Approximate Solutions of a Stochastic Variational Inequality Modeling an Elasto-Plastic Problem with Noise

An important issue in the failure analysis of some mechanical structures under seismic forcing is to estimate the statistics of the response of an elasto-perfectly plastic oscillator with noise. In a previous work, we noted that, due to noise, numerous and negligible plastic phases occur in the response on small time intervals at the instants of phase transition; therefore, we gave an empirical criterion based on mathematical tools related to a stochastic variational inequality (SVI) to remove this phenomenon from relevant statistics. Recently, we proposed an SVI with jumps to clearly separate the phases. We showed that the approximate solution converges, as the size of jumps ϵ goes to 0. The present work is in the straightforward line of these two studies. Here, an asymptotic study of ergodic measures of approximate solutions of the SVI is investigated by a combination of a theoretical and an empirical analysis. First, we show that these measures converge as ϵ goes to 0. Then numerical experiments on the convergence reveal, on the one hand, that the rate of convergence behaves as where A and B are positive numbers and, on the other hand, there exists an empirical small number ϵ⋆ for which the measures have numerically converged. In terms of engineering interests, we provide an interesting criterion using ϵ⋆ to calibrate the model with jumps. The model discards the negligible plastic phases and allows to estimate statistics for plastic deformations.