Thermally stressed state of solids of revolution with finite dimensions

The solution of the axisymmetric elasticity problem for an unparted hyperboloid of revolution is used to obtain the exact analytical solution of the reference problem of the thermally stressed state of an isotropic solid of revolution with finite dimensions and a transverse cross section of complex configuration.Kiev. Moscow. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 3–9, 1990.     

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.     

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.     

The first and second fundamental boundary-value problems of elasticity theory for a transversally isotropic paraboloid of revolution

We give an exact solution of the interior and exterior problems of elasticity theory for a transversally isotropic paraboloid of revolution in the case when the stresses prescribed on its surface or the displacements along one variable can be represented by a Hankel integral and those along the other variable can be expanded in a trigonometric series. It is assumed that the roots of the characteristic equation are of multiplicity greater than one.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 60–70.     

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.     

Equivalence of periodically layered and transversally isotropic media

The propagation of waves in isotropic media with homogeneous layers is considered. When the period decreases without bound and the number of periods grows without bound, these media go over into transversally isotropic media. The properties of the media obtained are investigated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 219–233, 1979.     

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.     

Long-time strength of isotropic polymers in plane stress state

A criterion of strength of isotropic polymers in the form of the critical work done by stress was postulated and shown, by statistical analysis, to be independent of the time and form of loading. The introduction of the time factor in the existing theories of strength of isotropic materials was postulated and experimentally verified with the aid of this criterion.Mekhanika Polimerov, Vol. 2, No. 6, pp. 837–844, 1966     

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 modeling of thermo-electro-magneto-mechanical solids at finite strains

A short overview of the mechanics of electromagnetic continua at finite strains is provided in this paper. In this framework objective thermodynamically-consistent constitutive relations are formulated for materials that exhibit thermo-electro-magnetomechanical coupling. In particular a model for an isotropic hyperelastic magnetic solid is considered, which finds application for magnetorheological elastomers (MRE). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the operation Alg Lat in finite dimensions

For any algebra $\text{U}$ of linear transformations on a finite dimensional complex vector space V, AlgLat $\text{U}$ denotes the set of transformations which leave invariant every invariant subspace of $\text{U}$. It is shown that $\text{AlgLat}\text{U}\text{=(}\text{R}\text{∩}\text{U}{}^{\mathrm{\perp }}\text{)}{}^{\mathrm{\perp }}$, where $\text{R}$ denotes the set of transformations of rank at most one and ^⊥ is an orthocomplement operation. This is used to calculate dimAlgLat$\text{U}$? dim$\text{U}$ when $\text{U}$ is the algebra generated by a nilpotent transformation.     

Dynamically possible finite deformations of isotropic,incompressible, elastic-inelastic solids with temperature independent response

Summary In isotropic, incompressible simple materials there are five known families of dynamically possible inhomogeneous finite deformations. It is shown that these deformations are also possible in isotropic, elastic-inelastic materials with temperature independent response. The method of solution is an inverse one. The deformation is specified precisely at the outset, and the problem of finding the surface tractions which are required in order to maintain the deformation is reduced to the problem of solving a system of ordinary differential equations.

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Zusammenfassung In isotropen inkompressiblen einfachen Substanzen mit Gedächtnis gibt es fünf Gruppen von dynamisch möglichen nicht-homogenen Deformationen. In der Arbeit wird gezeigt, dass diese Gruppen von Deformationen auch in isotropen inkompressiblen elastisch-inelastischen Substanzen dynamisch möglich sind. Die vorgeschlagene Methode stellt einen inversen Lösungstyp dar. Die Deformationen werden als gegebene Funktionen der Zeit vorausgesetzt, und das Problem der Berechnung der Oberflächenspannungen, welche diese Deformationen erzeugen, wird auf die Lösung eines Systems von gewöhnlichen Differentialgleichungen reduziert. Es wird auch gezeigt, dass die Anzahl der Gleichungen in diesem System für keine der erwähnten Gruppen von Deformationen 3+n übersteigt, wobein die Zahl der Strukturparameter ist, die die elastisch-inelastische Substanz charakterisieren.

     

On the finite field Kakeya problem in two dimensions

A two-dimensional Besicovitch set over a finite field is a subset of the finite plane containing a line in each direction. In this paper, we conjecture a sharp lower bound for the size of such a subset and prove some results toward this conjecture.     

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     

Structure of the frequency spectrum of nonaxisymmetric vibrations of a transversally isotropic hollow ball

We consider nonaxisymmetric vibrations of a hollow sphere made of a spherically Isotropic material. The original boundary-value problem is reduced to a system of ordinary differential equations by introducing new unknown junctions and expanding the sought solution in a series in spherical functions. The nonaxisymmetric problem of vibrations of a hollow ball decomposes into two independent problems that describe modes of first and second kind. The frequency spectrum for modes of first and second kind is analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 32–35, 1986.     

On the adoption of the Monte Carlo method to solve one-dimensional steady state thermal diffusion problems for non-uniform solids

The present paper is focussed on the investigation of the potential adoption of the Monte Carlo method to solve one-dimensional, steady state, thermal diffusion problems for continuous solids characterised by an isotropic, space-dependent conductivity tensor and subjected to non-uniform heat power deposition.