Method of singular integrodifferential equations in plane dynamic problems of fracture mechanics
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METHOD OF SINGULAR INTEGRODIFFERENTIAL EQUATIONS IN PLANE DYNAMIC PROBLEMS OF FRACTURE MECHANICS V. S. Kravets’
UDC 539.3
We develop a method for the solution of two-dimensional dynamic problems of the theory of elasticity for infinite bodies with smooth curvilinear cracks combining a modified method of finite differences in time with the method of singular integrodifferential equations in the space variables. Integral representations of the wave potentials are constructed and used to reduce the first boundary-value problem to the solution of systems of integrodifferential equations by the method of mechanical quadratures. The time dependences of the computed dynamic stress intensity factors at the tips of a rectilinear crack are analyzed for various impact and pulsed loads acting upon the crack lips. Keywords: dynamic plane problem, singular integrodifferential equations, finite differences, crack, stress intensity factor.
The nonstationary character of loading of load-carrying structural elements and their operation under the action of rapidly varying stresses form favorable conditions for the formation and propagation of cracklike defects. The subject of contemporary fracture mechanics is much broader [1] than the subject of fracture mechanics based on the model of propagation of the main crack, the notion of stress intensity factor (SIF) at the crack tip, and the criteria of unstable propagation of macrocracks [2–8]. The outlined idealized model does not cover numerous problems of dynamic fracture. However, this is practically the sole model capable of describing the process of propagation of fracture in structural elements on the macrolevel. The surveys of achievements in various fields of the dynamic fracture mechanics can be found in [9–12]. We focus our attention on a class of problems of determination of the time dependences of dynamic SIF for stationary cracks subjected to the action of impact and pulsed loads. These problems are characterized by a noticeable manifestation of the inertial effect when the behavior of the SIF as functions of time significantly differs from the character of changes in the dynamic load. Then the computed values of the SIF are used for the determination of the limiting equilibrium of bodies with stationary cracks, the character of propagation of nonstationary cracks, and the crack growth rate under cyclic loading. The solutions of these problems are quite complicated because the behavior of dynamic SIF has its own specific features depending on the physical and geometric parameters of the problem, the configuration of the body, the defects formed in the body and the mode of dynamic loading [2, 4, 7, 8, 10, 13, 14]. A fairly complete systematic presentation of the basic analytic methods of the dynamic theory of elasticity and fracture mechanics can be found in the monographs [2, 4, 7, 8, 15]. The analytic methods were, as a rule, developed for the canonical domains by using the Laplace and Fourier integral transformations with respect to time. The boundary-value problems in
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