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

采用复合渐近展开的方法，对弹性薄板中高频和低频的低维长波模型的初值问题进行了数学修正。该方法基于全三维理论的高频和低频近似的叠加。在主阶上，低频反对称和对称问题与经典的板弯曲和拉伸理论相吻合。高频近似没有静态对应，描述了与厚度共振频率一致的截止频率附近的长波运动。这项工作将有助于澄清板块动力学中众所周知的异常现象。这些问题的具体例子是[1]低频板弯曲近似中初始条件的数目与全三维理论不一致，以及[2]任何高阶板模型所使用的初始条件的选择，其特征是出现高阶时间导数。在首次建立了关于线各向同性弹性板和均匀初始数据的结果之后，将研究预应力、粘性和非均匀初始数据的影响。虽然该理论将应用于薄板，但该方法将在其他涉及薄域的领域中得到更广泛的应用。

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.