Iterative strategies based on constitutive dissipation

Abstract

FE formulation of solid mechanics problems typically require iterative strategies to solve the resulting nonlinear system. Basic strategies with prescribed load values, such as the Newton-Raphson method, are not valid in the presence of structural snap-back due to material or geometric non-linearity. In this case, the most common strategy is the cylindrical Arc-length method, which prescribes the norm of displacements as constraint, in order to obtain the load factor increment. However, because the constraint is quadratic, additional non-trivial criteria need to be introduced for choosing the appropriate solution, and the wrong choice may lead to potential spurious unloading among other problems. As a way to overcome these shortcomings, alternative methods have been proposed which are based on an energy dissipation constraint. Constitutive dissipation may be considered as an always increasing “time” parameter with the advantage that the corresponding constraint equation is always linear. Iterative methods based on a dissipation constraint can be found in the literature.]. But most of those formulations are based on specific constitutive models. In contrast, the present formulation is general in the sense that it is valid for any constitutive model, as long as the model subroutine provides, additional to the standard output, also the appropriate values of dissipation and dissipation derivatives. Some application examples in concrete structures with progressive fractures are also provided to illustrate the performance of the model proposed.