Abstract—In this article an isotropic linear elastic half-space with a cylindrical cavity of finite length is considered to be under the effect of a ring shape time-harmonic torsion force applied at an arbitrary depth on the surface of the cavity. The equation of equilibrium has been written in a cylindrical coordinate system. By means of Fourier cosine integral transform, the non-zero displacement component is obtained in the transformed domain. With the aid of the inversion theorem of the Fourier cosine integral transform, the displacement is obtained in the real domain. With the aid of boundary conditions, the involved boundary value problem for the fundamental solution is reduced to a generalized Cauchy singular integral equation. Integral representation of the stress and displacement are obtained, and it is shown that their degenerated form to the static problem coincides with existing solutions in the literature. Keywords—Cosine transform, Half space, Isotropic, Singular integral equation, Torsion I. INTRODUCTION NVESTIGATING half-spaces containing cylindrical cavities has been one of the great interests to many researchers. Analytical inspection of the dynamic interaction of piles with torsion moments and the pile cavities in a half- space is very important in many engineering structures such as wharves and other heavy structures. Pertaining to problems of this type, some approximate results were first obtained by for the case of hydrostatic pressure acting on an interval of an infinite cylindrical cavity extending through an infinite solid [1]. Treated the dynamic problem of a suddenly applied pressure over a finite interval of the cavity [2]. Because of the complexities encountered in the problem, the numerical results were presented only at large distance away from the location of pressure. Some interesting problems of determining the distribution of stress due to an exterior crack in an isotropic infinite elastic medium with a coaxial cylindrical cavity was studied by ([3, 4]). The response due to the application of static radial pressure and torsional ring load has been given by [5]. In addition, he proposed a quadrature method for evaluating of the singular solution for concentrated torsional and radial ring load acting on the wall of an infinite hole. Morteza Eskandari-Ghadi. Department of Engineering Science, Faculty of Engineering, University of Tehran. P.O.Box 11165-4563, Tehran, Iran (phone: +98-21-6111-2171; fax: +98-21-88632423; (e-mail: ghadi@ut.ac.ir) Mohammadreza Mahmoodian. Department of Civil Engineering, Science and Culture University, Tehran, P.O.Box: 13145-871, Tehran, Iran (e-mail: mohammadreza.mahmoudian@gmail.com) Parnes also considered the steady-state problem of the effect of a torsional line load, with a harmonic time dependency, applied on the surface of a bore [6]. In his paper, he compared the degenerated of his dynamic results with the static case. The solutions of the generalized problem associated with a finite cylindrical cavity in a half-space would be of even greater engineering interest and challenge. It has been found that the additional stiffness of the medium below the bottom of the hole can apparently lead to a noticeable change of the response in the upper region. [7] investigated the problem of torsional shear traction acting on an open finite cylindrical cavity in an isotropic half-space in a rigorous manner, and found the corresponding fundamental solution. They also mathematically examined the resulting load-induced as well as shape-induced singularities in the response. In the present paper, which is an extension of the work done in [7] for dynamic case, the elastodynamic response of an isotropic half-space containing a finite open cylindrical cavity under a torsional ring load at an arbitrary depth is considered. Owing to the particular topology of the domain, it is convenient to consider the response of the elastic solid in two separate regions, which have some continuity conditions. By considering the equation of motion in each region and with the aid of Fourier cosine integral transform, the non-zero displacement component is obtained in the transformed domain. By means of the inversion theorem for Fourier cosine integral transform and the displacement compatibility conditions, the governing equation is reduced to a generalized Cauchy singular integral equation. The equation is then investigated analytically and solved numerically. Integral representation of the dynamic stress and displacement are obtained and shown to be degenerated to known existing solutions in the literature. II. BOUNDARY VALUE PROBLEM AND THE SOLUTION An isotropic homogeneous linear elastic half-space is considered in cylindrical coordinate system (, ,) r z θ , with a depth-wise z-axis. A circular cylindrical cavity with radius 0 a > and length 0 l > , as shown in Fig. 1, is assumed to be in the medium. A known time-harmonic shear stress, * (, ) i t r z e ϖ θ τ μτ ϖ = , is considered to be applied on the wall of the cavity. Because of torsional symmetry, the displacement vector has only one non-vanishing component, which is M.Eskandari-Ghadi, M.Mahmoodian Analytical Solution of Time-Harmonic Torsional Vibration of a Cylindrical Cavity in a Half-Space I World Academy of Science, Engineering and Technology International Journal of Mechanical and Mechatronics Engineering Vol:6, No:1, 2012 184 International Scholarly and Scientific Research & Innovation 6(1) 2012 scholar.waset.org/1307-6892/5885 International Science Index, Mechanical and Mechatronics Engineering Vol:6, No:1, 2012 waset.org/Publication/5885