Behavior of the plastic deformation of an elasto-perfectly-plastic oscillator with noise

TL;DR

This paper proves that the variance of plastic and total deformation in an elasto-perfectly-plastic oscillator driven by white noise increases linearly over time, characterizing the drift coefficient and validating results with simulations.

Contribution

It provides a rigorous mathematical proof of the linear variance growth and characterizes the drift coefficient, advancing understanding of stochastic deformation behavior.

Findings

Variance of deformation increases linearly with time

The drift coefficient is explicitly characterized

Numerical simulations agree with theoretical predictions

Abstract

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.