Investigation of the three-dimensional stressed state in elastic plates with circular holes
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INVESTIGATION OF THE THREE-DIMENSIONAL STRESSED STATE IN ELASTIC PLATES WITH CIRCULAR HOLES V. P. Revenko
UDC 539.3
The stress-strain state (SSS) of an isotropic elastic plate weakened by a circular hole is determined in the three-dimensional statement. We propose a numerical-analytic method for the solution of three-dimensional problems of the theory of elasticity. The vector of displacements is represented via three independent harmonic functions. The spectral representation of the perturbed SSS is obtained in the form of a series in nonorthogonal eigenfunctions. The coefficients of the series are determined from the condition of the most exact validity of the boundary conditions on the surface of the hole. The estimate of the error of realization of the boundary conditions is obtained. The distribution of stresses is numerically analyzed depending on Poisson’s ratio and the radius of the hole. Keywords: vector of displacements, nonorthogonal eigenfunctions, Kirsch problem, quadratic form, elastic plate, numerical-analytic method, cylindrical hole.
The plane stressed state of a unilaterally stretched elastic plate weakened by a circular hole was found by Kirsch [1]. A long period of investigations in this direction led to the formation of an independent branch of the theory of elasticity dealing with the study of the influence of holes on the distribution of stresses in structural elements [2, 3]. As a result of the qualitative analysis of boundary-value problems for three-dimensional bodies [4], it is demonstrated that there is no asymptotic correspondence between the two- and three-dimensional problems of the theory of elasticity as the thickness of the plate increases. This is explained by the fact that, in the two-dimensional statement of the problem, it is impossible to take into account the influence of the ratio of the diameter of the hole to the thickness of the plate on the distribution of stresses. In [5, 6], the Kirsch problem was studied by the methods of the three-dimensional theory of elasticity without indicating the accuracy of evaluation of stresses. The case of a thick plate containing a circular hole was investigated in [7]. A theoretical approach to the determination of the three-dimensional SSS was proposed in [8]. In what follows, we compare the stresses obtained in the three-dimensional statement of the problem with the solution of the plane problem. In our numerical calculations, we use the numerical–analytic method [9, 10], which enables us to determine the three-dimensional distributions of the components of the stress tensor in the plate weakened by a hole with the required accuracy. Statement of the Problem Consider the three-dimensional stress-strain state (SSS) of an isotropic elastic plate with thickness 2h weakened by a circular cylindrical hole of radius R. The plate is loaded by stresses σ0 in the direction of the Ox1 -axis of a Cartesian coordinate system ( x1 , x2 , x3 ). Assume that the sides of the plate are parallel to the Pidstryhach Institute for Problems in Mechanics a
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