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.     

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 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 .     

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.     

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.     

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.

     

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.     

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.     

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.     

On the stability of equilibrium of continuous systems

Summary A sufficiency theorem for the stability of a linearly viscoelastic solid subjected to partial follower surface tractions is established. It is shown that an appropriately defined functional metric space must be introduced in order to formulate a well-posed problem. The usual energy method, if applicable, and the Galerkin method, if convergent, yield stability conditions only in a functional space whose metric is defined in an average

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Übersicht Es wird ein hinreichendes Kriterium für die Stabilität eines viskoelastischen festen Körpers abgeleitet, der Kräften unterworfen wird, die teilweise den Verformungen der Oberfläche folgen. Es wird gezeigt, daß ein geeignet definierter metrischer Funktionalraum eingeführt werden muß, um das Problem vernünftig formulieren zu können. Die übliche Energie-Methode, sofern sie anwendbar ist, sowie das GalerkinVerfahren, wenn es konvergiert, ergeben lediglich Stabilitätsbedingungen in einem Funktionalraum, dessen Metrik im Mittel definiert ist.

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This research was supported by the National Aeronautics and Space Administration under Grant NsG 605.     

Application of the Galerkin-Kantorovich-Dorodnitsyn method of integral relations to the solution of steady laminar boundary layers

Summary The Galerkin-Kantorovich-Dorodnitsyn N-parameter method of integral relations belongs in the same general category as those of Ritz and Galerkin. The method is employed to solve the steady, incompressible, two-dimensional flow of a laminar boundary layer. General Nth-order solutions are obtained for regions of accelerated flow and, with a modification to account for a possible separation point, for regions of retarded flow. The convergence and accuracy of the method are evaluated by comparing the solutions with a large number of similar and nonsimilar flow problems. The method permits a separation point to be approached as closely as desired without encountering numerical difficulties or requiring special continuation techniques.

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Übersicht Die N-Parameter-Methode der Integralbeziehungen von Galerkin, Kantorovich und Dorodnitsyn gehört zur gleichen Klasse wie die Verfahren von Ritz und Galerkin. Diese Methode wird benützt, um Lösungen für die stationäre, inkompressible, zwei-dimensionale Strömung einer laminaren Grenzschicht zu finden. Allgemeine Lösungen N. Ordnung werden für Gebiete mit beschleunigter Strömung erhalten und, mit gewissen Abänderungen zur Berücksichtigung eines möglichen Ablösepunktes, auch für Gebiete einer verzögerten Strömung. Die Konvergenz und die Genauigkeit der Methode werden durch Vergleich der Lösungen mit zahlreichen ähnlichen und nicht-ähnlichen Strömungsproblemen untersucht. Die Methode erlaubt eine beliebig weitgehende Annäherung an den Ablösepunkt, ohne daß numerische Schwierigkeiten auftreten oder besondere Grenzwertbetrachtungen erforderlich wären.

     

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     

A formulation of PANS capable of mimicking IDDES

The partially averaged Navier–Stokes (PANS) model, proposed in Girimaji (2006), allows to simulate turbulent flows either in RANS, LES or DNS mode. The PANS model includes $f_k$ which denotes the ratio of modeled to total kinetic energy. In RANS, $f_k=1$ while in DNS it tends to zero. In the present study we propose an improved formulation for $f_k$ based on the H-equivalence introduced by Friess et al. (2015). In this formulation the expression of $f_k$ is derived to mimic Improved Delayed Detached Eddy Simulation (IDDES). This new formulation behaves in a very similar way as IDDES, even though the two formulations use different mechanisms to separate modeled and resolved scales. They show very similar performance in separated flows as well as in attached boundary layers. In particular, the novel formulation is able to (i) treat attached boundary layers as properly as IDDES, and (ii) “detect” laminar initial/boundary conditions, in which case it enforces RANS mode. Furthermore, it is found that the new formulation is numerically more stable than IDDES.     

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.     

On network models of molten polymers: Loss of junctions due to stretching of material planes

Summary In a deforming continuum, material surfaces change their area. If material points are members of a surface which stretches during the deformation, these material points increase their relative distances. The continuum is modeled by a network of macromolecules which form temporary junctions between each other. Junctions on a stretching material surface increase their distance. As a consequence, the probability ofjunction breakage is increased.The stretching of material planes through a material point is described by means of an embedded coordinate system. The metric of this coordinate system is the Cauchy tensorC. Astretch measure _p is defined for arbitrary flows. The stretch measure is used to formulate a memory function in a rheological constitutive equation for large strain. Specific applications are made to unidirectional simple shear, uniaxial extension and biaxial extension.In a consecutive study, comparison with experiments will demonstrate to what degree the selected memory function can describe the actual behavior of the polymer.

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Zusammenfassung Materie-Flächen in einem Kontinuum ändern ihren Flächeninhalt, wenn sich das Kontinuum deformiert. Punkte auf einer Materie-Fläche vergrößern ihre Abstände, wenn die Fläche gedehnt wird. Das Kontinuum stellt man sich als ein Netzwerk von Makromolekülen mit temporären Knotenpunkten vor. Knotenpunkte in einer sich vergrößernden Fläche vergrößern während der Deformation ihren Abstand. Als Folge davon erhöht sich die Wahrscheinlichkeit, daß sich dieseKnoten lösen. Mit einem eingeprägten Koordinatensystem wird beschrieben, wie sich ebene Materie-Flächen, die durch einen Punkt der Materie gehen, vergrößern oder verkleinern. Mit der Metrik des Koordinatensystems, dem Cauchy-TensorC, wird ein Dehnungsmaß _p für Strömungen beliebiger Art definiert. Dieses Dehnungsmaß wird dazu verwendet, die Gedächtnisfunktion einer rheologischen Stoffgleichung für große Deformationen zu formulieren. Das Dehnungsmaß wird für besondere Strömungen wie die einfache Scherung, die einachsige Dehnung und die zweiachsige Dehnung berechnet.In einer späteren Arbeit soll durch Vergleich mit Experimenten gezeigt werden, inwieweit die gewählte Gedächtnisfunktion das Verhalten von Polymerschmelzen wiedergibt.

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On leave from the University of Stuttgart.

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With 2 figures     

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.