The study of the stress concentrations which exist at notches and cracks is of considerable importance in the understanding of the fracture of metals. The way in which these concentrations of stress are relieved by plastic deformation is not readily understood in terms of the classical theory of the elastic plastic solid since analytical solutions are obtained only in the simplest situations.
In this work a simple model of the relaxation process is considered, in which the crack and yielded regions are represented in terms of linear dislocation arrays. Alternatively the medium may be considered everywhere elastic and the cracks and yielded regions represented by arcs across which the stress is described and relative displacements are permitted.
First the relaxation from a sharp isolated crack in an infinite medium is treated for conditions 01" plane strain and antiplane strain. In antiplane strain this provides a model of the relaxation round a surface notch in a semi infinite medium. Simple expressions are obtained for the relation between the yield stress, the applied stress, the relative displacement in the crack tips and the extent of the plastic zones.
The effect of free surfaces or of neighbouring cracks is considered by expanding the analysis to consider an infinite periodic coplanar array of identical cracks. It is shown that the free surface causes plastic zones to spread more rapidly with increasing stress. The displacements for a given length of plastic zone are then reduced. If a critical displacement criterion is adopted for the initiation of fracture at a notch, then neglecting the effect of the free surface is shown to err on the safe side.
The effect of workhardening is also considered. An integral equation is obtained for the displacements and this is inverted numerically.
Finally a model of a tensile crack is treated in which the plastic zones from a single tip are represented by two linear arrays of dislocations inclined symmetrically to the plane of the crack. The applied tension is normal to the crack. Again this problem is treated numerically and preliminary calculations have been carried out to obtain the important relationship