A semi-analytical integration method for the numerical simulation of nonlinear visco-elasto-plastic materials

A semi-analytic integration method is proposed, which can be used in numerical simulation of the mechanical behavior of nonlinear viscoelastic and viscoplastic materials with arbitrary stress nonlinearity. The method is based upon the formalism of Prony series expansion of the creep response function and accepts arbitrary stress protocols as input data. An iterative inversion technique is presented, which allows for application of the method in routines that provide strain and require stress as output. The advantage with respect to standard numerical integration methods such as the Runge-Kutta method is that it remains numerically stable even for integration over very long time steps during which strain may change considerably due to creep or recovery effects. The method is particularly suited for materials, whose viscoelastic and viscoplastic processes cover a very wide range of retardation times. In the case of simulation protocols with phases of slowly varying stress, computation time is significantly reduced compared to the standard integration methods of commercial finite element codes. An example is given that shows how the method can be used in three dimensional (3D) constitutive equations. Implemented into a Finite Element (FE) code, the method significantly improves convergence of the implicit time integration, allowing longer time increments and reducing drastically computing time. This is shown in the case of a single element exposed to a creep and recovery cycle. Some simulations of non-homogeneous boundary value problems are shown in order to illustrate the applicability of the method in 3D FE modeling.