Stress intensity factors for cracks at the vertex of a rounded V-notch
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STRESS INTENSITY FACTORS FOR CRACKS AT THE VERTEX OF A ROUNDED V-NOTCH A. Kazberuk
UDC 539.3
The method of singular integral equations was applied to determine the stress intensity factors for a system of cracks emanating from the vertex of an infinite rounded V-notch subjected to symmetric loading. The numerical values were obtained for two cases—the case of a single crack and the case of a system of two cracks of equal length. The influence of the rounding radius of the vertex of the notch and its opening angle on the stress intensity factors at the crack tips was analyzed. The solution obtained as a result has a general nature—the stress intensity factors at the crack tip are expressed as a function of the V-notch stress intensity factor and, hence, this solution could be treated as an asymptotic relation for finite bodies with deep V-notches subjected to symmetric loads. Keywords: fracture mechanics, stress intensity factor, sharp and rounded V-notches, cracks, integral equations.
The problems of fracture mechanics concerning cracks emanating from the vertices of infinite V-notches are often used as asymptotic approximations for the solutions in finite fields. This model is appropriate because the length of a forming crack in the prefracture process is small as compared to the other dimensions of the body. In addition, a similar approach can be used to solve elastoplastic problems of fracture mechanics for bodies with V-notches within the scope of the model of plasticity strips [1, 2]. The assessment of the influence of the radius of the V-notch vertex on the stress intensity factors of the crack growing from this vertex is of high significance for the development of fracture criteria for solid bodies [3]. The problem of stress distribution in an elastic infinite wedge with symmetric crack at the vertex was considered by Doran [4], Ouchterlony [5], Smetanin [6], and others, by using the Mellin transform and Wiener–Hopf techniques. A close-type approximate solution for the problem of a symmetric sharp V-notch containing a crack at the vertex was obtained by Savruk and Rytsar [7] by the method of singular integral equations. For rounded V-shaped notches, Savruk and Kazberuk [8] deduced the relationship between the stress concentration at the notch vertex and the stress intensity factor for the corresponding sharp notch. In the present work, the system of arbitrarily many cracks located at the vertex of a rounded V-notch under tension is considered by using a similar method. This problem corresponds to the basic fracture model of notched bodies in plane states. System of Cracks at the Rounded Vertex of a V-Notch Consider an elastic infinite plate with rounded V-notch (Fig. 1). The notch is described by its angle 2β (0 ≤ 2β ≤ π). The neighborhood of the vertex of the notch has the shape of a circular arc whose radius of curvature is equal to ρ. Several cracks of length l k emanate from the vertex. The slope angle of the crack relative to the positive direction of the Ox-axis is denoted by αk . The solution of th
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