The semi-infinite region with a cut having a boundary cusp, that is, a point of selfcontact of the boundary, can serve as a model to investigate rupture under the local effect on the crack edges close to its tip. We solve the basic problems of the plane theory of elasticity for such regions. Using the hypothesis on the initial crack growth in the plane for which the normal rupture stress sigma_(beta) has the maximal value of the intensity coefficient [1], we obtain the equation to determine the direction of the crack growth from the boundary cusps of the regions under study. We consider the cases, when the constant pressures act on the regions of the cut sites on both sides of the cusp at the different arrangement of the pressure regions and various pressure values.
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