More accurate mathematical description in the assessment of plastic zone magnitude around the crack tip

Abstract

The thin infinite central cracked plate, made of ductile metallic material, is observed. The plate is loaded with uniformly distributed continuous loading, according to the Fig. 1a. The small plastic zones around crack tips will appear. It is assumed anisotropic and non-linear strain hardening of a plate material which can be good described by the Ramberg-Osgood equation. The investigations were carried out for the several discrete values of the strain hardening exponent n, i.e. for n = 3, 5, 7, 10, 25, 50 and 1000. The magnitude of plastic zone is investigated on the basis of the premises of the Dugdale´s cohesive model. In the frame of applied cohesive model, the problem is formulated fully exact. The analytical methods were applied. Our intention is to contribute in a more accurate mathematical description of the phenomena occurring within the cohesive zones and to solve the problems, in so called closed-form, without any additional assumptions, as for example, about small plastic zones (SSY), elastic-perfectly plastic material and so on. The commercial software Wolfram Mathematica 7.0 is used and the solutions are presented through the special, the Gamma and the Hypergeometric functions.