Influence of the Anisotropy of Materials on the Distribution of Stresses Near Parabolic Notches
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INFLUENCE OF THE ANISOTROPY OF MATERIALS ON THE DISTRIBUTION OF STRESSES NEAR PARABOLIC NOTCHES M. P. Savruk,1, 2, 3 A. Kazberuk,3 and L. I. Onyshko1
UDC 539.3
We consider the problems of distribution of stresses in an infinite anisotropic plane containing a parabolic notch for three main types of deformation in the case where the asymptotics of the stress field is specified at infinity (including stress intensity factors at the tip of the corresponding semiinfinite crack). The solutions of these problems are obtained with the help of the limit transition from the known analytic solutions obtained for an elliptic hole in the anisotropic plane for three types of loading at infinity (symmetric tension, transverse and longitudinal shear). These results generalize the well-known data on the distribution of stresses near narrow rounded notches in the isotropic plane and reflect the influence of anisotropy of the material on the stress concentration. Keywords: anisotropic and orthotropic planes, parabolic notches, plane and antiplane problems, stressed state.
The problem of stress concentration near a semiinfinite parabolic notch in the isotropic plane was investigated in [1, 2]. The case of a parabolic notch in the anisotropic plane was studied only under the conditions of symmetric tension [3]. In the present work, we generalize these results to the case of a parabolic notch in the anisotropic plane for three types of deformation: symmetric tension and transverse or longitudinal shear. The case of orthotropic plane containing a notch oriented in the direction of one of the axes of orthotropy of the material is considered in more detail. Symmetric Loading In a Cartesian coordinate system xOy , we consider an anisotropic plane containing an elliptic hole whose major a and minor b semiaxes lie on the Ox - and Oy -axes. The edge of the hole is free of external loads, and stresses σ ∞ y = p (Fig. 1а) are specified at infinity.
The analytic solution of the problem of tension of an anisotropic plate containing an elliptic hole at infinity by forces p perpendicular to the Ox -axis is represented in the form of two analytic functions Φ j (z j ) ( j = 1, 2) of complex variables z j = x + µ j y as follows [4–7]:
Φ1 (z1 ) = Φ10 (z1 ) + 1 2 3
pam2 ⎛ z ⎞ 1− 1 ⎟ , ⎜ ⎝ a + iµ1b R(z1 ) ⎠
Φ 2 (z2 ) = Φ 02 (z2 ) −
pam1 ⎛ z ⎞ 1− 2 ⎟ , ⎜ ⎝ a + iµ 2b R(z2 ) ⎠
(1)
Karpenko Physicomechanical Institute, Ukrainian National Academy of Sciences, Lviv, Ukraine. Corresponding author; e-mail: [email protected]. Politechnika Białostocka, Białystok, Poland.
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 51, No. 6, pp. 24–33, November–December, 2015. Original article submitted July 10, 2015. 1068-820X/16/5106–0773
© 2016
Springer Science+Business Media New York
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M. P. SAVRUK, A. KAZBERUK,
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AND
L. I. ONYSHKO
Fig. 1. Elliptic hole (a) and parabolic notch (b) in the anisotropic plane. where the functions Φ10 (z1 ) and Φ 02 (z2 ) are constant quantities that describe the homogeneous stressed state in the absence of the
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