Stressed state of a piezoceramic body containing a spheroidal inclusion

An exact solution is constructed to the electroelastic problem for a medium with a sealed-in spheroidal inclusion under the assumption that the medium and the inclusion are different piezoceramic, transversally isotropic bodies with coincident axes of anisotropy. The components of the associated electroelastic field of the inclusion are plotted as functions of the external load and of the geometry of the inclusion. Numerical values of the stresses and of the projection of the electric induction vector are given for the medium in the neighborhood of the inclusion. Universidade de Beira Interior, Portugal: Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; and Ukrainian State University of Food Technologies, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 28–42, 1999.     

The mixed boundary-value problem for a transtropic layer with a cavity

Using superposition of solutions and Fourier series we study the stressed state in a transversally isotropic layer with a cavity. We describe the results of numerical studies as functions of the elastic properties of the material, the geometric characteristics of the body, and the nature of the load variation. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995 pp. 8–15.     

The mixed problem for a transtropic cylinder under local heating of the surfaces

By the method of homogeneous solutions we study the stressed state of a transversally isotropic cylinder under local heating of the lateral surfaces. We describe the results of numerical studies as functions of the elastic properties of the material, the geometric characteristics, and the variation of the heating zone on one of the lateral surfaces of the cylinder. Five figures, 3 tables. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, Vol. 27, 1997, pages 3–10.     

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.     

Relationship of stress intensity factors for plane cracks in isotropic and transversally isotropic media

An interrelationship between the stress intensity factors for plane cracks in isotropic and transversally isotropic media is found for symmetric and obliquely symmetric force and symmetric temperature loadings. It is shown that the stress intensity factors for a plane crack in a transversally isotropic material can be obtained using standard formulas for the transition from the stress intensity factor for the corresponding problem in the case of an isotropic material. As examples, stress intensity factors are found for a circular crack, for a periodic system of circular cracks in a transversally isotropic material under force and temperature loading, etc. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 3–16, 1999.     

On the thermal stress state of transversally isotropic solids of revolution of finite dimensions

By solving the stationary thermoelastic problem for a transversally isotropic hyperboloid of revolution we conduct an analytic study of the thermal stress state of a finite solid of revolution with a meridional section of complicated shape and propose a way to solve the problem of the reliability of information obtained using the numerical methods developed and algorithms implemented on a computer. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 7–15.     

Thermally stressed state of an ellipsoidal inhomogeneity (inclusion) in the case of polynomial external force and temperature fields

The stress structure inside an ellipsoidal inhomogeneity (inclusion) is studied for polynomial force and temperature interactions of arbitrary order in an elastic isotropic medium. It is shown that with perfect mechanical and thermal contact between the phases, the stressed state within the inhomogeneity is also a polynomial in the cartesian coordinates whose order equals the maximum order of the polynomials for the external force and temperature fields. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 67–76, 1999.     

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.     

Apparatus and applications of a generalized Fourier method for transversally isotropic bodies bounded by a plane and a paraboloid of rotation

We construct new basic solutions of equilibrium equations of a transversally isotropic paraboloid of rotation and obtain summation theorems that express them in terms of cylindrical solutions for a half-space and vice versa. The problem of the action of an axial concentrated force on an elastic transversally isotropic half-space with a fixed inclusion in the form of a paraboloid of rotation is investigated. The problem is solved by a generalized Fourier method and is reduced to a system of integral equations with a Fredholm operator on the condition that intersections of the boundaries of the half-space and inclusion are absent. We consider dependences of stresses on the shape of the paraboloidal inclusion and on the distance between the boundary surfaces and analyze the results of the calculation.     

The stressed state in an infinite medium having a system of rectilinear thin elastic inclusions with cyclic symmetry

Using the machinery of complex variable theory we study the stressed state in an unbouned medium of cyclically located thin elastic inclusions of finite length. The problem is reduced to solving a system of two singular integro-differential equations. A numerical analysis is carried out for the stress intensity factors in the vicinity of the ends of the inclusions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 87–90.     

Mathematical model of an internal crack in an elastoplastic cylindrical shell

We propose a mathematical model that makes it possible to reduce the problem of the stressed state and limit equilibrium of a cylindrical anisotropic elastoplastic shell with internal crack to a system of nonlinear singular integral equations with discontinuous functions on the right-hand sides. We construct an algorithm for numerical solution of such systems together with the conditions of plasticity and boundedness of stresses near the crack. For a transversally isotropic shell, we carry out a numerical analysis of the dependence of the opening of the internal crack front on the load and geometric and mechanical parameters. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences L'viv. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 111–116, April–June, 1998     

Fracture of an isotropic medium strengthened with a regular system of stringers

An isotropic medium containing a system of foreign transverse rectilinear inclusions is considered. Such a medium can be interpreted as an infinite plate strengthened with a regular system of ribs (stringers) whose cross section is a very narrow rectangle. The medium is weakened by a periodic system of rectilinear cracks. The action of the stringers is re placed by unknown equivalent concentrated forces at the points of their connection with the medium. The boundary-value problem on equilibrium of the periodic system of cracks under the action of external tensile forces is reduced to a singular integral equation, from the solution of which the stress in tensity factors are found. The condition of limiting state of equilibrium of the cracks is formulated based on a criterion of brittle fracture. The stress state in the case where crack faces come into a partial contact is also considered. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 59–72, January–February, 2007.     

The stressed state of a transversally isotropic body with inner elliptic crack. A linearly varying pressure on the surface of the crack

An exact solution to the problem of the distribution of stresses in a transversally isotropic elastic body with inner elliptic crack is given. An assumption that the linearly varying pressure acts on the surface of the crack is made. The intensity coefficients of stresses at the vertex of the crack are found. Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp. 117–124. Original     

Thermoelastic state of a medium,containing a thermally insulated ellipsoidal cavity,when subjected to a uniform heat flow

This article examines a problem of static thermoelasticity for an isotropic medium with an ellipsoidal cavity. The orientation of the cavity relative to the heat flow is assumed to be arbitrary. The solution of the problem is constructed in closed form and is expressed through incomplete elliptic integrals of the first and second kinds. In cases when the ellipsoid degenerates into a spheroid or sphere, the stress-strain state can be written in elementary functions of the ellipsoidal coordinate.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 46–50, 1988.     

Thermally Stressed State of Bodies with Two Ellipsoidal Inclusions

Solutions are obtained for the interaction of two ellipsoidal inclusions in an elastic isotropic matrix with polynomial external athermal and temperature fields. Perfect mechanical and temperature contact is assumed at the phase interface. A solution to the problem is constructed. When the perturbations in the temperature field and stresses in the matrix owing to one inclusion are re-expanded in a Taylor series about the center of the second inclusion, and vice versa, and a finite number of expansion terms is retained, one obtains a finite system of linear algebraic equations in the unknown constants. The effect of a force free boundary of the half space on the stressed state of a material with a triaxial ellipsoidal inhomogeneity (inclusion) is investigated for uniform heating. Here it was assumed that the elastic properties of the inclusions and matrix are the same, but the coefficients of thermal expansion of the phases differ. Studies are made of the way the stress perturbations in the matrix increase and the of the deviation from a uniform stressed state inside an inclusion as it approaches the force free boundary.     

Antiplane strain of a multiconnected anisotropic body with longitudinal cavities and plane cracks

Applying the theory of generalized complex potentials and the method of least squares, we solve the problem of the stressed state of a multiconnected anisotropic body under an antiplane strain. The problem is reduced to a system of linear algebraic equations in the unknown constants that occur in the required functions. By numerical studies we exhibit the influence of the elastic and geometric characteristics on the stress distribution and the variation of the stress intensity factors in a cylinder with one or two cracks and in an infinite body with circular cavities and cracks. One figure. Six tables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25. 1995, pp. 56–62.     

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.     

Effect of the curvature at the corner of a rectangular inclusion on the state of stress of a piecewise homogeneous body subjected to longitudinal shear

Using the function theory, the article examines antiplane strain of an isotropic body with a rectangular isotropic elastic inclusion. The solution of the problem is reduced to a system of linear algebraic equations. The article presents a numerical analysis of the state of stress at the corner in dependence on the curvature at this point and on the elastic constants of the matrix and of the inclusion.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 69–71, 1988.     

The effective model of a stratified solid-fluid medium as a special case of the Biot model

The Biot model is compared with the effective model of a stratified periodic elastic-fluid medium, and the parameters of the transversally isotropic Biot model that turn it into the effective model are found. In the course of this comparison, some intermediate models, which are generalizations of the effective model and particular cases of the Biot model, are considered. For all of the models, the wave fronts excited by a point source are determined. The distinguishing feature of wave fronts in the intermediate models is the occurrence of double loops on some of them. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 172–195. Translated by L. Molotkov     

Interaction of a harmonic torsional wave with ring-shaped defects in an elastic body

We have solved the problem of determination of the stressed state in an isotropic elastic body near ring-shaped defects (a crack or a thin rigid inclusion) as a result of the action of a harmonic torsional wave. The method of solution is based on the use of discontinuous solutions of the equation of torsional vibrations and lies in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of angular displacement or tangential stress.