Axisymmetric interaction of a rigid disc with a transversely isotropic half‐space

Abstract

SUMMARYA theoretical formulation is presented for the determination of the interaction of a vertically loaded disc embedded in a transversely isotropic half-space. By means of a complete representation using a displacement potential, it is shown that the governing equations of motion for this class of problems can be uncoupled into a fourth-order partial differential equation. With the aid of Hankel transforms, a relaxed treatment of the mixed-boundary value problem is formulated as dual integral equations, which, in turn, are reduced to a Fredholm equation of the second kind. In addition to furnishing a unified view of existing solutions for zero and infinite embedments, the present treatment reveals a severe boundary-layer phenomenon, which is apt to be of interest to this class of problems in general. The present solutions are analytically in exact agreement with the existing solutions for a half-space with isotropic material properties. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are included for cases of different degrees of the material anisotropy and compared with existing solutions. Further numerical examples are also presented to elucidate the influence of the degree of the material anisotropy on the response.

Undrained behaviour of a non-homogeneous elastic medium: the influence of variations in the elastic shear modulus with depth

“…Eason et al (1955) developed a special numerical procedure to evaluate such infinite integrals, and further applications are investigated by Selvadurai & Rajapakse (1985) and Oliveira et al (2012). Owing to the oscillatory and singular nature of these integrands, an adaptive numerical procedure is used to enhance the accuracy of the numerical results; examples of such an application are given by Rahimian et al (2007) and Katebi et al (2010). For numerical evaluation of the integrals, the upper limit of integration is replaced by a finite value î 0 ; this limit is increased until a convergent result is obtained.…”

Rocking vibration of a rigid circular disc in a transversely isotropic full‐space

“…Eskandari-Ghadi [27] introduced a complete set of scalar potential functions for the elastodynamic problems related to transversely isotropic axially convex domain, which have been utilized to investigate the wave propagation and singularities in this study. Recently, Rahimian et al [28] solved the dual integral equations due to forced torsion vibration of a rigid circular disc attached on the surface of a transversely isotropic half-space, and Katebi et al [29] have made an in-depth investigation for the analytical solution of an axially symmetric interaction of a rigid disc with a transversely isotropic half-space in the static case.…”

The analysis of a rigid rectangular plate on a transversely isotropic multilayered medium

“…By applying double Fourier transforms to deduce a 2-D integral equation, Rahman [23] dealt with an elastic static problem concerning the normal shift of a rigid elliptical disc embedded in a transversely isotropic full-space. Katebi et al [24] researched the interaction of a vertically loaded disc embedded in a transversely isotropic half-space by deducing a pair of dual integral equations, which can be reduced to a Fredholm equation of the second kind. A review of their studies indicates that the existing solutions mainly concerns about a circle plate resting on or embedded in a transversely isotropic half-space in polar coordinates, while the solutions for a rectangular plate on a transversely isotropic multilayered medium are seldom reported.…”