Analysis of the stress-strain state of a non-shallow orthotropic cylindrical shell with an elliptical hole

Scientific-Production Organization Sistema, L'vov. Translated from Prikladnaya Mekhanika, Vol. 24, No. 4, pp. 57–63, Ap-il, 1988.     

Free vibrations of an unevenly tensioned tape

Vocational College at the Leningradskii Metallicheskii Zavod Industrial Association. Translated from Prikladnaya Mekhanika, Vol. 26, No. 4, pp. 72–79, April, 1990.     

On Full Two-Scale Expansion of the Solutions of Nonlinear Periodic Rapidly Oscillating Problems and Higher-Order Homogenised Variational Problems

We consider a scalar quasilinear equation in the divergence form with periodic rapid oscillations, which may be a model of, e.g., nonlinear conducting, dielectric, or deforming in a restricted way hardening elastic-plastic composites, with outer periodicity conditions of a fixed large period. Under some natural growth assumptions on the stored-energy function, we construct for uniformly elliptic problems a full two-scale asymptotic expansion, which has a precise double-series structure, separating the slow and the fast variables in all orders, so that its slowly varying part solves asymptotically an infinite-order homogenised equation (cf. Bakhvalov, N.S., Panasenko, G.P.: Homogenisation: Averaging Processes in Periodic Media. Nauka, Moscow, 1984 (in Russian); English translation: Kluwer, 1989), and whose higher-order terms depend on the higher gradients of the slowly varying part. We prove the error bound, i.e., that the truncated asymptotic expansion is higher-order close to the actual solution in appropriate norms. The approach is extended to a non-uniformly elliptic case: for two-dimensional power-law potentials we prove the non-degeneracy using topological index methods. Examples and explicit formulae for the higher-order terms are given. In particular, we prove that the first term in the higher-order homogenised equations is related to the first-order corrector to the mean flux, and has in general the form of a fully nonlinear operator which is quadratic with respect to its highest (second) derivative being a linear combination of the second minors of the Hessian with coefficients depending on the first gradient, and in dimension two is of Monge-Ampère type. We show that this term is present at least for some examples (three-phase power-law laminates).In the second part of the paper we extend to this nonlinear context some of the results previously developed by us in the linear case (Smyshlyaev, V.P., Cherednichenko, K.D. J. Mech. Phys. Solids 48, 1325–1357, 2000). In particular, we prove that the slowly varying part of the full asymptotic expansion is the rigorous asymptotics in all orders for the translationally averaged actual solution and flux, or in the sense of a higher-order version of the weak convergence. We then explore to what extent the method of variational truncation of the infinite-order homogenised equation, successfully implemented by us in the linear context in the previous work for constructing explicit higher-order homogenised equations, is extendable to the nonlinear regime. We propose a natural extension and prove that at least under some further natural non-degeneracy assumptions it has a solution (the existence), and that any such solution is close to the actual solution in appropriate norms.Acknowledgement We acknowledge the support of the Department of Mathematical Sciences, University of Bath, U.K. and partial support of the Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K. KDC also acknowledges the support of St. Johns College, Oxford, U.K. and thanks the Mathematical Institute, Oxford for its hospitality.     

On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process

The detailed analysis of the dynamical process of coin tossing is made. Through calculations, it is illustrated how and why the result is extremely sensitive to the initial conditions. It is also shown that, as the initial height of the mass center of the coin increases, the final configuration, i.e. head or tail, becomes more and more sensitive to the initial parameters (the initial velocity angular velocity, and the initial orientation), the coefficient of the air drag, and the energy absorption factor of the surface on which the coin bounces. If we keep the head upward initially but allow a small range for the change of some other initial parameters, the frequency that the final configuration is head, would be 1 if the initial height h of the mass center is sufficiently small, and would be clo to 1/2 if h is sufficiently large. An interesting question is how this frequency changes continuously from 1 to 1/2 as h increases. Detailed calculations show that such a transition is very similar to the transition from laminar to turbulent flows. A basic difference between the transition stage and the completely random stage is indicated: In the completely random stage, the deterministic process of the individual case is extremely sensitive to the initial conditions and the dynamical parameters, out the statistical properties of the ensemble are insensitive to the small changes of the initial conditions and the dynamical parameters. On the contrary, in the transition stage, both the deterministic process of the individual case and the statistical properties of the ensemble are sensitive to the initial conditions and the dynamical parameters. The mechanism for this feature of the transition stage is the existence of the long-train structure in the parameter space. The illuminations of this analysis on some other random phenomena are discussed.     

Second order phase equilibrium for classical bodies

Summary In this note the Author, recalling a previous work[15], gives a new formulation of second order phase equilibria for classical bodies such as those defined by Truesdell and Toupin in[8].The Author arrives at three equivalent systems of partial differential equations (generalized Ehrenfest equations), the conditions for whose integration are shown to be always satisfied.Finally, as particular cases, the equations ruling the phase equilibria for classical fluids and for n component-classical fluid mixtures are given.

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Sommario In questa nota l'Autore, rifacendosi ad un lavoro precedente[15], presenta una nuova formulazione degli equilibri di fase del secondo ordine per corpi classici come quelli definiti da Truesdell e Toupin in[8].L'Autore perviene a tre sistemi equivalenti di equazioni alle derivate parziali (equazioni di Ehrenfest generalizzate) dei quali viene dimostrata la integrabilità.Infine, come casi particolari, si ottengono le equazioni che governano l'equilibrio di fase per fluidi classici e per miscele fluide classiche ad n componenti.

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This work was supported by the Gruppo Nazionale per la Fisica Matematica of C.N.R.     

Über die Übertragungseinheiten und ein „Vierfelderdiagramm” zur Berechnung von Wärmeübertragern

Zusammenfassung Es wird dargelegt, wie man nach Einführung des bekannten Begriffes Übertragungseinheit und mit Hilfe eines Vierfelderdiagramms zu einer relativ einfachen und durchsichtigen Berechnung von Gleich- und Gegenstrom-Wärmeübertragern gelangt.

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On transfer units and a four-quadrant-diagram for the calculation of heat exchangers

It is shown that the introduction of the known concept of a transfer unit and the use of a four-quadrant diagram leads to a relatively simple and clear computation method for co-current and counter-current heat exchangers.

     

Invariant integrals in continuum mechanics

Presented at the International Conference on Contemporary Mathematical Problems in Mechanics and Their Applications (Moscow, November 11–15, 1987).     

Whose Thoughts Are They, Anyway? Dimensionally Exploding Bion's “Double-Headed Arrow” into Coadapting, Transitional Space

The physics and biology that found psychoanalysis account for discontinuous experience only in the presence of nonmeasurable, metaphysical operators; these include the ego and its subsystems as well as biological experience inherited through Lamarckian principles. Complex, self-organizing systems, however, can link biology to experience without metaphysics. They can also account for psychoanalytically relevant behaviors without appealing to stable internal representations. These behaviors include what W. R. Bion called transformation in O and its corollary, the appearance of the selected fact. By dimensionally exploding the double-headed arrow that he used to link the states Ps and D in his model for thinking (Ps D), we can generate a space that is, at once, psychoanalytically imaginal and dynamically coadapting. Isomorphic to D. W. Winnicott's transitional space, it is self-organizing. It is describable according to dynamics formulated by W J. Freeman, S. Kauffman and C. Langton and it can generate instantaneous conscious contents by way of a selective process analogous to spatio-temporal binding. As a whole, this model supports a clinical stance advanced by D. W. Winnicott as play, within transitional space.     

Some aspects of dilute suspension theory

A theory proposed by the author as representative of the flow of a general suspension contains three interaction forces, f, S and N. For a quasi-concentrated suspension and for a dilute suspension, N and S, N are omitted, respectively. For the latter special case, we treat diffusion of a fluid through an elastic solid. For a quasi-concentrated suspension, we show that F and S depend on the gradient of the motion gradient. We demonstrate the existence of interesting phenomena: non-simple behavior, dissipative effects, generalized lift and drag forces.Presented at the second conference Recent Developments in Structured Continua, May 23 – 25, 1990, in Sherbrooke, Québec, Canada.     

On linear versus nonlinear flow rules in strain localization analysis

Summary This note contains some remarks on the analysis of bifurcation phenomena, specifically strain localization (onset of a strain rate discontinuity), in small-deformation elastoplasticity. Nonassociative flow rules are allowed for to cover constitutive models frequently adopted for frictional (and softening) materials such as concrete. The conventional derivation of the localization criterion resting on an incrementally linear comparison material is critically reviewed and compared to the criterion resulting from actual nonlinear plastic flow laws.

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Sommario Si svolgono alcune considerazioni sui fenomeni di biforcazione in solidi elastoplastici in regime di piccole deformazioni (di linearità geometrica) e precisamente sul manifestarsi di localizzazioni intese come discontinuità nel campo delle deformazioni incrementali. Si considerano leggi nonassociate. Vengono così inclusi nella trattazione modelli costitutivi frequentemente adottati per descrivere il comportamento di materiali ad attrito interno e soggetti a danneggiamento (nel senso di degrado di rigidezze elastiche in seguito a deformazioni anelastiche), oltre che soggetti a manifestazioni di instabilità per incrudimento negativo (softening). Si esamina criticamente il criterio, frequentemente adottato in letteratura, di localizzazione fondato su materiale di confronto incrementalmente lineare e si fornisce, corredato da osservazioni comparative, il criterio che risulta dall'assunzione del modello di materiale incrementale non lineare

     

Studies of Mental “Noise”

Successive tests of subjects' performance in reaction time tasks were treated as time series and submitted to spectrum analysis. Rather than revealing the white noise expected by the view that variability is due to random error (equivalent power across frequency), the power spectra revealed colored noise. The slopes of the power spectra did not vary much for tasks differing in memory requirements. Spectrum analysis of time series from the logistic map also showed colored noise in regions on the edge of chaos, showing that the presence of colored noise in cognitive data need not oblige a theoretical account based on a complex, high-dimensional, system.     

Versuche zu einem „shear-rate-blockage“-Effekt am Protoplasma zentrifugierter Gewebe

Ohne ZusammenfassungI thank Mr.Scott Blair for his answer reserving the term dilatancy for cases of real dilatation. Now it's only to wish that this terminology is used according to his Report on the principles of rheological nomenclature (Amsterdam 1949).     

Visualization of fan-spreading in two-phase jet flows

Flow visualizations obtained in a two-phase jet flow with 80 m particles at a mass loading of 5% revealed the following.

Shock Waves in a Fluid with Bubbles Containing Evaporating Drops of a Liquefied Gas

The propagation and reflection of one-dimensional plane unsteady waves and pulses in a mixture of a fluid with two-phase bubbles containing evaporating drops is investigated. A significant effect of unsteady evaporation of the drops in the zone ahead of the shock wave on the wave propagation is demonstrated. The evaporation of the drops results in a pressure increase ahead of the wave and the shock wave as it were climbs to increasing pressure level. In contrast to bubbly fluids with single-phase bubbles, in a fluid with two-phase bubbles, at a fixed phase volume fraction, a decrease in bubble size results in an increase rather than a decrease of the oscillation amplitude. The wave reflection from a solid wall is essentially nonlinear and the maximum pressure attained at the wall is several times greater than the incident-wave intensity.     

First order phase equilibrium for classical bodies

Summary In this note the Author gives a general and unified treatment of first order phase equilibria for classical bodies like those considered by Truesdell and Toupin in[3].The Author reaches a system of partial differential equations (generalized Clapeyron equations) the conditions of whose solution are shown always to be satisfied.In particular, the Author derives the equations governing the polarized phase equilibrium for a fluid.Besides the equations ruling the phase equilibrium for a two phase n-component fluid mixture are given and the equivalence with the statical Gibbs-Duhem relation is shown.

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Sommario In questa nota l'Autore presenta una trattazione generale ed unificata degli equilibri di fase del primo ordine per corpi classici come quelli definiti da Truesdell e Toupin in[3].L'Autore perviene ad un sistema di equazioni alle derivate parziali (equazioni di Clapeyron generalizzate) del quale si dimostra la integrabilità.In particolare, si deducono le equazioni che governano gli equilibri di fase polarizzati.Inoltre si ottengono le equazioni che regolano l'equilibrio di fase per una miscela fluida a n componenti; in questo caso si dimostra l'equivalenza delle equazioni con la relazione statica di Gibbs-Duhem.

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This work was supported by the Gruppo Nazionale per la Fisica Matematica of C.N.R.     

Mathematics and Rheology

Summary The great advances made in Rheology during the last forty years owe much to the application of mathematics. But in some cases, there have been misunderstandings. As a result, rheologists have sometimes been unnecessarily restricted. Mathematics is limited only by self-consistency: Rheology deals with the physical world.In particular, three terms are discussed: infinity, zero and negative, all of which have at least two meanings which have been confused. Rheological phenomena cannot be explained merely by mathematical formulation. Unnecessary mathematics should be avoided but professional rheologists must not shirk mathematics.This paper was read in an abbreviated form, at aConference of the British Society of Rheology in April 1971 The Proceedings were not published.     

Conical flows in the neighborhood of breaks in the edges of surfaces separating cocurrent supersonic streams of perfect gas

An investigation is made into the conical flows which occur when a perfect (inviscid and nonheat conducting) gas flows over the terminal edges of surfaces with breaks separating an external and an internal flow with velocity vectors parallel to the line of intersection of the surfaces. Such flows are observed, in particular, in the neighborhood of breaks in the outlet edge of a nozzle of rectangular cross section with a straight or skewed exit plane under conditions of underexpanded flow of a supersonic jet into a cocurrent supersonic stream. By means of a linear analysis flow patterns corresponding to various flow interaction regimes and edge geometries are constructed and a law of similarity is formulated. The validity of the results thus obtained is confirmed by examples of the numerical solution of the complete nonlinear system of Euler equations. In this connection, within the framework of the approach outlined in [1], as a rule, together with the shocks and characteristic surfaces bounding the conical flow in question, the shear discontinuity separating the external and internal streams is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 119–127, January–February, 1935.     

Existence theorems for two-dimensional linear shell theories

We consider linearly elastic shells whose middle surfaces have the most general geometries, and we provide complete proofs of the ellipticity of the strain energies found in two commonly used two-dimensional models: Koiter's model and Naghdi's model.This work is part of the Project Junctions in Elastic Multi-Structures of the Program S.C.I.E.N.C.E. of the Commission of the European Communities (ContractnSC1 * 0473-C(EDB)).     

Heat and entropy in nonextensive thermodynamics

The concepts of the quantity of heat and work are deduced in the context of non-extensive statistical mechanics, following steps parallel to those employed in extensive statistical mechanics.Received: 31 January 2003, Accepted: 6 May 2003, Published online: 21 November 2003PACS: 05.70.-a, 05.30.-d, 05.90. + m     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.