Distribution of Stresses Near Angular Notches on an Orthotropic Elastic Plane Under Conditions of Antiplane Deformation
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DISTRIBUTION OF STRESSES NEAR ANGULAR NOTCHES ON AN ORTHOTROPIC ELASTIC PLANE UNDER CONDITIONS OF ANTIPLANE DEFORMATION M. P. Savruk,1, 2 L. Yo. Onyshko,1 and O. I. Kvasnyuk1
UDC 539.3
By using a solution of the antiplane eigenvalue problem of the theory of elasticity for an elastic orthotropic wedge, we construct the distribution of singular stresses and displacements in a neighborhood of the tip of the corresponding angular notch under the conditions of antiplane deformation. By the method of singular integral equations, we solve the corresponding problem for an orthotropic plane with semiinfinite angular rounded notch and establish the relationship between the stress intensity and stress concentration factors at the sharp and rounded tips of angular notches. Keywords: fracture mechanics, anisotropy, stress intensity factor, angular notch, antiplane deformation, method of singular integral equations.
Introduction Composite materials with high strength characteristics are extensively used in machine-building, shipbuilding, and aeronautical engineering. The materials are modeled by homogeneous orthotropic bodies. In recent years, the solutions of the problems of fracture mechanics for orthotropic bodies with sharp and rounded angular notches become quite urgent. The unified approach to the evaluation of stresses near sharp and rounded angular notches in isotropic bodies developed in [1–4] was extended to the corresponding plane problems for orthotropic materials [4–7]. This approach enables one to find the stress intensity factors (SIF) at the tip of a sharp notch via the stresses at the tips of the corresponding rounded angular notches, which can be found by using the available methods and, in particular, the method of singular integral equations (SIE) [8]. In what follows, we establish the relationship between the stress intensity and stress concentration factors for sharp and rounded angular notches in orthotropic bodies under the conditions of longitudinal shear (mode ІІІ deformation). Main Relations of the Antiplane Problem of the Theory of Elasticity for Orthotropic Media Consider the case of antiplane deformation (longitudinal shear) of an orthotropic body in a Cartesian coordinate system (x, y, z) . If the axis of a deformation is directed along the z -axis, then the components of the vector of elastic displacements can be represented in the form
u x = 0, 1 2
u y = 0,
u z = w = w(x, y) .
(1)
Karpenko Physicomechanical Institute, Ukrainian National Academy of Sciences, Lviv, Ukraine. Corresponding author; e-mail: [email protected].
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 55, No. 3, pp. 7–15, May–June, 2019. Original article submitted November 5, 2018. 1068-820X/19/5503–0299
© 2019
Springer Science+Business Media, LLC
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M. P. SAVRUK, L. YO. ONYSHKO,
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AND
O. I. KVASNYUK
According to the generalized Hooke’s law, the relationship between the nonzero components of strains ε yz and ε xz and stresses τ yz an d τ xz takes the form [9, 10]
2ε yz =
∂w = a44 τ yz , ∂y
2ε xz
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