On the uniqueness of a continuous solution of the problem of decay of an arbitrary discontinuity for a gradient system

S. K. Godunov has established that the Lagrange variational equations, the differential equations of crystal optics, belong to a class of gradient systems. The problem of the decay of an arbitrary discontinuity for this system is considered herein, and an example is constructed of the ambiguity of a continuous solution of this problem. Moreover, some sufficient conditions for uniqueness of the continuous solution are indicated.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 251–258, February, 1973.     

The symmetrically branched crack in an infinite elastic medium

Summary The plane elastostatic problem of a symmetrically branched crack in an infinite isotropic body loaded by normal stresses perpendicular to the main crack axis at infinity was studied by using the method of complex potentials. The problem was reduced to a system of three singular integral equations. By means of an approximation of the integrals through the Gauss and Lobatto numerical quadrature procedures, these singular integral equations were transformed into a system of linear equations, which can be readily solved. The stress intensity factors at the tips of the branched crack were computed directly from the solution of the above system of linear equations and were compared with the already existing experimental solutions.

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Zusammenfassung Es wird das ebene elastostatische Problem eines symmetrischen gegabelten Risses für den unendlichen, isotropen und mit senkrecht zur Riss-Hauptachse belasteten Körpers untersucht, und zwar unter Anwendung der Methode der komplexen Potentiale.Das Problem wird auf ein Systen von drei singulären Integralgleichungen reduziert und weiter auf ein System linearer Gleichungen transformiert, vermittelst einer Näherung der Integrale mit Hilfe des leicht lösbaren numerischen Quadraturverfahrens von Gauss und Lobatto. Die Spannungsintensitätsfaktoren in den Spitzen des gegabelten Risses werden rechnerisch ermittelt und mit experimentellen Ergebnissen verglichen.

     

Wave attenuation in an elastic medium

This article examines the propagation of viscoelastic (elastic) waves in a medium consisting of two layers of finite thickness. It is found that there is a mechanical effect manifest in the monotonic dependence of the damping factor on the parameters of the system. These dependences have distinct maxima and minima, thus making it possible to optimize the damping properties of systems by varying their geometric parameters.Translated from Dinamicheskie, Sistemy, No. 4, pp. 57–62, 1985.     

一类弹性梁的能量衰减估计

研究一类弹性梁的耗散性.首先应用非线性算子半群理论证明了系统的适定性;进而运用能量方法结合乘子技巧得到弹性梁系统的能量衰减估计.     

The poisson set of cracks in an elastic continuous medium

The method of effective (self-consistent) field is used for solving the problem of a random set of interacting cracks in an elastic medium. Construction of the first moment of solution is shown on the example of a medium containing Poisson set of plane elliptic cracks. Effective elastic constants of a medium with cracks are determined and the results obtained in the plane case are compared with published experimental data.     

The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.     

On the uniform decay in an integro-differential elastic equation

In this paper we consider an integro-differential elastic equation, in a bounded domain, and establish a uniform decay result under weaker conditions, on the relaxation function, than those usually found in the literature. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stressed state of a thin coating on an elastic inclusion of optimal shape in an elastic medium

We examine the properties of the stressed state of the thin intermediate zone separating a nonhomogeneous inclusion from the matrix. Explicit expressions are given for the components of the stress tensor in the thin coating depending on the properties of the phases and the load parameters.Simferopol' University. Translated from Dinamicheskie Sistemy, No. 10, pp. 26–30, 1992.     

The point moment at an arbitrary point of an elastic plane weakened by an elliptical hole

Expressions are obtained for the complex potentials characterizing the stress-strain state of an elastic plane with an elliptic hole subjected to a moment at an arbitrary point of the plane. The shear stresses at the edge of the hole are calculated. The stress intensity factors for the limiting case of a straight slit (a crack) are determined.     

Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear

We investigate the anti-plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. The problem is first reduced to a non-homogeneous Riemann–Hilbert problem, the solution of which can be obtained by evaluating the associated Cauchy integral. A set of linear algebraic equations is obtained from the compatibility condition imposed on the resulting analytic function defined in the inhomogeneity and its Faber series expansion. Each of the unknown coefficients in the corresponding analytic functions can then be uniquely determined by solving the linear algebraic equations, which are written concisely in matrix form. The resulting analytical solution is then used to quantify the displacement jump across the debonded section of the interface as well as the traction distribution along the bonded section of the interface. In addition, our solution allows us to obtain mode-III stress intensity factors at the two crack tips. The solution to the anti-plane problem of a partially debonded elliptical inhomogeneity containing a confocal crack is also derived using a similar method.     

The solution of the problem of the caustic in an elastic medium

The problem of the caustic in an elastic medium is solved by the method of local expansions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 255–268, 1974.     

High-frequency point sources in an inhomogeneous elastic medium

Higher approximations of ray asymptotics are investigated by the boundary-layer method. For sources that are naturally called a center of pressure and a center of rotation, direction diagrams are found for transverse and longitudinal waves, respectively, which are absent in a homogeneous medium.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 28–42, 1980.     

Reconstruction of cracks in an inhomogeneous anisotropic elastic medium

In this paper we give in two and three dimensions a reconstruction formula for determining cracks buried in an inhomogeneous anisotropic elastic body by making elastic displacement and traction measurements at the boundary. The information is encoded in the local Neumann-to-Dirichlet map. With the help of the Runge property, the local Neumann-to-Dirichlet map is connected to the so-called indicator function. This function can be expressed as an energy integral involving some special solutions, called reflected solutions. The heart of our method lies in analyzing the blow-up behavior at the crack of the indicator function, which is by no means an easy task for the inhomogeneous anisotropic elasticity system. To overcome the difficulties, we construct suitable approximations of the reflected solutions that capture their singularities. The indicator function is then analyzed by the Plancherel formula.     

Propagation of axially symmetric elastic vibrations along a waveguide in an elastic medium

The scatter equation is derived for axially symmetric (longitudinal and torsional) normal vibrations of an infinitely long elastic cylindrical waveguide embedded in an elastic medium. The low-frequency vibrations are analyzed thoroughly. It is shown that low-frequency longitudinal vibrations follow either of two branches of the scatter curve. Recommendations are made concerning the use of waveguide methods for studying the physicomechanical properties of polymers.     

The load transfer between two rigid spherical inclusions in an elastic medium

The paper is concerned with the load transfer problem between two rigid spherical inclusions in an elastic matrix. A reflection-type formula is developed which is accurate up to and including terms ofO(a/R)⁴, wherea is the maximum radius of the inclusions, andR is the centre-to-centre distance between the two inclusions. Asymptotic results are derived at near touching showing a weak logarithmic singularity in the load transfer. The results are verified by a direct numerical calculation using a boundary collocation method. The numerical method uses Kelvin's general solution as the basis functions for the approximate solution and is highly accurate and efficient, even at near touching.     

The strains and the energy in thin elastic shells of arbitrary shape for arbitrary deformation

Zusammenfassung Im ersten Teil (I) werden Formeln entwickelt, welche die kinetische und potentielle Energie dünner Schalen beliebiger Form explizit als Funktion eines beliebigen Verschiebungsfeldes ausdrücken. Dies geschieht auf rein analytische Weise ohne Zuhilfenahme geometrischer Darstellungen einzelner Schalenelemente. Von besonderem Wert für viele Anwendungen ist es, dass vollständige Freiheit in der Wahl der Flächenkoordinaten besteht.Im zweiten Teil (II) wird zunächst die Anwendung der Theorie auf zwei bekannte Beispiele gezeigt: 1. Platte in schiefwinkligen Koordinaten, 2. zylindrische Schale. Als neues Beispiel wird 3. die schraubenflächige Schale eingehend behandelt. Für einen besonderen Einspannungsfall wird die Frequenz der Grundschwingung als Funktion der Verwindung angenähert numerisch berechnet.     

A hollow ellipsoidal needle in an orthotropic elastic medium

The problem of the stress distribution on the surface of a hollow ellipsoidal needle in an orthotropic elastic medium and a homogeneous external field is solved. Explicit expressions are obtained for the stresses on the needle surface in terms of the elastic constants of the medium and parameters of the ellipsoid in a local system of coordinates connected to the normal to the surface at each point of the needle. The general solution of the problem of the stress concentration on an ellipsoidal inhomogeneity /1/ and the passage to the limit cases of an ellipsoidal cavity based on the presence of small parameters /2/ is used.     

On torsional loading in an axisymmetric micropolar elastic medium

Effects of torsional loading in an axisymmetric micropolar elastic half-space are studied. The components of microrotation displacement, force stress and couple stress are obtained for a half-space subjected to an arbitrary load produced by shearing stress. A special case of a particular type of twist has been discussed in detail for a specific model and the micropolar effects have been shown graphically.     

An equilibrium penny-shaped crack in an inhomogeneous elastic medium

The problem of a penny-shaped tensile crack in a continuously-inhomogeneous space is considered. The problem reduces to a dual integral equation for which an approximate analytical solution is constructed. It is proved that the approximate solution of the integral equation is asymptotically exact for both small and large values of the dimensionless geometric parameter of the problem. The accuracy of the solution obtained is investigated. Expressions are presented for the stress intensity factor, the energy of the opening of the crack, the displacements of its sides and the normal components of the stress tensor in the neighbourhood of its contour. In the numerical analysis of the solution of the problem, special attention is paid to analysing of the problem when the first derivative of the change in the elastic properties of the material changes sign.