Periodic System of Closely Located Holes in an Elastic Plane Under the Conditions of Antiplane Deformation
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PERIODIC SYSTEM OF CLOSELY LOCATED HOLES IN AN ELASTIC PLANE UNDER THE CONDITIONS OF ANTIPLANE DEFORMATION M. P. Savruk,1, 2 O. I. Kvasnyuk,1 and A. B. Chornenkyi1
UDC 539.3
We construct a singular integral equation of an antiplane periodic problem of the theory of elasticity for an isotropic plane weakened by smooth curvilinear holes. A numerical solution of the problem is obtained in the case of closely located holes with rounded V-shaped vertices made in the elastic plane under the conditions of uniform shear at infinity. On this basis, we determine the stress concentration factors at the rounded vertices of a bilateral V-shaped semiinfinite notch. By using the well-known relation between the stress intensity and stress concentration factors for sharp and rounded V-shaped notches, we perform the limit transition to a bilateral sharp V-shaped notch. The dependence of the stress intensity factor at the sharp vertices of a bilateral V-notch on its apex angle is determined. Keywords: antiplane problem of the elasticity theory, periodic system of curvilinear holes, bilateral V-shaped notch, stress intensity factors, method of singular integral equations.
In fracture mechanics, a significant attention is now given to the investigation of the processes of deformation and fracture of solids with sharp and rounded V-shaped notches [1–5]. The two-dimensional problems of the elasticity theory for bodies with V-shaped notches can be solved within the framework of a single approach developed in [1, 2, 5–10] for the modes I and ІІ of deformation, and in [11, 12] for the mode ІІІ of deformation. According to this approach, the stress intensity factors at the tip of a sharp notch are determined on the basis of the data on stress concentration of the tips of the corresponding rounded V-shaped notches whose radii of curvature are not too small. These data can be obtained by using various methods. The relationships between the stress intensity factors at the sharp tip of a semiinfinite V-shaped notch and the maximum stresses on the contour of the corresponding rounded V-shaped notch have been constructed. For bounded bodies, these relationships have the asymptotic character (as the radius of curvature at the tip of the notch tends to zero). This enables one to use these relations for the determination of the stress intensity factors at the tips of sharp notches according to the data on the concentration of stresses near rounded notches in the elastic bodies of any shape. In what follows, by the method of singular integral equations, we obtain a numerical solution of the antiplane problem for a periodic system of closely located curvilinear holes with rounded V-shaped tips. On this basis, we find the stress concentration factors at the tips of a two-sided V-shaped notch. By using the wellknown relations between the stress intensity and stress concentration factors for sharp and rounded V-shaped notches [11, 12], we establish the dependence of the stress intensity factors (SIF) at the tip of a two-sided sharp V-shaped notch on its
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