An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate

Rogerson, G. A.; Fu, Yibin

doi:10.1007/bf01187727

Cited by 56 publications

(33 citation statements)

References 11 publications

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“…Although pre-stress may well be introduced during the manufacturing process, it is the aforementioned load supporting scenario that mainly motivates this work. In view of the important applications of this particular material model within such a context, a number a recent studies have focused on various aspects of wave propagation and vibration in plates, half-spaces and layered structures composed of such material, see for example Dowaikh & Ogden (1991); Ogden & Roxburgh (1993); Ogden & Sotiropoulos (1995); Rogerson & Fu (1995);Rogerson (1997); Rogerson & Sandiford (1997).…”

Section: Introductionmentioning

confidence: 99%

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

Pichugin¹,

Rogerson²

2001

IMA Journal of Applied Mathematics

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Section: Introductionmentioning

confidence: 99%

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

Pichugin¹,

Rogerson²

2001

IMA Journal of Applied Mathematics

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“…[10]) that there exist two types of solutions for v and H(x2, k). The first type has H even in x2 and the corresponding solutions are called flexural waves:…”

Section: R~j-ofijmentioning

confidence: 99%

“…Numerical calculations of (2.13) and (2.15) given by Rogerson & Fu[10] show that the v 2 for the body wave modes is always positive, whilst for the Rayleigh-type modes, v 2 is non-negative only if 2[sinh(2kh) -2kh] ae=f. ~F(kh) forflexural waves,…”

mentioning

confidence: 99%

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

1995

J Elasticity

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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 th:'ee modes are found to _be g_iven by d2Al/dr 2 = -coAl -cllA112A1 -i3'IA2A3, dA2/dr -'~ 0'2~12~3 and dA3/dr = "y3AIA2, where a bar denotes complex conjugation, r is a slow time variable and co, cl, 71, "r2, 3' 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(¢) non-neutral modes may induce a much larger O(¢ 2/3) oscillation of 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.

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“…Dowaikh & Ogden (1990). Further description of the derivation of the equations (2.2)-(2.4) is given in Rogerson & Fu (1995).…”

Section: Governing Equationsmentioning

confidence: 99%

An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate

Pichugin

Rogerson

2002

Proc. R. Soc. Lond. A

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An asymptotically consistent two-dimensional theory is developed to help elucidate dynamic response in finitely deformed layers. The layers are composed of incompressible elastic material, with the theory appropriate for long wave motion associated with the fundamental mode and derived in respect of the most general appropriate strain energy function. Leading order and refined higher order equations for the mid-surface deflection are derived. In the case of zero normal initial static stress and in-plane tension, the leading order equation reduces to the classical membrane equation, with its refined counterpart also being obtained.The theory is applied to a one-dimensional edge loading problem for a semi-infinite plate. In doing so the leading order and higher order governing equations are used as inner and outer asymptotic expansions, the latter valid within the vicinity of the associated quasi-front. A solution is derived by using the method of matched asymptotic expansions.

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An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate

Cited by 56 publications

References 11 publications

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-off frequencies

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

An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate

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