New results on the semidiscrete Boltzmann equation for a binary gas mixture

Summary A plane semidiscrete model of the Boltzmann equation for a binary gas mixture with molecular collisions ruled by the hard-spheres interaction potential is described. After establishing a model, a theorem demostrating the global existence of mild solutions of the initial-value problem is given and the propagation of unidimensional shock waves examined.

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Sommario Si propone un modello semidiscreto piano dell'equazione di Boltzmann per una miscela binaria con collisioni molecolari soggette al potenziale di interazione delle sfere rigide. Costruito il modello, si dà un teorema di esistenza globale di soluzioni generalizzate per il problema di Cauchy, e si analizza la propagazione di onde d'urto unidimensionali.

     

A Stefan-type free boundary problem in a thermal switch

Summary We study a mathematical model of thermal switch in a glass under the Joule heating. We prove the existence of solutions for the free boundary problem related to the model. This problem is a Stefan-type problem with a source depending on the free boundary and the latent heat depending on the history of the free boundary.

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Sommario Si studia un modello matematico che descrive un fenomeno di cambiamento di fase in un vetro attraversato da una corrente elettrica. In particolare si dimostra l'esistenza di soluzioni per il problema a frontiera libera connesso con il modello. Questo problema è del tipo di Stafan, con sorgente funzione della frontiera libera e con calore latente che dipende dalla storia della frontiera.

     

Dynamic shakedown by modal analysis

Summary Dynamic shakedown of discrete elastic-perfectly plastic structures under a specified load history is studied using the dynamic characteristics of the structure provided by modal analysis. Several statical and kinematical theorems are presented, including lower and upper bound theorems for the minimum adaptation time of the structure. In the formulation of the kinematical theorems a crucial role is played by the appropriate definition of admissible plastic strain cycle.

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Sommario Si studia il problema dell'adattamento dinamico (shakedown) di una struttura discreta elasto-perfetta-mente plastica e soggetta ad una storia di carichi prestabilita, facendo uso a tale scopo delle caratteristiche dinamiche della struttura fornite dalla analisi modale. Vengono presentati svariati teoremi, sia di tipo statico che cinematico, tra cui taluni teoremi di delimitazione superiore ed inferiore del tempo minimo di adattamento. Nella formulazione dei teoremi cinematici ha un ruolo cruciale la corretta definizione di ciclo deformativo ammissibile.

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This paper is part of a research project sponsored by the National (Italian) Research Council, C.N.R., Group of Structural Engineering, and by the National Department of Education (M.P.I.).     

Gasdynamics of a plasma in a multiple-mirror magnetic trap with “point” mirrors

Equations are derived for the gasdynamics of a dense plasma confined by a multiple-mirror magnetic field. The limiting cases of large and small mean free paths have been analyzed earlier: ₀ and k, where is the length of an individual mirror machine, ₀ is the size of the mirror, and k is the mirror ratio. The present work is devoted to a study of the intermediate range of mean free paths ₀ k. It is shown that in this region of the parameters the process of expansion of the plasma has a diffusional nature, and the coefficients of transfer of the plasma along the magnetic field are calculated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 14–19, November–December, 1974.The authors thank D. D. Ryutov for the statement of the problem and interest in the work.     

The Kepler problem spinor regularization and quantization on the manifolds S2

Summary In a previous research we have shown that the KS-transformation, developed by Kustaanheimo and Stiefel for the regularization of the Kepler problem, may be interpreted as the correspondence which associates to each null 4-vector of the space of Minkowski a one-index spinor, defined up to a phase factor, and we have obtained a new form of the KS-transformation. In the present research we show that this formulation allows a straight derivation of the Hopf fibering of the sphere S³ (characterized by unit spinors) having the base space given by the section (sphere S²) of the light cone, and we show that the KS-transformation allows the quantization of the symplectic manifold S² in the sense of Souriau. The sphere S³ turns out to be a contact quantized manifold. The bilinear relation characteristic of the KS-theory and the column vectors of the KS-matrix are intimately related to the contact structure.

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Sommario In un precedente lavoro si è mostrato che la trasformazione KS, introdotta da Kustaanheimo e Stiefel per regolarizzare il problema di Keplero, è riconducibile alla ben nota corrispondenza fra vettori del cono isotropo dello spazio di Minkowski e spinori semplici, definiti a meno della fase, e si è pervenuti ad una nuova formulazione della KS. Nel presente lavoro si mostra come da tale formulazione scaturisca in modo naturale la fibrazione di Hopf della sfera S³ (caratterizzata dagli spinori unitari) avente quale base una sezione (sfera S²) del cono isotropo e si mette in luce come la trasformazione KS consenta di effettuare la quantizzazione della varietà simplettica S² nel senso di Souriau e di ottenere la sfera S³ quale varietà quantica di contatto. La relazione bilineare caratteristica della teoria KS ed i vettori colonna della matrice KS risultano intimamente legati alla struttura di contatto.

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Presented at the VI Congresso Nazionale dell'Associazione Italiana di Meccanica Teorica ed Applicata (AIMETA), Genova, October 1982. Work performed under the auspices of G.N.F.M. of the C.N.R. (Consiglio Nazionale delle Ricerche).     

Variational principles and differential inclusions for unilateral constraints in analytical mechanics

Summary The present paper deals with variational principles in terms of hemivariational inequalities and with differential inclusions for unilateral constraints in analytical mechanics. It is discussed how it is possible to describe one-sided constraints in a very general manner by the help of Clarke's generalized gradient.

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Sommario Questo scritto prende in esame i principi variazionali nella forma delle hermivariational inequalities e le inclusioni differenziali per vincoli unilaterali nella meccanica analitica. Si deduce come sia pissobile descrivere vincoli unilaterali, in modo generale facendo uso di gradienti generalizzati di Clarke.

     

Stochastic response of structures with small geometric imperfections

Summary A probabilistic model of the geometric imperfections of a real structure is proposed, in order to provide a general theory of the stochastic response of structures in presence of small random deviations from the perfect scheme. The main statistical measures of the stochastic response are derived and an application to the study of a particular conservative elastic system is developed.

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Sommario Si propone una teoria generale della risposta probabilistica di strutture, in presenza di piccole deviazioni aleatorie dei dati iniziali rispetto allo schema geometrico perfetto. Si deducono le principali proprietà statistiche della risposta della struttura a sollecitazioni esterne deterministiche, e si sviluppa una applicazione riguardante il comportamento aleatorio di un particolare sistema elastico conservativo.

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List of symbols element of the sample space of events - _kn random variables modelling the structural imperfections - P(o) probability density of random variables - random imperfection of the unloaded structure - u additional displacement of the loaded structure - u_o deterministic fundamental solution for the perfect structure - difference between the additional displacement of the loaded structure and the deterministic fundamental solution for the perfect structure - V₁=u₁ buckling mode of the perfect structure - _i intrinsic coordinates of the structure - suitable measure of the magnitude of the random imperfections - scalar geometric variable representing the internal product - random imperfection divided by - single scalar variable denoting the magnitude of the prescribed loads - potential energy of the structure - potential energy of the perfect structure - difference between and - _c lowest critical load - _s real local maximum for the magnitude of the prescribed loads - _c divided by _S - E{} expected value of a random variable - ² variance of a random variable - , random variables defined by Eq. (21)     

The Italian contribution to the mathematical theory of elasticity

Summary The contributions of Italian mathematical physicists to the mathematical theory of elasticity are reviewed, starting from the second half of the last century. The work of Piola, Betti, Beltrami, Volterra, Somigliana, Signorini and many other eminent scholars is briefly outlined, showing how Italian mathematics achieved a leading position, comparable to that unanimously accorded Italy in other areas of mathematics, in this field where theory is so close to application.

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Sommario Vengono passati in rassegna i contributi che i fisico-matematici italiani, a partire, circa, dall'inizio della seconda metà del secolo scorso, hanno dato alla teoria matematica dell'elasticità. L'opera di Piola, Betti, Beltrami, Volterra, Somigliana, Signorini e di tanti altri eminenti studiosi viene brevemente riassunta e posto in luce come la matematica italiana, anche in questo campo, cosi vicino alle applicazioni, abbia raggiunto posizioni di primato, comparabili a quelle che, unanimamente, sono state riconosciute all'Italia in altri settori della matematica.

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Invited paper.This article was taken from a lecture given by the author at the University of Palermo on December 16, 1977 and published in the Rendiconti del Circolo Matematico di Palermo (vol. XXVIII, sr. II). The editors of this publication have kindly given Meccanica permission to publish this abbreviated version of the lecture in English.     

A simplified flood routing model: variable prameter muskingum (VPM)

Summary Flood routing methods are numerical methods for estimating the movement of a flood wave along a channel reach, on the basis of the knowledge of the discharge hydrograph at the upstream end and of the hydraulic characteristics of the reach and, usually, in the hypothesis that no perturbation is coming from downstream (free boundary condition). The flood routing method wich is proposed is similar to the Muskingum one, but with variable and hydraulic parameters; it is able to estimate water levels too; is effective even if kinetic terms are not completely negligible; take advantage of the insignificance of the downstream condition and make it possible to obtain results starting upstream and proceeding downstream; for simplicity's sake, take advantage of the fact that the discharge loop of normal flood waves is quite small. Obtained results are much better that those obtainable from constant parameters methods and indeed, if the flood loop is less that 10%, very similar to those obtainable from more complex and time consuming models.

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Sommario I metodi di flood routing studiano la propagazione di un'onda di piena lungo un tratto di un corso d'acqua, assegnato l'andamento temporale della portata nella sezione di monte e le caratteristiche dell'alveo, e usualmente nell'ipotesi di assenza di perturbazioni provenienti da valle (condizione di valle passiva). Viene qui proposto un procedimento di flood routing, formalmente simile ad un Muskingum ma con i parametri variabili e calcolati per via idraulica; idoneo a stimare anche i livelli idrici; valido anche se i termini cinetici non sono del tutto trascurabili; che sfrutta l'irrilevanza della condizione di valle procedendo a cascata da monte a valle; che sfrutta, a vantaggio della semplicità, il fatto che per le normali onde di piena dei corsi d'acqua il cappio di portata è di dimensioni modeste. I risultati ottenuti sono molto migliori di quelli ottenibili con metodi a parametri costanti e, almeno per i casi in cui il cappio relativo è inferiore al 10%, paragonabili a quelli ottenuti con metodi molto più complessi ed onerosi.

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List of symbols x, t channel distance, starting upstream; time - z water surface height above datum - Q volumetric rate of discharge - P(x, z) steady rating curve - q=Q–P flood loop - g acceleration of gravity - A, B cross section wetted area and free surface width - I, S water surface slope and friction slope - c kinematic wave velocity - F Froude number - L,L ₀,L ₁,L ₂,L ₃ characteristic lengths of the channel - T=L/c characteristic time of the channel - D diffusion - p, l time and space steps - K, X Muskingum parameters - C ₁,C ₂,C ₃,C ₄ Muskingum coefficients - f _x=f/x,f _t=f/t etc. for the partial derivatives Paper presented at the First Italian Meeting of Computational Mechanics held in Milan, June 24–26, 1986.     

Thermal stresses in plane elasticity: Numerical solutions based only on harmonic functions

Summary A theoretical treatment is outlined allowing solution of thermal stress problems in plane elasticity by using only numerical methods suited to solving — in 2 D — the Laplace equation. Only one type of element matrix (supposing for the sake of simplicity F.E.M.s are used) and only one mesh would thus be required, both for the determination of the thermal field and of the displacement/stress field. The numerical solutions required in the plane domain of interest entail, consequently, only one variable per node in place of two. Even if numerous unit solutions are required in order to impose arbitrary boundary conditions, this reduction of nodal variables allows to spend less computation time in solving linear systems, at least for problems of a certain extent.

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Sommario Si delinea una tecnica che permette di risolvere i problemi di coazioni termiche in elasticità piana facendo uso solo di metodi numerici atti a risolvere l'equazione di Laplace. Un solo tipo di matrice degli elementi (nel caso si usi una formulazione a E. F.) e una sola reticolazione sono pertanto richiesti tanto per il problema termico come per quello elastico. Ne consegue altresì il vantaggio che le soluzioni numeriche richieste nel dominio di interesse cornportano una sola variabile per nodo, anzichè due; anche tenuto conto che sono richieste numerose soluzioni unitarie per poter imporre condizioni al contorno comunque definite, ne deriva per problemi di una certa ampiezza una riduzione del tempo di calcolo speso nella soluzione di sistemi.

     

The Kepler problem spinor regularization and pre-quantization of the negative-energy manifold

Summary We extend the results of our two foregoing papers to the whole phase space of the Kepler problem, by: i) developing a natural spinor extension of the KS-regulari-zation of the Kepler problem, due to Kustaanheimo and Stiefel, and ii) applying the results to the pre-quantization (in the sense of Souriau) of the negative-energy manifold, the energy levels satisfying a well-known condition. We show that the quantized manifold is diffeomorphic to the symplectic productS ₃×S ₃ of two 3-spheres of the spinor phase space, factored through a suitable equívalence relation.

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Sommario Estendendo i risultati ottenuti in due prece denti lavori all'intero spazio delle fasi del problema di Keplero, si perviene nella presente nota ad una naturale estensione spinorialedella trasformazione KS (introdotta da Kustaanheimo e Stiefel per regolarizzare il problema di Keplero) la quale consente di prequantizzare, nel senso di Souriau, la varietà delle orbite kepleriane isoenergetiche, con energia negativa spddisfacente ad una ben nota condizione. Si mostra che la varietà quantica associata è diffeomorfa al prodotto simplettico S ₃×S ₃,opportunamente quozientato, di due 3-sfere dello spazio spinoriale delle fasi.

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Work performed under the auspices of G.N.F.M. of the Italian Research Council (C.N.R.).     

Effect of blowing on the resistance of a sphere in laminar flow of viscous fluid

Using the method of matching asymptotic expansions [1–3], a stationary field of velocities is obtained in the vicinity of a sphere for Reynolds numbers R and R computed from the blowing velocity and the fluid flow, respectively; they satisfy the relations R² 1 and R 1. It is also shown that for intensive blowing (R 1), the resistive force is considerably smaller than that found by using the Stokes formula. For weak blowing the results are in good agreement with the solution of Oseen.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskai Fiziki, No. 3, pp. 110–114, May–June, 1972.     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

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

     

An open question regarding shear flow with suspended sediments

Summary This paper deals with the critical re-analysis of a series of experimental laboratory data (Coleman, 1981), concerning velocity profiles in a free surface stream, when suspended solid transport is present. The interpretation of results presented here is rather different from Coleman's original interpretation, which states that density stratification does not affect inner layer variables, e.g. von Karman constant. This difference is important, because Coleman's data are commonly cited as a proof of the above statement. In the present work the estimate of parameters is based on well known mathematical methods. Therefore the reliability of the results should be assured. So, when suspended sediment transport is present, von Karman constant variability seems to be an open question.

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Sommario In questa nota si effettua il riesame critico di una serie di dati sperimentali di laboratorio (Coleman, 1981), inerenti profili di velocità in una corrente a pelo libero in cui è presente un significativo trasporto solido in sospensione. Emerge una interpretazione duale rispetto a quella originale dello sperimentatore, comunemente accettata in letteratura come sostegno della ininfluenza della stratificazione sulle variabili di parete, principalmente sulla costante di von Karman. Nel presente lavoro la stima dei parametri è basata su semplici relazioni matematiche; ciò consente di considerare i risultati affidabili e di affermare che il problema è, quantomeno, aperto.

     

Boundary kinetic effects in the problem of slow flow over an evaporating heat-conducting spherical particle

It is shown that for sufficiently large values of the thermal conductivity of the condensed phase' as compared with the thermal conductivity of the vapor (/' Kn) the effects associated with the presence of a Knudsen layer on the evaporating surface must be taken into account in order to obtain a solution of the problem of a spherical particle in a slow (Re, 1) continuum (Knudsen number Kn 1) flow of its own vapor. The drag is calculated for various types of boundary conditions on the particle surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 134–139, March–April, 1987.In conclusion the authors wish to thank V. S. Galkin and M. N. Kogan for useful discussions.     

Historical deformation analysis of elastoplastic structures as a parametric linear complementarity problem

Summary Discrete models of elastoplastic beams and frames are considered. The plastic deformability laws (bending moments versus plastic rotations) are piecewise linearized. Plastic deformations are sought as they develop along a proportional loading process. The historical analysis in this sense, is shown to be ameneable to the solution of a parametric linear complementarity problem. Recent mathematical results on this problem, due to R. W. Cottle, are used to obtain numerical solutions. Extensions are pointed out to stepwise proportional loading paths, allowing for the irreversible nature of plastic deformations.

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Sommario Si studiano modelli discreti di telai elastoplastici. I legami tra momento flettente e rotazione plastica sono lineari a tratti. Si determina l'intera evoluzione delle deformazioni plastiche al crescere proporzionale e monotono dei carichi, formulando e risolvendo un problema parametrico lineare di complementarietà. La tecnica numerica alquanto efficiente adottata, è fornita da recenti risultati di R. W. Cottle. Il metodo è esteso a processi di caricamento proporzionali a stadi, tenendo conto, tra stadio e stadio, della irreversibilità della deformazione plastica.

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All other symbols are defined when used for the first time.     

Theory of plane flows of a readily conducting plasma in a pipe

Simplified equations are obtained describing slowly changing plane flows of a readily conducting quasineutral inviscid plasma in a pipe. The practically interesting case of flow in a channel with solid metal ideally conducting walls (electrodes) is analyzed. When the gas pressure is large by comparison with the magnetic pressure ( 1), the field and current distribution is determined by gas dynamic factors, and the solid electrodes perturb the longitudinal electric field in a skin of the flow, symmetrically on the two sides of the flow, leading to attenuation of the longitudinal electric field near the input to the pipe; we also consider problems in the motion of the plasma under ideal and under poor conductivity. In the converse limiting case ( 1), it is shown that as the motion of the plasma in the pipe accelerates near the anode, there is observed an increase in the intensity of the electric field which is sharply inhomogeneous in the transverse direction. The possibility of the plasma breaking away from the anode (the limiting regime) is indicated, this being accompanied by a divergence between the electron velocity and the velocity of the ions. A criterion is obtained for the breakaway of the plasma, and its possible connection with the occurrence of pre-anode explosions is noted. It is shown that for 1, Joule losses are small by comparison with the power in the charge and the magnitude of the losses is independent of the conductivity of the plasma.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 9–19, July–August, 1970.     

Heat transfer at a wedge in a hypersonic low-density gas stream

Results are presented of an approximate analysis of steady heat transfer on a sharp thin (sin < 1) strongly cooled (t_w 1) wedge, washed by a hypersonic (M 1) gas stream at zero angle of attack under almost free-molecule conditions. Dimensionless parameters on which the heat transfer depends have been established; approximate formulas for estimation purposes are given.The present analysis is not a rigorous quantitative theory and its results should be regarded only as estimates; its conclusions may be useful for experimental planning and for generalization of test data. The method used here is analogous to that applied in [1] in analyzing flow over a flat plate.     

On elastic-plastic structures with associated stress-strain relations allowing for work softening

Summary For the elementary (finite or infinitesimal) constituents of the structure are assumed generalized stress-strain relations which satisfy the condition of normality but which may exhibit work-softening, concavity of yield surfaces, variation of elastic coefficients with stress and/or plastic strain.The usual phenomena of geometric instability are excluded. Sufficient conditions are formulated for overall stability in spite of the presence of unstable elements, and for uniqueness of the incremental boundary-values problem. Conditions are discussed with a view to applications and expressed in terms of positive definiteness of appropriate quadratic forms.Finally, yield surfaces and flow laws for the structure are examined, and among other things their necessary association is shown.

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Sommario Per i costituenti elementari (finiti o infinitesimi) della struttura si assumono legami incrementali tra sforzi e deformazioni generalizzati che soddisfano alla condizione di normalità ma che per il resto sono generici, cioè tali da presentare eventualmente incrudimento negativo, concavità del campo elastico, variazione dei coefficienti elastici con gli sforzi e/o con le deformazioni plastiche.Esclusi per il sistema i fenomeni usuali di instabilità geometrica si formulano condizioni sufficienti per la stabilità del complesso nonostante la presenza di parti a funzionamento instabile, e per l'unicità del problema incrementale al contorno. Le condizioni sono discusse in vista delle applicazioni ed espresse in termini di definizione positiva di forme quadratiche opportune.Si esaminano infine le superfici di snervamento e le leggi di scorrimento per l'intera struttura e, tra l'altro, se ne dimostra la necessaria associazione.

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First published in Italian in Rendiconti dell'Istituto Lombardo Classe Scienze e Lettere, A 100, 1966.The present investigation has been promoted and financed by the Consiglio Nazionale delle Ricerche (C.N.R.) at the Istituto di Scienza delle Costruzioni of the Facoltà di Architettura, Politecnico of Milano. Part of it was carried out at Brown University (Providence R. I. - U.S.A.). The author acknowledges with tanks the NATO Fellowship assigned by the C.N.R. in 1964, the encouragement and kind hospitality he received from Professor D.C. Drucker, and the interesting discussions he had with Professor Drucker and Dr. A. C. Palmer.