a b s t r a c tThe torsional problem of a finite elastic cylinder with a circumferential edge crack is studied in this paper. An efficient solution to the problem is achieved by using a new form of regularization applied to dual Dini series equations. Unlike the Srivastav approach, this regularization transforms dual equations into a Fredholm integral equation of the second kind given on the crack surface. Hence, exact asymptotic expansions of the Fredholm equation solution, the stress intensity factor and the torque are derived for the case of a shallow crack. The asymptotic expansions are certain power-logarithmic series of the normalized crack depth. Coefficients of these series are found from recurrent relations. Calculations for a shallow crack manifest that the stress intensity factor exhibits the rather weak dependence upon the cylinder length when the torque is fixed and the triple length is larger than the diameter.
We consider the pressure of a plate on a half-space with a round cylindrical cavity. The surface of the cavity is reinforced by elastic elements that are modeled by very general operators. The problem is reduced to a Fredholm integral equation of second kind. A detailed study is made of the case of reinforcement described by the Winkler law. An approximate solution is obtained in the form of the asymptotics with respect to the radii of the plate and the cavity. One table. Bibliography: 3 titles.We study the pressure of a round convex plate on an elastic homogeneous and isotropic half-space z > 0 with the elastically reinforced cavity r = a, 0 _< z < ee. A plate with the equation z = f(r) is pressed on with axial force P. We assume that friction is absent in the region of contact. The displacements Ur and us of the surface of the cavity are certain operators on the stresses applied to this surface. For a large class of reinforcement models the connection of the displacements and strains r(z) = Trz(a,z), a(z) = ar(a,z) has the form 2#a-l~(a, ~)2#a-lgCr(a, () --/21(()~c(~) -/22(()fs(() + u*(~), in the space of Fourier cosine-and sine-transforms, where the elements of the functional matrix lij(~) are determined by the model of reinforcement, u*(() and w*(() are known functions connected with the boundary conditions in noncontact reinforcement surfaces and with the mass forces. As a preliminary step, assuming for the time being that the displacement of the entire surface z = 0 is known and neglecting friction, i.e., under the boundary conditions ~=~(~), a_
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