Solution of a Dynamic Antiplane Problem for a Body with Inclusion by the Method of Finite Time Differences
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SOLUTION OF A DYNAMIC ANTIPLANE PROBLEM FOR A BODY WITH INCLUSION BY THE METHOD OF FINITE TIME DIFFERENCES V. H. Popov 1 and O. P. Moiseenok 2
UDC 539.3
We solve a dynamic problem of antiplane deformation of a body with thin rigid striplike inclusion. The solution is based on the application of a modified method of finite time differences and the use of the discontinuous Helmholtz equation at each time node. We present the results of numerical analysis of the time dependence of the stress intensity factor for the inclusion for various loading modes.
Structures and machine parts fail as a result of the concentration of stresses not only near cracks but also near technological defects in the form of thin inclusions. The solution of dynamic problems of the mechanics of deformable bodies containing inclusions is based mainly on the application of the integral Laplace transformation with respect to time with subsequent numerical inversion. It is well known that this problem is actually not only mathematically complicated but also ill-posed. For this reason, even insignificant errors in the numerical determination of the images of the required quantities lead to considerable errors in finding the originals. This explains the appearance and development of new methods in which the procedure of integral transformation with respect to time is not used. Thus, as one of the methods of this sort, one can mention a modified method of finite time differences proposed earlier. The efficiency of this method is illustrated in [2] by the solution of a dynamic problem for a cracked body under the conditions of antiplane deformation. In the present work, we use the indicated method for the solution of a dynamic problem of antiplane deformation of a body containing a thin rigid inclusion. For the solution of this problem in the case of harmonic oscillations, see, e.g., [3–5]. However, the problem of investigation of stress concentration in the vicinity of a thin inclusion subjected to the action of impact or pulsed loads remains, in fact, open. Statement of the Problem Consider an elastic medium in the state of antiplane deformation containing a thin rigid striplike inclusion occupying a domain | x | ≤ a, | y | ≤ h / 2, h
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