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An Asymptotically Consistent Model for Long-Wave High-Frequency Motion in a Pre-Stressed Elastic Plate

Institute for Problems in Mechanics, Russian Academy of Sciences, Prospect Vernadskogo 101-1, Moscow 117526, Russia

E. V. Nolde

Institute for Problems in Mechanics, Russian Academy of Sciences, Prospect Vernadskogo 101-1, Moscow 117526, Russia

G. A. Rogerson

Department of Computer and Mathematical Sciences, University of Salford, Salford, M5 4WT, UK

A one-dimensional asymptotic model is derived to elucidate the effect of pre-stress on long-wave high-frequency two-dimensional motion in an incompressible elastic plate. Solutions for the leading-order displacements and pressure increment are derived in terms of the long-wave amplitude; a governing equation for which is derived from the second-order problem. This equation is shown to become elliptic for certain states of pre-stress. Loss of hyperbolicity is shown to be synonymous with the existence of negative group velocity at low wavenumber. A higher-order theory is constructed, with solutions obtained in terms of both the long-wave amplitude and its second-order correction. An equation relating these is obtained from the third-order problem. The dispersion relations derived from the one-dimensional governing equations are also obtained by expansion of the corresponding exact two-dimensional relations, indicating asymptotic consistency. The model is highly relevant for stationary thickness vibration of, or transient response to high-frequency shock loading in, thin-walled bodies and also fluid-structure interaction. These are areas for which the effects of pre-stress have previously largely been ignored.

Key Words: Asymptotics • pre-stress • elastic plates • dispersion

Mathematics and Mechanics of Solids, Vol. 7, No. 6, 581-606 (2002)
DOI: 10.1177/108128602029660


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