Issue 39

M. Shariati et alii, Frattura ed Integrità Strutturale, 39 (2017) 166-180; DOI: 10.3221/IGF-ESIS.39.17 167 thermal and/or mechanical loading [1]. FGMs can be used in the construction of tanks and cylindrical furnace such as cement kiln. Circumferential cracks are occasionally developed in cylindrical structures during service or production. These cracks are threats to the safety and reliability of these structures. Subsequent fracture and fatigue analysis of such cracks is of great interest, and requires the determination of stress intensity factors. The stress intensity factor (SIF) is an important parameter to determine the safety of a cracked part. Although several stress intensity factor handbooks have been published, the available solutions of stress intensity factors for pressure vessels are not always adequate for particular engineering applications. [2] Solutions to the problem concerning a circumferential crack in a circular cylinder made of homogeneous or composite materials are relatively few. Closed form stress intensity factor of an arbitrarily located inner-surface circumferential crack in an edge-restraint homogeneous cylinder under linear radial temperature distribution derived by Meshii and Watanabe [3]. Chen [4] evaluated stress intensity factors in a cylinder with a circumferential crack using the computing compliance and the finite difference methods to solve the boundary value problem. Lee [5] analyzed the stress distribution in a long circular cylinder of isotropic elastic material with a circumferential edge crack under uniform shearing stress and determined the stress intensity factor. Jones [6] applied the impulse response method to the analysis of the thermally striped internal surface of a hollow cylinder containing a circumferential crack on this surface. He calculated stress intensity factor and strain energy density factor ranges as functions of crack depth for various sinusoidal striping frequencies. Moulick and Sahu [2] derived weight functions for the surface and the deepest point of an internal semi elliptical crack in a thick-wall cylinder. Tran and Geniaut [7] developed an extended finite element method (X-FEM) axisymmetric model and employed it to compute stress intensity factors for cracked industrial specimens and components. They used the X-FEM model to assess the integrity and durability of a cracked rotor coil retaining ring during the power plant operation. Wu et al. [8] described a three-dimensional domain integral method for extracting mixed-mode stress intensity factors. Predan et al. [9] calculated the stress-intensity factor for the circumferential semi- elliptical surface cracks in a hollow cylinder cross section under torsion using a finite element technique. Sharma [10] et al. used the extended finite element method to evaluate the stress intensity factors of a semi-elliptical part through thickness axial/circumferential crack. The pipe or pipe-bend having a crack on the outer surface was subjected to internal pressure or opening bending moment in their research. Seifi [11] determined the stress intensity factors for internal surface cracks in autofrettaged functionally graded cylinders. Eshraghi and Soltani [12] obtained stress intensity factors for functionally graded cylinders with internal circumferential cracks using the weight function method for different combinations of cylinder geometry, crack depth, and material gradation. In this paper, two problems are considered; the first is the determination of the stress intensity factor ( K I ) of an inner penny-shaped crack in a FG rod loaded by uniform axial tension and the second problem is the calculation of the SIF for an inner circumferential crack in a FG thick walled cylinder under uniform axial tension. Both problems are solved under static and dynamic loading conditions. The extended finite element method is used in this study, to compute the stress and displacement fields necessary for determining the stress intensity factor. Stress intensity factors are obtained using the interaction integral method. The Newmark time integration scheme is used to solve the dynamical system of matrix equations obtained from the spatial discretization of equations of motion. The effects of crack radius, rotational speed of cylinders and material gradation on SIFs are studied. The programming is done in MATLAB. M ODELING OF FUNCTIONALLY GRADED CYLINDER n the present study, we assume that the material properties change along the radius of the cylinder and the volume fraction of inclusion ¬ i V follows a simple power function, i V r r R W R r R W P ¬ / , ¬¬¬¬¬¬¬¬ d d (2-1) where R and W are inner radius and thickness of cylinder, respectively. p is the power exponent determining the volume fraction profile. The volume fraction of matrix ¬ m V is obtained as below. m i V r V r ¬ ¬ 1 (2-2) I