Consistent time-cycle approximation for cyclic plasticity coupled to damage

Simulating cyclic loading of materials using conventional constitutive models in the time-domain is inherently expensive when it is desirable to trace the response under a large number of cycles. One way of circumventing this drawback is to introduce a coordinate transformation and substitute the cycle number for the physical time as the independent variable. Such an approach, however, requires that the evolution laws of the constitutive model are derived in terms of, e.g., amplitudes and mean values rather than the ”actual” state variables used in a conventional constitutive model. In this contribution, we propose a novel method of constructing consistent cyclic modeling based on conventional time-domain-models. The method relies on an exact difference equation in terms of individual cycles, where each time-cycle constitutes one discrete cycle step. Taking the exact difference equation as the starting point, we device a FE approximation in the cycle-domain, corresponding to the Quasi-Continuum Method in a spatial domain, whereby the difference equation is solved approximately. The resulting strategy, where the actual time history need be integrated for a few individual cycles only, allows for global error control. An adaptive algorithm, based on duality arguments, is then applied to adapt the finite element mesh in the cycle-domain with respect to a chosen error tolerance. In this manner, we resolve a minimal number of actual time-cycles in order to reach a prescribed accuracy. This paper gives the theoretical basis for the time-cycle approximation in terms of formulating the exact difference equation (in the cycle-domain) pertinent to a canonical time-dependent problem. Furthermore, we introduce the approximate solution of the cycle-domain problem.