Stochastic Variational Inequalities and Applications to Random Vibrations and Mechanical Structures

Abstract : This work is devoted to stochastic variational inequalities and their applications to random vibrations of mechanical structures. First, an effi cient method for obtaining numerical solutions of a stochastic variational inequality modeling an elasto-plastic oscillator with noise is considered. Since Monte Carlo simulations for the underlying stochastic process are too slow, as an alternative, approximate solutions of the partial di fferential equation de fining the invariant measure of the process are studied. Next, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal partial di fferential equations and local partial di fferential equations which can be interpreted with short cycles of the Markov process solution of the stochastic variational inequality. For engineering applications, we prove that plastic deformation for an elasto-perfectly-plastic oscillator has a variance which increases linearly with time and we characterize the corresponding drift coe cient by de ning long cycles behavior of the Markov process solution of the stochastic variational inequality. A major advantage of stochastic variational inequality is to overcome the need to describe the trajectory by phases (elastic or plastic). This is useful, since the sequence of phases cannot be characterized easily. However, it remains important to have informations on these phases. In order to reconcile these contradictory issues, we introduce an approximation of stochastic variational inequalities by imposing arti cial small jumps between phases allowing a clear separation of the phases. We prove that the approximate solution converges on any fi nite time interval, when the size of jump tends to 0. Then to study a more general case, a stochastic variational inequality is proposed to model an elasto-plastic oscillator excited by a filtered white noise. We prove the ergodic property of the process and characterize the corresponding invariant measure. This extends Bensoussan- Turi's method with a signi cant additional diffi culty of increasing the dimension. Finally, in a last chapter oriented to numerical experiments, we exhibit by probabilistic simulations the phenomenon of micro-elastic phases. The main di culty related to micro-elastic phasing is that they interfere on quantities of interest such that frequency of plastic deformations. An interesting criterion is provided which could be useful in engineering problems to discard micro-elastic phases and to evaluate statistics of plastic deformations of an elasto-plastic oscillator white noise excited.
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https://tel.archives-ouvertes.fr/tel-00653121
Contributor : Laurent Mertz <>
Submitted on : Monday, December 19, 2011 - 10:36:13 AM
Last modification on : Friday, March 22, 2019 - 1:31:45 AM
Long-term archiving on : Friday, November 16, 2012 - 3:55:32 PM

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  • HAL Id : tel-00653121, version 1

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Laurent Mertz. Stochastic Variational Inequalities and Applications to Random Vibrations and Mechanical Structures. Analysis of PDEs [math.AP]. Université Pierre et Marie Curie - Paris VI, 2011. English. ⟨tel-00653121⟩

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