Comparison is given of the accuracy of calculation of stress intensity factors at the crack tips by various methods when solving plane elasticity problems for bodies with cruciform and edge cracks. It is shown that, within the range of quadrature formulas for singular integrals discussed, the type of the formula chosen for the solution of an equation, if used correctly, affects negligibly the accuracy of the stress intensity factor evaluation at the crack tip, and in view of this a method is proposed based on simple relationships.
References are most significant in the development of methods for computing the stress state during mining along a coal seam, which may be considered as an elastic medium with a crack similar to a mine, instead of as a complex nonhomogeneous medium, consisting of a rock mass and a coal seam, part of which is removed. This elastic problem, as is well known, has several solutions depending upon the behavior near the ends of the crack (see [3]), and each solution has an analog in the mining problem. Problems dealing with the propagation of cracks in brittle bodies also reduce to consideration of an elastic medium with cracks.Investigations dealing with these questions consider basically simple problems, which have analytical solutions. There has been practically no work on the stress state in bodies with fissures or cracks near the boundaries oron the development of such cracks and their propagation along the surface, although these questions are of great practical interest. It is convenient to use the customary method of integral equations for solving these more complex problems. This paper examines the elastic problem for a halfplane with one or more cracks located along the straight, perpendicular boundary of the halfplane. A method quite similar to that described in [4] for the solution of problems in plane theory of elasticity for interacting regions is used for this.Let the elastic medium fill the lower half of the complex plane z= x+ iy with cuts along the imaginary axis. We will designate the series of cuts L 1, the area occupied by the elastic medium D v the entire halfplane D, and its boundary L 0. We will chose positive directions for the cuts arbitrarily; limiting values for L I for left and right functions contained in D l, will be designated plus (+) or minus (-) respectively.We will examine the primary basic problem, i.e. the problem with stresses given at the boundary. Its solution leads to determination of the functions r ~l(z) regular in D 1 satisfying the boundary conditions (see [3]):Here N i and T i are normal and tangential stresses given at the boundary of area D v for which N0--iT0= 0(1Ix) in the vicinity of a point at infinity. Also, for simplicity, we will assume 0,We will introduce the function g(t) on L 1 satisfying the relation
g (t) = r (t) --r (t) (4)and we will represent the functions which are being sought in the form dt.305
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