An empirical study on plastic deformations of an elasto-plastic problem with noise

• An elasto-plastic oscillator with noise is studied using its invariant measure.
• Due to the noise, transitions from plastic to elastic phases are not well defined.
• Then “ Enlarged plastic phases” are introduced to characterize plastic deformations.
• A criterion is given for their calibration, based on numerical solutions of PDEs.
• Statistical properties of the plastic deformations are then approximated.

Statistical properties of the plastic deformation related to an elastic perfectly plastic oscillator under standard white noise excitation are studied in this paper. Our approach relies on a stochastic variational inequality governing the evolution between the velocity and the non-linear restoring force. Bensoussan and Turi have shown that the solution is an ergodic Markov process. First, we exhibit, by means of probabilistic simulations, the phenomenon of micro-elastic phases which are small as well as numerous. The main difficulty related to this phenomenon is that the transitions between elastic and plastic phases are not well defined and quantities of interest such as frequency of plastic deformations cannot be characterized. Therefore, we investigate elastic phasing by means of the invariant probability measure of the problem. We present approximations of the probability density function of the elastic component and a similar expression to the Rice formula related to frequency of threshold crossings. These quantities are solutions of partial differential equations. Numerical experiments on these equations show that the non-linear restoring force tends to be highly distributed in the neighborhood of plastic thresholds. Finally, an interesting criterion is provided to discard micro-elastic phases and to evaluate statistics of plastic deformations which make sense for engineering purposes.