Fundamental solution of displacement equations for a transversely isotropic elastic medium

A fourth-order linear elliptic partial differential equation describing the displacements of a transversely isotropic linear elastic medium is considered. Its symmetries and the symmetries of an inhomogeneous equation with a delta function on the right-hand side are found. The latter symmetries are used to construct an invariant fundamental solution of the original equation in terms of elementary functions.     

Fundamental solution of a plane problem of elasticity theory for a composite anisotropic medium

The plane problem on the action of an arbitrarily oriented concentrated force, applied at some point of an elastic plane, composed of two different anisotropic half-planes, is considered. By a special choice of a particular solution the problem reduces to a well-known differential equation of the anisotropic theory of elasticity with discontinuous coefficients. The latter reduces, by the method of the integral Fourier transform, to the Riemann boundary value problem. Expressions for the stresses and displacement derivatives at an arbitrary point of the plane are obtained. The application of the obtained results is illustrated on the example of a problem on an elastic linear inclusion (strap).Translated from Dinamicheskie Sistemy, No. 4, pp. 40–45, 1985.     

Deformation theory for gas-impregnated transversely isotropic materials

A mathematical model is constructed for a gas-impregnated transversely isotropic material. The material is modeled by a three-phase solid at each point of which there is a solid phase, and free and sorbed gas. Gas movement proceeds in two ways: by a crack and macropores system (filtration flow), and through microcracks (diffusion flow).Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 93–97, 1989.     

The equations of the geometrically non-linear theory of elasticity and momentless shells for arbitrary displacements

To validate earlier results for the case of arbitrary deformations and displacements in orthogonal curvilinear coordinates, kinematic and static relations of the non-linear theory of elasticity are set up which, in the limit of small deformations, lead, unlike the known relations, to correct and consistent relations. The same relations are also constructed for momentless shells of general form for the case of arbitrary displacements and deformations on the basis of which the problem of the static instability of a cylindrical shell with closed ends, made of a linearly elastic material and under conditions of an internal pressure (the problem of the inflation of a cylinder), is considered. It is shown that, in the case of momentless shells, the components of the true sheat stresses are symmetrical, unlike the three-dimensional case. All the above-mentioned relations are constructed for the loading of deformable bodies both by conservative external forces of constant directions and, also, by two types of “following” forces.     

An asymmetric wave field in a transversely isotropic elastic medium

A transversely isotropic homogeneous elastic medium excited by a point force perpendicular to the anisotropic axis is considered. The wave field in this medium is constructed and investigated. The front sets of the SV and SH waves are in contact with one another at a point. The front sets in the vicinity of this point are investigated additionally. If we consider the SH wave (or the SV wave) separately, then a false plane front set arises in this region. In considering the SH and SV waves in combination, this false front set disappears. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 163–174.     

The relation between the equations of the two-dimensional theory of elasticity for anisotropic and isotropic bodies

The two-dimensional problem of the theory of elasticity for an isotropic body is reduced to the solution of the problem for an anisotropic body. Small additional terms are introduced into the biharmonic operator of the problem of the theory of elasticity of an isotropic body, so that the generalized biharmonic operator obtained has no multiple roots. The general solution is then represented in terms of a function of the generalized complex variables, and numerical investigations for isotropic and anisotropic bodies are carried out using the same algorithms. The effectiveness of such a replacement is demonstrated in numerical investigations for simplify connected and multiply connected regions of arbitrary section.     

Averaged systems of equations of the theory of elasticity in a medium with weakly compressible inclusions

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 126–133, January, 1992.     

Representation of the solutions of the equilibrium equations for non-thin transversely isotropic plates

A system of equilibrium equations for a transversely isotropic plate is derived by expanding the unknown functions in a Fourier series in Legendre polynomials of the thickness coordinate. It breaks up into two groups of equations corresponding to a generalized two-dimensional stressed state and bending of the plate. A method is presented for representing the general solution for the first group in an arbitrary approximation as biharmonic, irrotational, and vortical solutions. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 3–12, 1999.     

Initial supercritical behavior of buckled transversely isotropic elastic medium

The paper is concerned with a transversely isotropic homogeneous elastic medium subjected to uniform compression in the isotropy plane. The medium becomes unstable in the sense of Hadamard [1] at a definite level of initial strain. The critical strain is established to be uniquely determinate from the system of equations of bifurcation of equilibrium; however, there are many modes of buckling corresponding to this strain. A solution of the system of equations of bifurcation is built in the form of doubly periodic functions sinr ₁ x ₁sinr ₂ x ₂. The uncertainty of the mode of buckling consists in the fact that the wave numbers r ₁ and r ₂ remain arbitrary. In order to determine the relationship between the wave numbers we examine the initial supercritical behavior of the material. It turns out that the only possible modes are the chess-board mode (with r ₁ = r ₂) and the corrugation-type mode (when one of the wave numbers r ₁ or r ₂ vanishes). The initial supercritical equilibrium is shown as being stable.     

The general solution and reduction to diagonal form of a system of equations of linear isotropic elasticity

A simple representation is obtained of the general solution of the Lame system of equations for an isotropic material in the form of a linear combination of the first derivatives of the three functions satisfying three independent wave or harmonic (in the static case) equations. In the two-dimensional case of plane deformation, the found solution directly implies the Kolosov-Muskhelishvili representation of the displacements by two analytic functions of a complex variable. A formula for generation of new solutions is given.     

A tenth-order theory of stretching of transversely isotropic sheets

Summary The problem of determining three-dimensional Poisson's ratio effect corrections for two-dimensional plane stress theory is solved approximately by assuming equilibrium stress systems with parabolic variations in thickness direction of the primary face-parallel stresses. An application of a variational theorem for stresses and displacements is shown to lead to a tenth-order system of two-dimensional differential equations for stress measures and certain weighted averages of displacements components. It is further shown that the solution of the tenth-order system can be expressed in terms of a bi-harmonic function, in conjunction with the solutions of one second-order and one fourth-order differential equation, involving Laplace operators only.

---

Zusammenfassung Das Problem der drei-dimensionalen Korrekturen für die zwei-dimensionale Theorie des ebenen Spannungszustandes wird, durch die Anwendung eines Variationsprinzips für Spannungen und Verschiebungen approximativ auf ein zwei-dimensionales Problem von zehnter Ordnung reduziert. Es wird weiterhin gezeigt das die Lösung dieses Systems ausgedrückt werden kann durch eine Bi-Potential-Funktion, zusammen mit der Lösung einer Gleichung zweiter Ordnung, und der Lösung einer Gleichung vierter Ordnung, die beide nur Laplace Operatoren enthalten.

     

Integral equations of plane static boundary value problems in the moment elasticity theory of inhomogeneous isotropic media

Boundary value problems in the plane moment and simplified moment elasticity theory of inhomogeneous isotropic media are reduced to Riemann-Hilbert boundary value problems for a quasianalytic vector. Uniquely solvable integral equations over a domain are derived. As a result, weak solutions for composite inhomogeneous elastic media can be determined straightforwardly.     

An error bound in Reissner's theory of transversely isotropic plates

Summary The report shows that Nordgren's [3] bound on the error of solutions in Reissner's theory of anisotropic plates can be improved in the case of plates with large transverse shear deformability.

---

Zusammenfassung Diese Mitteilung beweist, daß Nordgren's [3] Fehlerabschätzung für Lösungen der Reissnerschen Theorie der anisotropen Platten für Platten mit schwacher Schiebungssteifigkeit der Querschnitte verbessert werden kann.

     

A general solution for three dimensional steady-state transversely isotropic hygro-thermo-magneto-piezoelectric media

In this article, we extract the general solution of three dimensional (3D) equations using potential theory method (PTM) for steady-state, transversely isotropic, hygro-thermo-magneto-piezoelectric media (HTMPM). The governing equations are simplified by introducing the displacement functions. A general solution is completely determined by advantage of the superposition principle and operator theory, which is connected in terms of two functions, fulfilling a second-order and twelfth-order homogeneous partial differential equation (PDE), separately. With the help of Almansi’s theorem, the general solution can be further shortened, which is stated by seven harmonic functions only. The acquired general solutions are straightforward structure and helpful in boundary value problems of HTMPM. Further, we apply the 3D fundamental solutions inside an infinite and on the surface of semi-infinite of a steady point heat source united with a steady point moisture source transversely isotropic HTMPM. Comprehensive and exact solutions are given in the form of elementary functions, which appear as a standard for various types of approximate solutions and numerical codes. Some numerical simulation is conducted based on the obtained general solutions.     

Fictitious domain method for a system of equations in elasticity theory

Two versions of the fictitious domain method are considered for the system of equilibrium equations of a nonhomogeneous anisotropic elastic solid with rigid clamping. The rate of convergence bounds are derived:.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 16–23, 1988.