On the basis of the improved theory of plates, which makes it possible to determine all the components of the stress tensor, we study the thermally stressed state of a finite round plate with a concentric inclusion of a different material. We obtain the exact analytic solution of the corresponding boundary-value problem for the system of singularly perturbed equations, valid for any ratios between the diameters of the plate and the inclusion. In the contact zones the stresses differ significantly (both quantitatively and qualitatively) from those predicted by the classical plate theories. The results obtained make it possible to draw a number of conclusions that are useful for estimating the strength of electrovacuum devices. Four figures. Bibliography: 5 titles.
By applying the improved theory of plates, which makes it possible to determine all components of the stress tensor, we study the stressed state of glass disks under axisymmetric bending. We obtain a closedform solution of the corresponding boundary-value problem for systems of singularly perturbed equations. It is shown that the size of the zone of "pure" bending--the domain with uniformly stretched surface-differs from those known in the literature. The results obtained make it possible to determine the optimal geometric parameters of the punch-glass disk-support system in strength testing of the glass by the method of axisymmetric bending.
539.377On the basis of the revised theory of plates, which makes it possible to determine the components of the stress tensor, we study the thermostressed state of an unbounded plate stiffened by a rod-ring. In the stiffening and in the contact zone of the plate the stresses differ significantly, both quantitatively and qualitatively, from those predicted by the classical theories of elasticity.A considerable number of papers have been devoted to the problem of determining the stress state of stiffened shells and plates. The main results and a rather complete bibliography can be found in the monographs [1,3,4,8]. (We note that the majority of these studies were carried out in L'vov.) However all of these studies were based on a theory of plates and shells that does not describe the three-dimensional effects that can occur in the contact zones. An analysis of the thermal stresses in plates with an inhomogeneous inclusion [2,7] based on the revised theory [6] gives reason to expect a significant contribution from these effects in the case of stiffening.We consider a uniformly heated infinite plate of thickness 2h stiffened along the curve r = rl by a rod-ring of the same thickness and made of a different material. Determining the stress-strain state of such a system using the revised theory, which makes it possible to determine all the components of the stress tensor [6], reduces to solving the following system of singularly perturbed differential equations for the functions w and ~P:Here t is the temperature, r is the radial coordinate, =45-r~rr r , ai=(1-2-)at is the coefficient of linear thermal expansion, and L, is the Poisson coefficient. We introduce the following notation for the Kelvin functions [5]:fl(x) = berx, f2(x) = beix, f3(x) = kerx, f4(x) = keix,We then write the general solution of the system (1) in the formi=l where T/= ?~E -1 and (..The boundary conditions (the absence of stresses at infinity and on the surface r = r0, where ro is the interior radius of the rod-ring) and the coupling conditions (continuity of the stresses ar and ~r~ and
We carry out a revised analysis of the concentration of thermal stresses in a shallow spherical shell with a small foreign inclusion. We show that near the interface of the materials the stressed state has a sharply expressed three-dimensional character; thus the stresses differ significantly (both qualitatively and quantitatively) from the shells predicted by the standard thearies_.The study of thermal stresses in shells with a foreign inclusion is of great value in applications, in particular in connection with the manufacture, testing, and use of electrovacuum devices. A bibliography devoted to this problem can be found in the monograph of Podstrigach, Kolyano, and Semerak [2]. However, nearly all the results known in the literature on this problem were obtained by applying shell theories that do not provide a sufficiently complete description of the effects connected with the three-dimensional nature of the stressed state in the contact zones.At the same time the study of the thermally stressed state of plates with foreign inclusions carried out using the revised theory of plates [6] has shown [3,7] that in the contact zones the stressed state differs from that predicted by the standard theories of plates. These effects have a boundary-layer character and arise at a distance from the contact surface commensurate with the thickness of the plate. The presence of such effects caused by the foreign matter, should be observed also in thin-walled shell structures.In the present paper we conduct a revised analysis of the thermally stressed state in the contact zone of a shell as follows. From the theory of shells we determine the thermally stressed state of a shell containing a round inclusion. On the basis of a parametric analysis of this solution we identify a contact zone of the shell that can be replaced by an annular plate without any significant distortion of the contact stresses. Then, using the revised theory of plates, which makes it possible to determine all the components of the stress tensor [6], we determine the stressed state of a circular plate with a foreign inclusion; the boundary conditions for this problem are formulated by applying the solution obtained from shell theory.Consider a uniformly heated shallow spherical shell containing a small foreign inclusion of radius r0. The thickness of the shell and the inclusion are the same and equal to 2h. For the shallow spherical shell we take the initial data to be the following system of key equations [4]:Here ~ is the stress function, w is the deflection, T is the temperature, R is the radius of the shell, E is the Young's modulus, v is the Poisson coefficient, ~ is the temperature coefficient of linear expansion, and A is the Laplacian. We introduce the following notation for the Kelvin functions [5]:~2i_1(X) = f2i(X), ~2i(X)=--f2i_l(Z), i=l,2.Then the general solution of the system (1) can be represented as
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