Torsion of Transversely Isotropic Plate with a Non-Circular Cylindrical Hole
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International Applied Mechanics, Vol. 56, No. 4, July, 2020
TORSION OF TRANSVERSELY ISOTROPIC PLATE WITH A NON-CIRCULAR CYLINDRICAL HOLE I. Yu. Khoma1 and O. A. Strygina2
The problem of the stress state of an unbounded transversely isotropic plate with a noncircular cylindrical hole is solved by expanding functions into Fourier–Legendre series in the thickness coordinate and using the boundary shape perturbation method. The surface of the hole is free from external forces, and the plate is subject to constant torques at infinity. The stress distribution in the vicinity of the hole with elliptical boundary or triangular boundary with rounded corners in the midsurface is analyzed. The dependence of stresses on the relative thickness of the plate and elastic constants is established. Keywords: infinite transversely isotropic plate, torsion of the plate, stress state, noncircular hole, elliptical boundary, triangular boundary Introduction. Many publications are devoted to the stress concentration near holes and cavities in elastic bodies [6–9, 13, 16–19]. Various methods are used to solve boundary-value problems for plates weakened by noncircular (curvilinear) cavities and holes. Based on homogeneous solutions, in [7, 12], a method was proposed for reducing boundary-value problems to a governing system of singular integral equations. The stress state of an elliptical cylindrical shell of variable thickness depending on the geometric parameters was analyzed in [14]. The effect of the orthotropy of the material on the stress state of a quadrangular plate of various shapes was studied in [15]. In [20, 21], finite element discretization is applied to the study of the stress concentration in a plate with a circular and elliptical hole. To determine the stress state of a transversely isotropic plate in [3, 17], the method of expanding the unknown functions into Fourier–Legendre series [1, 11] is used together with the boundary-shape perturbation method [2, 5]. Using the method described in [17], the stress distribution around a noncircular hole (with a square and triangular boundary in the midsurface) was analyzed for a splitting force set on the boundary surface: a pair of forces tending to stretch or compress the plate in thickness. This paper presents a solution to the problem of the stress state of an unbounded transversely isotropic plate with a noncircular cylindrical hole subject to a torque at infinity. 1. Problem Formulation and Basic Equations. Consider a transversely isotropic plate of constant thickness 2h. The midsurface S of the plate coincides with the isotropy plane. We introduce a Cartesian coordinate system Ox1 , x 2 , x 3 and assume that the coordinates x1 , x 2 are located on the plane S and x 3 Î [ -h, h]. The plate is weakened by a noncircular cylindrical hole L ´ [ -h, h], the boundary Lof the hole differs slightly from a circle of radius R. The hole surface is free from external forces, and at (1)¥ 1) ¥ infinity, the plate is subject to constant torques s12 = s (21 = H ( H = const ). To solve the pr
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