Journal of Elasticity

, Volume 41, Issue 1, pp 13–37 | Cite as

Resonant-triad instability of a pre-stressed incompressible elastic plate

  • Yibin Fu
Article
Received:
Revised:

Abstract

The dynamic stability properties of a pre-stressed incompressible elastic plate are studied in this paper with respect to perturbations in the form of one near-neutral mode and two non-neutral modes interacting resonantly. The pre-stresses are assumed to be an all-round pressure. With the aid of a novel derivation procedure, the evolution equations governing the scaled amplitudes of the three modes are found to be given by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaCa% aaleqabaGaaGOmaaaakiaadgeadaWgaaWcbaGaaGymaaqabaGccaGG% VaGaamizamaaBaaaleaacqaHepaDdaahaaadbeqaaiaaikdaaaaale% qaaOGaeyypa0JaeyOeI0Iaam4yamaaBaaaleaacaaIWaaabeaakiaa% dgeadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGJbWaaSbaaSqaai% aaigdaaeqaaOGaaiiFaiaadgeadaWgaaWcbaGaaGymaaqabaGccaGG% 8bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyAaiabeo7aNnaaBa% aaleaacaaIXaaabeaakiqadgeagaqeamaaBaaaleaacaaIYaaabeaa% kiqadgeagaqeamaaBaaaleaacaaIZaaabeaaaaa!5308!\[d^2 A_1 /d_{\tau ^2 } = - c_0 A_1 - c_1 |A_1 |^2 - i\gamma _1 \bar A_2 \bar A_3 \], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadg% eadaWgaaWcbaGaaGOmaaqabaGccaGGVaGaamizaiabes8a0jabg2da% 9iabeo7aNnaaBaaaleaacaaIYaaabeaakiqadgeagaqeamaaBaaale% aacaaIXaaabeaakiqadgeagaqeamaaBaaaleaacaaIZaaabeaaaaa!4324!\[dA_2 /d\tau = \gamma _2 \bar A_1 \bar A_3 \] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadg% eadaWgaaWcbaGaaG4maaqabaGccaGGVaGaamizaiabes8a0jabg2da% 9iabeo7aNnaaBaaaleaacaaIZaaabeaakiqadgeagaqeamaaBaaale% aacaaIXaaabeaakiqadgeagaqeamaaBaaaleaacaaIYaaabeaaaaa!4325!\[dA_3 /d\tau = \gamma _3 \bar A_1 \bar A_2 \], where a bar denotes complex conjugation, τ is a slow time variable and c0, c1, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiaaigdaaeqaaaaa!387B!\[\gamma _1 \], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiaaikdaaeqaaaaa!387C!\[\gamma _2 \], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiaaiodaaeqaaaaa!387D!\[\gamma _3 \] are real constants. These equations are solved exactly for the special case when A2 and A3 have constant amplitudes but time-dependent phases. A series of new post-buckling states, which does not exist when the perturbation is monochromatic, are found. We show that two nonneutral modes can interact resonantly to produce a much larger near-neutral mode, and in particular, two O(ε) non-neutral modes may induce a much larger % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiaacI% cacqaH1oqzdaahaaWcbeqaamaalyaabaGaaGOmaaqaaiaaiodaaaaa% aOGaaiykaaaa!3B87!\[O(\varepsilon ^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} )\] oscillation or static post-buckling state. In this sense, resonant-triad interaction is a powerful mechanism in producing high levels of strain and stress in a pre-stressed elastic plate.

Key words

instability resonance post-buckling 
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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Yibin Fu
    • 1
  1. 1.Department of MathematicsUniversity of ManchesterManchesterU.K.

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