An asymptotic analysis of initial-value problems for thin elastic plates

Abstract

An asymptotic technique is developed to analyse initial-value problems in application to the three-dimensional theory of thin elastic plates. Various sets of long-wave initial data are considered, with arbitrary distribution along the plate thickness. To account for the initial data, composite asymptotic expansions are employed, utilizing both two-dimensional low-frequency and high-frequency models. The former correspond to the classical theories of plate bending and extension, and their refinements, whereas the latter are associated with the long-wave motions occurring in the vicinities of thickness stretch and shear resonance frequencies. Six cases of iteration process are revealed depending on the symmetry of the initial data and their thickness variation. For each case, approximate two-dimensional initial conditions are derived, including higher-order corrections for the two-dimensional low-frequency refined plate theories. The validity of the proposed approach is justified by comparison with the exact solution of the model plane problem for initial data with uniform distribution along the thickness and sinusoidal distribution along the mid-plane. The methodology of the paper has potential for more general initial-value problems specified over a narrow domain, including many of those commonly met in physical applied mathematics.