a* ( n z k ; 'z k , z ) b* ( n z k ; 'z k 1 , z ) A( z ) a ( n z k , z ) b( n z k , z ), z Z. 198) is linear in incremental displacement for a given displacement variation. This linearized equation is solved iteratively with respect to incremental displacement until the residual forces (the right side of the equation) vanish at each time step. 4 Nonlinear Dynamic Problems When a time-dependent load and/or a boundary condition is applied to the structure, the transient response becomes important.
E equals one for kinematic hardening and zero for isotropic hardening. 93) can be explicitly solved without iteration. For example, when the status of a material is plastic, the Jˆ is obtained by Jˆ A 1 A tr Ș 2 3 N ( n ep ) 2 2P H . 2 Finite Rotation with Objective Integration For a finite rotational problem, objective rate tensors have to be used to describe the motion of the structure and to obtain the material frame independent results. Although an extensive amount of research has been done pertaining to objective rates, only corotational Cauchy stress will be introduced here.
159) as K ^w [ H (:)] 1 N w *g 0 and ( x xc ([ c ))T en t 0 on * c `. 167) The existence and uniqueness of the solution to the variational inequality has been extensively studied for linear elastic material  and . 166) for the nonlinear elastic problem has been proved by Ciarlet  for a polyconvex strain energy function. However, from an engineering point of view, it is not convenient to solve the variational inequality directly without mentioning the construction of a test function on constraint set K.