Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

Numerical validation of the shear compression specimen. Part II: Dynamic large strain testing

Part I of this work addressed quasi-static loading of the shear compression specimen (SCS), which has been especially developed to investigate the shear dominant response of materials at various strain rates. The stress and strain states were characterized numerically. Approximations were presented to reduce the measured load,P, and displacement,d, into equivalent stress and strain . This paper addresses dynamic loading of the SCS. Several simulations were made for representative materials, whose stress-strain behavior is assumed to be rate-independent. The results show that stress wave loading induces strong oscillations in theP-d curve. However, the curve remains smooth in the gage section. The oscillations are about the quasistatic load values, so that with suitable filtering of the dynamicP-d curves, the quasi-static ones are readily recovered. Consequently, the approach that was developed for quasi-static loading of the SCS is now extended to dynamic loading situations. The average strain rate is rather constant and scales linearly with the prescribed velocity. As the plastic modulus becomes smaller, the strain rate reaches higher values. Friction at the end pieces of the specimen is also investigated, and shown to have a small overall influence on the determined mechanical characteristics. This paper thereby confirms the potential of the SCS for large strain testing of materials, using a unified approach, over a large range of strain rates in a seamless fashion.     

Simulation of viscoelasticity for one case of material testing

Conclusions Similariry conditions have been established on the basis of which the viscosity can be simulated in testing viscoelastic materials for tension (compression) under hydrostatic pressure. It has been shown that criteria and account for the effect of viscosity, while the II number accounts for the effect of pressure. The criterion is, in form, identical to the analogous parameter in the theory of non-Newtonian fluid flow. It has been shown, furthermore, that criterion is the monodromic version of criterion (the similarity number). When P=0 or P is very small and the II number degenerates, then only criterion or criterion should be used.Institute of Problems in Mechanics, Academy of Sciences of the USSR, Moscow. Translated from Prikladnaya Mekhanika, Vol. 11, No. 6, pp. 109–114, June, 1975.     

On the continuum modelling of porous media containing fluid: a molecular viewpoint with particular attention to scale

Mass conservation and linear momentum balance relations for a porous body and any fluid therein, valid at any given length scale in excess of nearest-neighbour molecular separations, are established in terms of local weighted averages of molecular quantities. The mass density field for the porous body at a given scale is used to identify its boundary at this scale, and a porosity field is defined for any pair of distinct length scales. Specific care is paid to the interpretation of the stress tensor associated with each of the body and fluid at macroscopic scales, and of the force per unit volume each exerts on the other. Consequences for the usual microscopic and macroscopic viewpoints are explored.Nomenclature material system; Section 2.1. - porous body (example of a material system); Sections 2.1, 3.1, 4.1 - fluid body (example of a material system); Sections 2.1, 3.1, 4.1 - weighting function; Sections 2.1, 2.3 - ,h weighting function corresponding to spherical averaging regions of radius and boundary mollifying layer of thicknessh; Section 3.2 - Euclidean space; Section 2.1 - V space of all displacements between pairs of points in; Section 2.1 - mass density field corresponding to; (2.3)₁ - ^P , ^f mass density fields for , ; (4.1) - P momentum density field corresponding to; (2.3)₂ - v velocity field corresponding to; (2.4) - S _r (X) interior of sphere of radiusr with centre at pointx; (3.3) - boundary ofany region - region in which ^p > 0 with = ,h; (3.1) - subset of whose points lie at least+h from boundary of ; (3.4) - abbreviated versions of ; Section 3.2, Remark 4 - strict interior of ; (3.7) - analogues of for fluid system ; Section 3.2 - general version of corresponding to any choice of weighting function; (4.6) - interfacial region at scale; (3.8) - ₀ scale of nearest-neighbour separations in ; Section 3.2. Remark 1 - porosity field at scales ( ₁; ₂); (3.9) - pore space at scales ( ₁; ₂); (3.12)     

Interval stochastic matrices: A combinatorial lemma and the computation of invariant measures of dynamical systems

The concept of an interval stochastic matrix is introduced. We prove a combinatorial theorem which describes the network flow associated with an interval matrix. The semi-invariant vectors of are characterized in terms of eigenvectors with unit eigenvalue of stochastic matrices . These results are then applied to the approximation and machine computation of invariant measures of dynamical systems.Funded under Australian Research Council Grant A 8913 2609.     

Druckabhängigkeit der Viskosität einiger Polyolefinschmelzen

Zusammenfassung Es wird die Druckabhängigkeit des nicht -Newtonschen Fließverhältens von Polyolefinschmelzen (Hochdruck-, Niederdruck-,Phillips-Polyäthylen und Polypropylen) experimentell untersucht und gefunden, daß der durch Gl. [1] definierte Druckkoeffizient mit zunehmender Deformationsgeschwindigkeit kleiner wird und dabei die (im einzelnen in der Tabelle 1 angeführten) Werte annimmt. Der Druckkoeffizient der Polyolefinschmelzen ist ebenso wie für vieleNewtonsche Flüssigkeiten bis 2000 kp cm^–2 unabhängig vom Druck, er wird mit zunehmender Temperatur kleiner und nimmt mit zunehmender Verzweigung zu. Die Meßergebnisse werden mit Hilfe eines Aufweitungsvolumens interpretiert. Es wird gezeigt, daß eine Deutung des Fließverhaltens von Polyäthylen durch das freie Volumen allein nicht möglich ist.

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Summary The influence of pressure of the non-Newtonian flow behaviour of polyolefin melts (Low- and High density Polyethylene,Phillips-Polyethylene and Polypropylene) was investigated. The results are: the coefficient of pressure as defined by eq. [1], decreases with increasing shear rate and reaches the values given in table 1 . The pressure coefficient of polyolefin melts does not depend on pressure up to 2000 kp cm^–2. As observed with manyNewtonian fluids, decreases with increasing temperature and increases with the degree of branching. The experimental results are explained by means of a so called volume of expansion. It has been shown that it is impossible to explain the flow behaviour of polyethylene exclusively with the free volume.

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Für die Diskussion und Förderung dieser Arbeit danke ich Herrn Professor Dr.K.-H. Hellwege und Herrn Dr.W. Knappe.     

Multiaxial elongation of polyisobutylene with various and changing strain rate ratios

The multiaxial elongational rheometer equipped with rotary clamps is modified such that in addition to simple, equibiaxial and multiaxial elongations also tests with new modes of elongation can be performed. As an example, polyisobutylene is elongated with a ratio of the principal strain rates of and magnitudes of the maximum strain rate , 0.04 and 0.08 s^–1. As a test result, the first elongational viscosityµ ₁ (t) is obtained which follows closely the linear viscoelastic shear viscosity . In contrast, the second elongational viscosityµ ₂ (t) remains below . By means of a further modification of the rheometer, the test modes can be varied during the deformation period. This allows one to investigate the influence of a well-defined rheological pre-history on the following rheological behaviour. As an example a variation ofm = 0.5 2 was performed. The measured normal-stress differences superpose from the single steps of deformation similar to the linear viscoelastic prediction.Dedicated to Prof. F. R. Schwarzl on the occasion of his 60th birthday     

Dimensional analysis of pore scale and field scale immiscible displacement

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a macroscopic capillary number which differs from the usual microscopic capillary number Ca in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. can be related to the microscopic capillary number Ca and the LeverettJ-function. Previous dimensional analyses contain a tacit assumption which amounts to setting = 1. This fact has impeded quantitative upscaling in the past. Our definition for , however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at 1. The length scale related difference between the macroscopic capillary number for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

Grundzüge einer Stabilitätstheorie für elastische Systeme unter nichtkonservativer Belastung

Zusammenfassung Um den Einfluß von nichtkonservativen Kräften zu erfassen, muß die Stabilitätstheorie nichtkonservativer elastischer Systeme auf dem Prinzip der virtuellen Verrückung aufgebaut werden, das von genügender Allgemeinheit ist. Als mathematische Methode steht nach Klärung von Konvergenzfragen das Galerkinsche Verfahren zur Verfügung, das dem ursprünglichen Problem der Stabilität elastischer Systeme das Problem der Stabilität der Bewegung eines mechanischen Systems von unendlich vielen Freiheitsgraden als gleichwertig zuordnet und die Algebraisierung des Stabilitätsproblems erlaubt. Damit wird die Ljapunowsche Theorie anwendbar. Als Wichtigstes ist die Berechtigung der Linearisierung zu untersuchen, wozu der Einfluß der Dämpfung festzustellen ist. Es ergibt sich, daß bei Berücksichtigung von Dämpfung die Stabilität stets asymptotisch ist, so daß die Linesarisierung gerechtfertigt wird. Darüber hinaus zeigt es sich, daß die Struktur der Matrizen und von entscheidender Bedeutung für die Zuständigkeit eines der in Frage kommenden Stabilitätskriterien ist: Ist symmetrisch und eine beliebige positive Diagonalmatrix bzw. schiefsymmetrisch, oder symmetrisierbar bzw. unsymmetrisch und noch gleichzeitig (die Dämpfungskoeffizienten alle gleich), so wirkt in diesen Fällen die Dämpfung günstig, und die Stabilitätskriterien des ungedämpften Vorganges, das statische bzw. das kinetische, bleiben daher hinreichend. Man darf dann auch ohne Dämpfung rechnen. Ist dagegen unsymmetrisch und eine positive Diagonalmatrix mit verschiedenen Elementen, so darf in solch einem Fall nicht ohne Dämpfung gerechnet werden, weil gegenüber dem ungedämpften Vorgang ein Sprung in den Stabilitätsbedingungen auftritt, durch welchen das Ergebnis der Rechnung ohne Dämpfung in Frage gestellt wird.     

Generalized diagonalization of matrices over quaternion field

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Singular Perturbation of Boundary Value Problem for Quasilinear Third-Order Ordinary Differential Equations Involving Two Small Parameters

ＩｎｔｒｏｄｕｃｔｉｏｎＡｂｏｕｔｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｓｅｃｏｎｄ＿ｏｒｄｅｒｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｗｏｒｋｓｏｆａｌａｒｇｅｎｕｍｂｅｒｗｅｒｅｃｏｎｓｉｄｅｒｅｄｏｎｌｙｆｏｒｔｈｅｃａｓｅｏｆｉｎｖｏｌｖｉｎｇｏｎｅｓｍａｌｌｐａｒａｍｅｔｅｒ[1- 4].Ｏｎｌｙａｆｅｗｗｏｒｋｓｗｅｒｅｃｏｎｓｉｄｅｒｅｄｆｏｒｔｈｅｃａｓｅｏｆｉｎｖｏｌｖｉｎｇｔｗｏｓｍａｌｌｐａｒａｍｅｔｅｒｓ[5 - 8]…     

Der selbsterregte unsymmetrische Kreisel

Zusammenfassung Wir können alle bisherigen Erkenntnisse über die Selbsterregung des unsymmetrischen Kreisels dahin zusammenfassen, daß — abgesehen von den trivialen Fällen der Drehungen um jede der drei Hauptachsen und abgesehen von den permanenten Drehungen um beliebige Achsen — zu jeder achsenfesten zeitunabhängigen Selbsterregung achsenbewegliche Drehungen gehören, wie wir solche in Ziff. 2 bis 7 untersucht haben; ferner daß es, wieder abgesehen von jenen trivialen Fällen, nur den einzigen Typ der achsenfesten zeitabhängigen Selbsterregung von Ziff. 9 gibt, welche eine achsenfeste zeitabhängige Drehung zu beeinflussen, wenn auch weder zu erzeugen noch zu vernichten vermag; endlich daß der Momentvektor der Selbsterregung, welcher eine beliebige achsenfeste Drehung erzeugen oder vernichten oder in beliebiger Weise beeinflussen soll, im allgemeinen (also abgesehen von den trivialen Fällen und von dem Fall in Ziff. 9) zeitabhängig in einer körperfesten Ebene wandern muß. Die achsenbeweglichen Drehungen, welche von achsenfester zeitunabhängiger Selbsterregung erzeugt oder unterhalten werden, sind je nach der Lage (und Größe) des Momentes der Selbsterregung periodisch hinsichtlich der Bewegung des Drehvektors im Kreisel oder asymptotisch mit unbeschränkt wachsendem Drehvektor oder asymptotisch mit einem Drehvektor , der sich einem permanenten Drehvektor (von endlichem Betrag) annähert (oder auch sich von ihm entfernt).     

The flow regime in the turbulent near wake of a hemisphere

The work presented is a wind tunnel study of the near wake region behind a hemisphere immersed in three different turbulent boundary layers. In particular, the effect of different boundary layer profiles on the generation and distribution of near wake vorticity and on the mean recirculation region is examined. Visualization of the flow around a hemisphere has been undertaken, using models in a water channel, in order to obtain qualitative information concerning the wake structure.List of symbols C _p pressure coefficient, - D diameter of hemisphere - n vortex shedding frequency - p pressure on model surface - p ₀ static pressure - Re Reynolds number, - St Strouhal number, - U, V, W local mean velocity components - mean freestream velocity inX direction - U ^* shear velocity, - u, v, w velocity fluctuations inX, Y andZ directions - X Cartesian coordinate in longitudinal direction - Y Cartesian coordinate in lateral direction - Z Cartesian coordinate in direction perpendicular to the wall - it* boundary layer displacement thickness, - diameter of model surface roughness - elevation angleI - O boundary layer momentum thickness, - _w wall shearing stress - dynamic viscosity of fluid - density of fluid - streamfunction - _x longitudinal component of vorticity, - _y lateral component of vorticity, - _z vertical component of vorticity, This paper was presented at the Ninth symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984     

Das Dispersionsmodell

Zusammenfassung Ein Vergleich im Frequenzbereich zeigt, daß bei der Berechnung der Verweilzeitverteilung mit dem Dispersionsmodell das endlich begrenzte System für Péclet-Zahlen Pe > 10 mit guter Näherung durch ein einseitig unbegrenztes System und für Pe > 50 durch ein beidseitig unbegrenztes System ersetzt werden kann.

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The dispersion model. A comparison of approximations

A comparison in the frequency domain shows that for the determination of the residence time distribution with the dispersion model the finitely restricted system may be substituted with good approximation for Peclet numbers Pe > 10 by a one-side unrestricted system and for Pe > 50 by a both-side unrestricted system.

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Bezeichnungen A Rohrquerschnitt - A=A mittlerer Strömungsquerschnitt in der Schüttschicht - Konzentration (Partialdichte) der Bezugskomponente i - Bezugskonzentration nach Gl. (5) - c_i Konzentration (Dichte) der reinen Bezugskomponente i - D axialer Dispersionskoeffizient - E Fehler im Frequenzbereich nach Gl. (36) - G(,) Übertragungsfunktion - G(,i) Frequenzgang - L Länge der Schüttschicht - m Masse - Massenstrom - Péclet-Zahl - s Laplace-Variable - t Zeit - t Impulsbreite - v Strömungsgeschwindigkeit im leeren Rohr - mittlere axiale Strömungsgeschwin digkeit in der Schüttschicht - V=AL Zwischenraumvolumen der Schüttschicht - x Ortskoordinate - (t) Dirac-Stoss - Porosität - dimensionslose Zeit - dimensionslose Konzentration - Laplace-Transformierte der Konzentration - Fourier-Transformierte der Konzentration - dimensionslose Ortskoordinate - =s dimensionslose Laplace-Variable - mittlere Verweilzeit - Kreisfrequenz - = dimensionslose Kreisfrequenz Indices A Ausgang - D Dispersion - E Eingang - i Bezugskomponente - K Konvektion Mitteilung Nr. 44 des Institutes für Mess-und Regel-technik der Eidgenössischen Technischen Hochschule Zürich (Vorsteher: Prof. Dr. P. Profos)     

Rheology of concentrated disperse systems III. General features of the proposed non-newtonian model. Comparison with experimental data

Summary A non-newtonian viscosity equation where is the volume concentration and is an intrinsic viscosity, function of a relative shear rate and being structural parameters, has been proposed in a previous paper (1). From empirical grounds, the valuep = 1/2 holds for a large class of systems, like suspensions of rodand disc-shaped particles. In the high shear rate limit, aCasson law-type is recovered and discussed, especially the concentration dependence of the yield stress. However, the latter disappears in the low shear limit, and must be considered as a pseudo-yield stress. Good agreement is found in this low shear limit with some theoretical results ofBueche for polymers. More generally, the viscosity equation displays pseudo-plastic behaviour and fitting it on experimental data allows the determination of the structural parameters. Some examples (especially Red Blood Cell suspensions and Blood) are studied and support the model. Nevertheless, for spherical particle suspensions, the best fitting is reached forp = 1. Accurate values of particle diameters can be deduced from the structural parameter , in this case.

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Zusammenfassung In einer vorangegangenen Arbeit (1) wurde eine Viskositätsgleichung für eine nicht-newtonsche Flüssigkeit von der Form vorgeschlagen, worin die Volumenkonzentration und eine Grenzviskosität bedeutet; die letztere stellt eine Funktion der relativen Schergeschwindigkeit dar, die Konstantenk ₀,k und bezeichnen Strukturparameter. Empirisch wird gefunden, daß für eine große Klasse von Systemen, wie z. B. stäbchen- und scheibchenförmigen Teilchen,p = 1/2 gilt. In der Grenze hoher Schergeschwindigkeiten wird ein Verlauf gemäß einer Casson-Gleichung gefunden und diskutiert, insbesondere bezüglich der Konzentrationsabhängigkeit der Fließspannung. Allerdings verschwindet diese in der Grenze niedriger Schergeschwindigkeiten und muß daher als Pseudo-Fließspannung betrachtet werden. In diesem Grenzfall wird eine gute Übereinstimmung mit theoretischen Voraussagen vonBueche an Polymeren gefunden. Ganz allgemein beschreibt die obige Viskositätsgleichung ein pseudoplastisches Verhalten, und ihre Anpassung an experimentelle Werte erlaubt die Bestimmung der Strukturparameter. Einige Beispiele, insbesondere Suspensionen von roten Blutkörperchen und Blut, werden untersucht und bestätigen das Modell. Allerdings erhält man bei Suspensionen kugelförmiger Teilchen die beste Anpassung fürp = 1. In diesem Fall kann man mit Hilfe des Strukturparameters genaue Werte der Teilchendurchmesser bestimmen.

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With 5 figures and 3 tables     

Effect of a Short Roughness Strip on a Turbulent Boundary Layer

The response of a turbulent boundary layer to a short roughness strip is investigated using laser Doppler velocimetry (LDV) and laser induced fluorescence (LIF). Skin friction coefficients are inferred from accurate near-wall measurements. There is an undershoot in , where is the undisturbed smooth wall skin friction coefficient, immediately after the strip. Downstream of the strip, overshoots before relaxing back to unity in an oscillatory manner. The roughness strip has a major effect on the turbulent stresses ; these quantities increase, relative to the undisturbed smooth wall, in the region between the two internal layers originating at the upstream and downstream edges of the strip. The increase in the ratio suggests a decrease in near-wall anisotropy. From the flow visualizations, it is inferred that streamwise vortical structures are weakened immediately downstream of the strip. Consistently, streamwise length scales are also reduced; direct support for this is provided by measured two-point velocity correlations.     

Transient response of a finite crack in an orthotropic strip under the in-plane shear impact

The transient response of a central crack in an orthotropic strip under the in-plane shear impact loading is studied by using the dual integral equation method proposed by Copson and Sih. The general formula for the shear stress intensity factor near the crack tip is derived. Numerical results of with in various cases are obtained by solving the second kind Fredholm integral equation and by performing the inverse Laplace transform.     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

Single droplet combustion in a gravitational environment

A theoretical analysis is developed to study the combustion characteristics of a fuel droplet in a gravitational field. The normalized governing system consists of the complete conservation equations inr-z coordinates and includes finite-rate global kinetics. The Clausius-Clapeyron law is applied at the liquid-vapor interface to describe the evaporation process. A modified body-fitted grid generation technique is used to handle irregular boundaries. The effects of changing the droplet diameter and the gravity level are investigated. Under the variation of droplet diameter, flame structures, including isotherms, flame shape, velocity vector field, and mass burning rate are studied in detail. The predicted results exhibit good agreement with experimental data. When the gravity level increases, the computed results show that the flame shape is sensitive to variation in gravity. A simple correlation, , is found. Within the elevated gravity domain of experiment, the computed data agree well with measurements obtained by Okajima and Kumagai [10].In einer theoretischen Studie wird die Verbrennungscharakteristik eines Ethanoltröpfchens in einem Gravitationsfeld untersucht. Die normierten Grundbeziehungen umfassen die vollständigen Erhaltungsgleichungen inr-z-Koordinaten unter Einschluß einer Globalkinetik endlicher Umsatzrate. Das Clausius-Clapeyronsche Gesetz wird an der Flüssigkeits-Dampf-Phasengrenze angesetzt, um den Verdampfungsprozeß zu beschreiben. Eine spezielle Gittergenerierungstechnik ermöglicht die Behandlung irregulärer Berandungen. Die Einflüsse veränderlichen Tropfendurchmessers bzw. variablen Gravitationsniveaus werden untersucht. Bei variablem Tröpfchendurchmesser erfolgt eine Detailuntersuchung der Flammenstrukturen nach Isothermenfeld, Flammenform, Feld der Geschwindigkeitsvektoren und Verbrennungsrate. Die vorausberechneten Ergebnisse stimmen gut mit experimentellen Daten überein. Bei Anstieg des Gravitationsniveaus zeigen die Rechnungen eine empfindliche Veränderlichkeit der Flammenform. Es läßt sich eine einfache Beziehung aufstellen. Im Bereich höheren Gravitationspegels stimmen die berechneten Werte gut mit den Messungen von Okajima und Kumagai [10] überein.Financial support of this research by the National Science Council of the R.O.C., under project NSC 81-0401-E-009-531 is greatly appreciated. The authors also wish to express their gratitude to National Chiao Tung University for providing computer facility.