  Earlier works in engineering, partly experimental, partly computational have
revealed that asymptotically, when the excitation is a white noise, plastic
deformation and total deformation for an elasto-perfectly-plastic oscillator
have a variance which increases linearly with time with the same coefficient.
In this work, we prove this result and we characterize the corresponding drift
coefficient. Our study relies on a stochastic variational inequality governing
the evolution between the velocity of the oscillator and the non-linear
restoring force. We then define long cycles behavior of the Markov process
solution of the stochastic variational inequality which is the key concept to
obtain the result. An important question in engineering is to compute this
coefficient. Also, we provide numerical simulations which show successful
agreement with our theoretical prediction and previous empirical studies made
by engineers.
