An analytical approach for the growth rate of the variance of the deformation related to an elasto-plastic oscillator excited by a white noise

For a few decades ago, there has been a huge amount of studies on plastic deformation of elasto-plastic oscillators in the engineering literature. In one of our recent works [5], we introduced a novel notion of long cycle behavior of the Markovian solution of the corresponding stochastic variational inequality of an elasto-perfectly-plastic oscillator, which can characterize in a probabilistic framework the variance of the plastic deformation. In this paper, we provide an analytical formula for the characteristic function of the probability distribution of the plastic deformation on long cycles; from our result we also derive a deterministic representation of the variance of the plastic deformation on long cycles. In addition, numerical experiments are carried out in support of our theoretical prediction.