Dynamic-mechanical properties of a bitumen-silica composite

Summary The dynamic-mechanical behaviour of bitumensilica composites is described by a linear biparabolic model. Its mathematical expression allows the calculation of the mean relaxation times () either at different temperatures and given filler contents or for diverse filler contents () at imposed temperatures. At fixed filler concentration and within restricted temperature domains, obeys Arrhenius' law. The activation energies are respectively close to 10 kcal/mole (creep) and 30 kcal/mole (glass-transition). varies exponentially with. The mathematical treatment of the expressions ofE , as a function of temperature and of, leads to a general equation relating the complex modulus to temperature, frequency and filler content. A unique master curve, accounting for the viscoelastic behaviour of the composites, in limited ranges, can thus be constructed.

---

Zusammenfassung Das dynamisch-mechanische Verhalten von Bitumen-Siliziumdioxyd-Zusammensetzungen kann durch ein lineares biparabolisches Modell beschrieben werden. Sein mathematischer Ausdruck erlaubt die Ausrechnung der mittleren Relaxationszeiten () entweder für verschiedene Temperaturen bei gegebenem Füllstoffgehalt oder für unterschiedliche Siliziumdioxydmengen () bei bekannter Temperatur. Für einen bestimmten Füllstoffgehalt folgt in einem beschränkten Temperaturbereich dem Arrheniusschen Gesetz. Die Aktivierungsenergien betragen näherungsweise 10 kcal/Mol (Fließprozeß) bzw. 30 kcal/Mol (Glasübergang). ändert sich exponentiell mit. Die mathematische Umformung der Ausdrücke fürE und als Funktion der Temperatur und des Parameters ergibt eine allgemeine Gleichung, die den komplexen Modul mit der Temperatur, der Frequenz und dem Füllstoffgehalt verknüpft. Man kann eine einzige Masterkurve bilden, die das viskoelastische Verhalten der Zusammensetzungen zumindest in begrenzten Bereichen beschreibt.

---

Résumé Le comportement mécanique dynamique des composites à base de bitume et de silice peut être décrit par un modèle biparabolique linéaire. L'expression mathématique permet le calcul des temps moyens () de relaxation d'une part aux différentes températures, à taux de charge donné, et d'autre part pour diverses valeurs des taux de charge (paramètre) à température imposée. A taux de charge donné, et pour des domaines de température restreints, suit la loi d'Arrhénius. Les énergies apparentes d'activation sont respectivement voisines de 10 kcal/mole (processus de fluage) et de 30 kcal/mole (passage à l'état vitreux). Avec, varie exponentiellement. L'évaluation mathématique deE , de en fonction deT et de conduit à une expression générale du module complexe en fonction de la température, de la fréquence et du taux de charge. On peut donc construire une courbe maitresse unique qui décrit entièrement, mais dans des domaines restreints, le comportement viscoélastique des composites.

---

---

With 6 figures     

Shear viscosity behavior of highly concentrated suspensions at low and high shear-rates

A method is proposed for determining the shear viscosity behavior of highly concentrated suspensions at low and high shear-rates through the use of a formulation that is a function of three parameters signifying the effects of particle size distribution. These parameters are the intrinsic viscosity [], a parametern that reflects the level of particle association at the initiation of motion and the maximum packing concentration _m. The formulation reduces to the modified Eilers equation withn = 2 for high shear rates. An analytical method was used for the calculation of maximum packing concentration which was subsequently correlated with the experimental values to account for the surface induced interaction of particles with the fluid. The calculated values of viscosities at low and high shear-rates were found to be in good agreement with various experimental data reported in literature. A brief discussion is also offered on the reliability of the methods of measuring the maximum packing concentration. _r = /₀ relative viscosity of the suspension - volumetric concentration of solids - k _n coefficient which characterizes a specific effect of particle interactions - _m maximum packing concentration - _r,0 relative viscosity at low shear-rates - [] intrinsic viscosity - n, n parameter that reflects the level of particle interactions at low and high shear-rates, respectively - _r, relative viscosity at high shear-rates - (_m)_s, (_m)_i, (_m)_l packing factors for small, intermediate and large diameter classes - v _s, v_i, v_l volume fractions of small, intermediate and large diameter classes, respectively - _si, _sl coefficient to be used in relating a smaller to an intermediate and larger particle group, respectively - _is, _il coefficient to be used in relating an intermediate to a smaller and larger particle group, respectively - _ls, _li coefficient to be used in relating a larger to a smaller and intermediate particle group, respectively - _m0 maximum packing concentration for binary mixtures - _m,e measured maximum packing concentration - _m,c calculated maximum packing concentration     

Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

Injectivity and self-contact in nonlinear elasticity

Let be a bounded open connected subset of ³ with a sufficiently smooth boundary. The additional condition det dx vol () is imposed on the admissible deformations : ¯ of a hyperelastic body whose reference configuration is ¯. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Ball's theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W ^1,p (;³), p > 3, that is injective almost everywhere.     

On a singular integral equation in the theory of radiative transfer

Summary A singular integral equation arising in the theory of radiative transfer with polynomial scattering indicatrices is studied in this paper. In the case of a semi-infinite atmosphere, brightness coefficients and auxiliary functions satisfy this equation when the free term is suitably specified. The general solution of the singular equation is given a closed form in terms of its moments _n and of the fundamental solution of a homogeneous Hilbert problem. The _n's (n 1) are seen to satisfy a set of linear equations, and ₀ turns out to be simply related to the subsequent _n's.

---

Sommario In questo articolo viene studiata un'equazione integrale singolare che si presenta nella teoria del trasporto radiativo con indicatrici di scattering polinomiali. I coefficienti di riflessione e trasmissione e le funzioni ausiliarie di un'atmosfera semiinfinita soddisfano questa equazione, con opportune determinazioni del termine noto. La soluzione generale dell'equazione viene espressa in forma chiusa mediante i suoi momenti _n e mediante la soluzione fondamentale di un problema omogeneo di Hilbert. Le _n (n 1) soddisfano a loro volta un sistema di equazioni lineari, e ₀ risulta legato alle successive _n da una semplice relazione.

---

---

This work was done under the auspices of the CNR Research Groups.     

Gaussian Closure of One-Dimensional Unsaturated Flow in Randomly Heterogeneous Soils

We propose a new method for the solution of stochastic unsaturated flow problems in randomly heterogeneous soils which avoids linearizing the governing flow equations or the soil constitutive relations, and places no theoretical limit on the variance of constitutive parameters. The proposed method applies to a broad class of soils with flow properties that scale according to a linearly separable model provided the dimensionless pressure head has a near-Gaussian distribution. Upon treating as a multivariate Gaussian function, we obtain a closed system of coupled nonlinear differential equations for the first and second moments of pressure head. We apply this Gaussian closure to steady-state unsaturated flow through a randomly stratified soil with hydraulic conductivity that varies exponentially with where =(1/) is dimensional pressure head and is a random field with given statistical properties. In one-dimensional media, we obtain good agreement between Gaussian closure and Monte Carlo results for the mean and variance of over a wide range of parameters provided that the spatial variability of is small. We then provide an outline of how the technique can be extended to two- and three-dimensional flow domains. Our solution provides considerable insight into the analytical behavior of the stochastic flow problem.     

Ergodic Properties of Weak Asymptotic Pseudotrajectories for Semiflows

It is known that various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes can be analyzed by relating them to an appropriately chosen semiflow. Here, we introduce the notion of a stochastic process X being a weak asymptotic pseudotrajectory for a semiflow and are interested in the limiting behavior of the empirical measures of X. The main results are as follows: (1) the weak* limit points of the empirical measures for X axe almost surely -invariant measures; (2) given any semiflow , there exists a weak asymptotic pseudotrajectory X of such that the set of weak* limit points of its empirical measures almost surely equal the set of all ergodic measures for ; and (3) if X is an asymptotic pseudotrajectory for a semiflow , then conditions on that ensure convergence of the empirical measures are derived.     

Hyperbolic phenomena in a strongly degenerate parabolic equation

We consider the equation u _t =((u) (u _x )) _x , where >0 and where is a strictly increasing function with lim _s = <. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u _t = ((u)) _x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.     

Determination of water retention in stratified porous materials

Predicted and measured water-retention values,(), were compared for repacked, stratified core samples consisting of either a sand with a stone-bearing layer or a sand with a clay loam layer in various spatial orientations. Stratified core samples were packed in submersible pressure outflow cells, then water-retention measurements were performed between matric potentials,, of 0 to -100 kPa. Predictions of() were based on a simple volume-averaging model using estimates of the relative fraction and() values of each textural component within a stratified sample. In general, predicted() curves resembled measured curves well, except at higher saturations in a sample consisting of a clay loam layer over a sand layer. In this case, the model averaged the air-entry of both materials, while the air-entry of the sample was controlled by the clay loam in contact with the cell's air-pressure inlet. In situ, avenues for air-entry generally exist around clay layers, so that the model should adequately predict air-entry for stratified formations regardless of spatial orientation of fine versus coarse layers. Agreement between measured and predicted volumetric water contents,, was variable though encouraging, with mean differences between measured and predicted values in the range of 10%. Differences in of this magnitude are expected due to variability in pore structure between samples, and do not indicate inherent problems with the volume averaging model. This suggets that explicit modeling of stratified formations through detailed characterization of the stratigraphy has the potential of yielding accurate() values. However, hydraulic-equilibration times were distinctly different for each variation in spatial orientation of textural layering, indicating that transient behavior during drainage in stratified formations is highly sensitive to the stratigraphic sequence of textural components, as well as the volume fraction of each textural component in a formation. This indicates that prolonged residence times of water, nutrients, and pollutants are likely within finer-textured layers, when conditions have resulted in drainage of underlying coarser-textured strata.     

Existence of the vortex solutions of some semilinear elliptic equations

The purpose is to extend the existence result of vortex solutions to semilinear elliptic equations for a large class of nonlinearities. M. I. Weinstein used variational techniques to show the existence of nodal solutions for the specific nonlinear term f(¦¦)=(1–¦¦²). An ordinary differential equation phase space setting is used to show the unique transverse intersection of unstable and stable manifolds which contain the solutions satisfying the necessary boundary conditions under certain assumptions on the nonlinearity.     

The third-normal stress difference in entangled melts: Quantitative stress-optical measurements in oscillatory shear

Stress-optical measurements are used to quantitatively determine the third-normal stress difference (N ₃ = N ₁ + N ₂) in three entangled polymer melts during small amplitude (<15%) oscillatory shear over a wide dynamic range. The results are presented in terms of the three material functions that describe N ₃ in oscillatory shear: the real and imaginary parts of its complex amplitude ₃ ^* = ₃ - i ₃ , and its displacement ₃ ^d . The results confirm that these functions are related to the dynamic modulus by ² ₃ ^* ()=(1-)[G ^*())– G ^*(2)] and ² ₃ ^d ()=(1- )G() as predicted by many constitutive equations, where = –N ₂/N ₁. The value of (1-) is found to be 0.69±0.07 for poly(ethylene-propylene) and 0.76±0.07 for polyisoprene. This corresponds to –N ₂/N ₁ = 0.31 and 0.24±0.07, close to the prediction of the reptation model when the independent alignment approximation is used, i.e., –N ₂/N ₁ = 2/7 – 0.28.     

Nonlinear static behavior of a cantilever subjected to an inclined follower force

The nonlinear static behavior of a linearly elastic cantilever subjected to a nonconservative force of the follower type is formulated and examined. The formulation allows for finite rotations with small strains (the elastica). Exact solutions are found. The investigation is greatly facilitated by means of a phase plane analysis in which the phase plane variables are related to slope angle and bending moment. Some of the interesting and unusual effects occurring in this system are discussed and illustrated with a set of deflection curves for a typical case.Nomenclature x, y coordinates of a point on the deformed elastic axis - slope angle - s arc length - L length of beam (assumed constant) - x _L , y _L , _L values of x, y, at s=L - P applied force - constant angle between P and end tangent - angle between P and the horizontal - EI beam stiffness (assumed constant) - u, dimensionless variables defined by (7) and (8) - c ² load parameter defined by (10) - k, transformation parameters defined by (21) - F(, k), E(, k) elliptic integrals of the first and second kind - argument of elliptic integrals - ₀, ₁ values of at u=0 and u=1 - m, n positive integers - N mode number     

Der Spannungs- und Verschiebungszustand drehsymmetrischer Membrane mit beliebig gekrümmter Meridiankurve

Zusammenfassung Die Einführung von Zylinderkoordinaten (x, r, ) in die Gleichgewichtsbedingungen der Schnittkräfte bzw. in die Beziehungen zwischen Verzerrung und Verschiebungen am differentialen Schalenabschnitt ermöglicht die Berechnung des Spannungs- und Verschiebungszustandes von drehsymmetrischen Membranen mit beliebig gekrümmter Meridiankurve auf die Integration einer einfachen, linearen partiellen Differentialgleichung zweiter Ordnung für eine charakteristische FunktionF bzw. zurückzuführen. Eine geschlossene Lösung und damit eine Darstellung der Schnittkräfte und Verschiebungen durch explizite Formeln ist bei harmonischer Belastung cosn für zwei Funktionsgruppen=x ² und=x ^–3 möglich. Im Sonderfall der drehsymmetrischen und der antimetrischen Belastung mitn=0 undn=1 gelten die Gleichungen der Schnitt- und Verschiebungsgrößen für eine beliebige Meridianfunktion=(). Die Betrachtungen der Randbedingungen offener Schalen bei harmonischer Belastung geben über die infinitesimalen Deformationen einer drehsymmetrischen Membran mit überall negativer Krümmung Aufschluß.     

Die Analogie zwischen den Verschiebungen und den Spannungsfunktionen in der Biegetheorie der Kreiszylinderschale

Zusammenfassung Für die Kreiszylinderschale wurde eine Biegetheorie aufgestellt, in der die Gleichgewichtsbedingungen (unter Voraussetzung der Symmetrie des Momententensors M _ik ) durch drei Spannungsfunktionen ₁, ₂, ₃ exakt erfüllt sind. Bei der Definition der Deformationsgrößen und der Einführung der Elastizitätsgesetze war die Reißner-Meißnersche Theorie der symmetrisch belasteten Rotationsschale das Vorbild. Die drei Differentialgleichungen für die Verschiebungen _{1 2}, ₃ unterscheiden sich von den drei Differentialgleichungen für die Spannungsfunktionen ₁, ₂, ₃ formal nur im Vorzeichen der Poissonschen Querkontraktionsziffer v. Die beiden Differentialgleichungen achter Ordnung, die man nach Eliminationsprozessen sowohl für ₃ als auch für ₃ erhält, unterscheiden sich nicht mehr voneinander. So trifft man bei der Zylinderschale die Timpe-Wieghardtsche Analogie zwischen Durchbiegung ₃ der Platte und Airyscher Spannungsfunktion ₃ der Scheibe wieder.Es konnte ferner gezeigt werden, daß unsere neue Biegetheorie der bekannten Flüggeschen Theorie an Genauigkeit nicht nachsteht.Es ist wohl nicht zu bezweifeln, daß auch bei Schalen beliebiger Gestalt unsere Analogie vorhanden ist. Sie scheint uns wertvoll als Ordnungsprinzip inmitten der Fülle von Gleichungen, die nun einmal zu einer Schalentheorie gehören.Die Formulierung des Schalenproblems mit Hilfe der drei Spannungsfunktionen ₁, ₂, ₃ wird sich immer dann empfehlen, wenn die Randbelastung vorgegeben ist. Denn dann lassen sich die Randbedingungen in den Spannungsfunktionen übersichtlicher formulieren als in den Verschiebungen. Auch die Gewißheit, daß selbst durch radikales Streichen lästiger Glieder in den Differentialgleichungen der Spannungsfunktionen die Gleichgewichtsbedingungen nicht verletzt werden, mag manchem Rechner angenehm sein.     

Dynamic-mechanical properties of a bitumen-silica composite

Summary Dynamic-mechanical properties of bitumen-silica composite materials, measured at room temperature, do not vary with the volume ratio () in a simple manner as do usual bituminous concretes. However,E is a linear function of the interfacial area () between the filler and the binder per unit volume. ThusE = E ₀ +a(), wherea is a constant related to the storage modulus, in the absence of voids and with a void ratio factor. The loss moduli, plotted against () go through a maximum in a similar way as when plotted versus decreasing temperatures.

---

Zusammenfassung Die bei Zimmertemperatur gemessenen dynamischmechanischen Eigenschaften von Bitumen, welches mit Siliziumdioxid gefüllt worden ist, ändern sich nicht in einer so einfachen Weise mit dem Volumenanteil des Füllstoffs, wie es bei gewöhnlichen Asphaltbetonen der Fall ist. Dagegen ergibt sich der SpeichermodulE als eine lineare Funktion der Grenzfläche zwischen Bindemittel und Füllstoff pro Volumeneinheit:E = E ₀ +a(), wobeia eine Konstante bedeutet, die zum einen von dem Speichermodul bei Abwesenheit von Hohlräumen und zum andern von einem durch solche Hohlräume bedingten Faktor abhängt. Der Verlustmodul als Funktion von zeigt ein Maximum ähnlich wie bei der Auftragung gegen die Temperatur.

---

Résumé Les propriétés mécaniques dynamiques, mesurées à la température ambiante, d'un système composite bitume-silice, n'évoluent pas simplement avec la fraction volumique de charges comme dans le cas des enrobés usuels. En effet, le module de conservationE varie linéairement avec l'aire de l'interface liant-charge par unité de volume (). Ainsi la relation suivante a pu être mise en évidence:E = E ₀ +a(), la constantea étant fonction d'un module indépendant du taux de vide et d'un terme relié à ce dernier. Si augmente, les modules de perte passent par une valeur maximale. Ces variations sont semblables à celles que l'on aurait si l'on portait ces modules en fonction de la température.

---

---

With 9 figures and 2 tables     

Perturbation analysis for periodic heat transfer in radiating fins

A perturbation analysis is presented for periodic heat transfer in radiating fins of uniform thickness. The base temperature is assumed to oscillate around a mean value. The perturbation expansion is carried out in terms of dimensionless amplitude of the base temperature oscillation. The zero-order problem which is nonlinear, and corresponds to the steady state fin behaviour, is solved by quasilinearization. A method of complex combination is used to reduce both the first and the second order problems to two, coupled linear boundary value problems which are subsequently solved by a noniterative numerical scheme. The second-order term is composed of an oscillatory component with twice the frequency of base temperature oscillation and a time-independent term which causes a net change in the steady state values of temperature and heat transfer rate. Within the range of parameters used, the net effect is to decrease the mean temperature and increase the mean heat transfer rate. This is in constrast to the linear case of convecting fins where the mean values are unaffected by base temperature oscillations. Detailed numerical results are presented illustrating the effects of fin parameter N and dimensionless frequency B on temperature distribution, heat transfer rate, and time-average fin efficiency. The time-average fin efficiency is found to reduce significantly at low N and high B.

---

Störungsanalyse für periodische Wärmeübertragung an Strahlungsrippen

Zusammenfassung Eine Störungsanalyse wird für periodische Wärmeübertragung in Strahlungsrippen gleicher Dicke vorgelegt. Die Fußtemperatur wird als um einen Mittelwert schwingend angenommen. Die Störungsentwicklung wird in Termen einer dimensionslosen Amplitude e dieser Schwingung angesetzt. Das Problem nullter Ordnung, das nichtlinear ist und dem stationären Verhalten der Rippe entspricht, wird durch Quasilinearisierung gelöst. Eine Methode der komplexen Kombination wird angewandt, um die Probleme erster und zweiter Ordnung auf zwei gekoppelte Grenzwertprobleme zu reduzieren, die nacheinander nach einem nichtiterativen Schema gelöst werden. Der Term zweiter Ordnung besteht aus einer Schwingungskomponente mit der doppelten Frequenz der Schwingung der Fußtemperatur und einem zeitunabhängigen Term, der eine Nettoänderung der stationären Werte der Temperatur und der Wärmeübertragung verursacht. Im verwendeten Bereich der Parameter tritt eine Abnahme der mittleren Temperatur und eine Zunahme der mittleren Wärmeübertragung auf. Das steht im Gegensatz zum linearen Fall der Konvektionsrippe, bei dem die Mittelwerte durch Schwingungen der Fußtemperatur nicht beeinflußt werden. Detaillierte numerische Ergebnisse zeigen die Einflüsse des Rippenparameters N und der dimensionslosen Frequenz B auf Temperatur Verteilung, Wärmeübertragung und zeitliches Mittel des Rippengütegrades. Dieses zeitliche Mittel nimmt merklich ab bei kleinem N und hohem B.

---

Nomenclature b fin thickness - B dimensionless frequency, L²/ - E emissivity - f₀, f₁ functions of X - g₀, g₁, g₂ functions of X - h₀, h₁, h₂ functions of X - k thermal conductivity - L fin Length - N fin parameter, 2EL²T_bm/bk - q heat transfer rate - Q dimensionless heat transfer rate, qL/kbT_bm - t time - T temperature - T_b fin base temperature - T_S effective sink temperature - T_bm mean fin base temperature - x axial distance - X dimensionless axial distance, x/L - dimensionless amplitude of base temperature (s. Eq.2) - thermal diffusivity - instantaneous fin efficiency - time-average fin efficiency - _ss steady state fin efficiency - dimensionless temperature, T/T_bm - ₀ zero-order approximation - ₁ first-order approximation - ₂ second-order approximation - _2s steady component of ₂ - , ₁, ₂ constants - complex function of X - ₁ real part of - ₂ imaginary part of - complex function of X - ₁ real part of Y - ₂ imaginary part of - dimensionless time, t/L² - frequency of base temperature oscillation     

Dissipation effects on MHD free convection flow past a semi-infinite vertical plate

Viscous and Joule dissipation effects are considered on MHD free convection flow past a semi-infinite isothermal vertical plate under a uniform transverse magnetic field. Series solutions in powers of a dissipation number (=gx/c _p) have been employed and the resulting ordinary differential equations have been solved numerically. The velocity and temperature profiles are shown on graphs and the numerical values of ₁(0)/₀(0) (, temperature function) have been tabulated. It is observed that the dissipation effects in the MHD case become more dominant with increasing values of the magnetic field parameter (=M ²/(Gr _x /4)^1/2) and the Prandtl number.     

Die Torsionsspannungen in Wellen mit tiefen Längsnuten

Übersicht MitF(x, y) als Spannungsfunktion einer Welle ohne Nut und(, y) als Potentialfunktion des Quelle-Senke-Systems erhält man Spannungsfunktionen(, y) =F(x, y) –(, y) für Wellen mit tiefen Längsnuten. Es wird gezeigt, daß sich damit die Schubspannungen in den Läufern von Schraubenverdichtern ermitteln lassen.

---

Shearing stresses in shafts with deep longitudinal grooves

Summary The stress functions(, y) of shafts with deep longitudinal grooves may be represented by(, y) =F(x, y) –(, y) whereF(x, y) is the stress function of a cylindrical shaft without grooves and(, y) denotes the potential function of the source-sink system. It is shown that the shearing stresses in rotors of screw-compressors may be obtained in this way.

     

Multiple scattering of anisotropic radiation through a layer of clouds

A method by which the transport of anisotropic radiative multiple scattering can be predicted is developed in this paper. A one-dimensional integral intensity model and a three-dimensional diffusion intensity model are both constructed. The former provides a closed-form solution, while the latter involves successive approximation and Gauss's quadrature. On the basis of these methods, the reflection and transmission of solar radiation in a homogeneous cloud layer are computed. The results differ from those for isotropic and Rayleigh scattering assumptions and illustrate the effects on transmission and reflectivity of optical thickness, wavelength, incidence angle, and albedo of single scattering.Nomenclature D ⁺ transmitted diffusion radiation intensity [W/cm² sr · m] - D ^– reflected diffusion radiation intensity [W/cm² sr · m] - I pencil of radiation or specific intensity [W/cm² sr · m] - I ₀ solar irradiance [W/cm²] - K extinction cross-section or total cross-section, + - s (u, ), unit scattered radiation vector - s ₀ (u ₀, ₀), unit incident radiation vector - t optical thickness - u cosine of the viewing angle, , which is measured from the vertical - u ₀ cosine of the angle of incident, ₀, which is measured from the vertical - absorption cross-section - scattering function - absorption coefficient - scattering angle, s · s ₀ - scattering cross-section - scattered azimuthal angle - ₀ incident azimuthal angle - a sphere - a solid angle - ₀ albedo of single scattering,      

Saint-Venant's principle in nonlinear plane elasticity with sufficiently small strains

A homogeneous, isotropic cylinder in an equilibrium state of plane strain, whose cross-section is a rectangle R : [0 < y ₁ < 2L; 0 < y ₂ < h] with h/L 1, is considered. There are no body forces and the long sides are stress free. At y ₁ = 0 and y ₁ = 2L, there are arbitrary loadings, each statically equivalent to a uniformly distributed tensile or compressive stress c. Within the theory of nonlinear elasticity and with the strains and strain gradients assumed to be sufficiently small (but with no such assumptions on the displacement gradients), it is proved that if (,=1,2) represents the Cauchy stress tensor and the Kronecker delta, then |–c₁₁| decays exponentially to zero in R with distance from the nearer end, and the decay constant depends only upon the material but is independent of L.