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

Oleh:

Kaplunov, J. D. ; Nolde, E.V. ; Rogerson, G.A.

Jenis: Article from Journal - ilmiah internasional
Dalam koleksi: Mathematics and Mechanics of Solids vol. 7 no. 6 (Dec. 2002), page 581-606.
Topik: Asymptotics; pre-stress; elastic plates; dispersion
Fulltext: 581MMS76.pdf (544.35KB)

Isi artikel

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 wavenumbet 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.

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