Justification and refinement of initial value problems for long wave models in thin structures

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The method of compound asymptotic expansions will be used to formulate mathematically correct initial value problems for lower dimensional long wave models, both high and low frequency, in thin elastic plates. The methodology is based upon superposition of high and low frequency approximations of the full three-dimensional theory. In the leading order, the low frequency anti-symmetric and symmetric problems coincide with the classical theories of plate bending and extension. The high frequency approximation has no static counterpart and describes long wave motion in the vicinity of the cut-off frequencies, coinciding with the thickness resonance frequencies. The work will enable clarification of well-known anomalies in plate dynamics. Specific examples of these are [1] the non-coincidence of the number of initial conditions in the low frequency plate bending approximation with the full three-dimensional theory, and [2] the choice of initial conditions utilised for any higher order plate models, characterised by the occurrence of high order time derivatives. After first establishing results in respect of a linear isotropic elastic plate and homogeneous initial data, the effects of pre-stress, viscosity and non-homogeneous initial data will be investigated. Although application of the theory will be carried out in respect of thin plates, the methodology will have wider applications within other areas involving thin domains.