An Exact Stiffness Method for Elastodynamics of a Layered Orthotropic Half-Plane

Abstract

A method is presented in this paper to compute displacements and stresses of a multilayered orthotropic elastic half-plane under time-harmonic excitations. The half-plane region under consideration consists of a number of layers with different thicknesses and material properties. Exact layer stiffness matrices describing the relationship between Fourier transforms of displacements and tractions at the upper and bottom surface of each layer are established explicitly by using the analytical general solutions for displacements and stresses of a homogeneous orthotropic elastic medium. The global stiff ness matrix which is also symmetric and banded is assembled by considering the traction continuity conditions at the interface between adjacent layers of the multilayered half-plane. The numerical solution of the global stiffness equation results in the solutions for Fourier transform of displacements at layer interfaces. Thereafter displacements and stresses of the multilayered plane can be obtained by the numerical integration of Fourier integrals. Only negative exponential terms of Fourier transform parameter are found to appear in the elements of layer stiffness matrices. This ensures the numerical stability in the solution of the global stiffness equation. In addition, the size of the final equation system is nearly onehalf of that corresponding to the conventional matrix approach for layered media based on the determination of layer arbitrary coefficients. The present method provides accurate solutions for both displacements and stresses over a wide range of frequencies and layer thicknesses. Selected numerical results are presented to portray the influence of layering, material orthotropy, and frequency of excitation on the response of five layered systems. Time-domain solutions are also presented to demonstrate the features of transient surface displacements due to a surface loading pulse applied to layered orthotropic half-planes.