On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

The effect of extension on the structure of filaments of 6.6-nylon

Modifications to the structure of filaments of 6.6 nylon, due to extension at various rates, have been investigated by optical and X-ray methods. Stable changes in the density and the principal X-ray spacings have been observed together with a reversible transition, during extension, of the kind noted byBrill.

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Zusammenfassung Strukturänderungen an 6.6-Nylonseide, hervorgerufen durch Verstrecken mit verschiedenen Geschwindigkeiten, werden mittels optischer und röntgenographischer Methoden untersucht. Es werden während der Verstreckung bleibende Veränderungen der Dichte und der Hauptebenenabstände sowie einBrillscher umkehrbarer Übergang beobachtet.

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Résumé Moyennant des méthodes optiques et radiologiques on a etudié des modifications de la structure des fils de nylon 6.6 entraînées á plusieurs vitesses d'allongement. On a observé des changements définitifs de la densité et des distances réticulaires principales ainsi qu'une transition reversible, du type constaté parBrill, au cours de l'allongement.

     

Investigation of the stability problems of elastic bodies using the method of mathematical theory of elasticity

In this paper, using the equilibrium equations and boundary conditions of elastic stability problem of and the method of mathematical theory of elasticity, we solve some elastic stability problems, which were studied by Ишлынский^[2] and Войцеховская^[3,4],and obtained more reasonable results than theirs.     

Asymptotic Behaviour of a Pile-Up of Infinite Walls of Edge Dislocations

We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions x _i > 0 of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x = 0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number of walls, n, tends to infinity, and characterise this behaviour in terms of Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter β _n , corresponding to ${\beta_n \ll 1/n, 1/n \ll \beta_n \ll 1}$ , and ${\beta_n \gg 1}$ , and the two critical regimes β _n ~ 1/n and β _n ~ 1. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter β _n .     

On the orders of the best approximations of integrals of functions by integrals of rank σ

We study the values e _σ(f) of the best approximation of integrals of functions from the spaces L _p (A, dμ) by integrals of rank σ. We determine the orders of the least upper bounds of these values as σ → ∞ in the case where the function ƒ is the product of two nonnegative functions one of which is fixed and the other varies on the unit ball U _p (A) of the space L _p (A, dμ). We consider applications of the obtained results to approximation problems in the spaces S _p ^ϕ. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 4, pp. 528–559, October–December, 2007.     

Asymptotic properties of solutions of systems of nonlinear functional differential equations of neutral type

We establish sufficient conditions for systems of nonlinear functional differential equations of neutral type to have solutions that are continuously differentiable and bounded for t ∈ ℝ (together with their first derivatives) and investigate the asymptotic properties of these solutions. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 20–26, January–March, 2009.     

Application of complex path-independent integrals to problems of bending of thin elastic plates

Summary Complex path-independent integrals have been already widely applied to problems of plane and antiplane elasticity for the determination of a variety of quantities of interest including stress intensity factors, loading intensities and the positions of geometrical characteristic lengths of singularities in the elastic field (like cracks, holes and inclusions). In this paper, we show that the same results apply also to the case of problems of thin isotropic elastic plates under bending where the complex-variable formulation is also valid. We make reference to the experimental methods which are appropriate for these integrals in an engineering environment and, finally, we apply this approach to the location of a circular hole in the problem of bending of a thin plate. Numerical results are also presented.

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Die Anwendung komplexer, wegunabhängiger Integrale bei Biegeproblemen dünner elastischer Platten

Übersicht Komplexe, wegunabhängige Integrale wurden schon häufig bei ebenen und antiplanaren Elastizitätsproblemen benutzt, etwa zur Ermittlung von Spannungskonzentrationsfaktoren, Lastintensitäten und der Position charakteristischer geometrischer Größen von Singularitäten im elastischen Feld (wie bei Rissen, Löchern und Einschlüssen). Es wird hier gezeigt, daß diese Ergebnisse auf Biegeprobleme dünner, isotrop-elastischer Platten übertagbar sind, da die Formulierung mit komplexen Variablen ebenfalls gültig ist. Auf die experimentellen Methoden, die für die ingenieurmäßige Ausnutzung geeignet sind, wird hingewiesen und dies wird erläutert anhand der Lokalisierung eines Kreislochs in einer gebogenen dünner Platte; numerische Resultate dazu werden angegeben.

     

Experimental investigations of the stability limit of the helical flow of pseudoplastic liquids

The stability of the laminar helical flow of pseudoplastic liquids has been investigated with an indirect method consisting in the measurement of the rate of mass transfer at the surface of the inner rotating cylinder. The experiments have been carried out for different values of the geometric parameter = R ₁/R ₂ (the radius ratio) in the range of small values of the Reynolds number,Re < 200. Water solutions of CMC and MC have been used as pseudoplastic liquids obeying the power law model. The results have been correlated with the Taylor and Reynolds numbers defined with the aid of the mean viscosity value. The stability limit of the Couette flow is described by a functional dependence of the modified critical Taylor number (including geometric factor) on the flow indexn. This dependence, general for pseudoplastic liquids obeying the power law model, is close to the previous theoretical predictions and displays destabilizing influence of pseudoplasticity on the rotational motion. Beyond the initial range of the Reynolds numbers values (Re>20), the stability of the helical flow is not affected considerably by the pseudoplastic properties of liquids. In the range of the monotonic stabilization of the helical flow the stability limit is described by a general dependence of the modified Taylor number on the Reynolds number. The dependence is general for pseudoplastic as well as Newtonian liquids.Nomenclature C _i concentration of reaction ions, kmol/m³ - d = R ₂ –R ₁ gap width, m - F _M () Meksyn's geometric factor (Eq. (1)) - F ₀ Faraday constant, C/kmol - i _l density of limit current, A/m³ - k _c mass transfer coefficient, m/s - n flow index - R ₁,R ₂ inner, outer radius of the gap, m - Re = V _m ·2d·/µ _m Reynolds number - Ta _c = _c ·d^3/2·R ₁ ^1/2 ·/µ _m Taylor number - Z _i number of electrons involved in electrochemical reaction - = R ₁/R ₂ radius ratio - µ apparent viscosity (local), Ns/m² - µ _m mean apparent viscosity value (Eq. (3)), Ns/m² - µ _i apparent viscosity value at a surface of the inner cylinder, Ns/m² - density, kg/m³ - _c angular velocity of the inner cylinder (critical value), 1/s     

The effect of random step changes of system parameters on the stability of steady vibration of nonlinear systems

Summary The author proposes a method of analysing the effect of random step changes of system parameters on the stability of steady solution of a nonlinear system in case this steady solution is not the only one. The method is based on the assumption that the time between successive step changes is comparatively long against the period of steady vibration so that transients become stabilized after each parameter change. The parameter is free to vary on a finite interval of values.

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Übersicht Es wird ein Verfahren vorgeschlagen, um die Auswirkung von zufälligen Sprungstörungen der Systemparameter auf die Stabilität von stationären Lösungen nichtlinearer Systeme zu untersuchen. Dabei wird vorausgesetzt, daß es mehrere stationäre Lösungen gibt. Es wird angenommen, daß die Zeit zwischen den nacheinander folgenden Sprungstörungen vergleichsweise lang gegenüber der Periode der stationären Schwingung ist, so daß die Einschwingvorgänge nach jeder Parameteränderung abgeklungen sind. Der Parameter kann sich dabei in einem endlichen Wertcbcreich ändern.

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Herrn Professor Dr. K. Klotter zum 70. Geburtstag gewidmet.     

Boundaries of the plasma-free regime of electroexplosion of a foil

Results of the experimental studies of the commutation characteristics of an electroexplosive breaker upon heating of a copper foil by a current pulse of duration τ≈1msec are given. The dependences of the electric-field strength and the specific power of the circuit breaker during an electroexplosion of the foil on the specific power of the source P_s are found and it is shown that the transition of the plasma-free regime of electroexplosion of the foil to a plasma regime occurs for P_s=30–40GW/g. Institute of Experimental Physics, Sarov 607190. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 14–20, January–February, 2000.     

Rheology of dispersant-free aqueous slurries of carbon black

The rheological characteristics of dispersant-free aqueous slurries of carbon blacks have been studied over a range of concentrations of 4 to 18%. Single pass measurements were carried out with a falling plunger viscometer in order to determine as nearly as possible, the properties of the undisturbed slurries. The slurries exhibit plastic or pseudoplastic flow behavior but are extremely sensitive to conditions of preparation. Slurries can be thickened by the application of high shear and thinned by the application of low shear. These changes are not spontaneously reversible on standing. The rheology of slurries of different carbon blacks depends on their particle size, structure, and surface chemistry. These results are interpreted in terms of a model involving small anisometric aggregates which can associate to form either a network or else a system of large, spherical flocs.

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Zusammenfassung Die rheologischen Eigenschaften von dispersionsmittelfreien wäßrigen Rußschlämmen wurden über einen weiten Konzentrationsbereich von 4 bis 18% untersucht. Um so nahe wie möglich die Eigenschaften der ungestörten Schlämme zu erfassen, wurden einmalige Messungen mit einem Fallzylinderviskosimeter vomLawaczek-Typ ausgeführt. Die Schlämme, die ein plastisches oder pseudoplastisches bzw. strukturviskoses Fließverhalten aufweisen, sind höchst empfindlich gegenüber der Art der Vorbereitung.Die Schlämme können durch Anwendung hoher Scherbeanspruchung eine Viskositätserhöhung zeigen und wider Erwarten durch geringe Scherbeanspruchung eine Viskositätserniedrigung. Diese Änderungen sind beim Stehen nicht spontan reversibel. Es wird dargestellt, daß die Rheologie der Schlämme verschiedener Rußarten von ihrer Teilchengröße, ihrer Struktur und den chemischen Oberflächeneigenschaften abhängt.Diese Ergebnisse werden interpretiert mittels einer Modellvorstellung, die kleine anisometrische Aggregate voraussetzt, welche zu einem Netzwerk assoziieren oder irgendein System von großen kugelförmigen Flocken bilden können.

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Presented at the Southwest-Southeast Regional Meeting of the American Chemical Society (December 7–9, 1961, New Orleans, La.).     

Investigation of the Structure of the Set of Continuous Solutions of Systems of Linear Difference Equations with Continuous Argument

We obtain conditions for the existence of continuous and N-periodic solutions, where N is a positive integer number, for systems of linear difference equations with continuous argument and investigate the structure of the set of these solutions. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 351–359, July–September, 2005.     

Explicit integration of one problem of motion of the generalized Kowalevski top

In the problem of motion of the Kowalevski top in a double force field the four-dimensional invariant submanifold of the phase space was pointed out by [Kharlamov, M.P., 2002. Mekh. Tverd. Tela 32, 33–38]. We show that the equations of motion on this manifold can be separated by the appropriate change of variables, the new variables s₁, s₂ being elliptic functions of time. The natural phase variables (components of the angular velocity and the direction vectors of the forces with respect to the movable basis) are expressed via s₁, s₂ explicitly in elementary algebraic functions.     

Sufficient Conditions for the Existence of Bounded Solutions of Systems of Nonlinear Difference Equations

We obtain sufficient conditions for systems of nonlinear difference equations x(n + 1) = A(x(n))x(n) + f(n), n ∈ ℤ, where A(x) is a matrix function continuous on ℝ ^m , to have solutions in the space of bilateral number sequences. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 165–173, April–June, 2005.     

Study on dimensionless criterion of fracture of closed pipe due to freezing of water

The pressure rise due to freezing of water enclosed in pipes has been studied, by means of theoretical computation and experiments, under the various heat transfer conditions and ambient temperatures, taking deformation of pipe walls into account. It has been found that the presented models well represent the actual phenomena and dimensionless parameters proposed are effective for correlating the pressure rises, the conditions of heat transfer, dimensions of the pipes. The tangential stress has been analyzed in dimensionless form to clarify its dependabilities on the modified Biot number and the ratio of thickness of a pipe wall and its radius. It has also shown thatF _crit, the critical Fourier number, can be considerably increased by introducing a void area at the center of pipes and the criterion of freezing fracture of pipes can be expressed as a combination of the dimensionless parameters presented in the report.

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Studie über ein dimensionsloses Kriterium zur Vorhersage des Berstens eines geschlossenen Rohres durch frierendes Wasser

Zusammenfassung Es wurde der Druckanstieg bei Einfrieren eines in Rohren eingeschlossenen Wassers experimentell und mittels theoretischer Berechnungen bei unterschiedlichen Werten von Wärmübergang und Umgebungstemperatur und unter Berücksichtigung der Verformung der Rohrwand berechnet. Dabei konnte festgestellt werden, daß das vorgestellte Modell die tatsächlichen Phänomene gut wiedergibt und die vorgeschlagenen dimensionslosen Parameter für die Berechnung des Druckanstieges und die Berücksichtigung der Wärmeübergangsbedingungen geeignet sind. Die Tangentialspannung wurde in dimensionsloser Form analysiert, um die Abhängigkeit von der modifizierten Biot-Zahl und von dem Verhältnis von Wandstärke zu Radius des Rohres zu klären. Es wird auch gezeigt, daß die kritische modifizierte Fourier-Zahl beträchtlich erhöht werden kann, wenn man eine Gaszone im Zentrum des Rohres einführt. Weiterhin wird gezeigt, daß das Kriterium für den Einfrierbruch des Rohres als Kombination dimensionsloser Parameter ausgedrückt werden kann.

     

Kinds of local self-similarity of the velocity field of prewall turbulent flows

An analysis of dimensionalities and an approach used by Millikan [1] in analysis of mean motion are applied to investigation of the pulsational motion of three types of prewall flows of an incompressible liquid, i.e., in a boundary layer with longitudinal flow around a plate, in a round tube, and in a flat channel. It is shown that with sufficiently large Reynolds numbers there exists an interval of distances from the wall x₂, within which the integral one-point correlations and the narrow-band one-point correlations _jk do not depend on x₂. In frequency space, there exists a hyperbolic interval in which _jk=A_jku ²f^-1. Here A_jk=const; u is the dynamic velocity; and f is the frequency. It is also shown that, from the point of view of the mean motion, a distinction must be made between Kármán turbulent flow with rather large Reynolds numbers and non-Kármán flow with small, but turbulent Reynolds numbers. In the latter case, the coefficients in the logarithmic profiles of the velocity and in the law of the resistance depend on the Reynolds number. The article gives an evaluation of the Reynolds number, which can be assumed to be rather large.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 35–42, November–December, 1976.The author considers it his pleasant duty to express his indebtedness to M. A. Kashina for furnishing experimental data.     

A property of the set of Rivlin-Ericksen tensors

It is shown that if a motion is such that the first two Rivlin-Ericksen tensors A ₁ and A ₂ have the same unit proper vectors, and A₁ has distinct proper numbers, then all the Rivlin-Ericksen tensors A ₁, A ₂, ..., A _n, ... have the same proper vectors. If further this motion is steady and isochoric, then the product of velocity components corresponding to the unit proper vectors, assumed to be non-vanishing, and the product of the abnormalities of the vector fields of the proper vectors, also assumed to be non-vanishing, each bear a constant value along a streamline. If A ₁ has a pair of equal proper numbers the first statement remains true, but the results for steady isochoric motions will not necessarily hold.     

Some specific features of high-temperature deformation of materials

Experimental results for high-temperature deformation of an iron-based structural material in the temperature ranges including the points of the Fe_α→Fe_β→Fe_ψ transition are given. It is shown that the strain-strength properties of the material change nonmonotonically on the interval 700°C<T<1000°C and that the internal phase-structural changes exert an effect on the thermal effects. Approximating dependences that permit one to describe deformation processes in the indicated temperature ranges upon uniaxial loading are proposed. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 152–156, November–December, 1999.     

Approximate determinations of the center of shear by use of the St. Venant solution for the flexure problem of plates of variable thickness

Summary We extend a recent analysis of the shear center problem in the context of the linear theory of transverse bending of orthotropic elastic plates by an explicit comparison of results based on several differing definitions of the center of shear within the framework of the St. Venant flexure solution for the problem of orthotropic plates. We also consider transverse bending of anisotropic plates where we find a significant aspect ratio effect not present in the orthotropic, case, and the shear center problem for plates with combined midplane bending and stretching as a consequence of constitutive coupling.

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Näherungsweise Bestimmungen des Schubmittelpunks für Platten variabler Dicke im Rahmen der St. Venant'schen Querkraft-Biegungslösung

Übersicht Eine kürzliche Analyse des Schubmittelpunktproblems als ein Problem der linearen Theorie orthotroper elastischer Platten wird erweitert durch einen Vergleich von Resultaten auf Grund von verschiedenen Definitionen des Schubmittelpunkts im Rahmen des St. Venant'schen Ansatzes für das Querkraft-Biegeproblem. In Ergänzung dieser Betrachtungen wird eine Bestimmung des Schubmittelpunkts anisotroper Platten ausgeführt, welche, im Unterschied zum orthotropen Fall, einen ausgeprägten Einfluß der Plattenlänge zeigt. Weiterhin behandelt wird das Schubmittelpunktproblem für Platten, für welche Biegung und Streckung durch das Stoffgesetz gekoppelt sind.

     

Flow birefringence of dilute solutions of polyethyleneoxide of high molecular weight at high rates

Summary Birefringence and orientation angle of solutions of Polyox WSR 301 in the concentration range of 50 to 1350 ppm have been measured up to 8000 s^–1. A few marked differences with ordinary flexible polymers have been evidenced: (i) a very low value of the extinction angle at high shear rate, which could be attributed to an unusually high molecular weight; (ii) the independence of this angle on the concentration, which is due to the variation of the flexibility with the concentration, as shown by the study of the concentration dependence of the segmental optical anisotropy; (iii) the concentration dependence of the elongation shows a plateau value forc 150 ppm, which may come from a concentration dependent and reversible association process, as shown from reduced viscosity measurements.

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Zusammenfassung Es wurden die Doppelbrechung und der Orientierungswinkel von Polyox-WSR 301-Lösungen im Konzentrationsbereich 50–1350 ppm bis zu Schergeschwindigkeiten von 8000 s^–1 gemessen. Dabei wurden einige ausgeprägte Unterschiede gegenüber normalen flexiblen Polymeren gefunden: (i) ein sehr niedriger Wert des Auslöschwinkels, was auf ein ungewöhnlich hohes Molekulargewicht schließen läßt: (ii) die Unabhängigkeit dieses Winkels von der Konzentration, was von einer Änderung der Flexibilität mit der Konzentration herrührt, wie sie durch die Untersuchung der Konzentrationsabhängigkeit der optischen Anisotropie der Segmente aufgezeigt wird; (iii) die Existenz eines Plateaus in der Konzentrationsabhängigkeit der Streckung beic 150 ppm, welches auf einen konzentrationsabhängigen reversiblen Assoziationsvorgang hindeutet, der auch durch Messung der reduzierten Viskosität nahegelegt wird.

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With 5 figures