A technique combining the advantages of conformal mapping and boundary collocation arguments for calculating stress intensity factors for cracks in plane problems is described. The difficulty of finding the mapping function on a rigidly prescribed parameter region is avoided at the expense of using boundary collocation methods on part of the boundary. Conventional collocation arguments are modified by prescribing stress, force, and moment conditions in a least-square collocation sense. These pseudo-redundant conditions provide a reasonable basis for estimation of the effects of inaccuracy of the boundary conditions. The technique is applied to the problem of a circular disk with an internal crack under a loading of external hydrostatic tension. In~oducfionMethods for calculating stress intensity factors for cracks have important applications in fracture mechanics. For plane problems, the authors have previously developed the Muskhelishvili conformal mapping method [ 1] into an effective technique for a certain class of crack problems [2], [3]. At about the same time, boundary collocation methods were reintroduced and applied to crack problems by several authors [4], [5].The boundary collocation methods depend on the selection of a class of stress functions satisfying the loading conditions on the crack and then matching boundary conditions at selected points on the remaining portion of the boundary. The computational simplicity of this approach is attractive; on the other hand, serious difficulties arise in assessment of accuracy. Convergence to the "correct" solution must be assessed on the basis of estimating the effect of the off-point residual errors in the boundary conditions. In fact, "apparent" convergence to incorrect values is quite possible if, for example, the class of stress functions chosen were incomplete.The mapping technique has its difficulties too. It is frequently very difficult to find accurate polynomial mappings of the physical region onto a suitable parametric region. On the other hand, the ensuing stress analysis is well understood. Assessment of accuracy is related to the approximation of geometry rather than the effect of residual errors in the boundary conditions.The technique proposed in this paper is a natural compromise of the two methods. The simple form of a mapping function carrying a circle and its exterior in the parameter plane into a crack and its exterior, respectively, will be used. The remaining portion of the boundary in the physical plane will correspond to a directly calculable curve in the auxiliary plane. The continuation arguments of Muskhelishvili are then employed to describe stress functions with, e.g., "traction-free" conditions on the crack. Collocation methods can then be introduced to satisI~¢ the conditions on the remaining portions of the boundary.This plane eliminates the difficulty of finding accurate polynomial approximations of the exact geometry in terms of a rigidly specified parameter domain. On the other hand, much of the mathematical insight provided...

A partitioning plan combined with the modified mapping-collocation method is presented for the solution of awkward configurations in two-dimensional problems of elasticity. It is shown that continuation arguments taken from analytic function theory can be applied in the discrete to “stitch” several power series expansions of the stress function in appropriate subregions of the geometry. The effectiveness of such a plan is illustrated by several numerical examples.

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The stress intensity factors for single edge cracks in rectangular tensile sheet are studied by using a complex variable form of analysis. The results for deep cracks show a considerable Increase in the intensity factors as compared to the case of symmetric edge cracks. Accurate numerical values of the stress intensity factors for the appropriate range of parameters are included.

Stress-concentration factors are analytically determined for relatively deep U-shaped edge notches of constant radii in a semi-infinite sheet with uniaxial tension applied parallel to the sheet edge. Because a strong geometric similarity exists between a U-notch and a semi-elliptical notch of the same tip radius, a comparison of such analytical data is effected. Stress-concentration factors for U-notches in a semi-infinite sheet are also compared with other comparable analytical and experimental data available in the literature.Finite-width correction factors are applied to the results of the semi-infinite-sheet analysis to provide practical design data. These data are compared with experimental results provided by a number of investigators. Notation b CwHalf-width of a finite-width plate. Finite-width correction factor = Kr,/KI, = k,/k,.