Stress-Strain State of a Composite Plate Under the Action of a Transient Movable Load

The transient stress-strain state and normal displacements in a thin elastic unrestricted composite plate of constant thickness under an unsteady pressure, in particular, with a movable “load spot,” is investigated for the problem of a tangential impact on the plate. The approach to solving the problem is based on Green’s function method and the superposition principle, according to which the desired solution is related to the load by means of an integral operator of convolution type in spatial variables and time. The kernel of this operator is Green’s function, which is a transient fundamental solution to the function of normal deflections of the plate from the action of a single concentrated normal force momentum applied to some point on its surface. To find the Green’s function, the integral Laplace transform in time and a two-dimensional integral Fourier transform in coordinates are used. The original integral Laplace transform was found analytically, but for the inverse two-dimensional integral Fourier transform, a numerical method for integrating rapidly oscillating functions was used. The fundamental solution obtained made it possible to represent the sought-for function of transient normal displacements in the form of a triple convolution of Green’s function with the function of transient pressure distributed over a rectangular area with a time-varying amplitude and impact boundaries. To calculate the convolutions integrals, the numerical method of rectangles is used. Employing the function of normal displacements, the transient stress-strain state of a composite unbounded Kirchhoff plate is found and investigated. In this case, reduced engineering constants are used, which are calculated from the generalized stiffness of the laminated material. As an example, the space-time dependences of the transient deflection are constructed and distributions of stresses and strains in the upper layers of a polymer composite plate with a symmetric reinforcement scheme with respect to the median plane under the influence of a time-varying pressure distributed over a movable rectangular load spot are obtained.