Dysypacja Energii w Ośrodku Sprężysto-Plastycznym Wywołana Występowaniem Szczeliny Osiowo-Symetrycznej

A penny-shaped crack in a material which is ideally elastic-plastic has been envisaged under the assumption that the plastic zone constitutes a very layer surrounding the crack. The Dugdale hypothesis has been adapted and thus the problem has been reduced to that for an elastic semi- space. The expressions for the length of plastic zone as a function of load, obtained in [18] are now used in further calculations. The entire energy absorbed in the process of creation of the new surface is here ascribed to the work expanded in the irreversible plastic deformation I, the work of cohesive forces being neglected. The displacements of crack faces are calculated as well as the plastic energy dissipation. The fracture criterion and plasticity condition of Huber-Mises-Hencky are discussed. The shape of the crack, obtained in this paper, differs considerably from that predicted by the theory of elasticity, particularly at the crack tip. The differences in the values of the critical pressure calculated from Griffith-Sack-Sneddon formula and those obtained by use of the equations derived here, are also significant. It is shown that for macro-cracks when crack radius 1 is not too small the following formula holds:

〖P_erit=[πE(dW〗_p/dA)_crit/2(1-v^2)l]^(1/2)

which agrees with the Orowan-Irwin modification of Griffith's theory. The symbol (dWp/dA) crit denotes the plastic work per unit area of a new surface, dissipated in the course of loading before the fracture.
The results of the present paper hold for ductile materials possessing the plastic region localized along the symmetry axis of the crack, e.g. for the so-called «quasi-brittle» solids. Two schemes of loading are considered: 1) pressure applied on crack surfaces and 2) pressure applied at infinity, attention being paid to the slightly different mechanism of fracture in both the cases.