Limit equilibrium of a closed transversally isotropic cylindrical shell with a nonthrough longitudinal crack

In the context of an analog of the Leonov-Panasyuk-Dagdeil model we consider the problem of limit equilibrium of a nonshallow transversally isotropic cylindrical shell weakened by a nonthrough surface longitudinal crack. Based on the equations that take account of the initial stresses, we reduce the problem to a system of two singular integral equations with unknown limits of integration. We carry out a numerical analysis of the dependence of the opening of the edges of the crack on the load and the geometric and physico-mechanical parameters of the shell. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 31–36.     

Investigation of frequencies of natural vibrations of a transversally isotropic cylindrical panel with a circular hole

We consider the problem of natural vibrations of a hingedly supported transversally isotropic cylindrical panel with a circular hole. The deformation of the shell is described by modified equations of the Timoshenko theory of shells. The numerical solution of the problem is constructed by the indirect method of boundary integral equations based on the sequential representation of Green functions.     

Thermal stresses in a plate with transversely isotropic material

In this paper the thermal stresses in a plate with transversely isotropic material have been obtained by the method of Hankel transforms. Three cases of surface temperature over a circular region of exposure with flux of heat, paraboloidal distribution and constant temperature with surface radiation have been considered. Numerical results are presented for the case of surface radiation.     

Thermoelastic strain of a transversally isotropic cone

The exact solution to the problem of statical thermoelasticity for a transversally isotropic cone in the case where the characteristic equation has multiple roots, the surface of the cone is free of the action of external forces, and the temperature on the boundary can be represented in one variable by the Mellin integral and in a second variable, by a trigonometric series, is presented. Bibliography: 6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 23–32.     

Forced vibration of an initially stressed thick rectangular plate made of an orthotropic material with a cylindrical hole

The forced vibration of an initially statically stressed rectangular plate made of an orthotropic material is studied. The plate is simply supported along all its edges and contains an internal across-the-width cylindrical hole of rectangular cross section with rounded corners. The initial stresses are created by uniformly distributed normal forces applied to opposite end faces of the plate. Because of the hole, these stresses are not uniform in the plate and significantly affect the stress field caused by additional time-harmonic dynamical forces acting on the upper face of the plate. Hence, for solving the boundary-value problem considered, the superposition principle is unsuitable. Therefore, our investigations are carried out within the framework of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The corresponding boundary- value problems on determining the initial and additional, dynamical stress states are solved by using the three-dimensional finite-element method. Numerical results on stress concentrations around the cylindrical hole and the fundamental frequencies, and on the influence of the initial stresses on the frequencies are presented.     

Solution of the equations of a transversally isotropic cylindrical shell taking account of deformations caused by physico-chemical processes

Using the operator method we reduce the system of resolvent differential equations for a transvers ally isotropic cylindrical shell that takes account of the deformations caused by physicochemical processes to a single equation of tenth order. We construct the Green's function of this equation for the case of an infinite shell.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 80–84.     

Antiplane deformation of an isotropic massif with a curvilinear cavity

We solve the problem of antiplane deformation for an infinite isotropic massif with a curvilinear cavity in which a harmonic shear wave is propagating. The solution of the problem, which is based on the application of the theory of functions of a complex variable, is reduced to finding unknown constants from a system of linear algebraic equations. Numerical results are given for a circular cavity. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 115–119.     

Stability of transversally isotropic shells coupled with an elastic foundation

The stability of shells coupled with an elastic Winkler foundation is investigated. It is assumed that the shell is made of a material (glass-reinforced plastic) with low resistance to shear, as a result of which generalized theories that take transverse shear strains into account [1–4] must be used in the stability calculations. The solution obtained is compared with the corresponding solution obtained on the basis of the classical Kirchhoff-Love theory [8].Lvov Polytechnic Institute. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 669–673, July–August, 1969.     

On the stability of a compressible Rivlin's cube made of transversely isotropic material

^** Email: kostas.soldatos{at}nottingham.ac.uk This paper considers a unit cube made of a compressible, transverselyisotropic elastic material, with the direction of transverseisotropy being aligned normal to one pair of the cube's faces,and investigates the stability of a dilatation equilibrium stateof that cube, with respect to superposed pure homogeneous deformationswith principal directions parallel to the cube edges. This dilatation,intermediate equilibrium state (state I) of the cube is assumedattainable in two different ways. Accordingly, in what is termedas the ‘principal stability problem’ under investigation,state I is considered to be that of uniform dilatation, whichis attained upon loading normally and uniformly the oppositefaces of the unit cube with certain pairs of equal and oppositelydirected forces having appropriately selected magnitudes. Inwhat is termed as the ‘modified stability problem’under investigation, the same compressible, transversely isotropicunit cube is loaded uniformly by three identical pairs of equaland oppositely directed forces acting normally to its faces,and, hence, it attains in state I the shape of a certain rectangularparallelepiped. The necessary and sufficient conditions forstability of state I of the cube deformation are obtained inthe form of three inequalities, which are found to hold regardlessof whether the intermediate equilibrium state I is that of uniformdilatation or that of the aforementioned rectangular parallelepiped(non-uniform dilatation). These, however, lead to quite differentspecific results and conclusions when applied in connectionwith, first, the principal and, then, the modified stabilityproblem of a unit cube made of a particular type of a transverselyisotropic extension of the Blatz–Ko (isotropic) material.     

On the stressed state of a non-thin transversally isotropic spherical shell with a circular hole

On the basis of a generalized theory constructed using the Fourier-series expansion of the unknowns in Legendre polynomials of the thickness coordinate we give a representation of the general solution of the equilibrium equations of a transversally isotropic spherical shell for an arbitrary approximation. On this basis we study the problem of the stressed state of a shallow spherical shell with a circular cavity on whose boundary surface there are tangential stresses varying nonlinearly over the thickness. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 19–24.     

The stressed state of a transversally isotropic medium with a spheroidal inclusion under nonideal mechanical contact

We study the problem of the stressed state of a transversally isotropic medium containing a foreign inclusion in the form of a prolate spheroid under an arbitrary homogeneous stress field at infinity. On the “medium-inclusion” interface there is slipping without flaking. The stressed state is constructed in the medium and in the inclusion using the exterior and interior problems for a prolate spheroid on the basis of potential functions. The solution of the problem is reduced to studying infinite systems of linear algebraic equations. The results of numerical studies are shown as graphs that describe the stress distribution in both the transversally isotropic medium and in the inclusion under various boundary conditions. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 15–26.     

Singular integral relations and equations for a piecewise homogeneous transversally isotropic space with interphase defects

Using the constructed discontinuous solution for a piecewise homogeneous transversally isotropic space, we have obtained two-dimensional singular integral relations, which connect the jumps and sums of components of the stress tensor and displacement vector. These relations enable one to reduce the problems of interphase defects of arbitrary nature directly to systems of two-dimensional singular integral equations, whose kernels can be written in explicit form.     

On the problem of axial tension of a circular inhomogeneous transversally isotropic shell

We consider the problem of axial tension of a circular inhomogeneous cylindrical shell by longitudinal forces. The determination of the stress-strain state of the shell is performed on the basis of a refined theory. The possibility of losing local stability under axial tension is discussed.     

Propagation and interaction of short waves in a homogeneous transversally isotropic elastic medium

The Cauchy problem for the equations of motion of a homogeneous transversally isotropic elastic medium is considered. For its solution, a short-wavelength asymptotic expansion is constructed, which is also applicable near specific directions. The resonance set, i.e., the set of points at which the ray expansion cannot be used is described.