Long wave asymptotic integration of the governing equations for a pre-stressed incompressible elastic layer with fixed faces
✍ Scribed by E.V. Nolde; G.A. Rogerson
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 284 KB
- Volume
- 36
- Category
- Article
- ISSN
- 0165-2125
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✦ Synopsis
A model for long wave motion in the vicinity of the cut-off frequencies of a pre-stressed incompressible elastic layer is constructed. In contrast with most earlier studies, the faces of the layer are assumed fixed, rather than free, and in consequence there is no fundamental mode. Appropriate asymptotically approximate equations are derived and integrated through a systematic perturbation process. Models are presented for both anti-symmetric and symmetric motion. In the former case, leading-order solutions for displacement components and pressure increment are found in terms of the long wave amplitude, with an equation for this essential parameter obtained from the second-order problem. In the symmetric case, the essential parameter is the incremental pressure and the cut-off frequencies are defined through a transcendental equation of the form tan Λ = Λ. In both cases the governing equation for the appropriate essential parameter may become elliptic for certain states of pre-stress. The paper also details the derivation of higher-order theories.