Stressed State of an Orthotropic Plane with Two-Section Kinked Crack Under Antiplane Deformation
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STRESSED STATE OF AN ORTHOTROPIC PLANE WITH TWO-SECTION KINKED CRACK UNDER ANTIPLANE DEFORMATION M. P. Savruk,1, 2 L. I. Onyshko,1 O. I. Kvasnyuk,1 and N. M. Bida1
UDC 539.3
The method of singular integral equations is used to obtain the solution of the antiplane problem of fracture mechanics for an orthotropic plane with two-section kinked crack with regard for the specific features of stresses at the tips of angular cracks. For this purpose, we apply the unified approach developed earlier to the solution of the problems of stress concentration in isotropic bodies with pointed and rounded notches. Keywords: elasticity theory, orthotropy, stress intensity factor, two-section kinked crack, antiplane deformation, method of singular integral equations.
Introduction The solution of singular integral equations (SIE) of the boundary-value problems of elasticity theory for anisotropic bodies containing nonsmooth curvilinear cracks should be performed with regard for the singularities formed at the angular points and, thus, encounters serious mathematical difficulties. This explains the necessity of application of approximate methods. In [1–4], a unified approximate approach was developed for the solution of two-dimensional problems of the theory of elasticity and fracture mechanics for isotropic bodies with sharp and rounded angular notches. In the present work, this approach is applied to the solution of the antiplane problem of elasticity theory for an orthotropic body. We find the solution of the problem of shear at infinity for an orthotropic space containing a load-free two-section kinked crack by the limit transition from the solution of the problem posed for the corresponding rounded crack as the radius of rounding at the notch tip tends to zero. Formulation of the Problem First, we consider an orthotropic plane in a Cartesian coordinate system xOy containing a kinked crack rounded at the notch tip whose contour consists of two identical rectilinear segments, which form an angle 2β . These segments are connected by a circular arc of radius ρ . The center of the circle lies on the Ox -axis at the point x = l(cos β − ε)/sin β , where ε = ρ/l is the relative radius of curvature at the rounded tip of the angular crack, 2l is the total crack length, and 2a is the distance between the crack tips. The crack tips A and B are located on the Oy -axis symmetrically about the Ox -axis (Fig. 1). 1 2
Karpenko Physicomechanical Institute, Ukrainian National Academy of Sciences, Lviv, Ukraine. Corresponding author; e-mail: [email protected].
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 56, No. 2, pp. 7–13, March–April, 2020. Original article submitted June 20, 2019. 1068-820X/20/5602–0143
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Springer Science+Business Media, LLC
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Fig. 1. A two-section crack rounded at the notch tip. At infinity, the body is subjected to the action of shear by stresses τ ∞ yz = τ and the faces of smooth curvi-
linear crack are free of external loads:
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