Long cycle behavior of the plastic deformation of an elasto-perfectly-plastic oscillator with noise

Abstract

For decades, a vast amount of research effort in experimental engineering together with numerical simulations has been devoted to the study of the plastic deformation and total deformation of elasto-perfectly-plastic (EPP) oscillators. All of these results reveal that both the plastic and total deformations of an EPP oscillator, being excited by a white noise, have variances that increase linearly with time and share a common asymptotic growth rate. Before our present work, there was no apparent theoretical justification on this empirical observation. In this paper, we use a stochastic variational inequality (SVI) for the modeling of the evolution between the velocity of an EPP oscillator and its non-linear restoring force; and this modeling has already been justified in some previous works of the authors. By introducing the novel notion of long cycle behavior of the Markovian solution of the corresponding SVI, we first establish a mathematical explanation for the empirical observation and characterize the mentioned asymptotic growth rate in terms of certain stopping times read off from the trajectory; secondly, we show an effective method on computing this asymptotic growth rate, which has been a long lasting challenging question to engineers. Finally numerical simulation is provided to illustrate the notable agreement between our theoretical prediction and empirical studies in the engineering literature.