Transient Deformation of Anisotropic Timoshenko’s Plate

Abstract

In this paper, we will present an approach to constructing of dynamical spatial Green’s function (elementary solutions, dominant function) for a thin infinite elastic plate of constant thickness. The plate material is anisotropic with a single plane of symmetry, geometrically coinciding with plate’s middle plane. The Timoshenko theory was used for describing the plate movement. Transient spatial Green’s functions for normal displacements and angles of orthogonal alteration to middle surface before deformation of material fiber are built in the Cartesian coordinate system. To construct Green’s function, direct and inverse Laplace and Fourier integral transformations are applied. The originals of Laplace Green’s functions were analytically found with the theorem of residues. To construct Fourier originals, a specific method was used based on Fourier series transformation inversion integral connection with Fourier series on a variable interval. Green’s function found for normal displacement made it possible to represent the normal transient function as three-fold convolution of Green function with distant load function. The functions of normal distant displacements were constructed in case of the impact of transient total loads concentrated and distributed across rectangular courts. The numerical method of rectangles was used to calculate the convolution integrals. The influence of the concentrated load speed on transient normal displacements of the anisotropic plate was analyzed. As a verification of constructed transient spatial Green’s functions, the results of numerical solutions were compared with the results found using known transient Green’s functions for isotropic thin elastic rectangular simply supported Timoshenko’s plate which solutions are constructed using Laplace integral transformation in time and its decomposition into Fourier series on coordinates. Besides, its confidence was proved analyzing the nature of waves in anisotropic, orthotropic and isotropic plate, found in the process of numerical calculations. The results are represented as diagrams. Examples of calculations are given.