Mathematical Problems in Mechanics
Long cycle behavior of the plastic deformation of an elasto-perfectly-plastic oscillator with noise
Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 853-859.

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.

Des résultats expérimentaux en sciences de lʼingénieur ont montré que, pour un oscillateur élasto-plastique-parfait excité par un bruit blanc, la déformation plastique et la déformation totale ont une variance, qui asymptotiquement, croît linéairement avec le temps avec le même coefficient. Dans ce travail, nous prouvons ce résultat et nous caractérisons le coefficient de dérive. Notre étude repose sur une inéquation variationnelle stochastique gouvernant lʼévolution entre la vitesse de lʼoscillateur et la force de rappel non-linéaire. Nous définissons alors le comportement en cycles longs du processus de Markov solution de lʼinéquation variationnelle stochastique qui est le concept clé pour obtenir le résultat. Une question importante en sciences de lʼingénieur est de calculer ce coefficient. Les résultats numériques confirment avec succès notre prédiction théorique et les études empiriques faites par les ingénieurs.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.09.020
Bensoussan, Alain 1, 2, 3; Mertz, Laurent 4; Yam, S.C.P. 4

1 International Center for Decision and Risk Analysis, School of Management, University of Texas at Dallas, Box 830688, Richardson, TX 75083-0688, USA
2 Graduate School of Business, The Hong Kong Polytechnic University, Hong Kong
3 Graduate Department of Financial Engineering, Ajou University, Suwon 443-749, Republic of Korea
4 Department of Statistics, The Chinese University of Hong Kong, NT, Shatin, Hong Kong
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Bensoussan, Alain; Mertz, Laurent; Yam, S.C.P. Long cycle behavior of the plastic deformation of an elasto-perfectly-plastic oscillator with noise. Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 853-859. doi : 10.1016/j.crma.2012.09.020. http://www.numdam.org/articles/10.1016/j.crma.2012.09.020/

[1] Bensoussan, A.; Turi, J. Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators, Applied Mathematics and Optimization, Volume 58 (2008) no. 1, pp. 1-27

[2] Bensoussan, A.; Mertz, L.; Pironneau, O.; Turi, J. An ultra weak finite element method as an alternative to a Monte Carlo method for an elasto-plastic problem with noise, SIAM J. Numer. Anal., Volume 47 (2009) no. 5, pp. 3374-3396

[3] L. Borsoi, P. Labbe, Approche probabiliste de la ruine dʼun oscillateur élasto-plastique sous séisme, in: 2ème colloque national de lʼAFPS, 18–20 April, 1989.

[4] Karnopp, D.; Scharton, T.D. Plastic deformation in random vibration, The Journal of the Acoustical Society of America, Volume 39 (1966), pp. 1154-1161

Cited by Sources:

This research in the Note was supported by WCU (World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-20007) and by the Research Grants Council of HKSAR (PolyU 5001/11P). This research was partially supported by a grant from CEA, Commissariat à lʼénergie atomique and by the National Science Foundation under grant DMS-0705247. A large part of this work was completed while the second author was visiting the University of Texas at Dallas and the Hong-Kong Polytechnic University. We wish to thank warmly these institutions for the hospitality and support. The third author also expresses his gratitude to the generous support from HKGRF 502408.