Torsional oscillations of a finite elastic cylinder containing an outer circular crack

Abstract

UDC 539.3 We determine the stressed state of a finite cylinder containing an outer circular crack in the process of tor- sional oscillations. The oscillations are caused by the action of a harmonic torsional moment upon a rigid circular plate attached to one end of the cylinder. The problem is reduced to a Fredholm integral equa- tion of the second kind for the unknown stresses in the plane of the crack. The elements of machine parts and building structures often have the shape of cylinders. Cracks in these elements significantly deteriorate their characteristics and may lead to failures, especially under the conditions of dynamic loading. Outer circular cracks are often formed in the process of operation of these products and struc- tures. Hence, the analysis of the stressed state of cylindrical bodies with defects of this sort plays an important role in contemporary investigations. The procedure of theoretical determination of the stressed state of cylinders weakened by cracks requires the solution of boundary-value problems for the corresponding partial differential equations. However, the exact solution of these equations is not found up to now. Hence, as a possible ap- proach to the solution of these problems, we can mention the application of direct numerical methods (the meth- ods of finite differences and finite elements) (1, 2). A method based on the reduction of the analyzed boundary- value problems to integral equations (as a rule, singular) was proposed in (3-6). Note that the major part of works was devoted to the investigation of the equilibrium of finite and infinite cylinders with circular cracks, whereas the number of available theoretical studies dealing with the analysis of the stressed state of cylinders with circumferential cracks under the conditions of dynamic loading is much smaller. As a rule, the researchers study cylinders of infinite length (7-11). In recent years, it has been pro- posed to use mixed numerical-experimental methods for the evaluation of the stress intensity factors in cylindri- cal bodies with outer circular cracks. Thus, the case of dynamic tension was analyzed in (12) and the case of dynamic torsion of cylindrical specimens with cracks of this sort was investigated in (13). However, all these methods, just as the available experimental procedures, have serious disadvantages connected with the fact that the experiments performed for each specific specimen require the application of complex measuring equipment. This complicates the analysis of the influence of the geometric sizes of a given cylinder and a crack on the val- ues of the stress intensity factors. Hence, the problem of theoretical determination of the stress intensity factors in cylinders weakened by cracks under the conditions of dynamic loading remains challenging up to now. In what follows, we propose the solution of the problem of determination of the stressed state of a cylinder weakened by an outer circular crack under the conditions of torsional oscillations.