Tangent modulus tensor in plasticity under finite strain

Summary The tangent modulus tensor, denoted as , plays a central role in finite element simulation of nonlinear applications such as metalforming. Using Kronecker product notation, compact expressions for have been derived in Refs. [1]–[3] for hyperelastic materials with reference to the Lagrangian configuration. In the current investigation, the corresponding expression is derived for materials experiencing finite strain due to plastic flow, starting from yield and flow relations referred to the current configuration. Issues posed by the decomposition into elastic and plastic strains and by the objective stress flux are addressed. Associated and non-associated models are accommodated, as is plastic incompressibility. A constitutive inequality with uniqueness implications is formulated which extends the condition for stability in the small to finite strain. Modifications of are presented which accommodate kinematic hardening. As an illustration, is presented for finite torsion of a shaft, comprised of a steel described by a von Mises yield function with isotropic hardening.Notation B strain displacement matrix - C=F ^T F Green strain tensor - compliance matrix - D=(L+L ^T )/2 deformation rate tensor - D fourth order tangent modulus tensor - tangent modulus tensor (second order) - d VEC(D) - e VEC() - E Eulerian pseudostrain - F, F _e ,F _p Helmholtz free energy - F=x/X deformation gradient tensor - f consistent force vector - residual function - G strain displacement matrix - h history vector - h time interval - H function arising in tangent modulus tensor - I, I ₉ identity tensor - i VEC(I) - k ₀,k ₁ parameters of yield function - K _g geometric stiffness matrix - K _T tangent stiffness matrix - k _k kinematic hardening coefficient - J Jacobian matrix - L=v/x velocity gradient tensor - m unit normal vector to yield surface - M strain-displacement matrix - N shape function matrix - n unit normal vector to deformed surface - n ₀ unit normal vector to undeformed surface - n unit normal vector to potential surface - r, R, R ₀ radial coordinate - s VEC() - S deformed surface - S ₀ undeformed surface - t time - t, t ₀ traction - t VEC() - VEC( ) - t VEC() - t _r reference stress interior to the yield surface - t t–t _r - T kinematic hardening modulus matrix - u=x–X displacement vector - U permutation matrix - v=x/t particle velocity - V deformed volume - V ₀ undeformed volume - X position vector of a given particle in the undeformed configuration - x(X,t) position vector in the deformed configuration - z, Z axial coordinate - vector of nodal displacements - =(F ^T F–I)/2 Lagrangian strain tensor - history parameter scalar - , azimuthal coordinate - elastic bulk modulus - flow rule coefficient - twisting rate coefficient - elastic shear modulus - iterate - Second Piola-Kirchhoff stress - Cauchy stress - Truesdell stress flux - deviatoric Cauchy stress - Y, Y yield function - residual function - plastic potential - X, Xe, Xp second order tangent modulus tensors in current configuration - X, Xe, Xp second order tangent modulus tensors in undeformed configuration - (.) variational operator - VEC(.) vectorization operator - TEN(.) Kronecker operator - tr(.) trace - Kronecker product     

Decreasing of the initial shear modulus with increasing axial strain explained by means of a plastic-hypoelastic model

Summary The purpose of this work is to examine in detail the possibility to explain the decreasing of the initial shear modulus with increasing axial strain, observed first by Feigen, by means of the plastic-hypoelastic stress-strain relation suggested by Lehmann and by the author of the present paper.Notations _ij components of the infinitesimal strain tensor dilatation - strain deviator - _ij components of the stress tensor - spherical part of the stress tensor - stress deviator - ²= _{ij ij} second invariant of the stress deviator - = ₃₃ axial strain - e= ₁₃ shear component of the strain tensor - =2 ₁₃ shear strain - = ₃₃ axial stress - s= ₁₃ shear stress - T _ij components of Cauchy's stress tensor - F _ij components of the deformation gradient - L _ij components of the velocity gradient (Eulerian coordinates) - components of the rate of deformation tensor - components of the spin tensor - components of the rate of deformations deviator - components of Cauchy's stress deviator - T=T ₃₃ axial Cauchy's stress With 7 Figures     

Viscous response of an anchored spinning liquid column to various axial excitations

Summary For a solidly rotating viscous cylindrical liquid column of finite length the response to axial synchronous, counter- and one-sided excitation is determined for anchored contact lines at the disc-rim. For a rotating column additional responses of inertial waves (hyperbolic range) appear for < 2 ₀, while in the elliptic range < 2 ₀ the sloshing response occurs. The various responses for the free surface displacement have been numerically evaluated. Only in the one-sided exitation case all resonance peaks appear, while for synchronous excitation only the odd resonances and for counter-excitation only the even resonance peaks occur.Notation a radius of column - h length of liquid bridge - I _n modified Bessel function - p liquid pressure - r, ,z cylindrical polar coordinates - t time - u, v, w velocity distribution - Weber number - z ₀ excitation amplitude - liquid density - surface tension - surface tension parameter - Ohnesorge number - liquid surface displacement - kinematic viscosity - ₀ rotational speed - dimensionless rotational speed - forcing frequency - dimensionless forcing frequency - dimensionless forcing frequency for non-viscous liquid - a= root of bi-cubic Eq.(33) - root of bi-cubic Eq.(33)     

Higher approximations for supersonic flow past slowly oscillating bodies of revolution

Summary Supersonic flow past slowly oscillating pointed bodies of revolution is studied. Starting from the complete nonlinear potential equation an elementary linearized solution is discussed and it is shown how this solution together with the method of matched asymptotic expansions can be used to derive an elementary second-order slender body theory. This approach is further demonstrated for the oscillating cone and its range of validity is evaluated by comparison with other theoretical methods.

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Zusammenfassung Es wird die Überschallströmung um langsam schwingende spitze Rotationskörper untersucht. Ausgehend von der vollständigen nichtlinearen Potentialgleichung wird zuerst eine elementare linearisierte Lösung besprochen und gezeigt, wie diese Lösung im Verein mit der Method of matched asymptotic expansions zur Herleitung einer elementaren Schlankkörpertheorie zweiter Ordnung verwendet werden kann. Die Theorie wird am Beispiel des schwingenden Kegels näher erläutert und mit anderen Methoden verglichen.

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Symbols a Velocity of sound - c _N Normal force coefficient - Damping coefficient - F _(x) Dipole distribution - k Reduced frequency - M Mach number - R (x) Meridian profile - t Time - x, r, Cylindrical coordinates - - Ratio of specific heats - Amplitude of oscillation - Thickness ratio - Perturbation potential - Zero angle of attack potential - æ - Velocity potential - Out-of-phase potential - - In-phase potential - - Source coordinate With 4 Figures     

Die Entwicklung von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkaven Wänden

Zusammenfassung Messungen des Anwachsens von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkav gekrümmten Wänden (Görtler-Taylor-Wirbel) ergaben drei deutlich getrennte Bereiche: Es traten zunächst Wirbel mit der Wellenläge 0,9 auf (=Grenzschichtdicke, =Höhe einer Zelle, die zwei gegensinnig drehende Wirbel enthält). Je nach Größe der mit der Verdrängungsdicke ₁ der Grenzschicht gebildeten Reynolds-Zahl erschienen dann kurze Zeit später Wirbel mit 2,5, wenn war. Im Bereiche dagegen traten stattdessen bei den hier durchgeführten Versuchen immer Wirbel mit der Wellenlänge 6,5 auf. Bei werden die ersten Tollmien-Schlichting-Wellen mit der Wellenlänge _TS 6· angefacht. In ihren wandnahen Bereichen der Wellentäler könnten sich dann die oben genannten Längswirbel der Wellenlänge 6,5· ausbilden, die die zwei-in eine dreidimensionale Störung allseits gleicher Größenordnung verwandeln können.

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The development of longitudinal vortices in boundary layers growing with time along concave walls

Summary Measurements of the growth of longitudinal vortices in boundary layers growing with time along concave walls (Görtler-Taylor vortices) rendered three distinctly separated regions. First, vortices with a wave-length 0.9 appeared (-boundary layer thicness, =height of a cell containing two counterrotating vortices). Then, depending on the Reynolds number R _{a 1}/v ₁=displacement thickness), vortices with 2.5 appeared shortly afterwards, provided . In the region , however, the wave-length was 6.5. For the first Tollmien-Schlichting waves with _TS 6 were excited. In the wave-throughs close to the wall the abovementioned longitudinal vortices with wave length 6.5 may then be formed. This might transform the two-dimensional into a three-dimensional flow of equal order of magnitude in all directions.

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Zeichenerklärungen R _a Innenradius - Re _a Reynolds-Zahl gebildet mit dem InnenradiusR _a - Reynolds-Zahl gebildet mit der Verdrängungsdicke ₁ - kritische Taylor-Zahl - h Standhöhe der Flüssigkeit im Zylinder - t Zeit - z Anzahl - Steigungswinkel der Geraden - Grenzschichtdicke - ₁ Verdrängungsdicke - Wellenlänge (enthält ein gegensinnig rotierendes Längswirbelpaar) - v kinematische Zähigkeit - Winkelgeschwindigkeit Indizes K Knickpunkt der Geradensteigung - L unterhalb des Knickpunktes der Geradensteigung - TS Tollmien-Schlichting - e Einsatz der Wirbelentstehung     

Slip flow past an approximate spheroid

Summary Creeping axisymmetric slip flow past a spheroid whose shape deviates slightly from that of a sphere is investigated. An exact solution is obtained to the first order in the small parameter characterizing the deformation. As an application, the case of flow past an oblate spheroid is considered and the drag experienced by it is evaluated. Special well-known cases are deduced and some observations made.Notation A _n, B_n, C_n, D_n, E_n, F_n, b₂, d₂ Constants - a, b radii of spheres - coefficient of sliding fraction - D drag - , _m parameters characterizing the deformation of the sphere - c a(1+) - viscosity coefficient - - dimensionless coordinate - I _n Gegenbauer function - P _n Legendre function - Stream function - U stream velocity at infinity     

On the motion of a curved and twisted rod

Summary In this paper, the equations of motion of a curved and twisted rod are derived from the basic principles of dynamics. The set of equations gives the extensional, flexural and torsional motions of the rod. The coupling among these types of motion due to the curvature and tortuosity of the rod is shown explicitly in the case of a helical spring. By successive simplification of the equations, the equations of motion of a circular ring and those of a straight rod are obtained. In this respect, the derived equations can be considered as a generalization of the elementary theories of rod in extensional, torsional and flexural motion.The dispersion relation of a helical spring is calculated for the two lower frequency modes. It is shown that the frequency-wave length relationship is not monotonically decreasing as in the cases of uncoupled flexural or torsional motion. Finally, frequencies are calculated based on the approximate frequency expression ofLove to show that Love's frequency expression for a helical rod is accurate.

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Zur Bewegung eines gekrümmten und verdrillten Stabes

Zusammenfassung In dieser Arbeit werden die Bewegungsgleichungen eines gekrümmten, verdrillten Stabes von den Grundgleichungen der Dynamik hergeleitet. Dieses System von Gleichungen legt die Bewegung des Stabes durch Längsdehnung, Biegung und Torsion fest. Die Kopplung dieser Bewegunsarten, durch Krümmung und Verdrillung, wird für den Fall einer Spiralfeder explizit gezeigt. Durch Vereinfachung der Gleichungen werden die Bewegungsgleichungen des Kreisringes und die des geraden Stabes erhalten. In dieser Hinsicht können die hergeleiteten Gleichungen als Verallgemeinerung der elementaren Theorein der Bewegung eines Stabes betrachtet werden.Die Dispersionsgleichung der Spiralfeder wird für die beiden niedrigsten Frequenzen berechnet. Es wird gezeigt, daß das Verhältnis Frequenz-Wellenlänge nicht wie in den Fällen der ungekoppelten Biege-oder Torsionsbewegung monoton abnimmt. Abschließend werden die Frequenzen nach dem Loveschen Näherungsausdruck berechnet, um zu zeigen, daß die Genauigkeit dieses Ausdruckes für Spiralfedern gut ist.

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Nomenclature position vector of a point on axis of curved rod - position vector relating any pointB in a plane perpendicular to the axis of the rod to pointP where plane cuts axis of rod - R modulus of - linear momentum vector - density - axial force vector = - axial force vector at origin - transverse shear force vector - transverse shear force vector at origin - moment vector - moment vector at origin - s distance measured along axis of rod - t time - a radius of gyration of rod for circular sections - l total length of spring along axis - A cross-sectional area of rod - s/a — normalized coordinates measured along axis of rod - applied load vector - angular momentum vector - trihedral of space curve-unit vectors in tangential, normal and binormal directions - _t axial rotation - displacement vector - normalized displacement vector= - ₀,₀ curvature and tortuosity of rod - , normalized curvature and tortuosity of rod - , circular frequency and normalized frequency respectively - E, G Young's modulus and shear modulus - v Poisson's ratio - Wave number - pitch angle of helical spring - d wire diameter of helical spring - R coil radius of helical spring - N number of turns of helical spring With 5 Figures     

On the torsional waves in an elastic bar with long-range cohesive forces

Summary With reference to some of the results obtained in [1], the equation of motion and the equations of nonlocal stress components in the form of Kroener-Eringen are transformed into the Fourier space. A lengthy calculation using a separable form of the nonlocal elastic moduli leads to the governing equation of the problem with two types of solutions. In each case the nonlocal phase velocity and, respectively, the nonlocal group velocity of the very short waves turn out to be by about 36% less than the velocities of their classical counterparts.Notation a atomic spacing - c ₂ classical speed of shear waves - c _2nonloc nonlocal speed of shear waves - c _a2nonloc,c _b2nonloc nonlocal speeds in rods - k wave number - r radius vector - R radius of the bar - v hoop displacement - Fourier transform ofv - V volume of the rod - z axial coordinate - delta sequence - , Lamé's constants - , nonlocal moduli - transform of - A wave length - * mass density - frequency     

Damped response of an axially excited rotating liquid bridge in zero-gravity

Summary The response of a solidly rotating finite liquid bridge due to axial excitation exhibits for frictionless liquid at the resonances singularities. For the experimenter in a spacelabmission the actual resonance amplitude is of quite some importance. For this reason damping, that has to be measured in ground tests, has been introduced into the results of the response.Notation a radius of the liquid bridge - h length of the liquid bridge - I ₀,I ₁ modified Besselfunctions - J ₀,J ₁ Besselfunctions - r, ,z polar coordinates - t time - excitation amplitude - elliptic case - hyperbolic case - abbreviation - damping factor of liquid - (z, t) free surface displacement - =² – ² surface tension - surface tension - liquid density - ₀ rotational speed of liquid bridge - forcing frequency of axial excitation - natural frequency of liquid bridge With 2 Figures     

Response of a spinning liquid column to pitch excitation

Summary The response of a solidly rotating liquid bridge consisting of inviscid liquid is determined for pitch excitation about its undisturbed center of mass. Free liquid surface displacement and velocity distribution has been determined in the elliptic (>2₀) and hyperbolic (<2₀) excitation frequency range.List of symbols a radius of liquid column - h length of column - I ₁ modified Besselfunction of first kind and first order - J ₁ Besselfunction of first kind and first order - r, ,z cylindrical coordinates - t time - u, v, w velocity distribution in radial-, circumferential-and axial direction resp. - mass density of liquid - free surface displacement - velocity potential - ₀ rotational excitation angle - ₀ velocity of spin - forcing frequency - _1n natural frequency - surface tension - acceleration potential - for elliptic range >2₀ - for hyperbolic range >2₀     

Natural damped frequencies and axial response of a rotating finite viscous liquid column

Summary For a finite solidly rotating cylindrical liquid column the damped natural axisymmetric frequencies have been determined. The liquid was considered incompressible and viscous. The cases of freely slipping edges and that of anchored edges have been treated. It was found that instability appears in a purely aperiodic root for the spinning liquid bridge. This is in contrast to the instability appearing in the damped oscillatory natural frequency of a nonspinning liquid column at . The spinning viscous liquid column exhibits the same instability as the frictionless liquid. It appears at for axisymmetric oscillations.List of symbols a radius of liquid column - I _m modified Bessel function of first kind and orderm - s complex frequency ( ) - r, ,z polar cylindrical coordinates - p pressure - t time - u, v, w radial-, azimuthal- and axial velocities of liquid, respectively - Weber number - h height of liquid column - dynamic viscosity of liquid - v kinematic viscosity of liquid (v=/) - density of liquid - surface tension of liquid - _r , _rz shear stress - (r, z, t) circulation - (r, z, t) streamfunction - ₀ angular velocity of liquid column about the axis of symmetry - (,t) free surface displacement     

Bounding-surface plasticity: Stress space versus strain space

Summary A bounding-surface plasticity model is formulated in stress space in a general enough manner to accommodate a considerable range of hardening mechanisms. Conditions are then established under which this formulation can be made equivalent to its strain-space analogue. Special cases of the hardening law are discussed next, followed by a new criterion to ensure nesting. Finally, correlations with experimental data are investigated.Notation (a) centre of the stress-space (strain-space) loading surface; i.e., backstress (backstrain) - ^* (a ^*) centre of the stress-space (strain-space) bounding surface - (a ) target toward which the centre of the stress-space (strain-space) loading surface moves under purely image-point hardening - (b) parameter to describe how close the loading surface is to nesting with the bounding surface in stress (strain) space; see (H10) - (c) elastic compliance (stiffness) tensor - (d) parameter to describe how close the stress (strain) lies to its image point on the bounding surface; see (H10) - (D) generalised plastic modulus (plastic compliance); see (1) - function expressing the dependence of the generalised plastic modulus on (plastic complianceD ond) - ^* (D ^*) analogue to (D) for the bounding surface - function expressing the dependence of ^* on (D ^* ond) - () strain (stress) - ' (') deviatoric strain (stress) - ^P ( ^R ) plastic strain (stress relaxation); see Fig. 1 - () image point on the bounding surface corresponding to the current strain (stress) - _iso (f _iso) at the point of invoking consistency, the fraction of local loading-surface motion arising from a change of radius; i.e., fraction of isotropic hardening in the stress-space theory - _kin (f _kin) at the point of invoking consistency, the fraction of local loading-surface motion arising from a change in the backstress (backstrain); i.e., fraction of kinematic hardening in the stress-space theory - _nor (f _nor) at the point of invoking consistency, the fraction of backstress (backstrain) motion directed toward the image stress (strain); i.e., the image-point fraction of the kinematic hardening in the stress-space theory - _ima (f _ima) at the point of invoking consistency, the fraction of backstress (backstrain) motion directed toward the image stress (strain); i.e., the image-point fraction of the kinematic hardening in the stress-space theory - function relating _iso to , , and (f _iso tob,d, andl) - function relating _kin to , , and (f _kin onb,d, andl) - function relating _nor to , , and (f _nor onb,d, andl) - function relating _ima to , , and (f _ima onb,d, andl) - the fraction of outwardly normal bounding-surface motion at the Mróz image point which arises from a change of radius - the fraction of outwardly normal bounding-surface motion at the Mróz image point which arises from a change in the centre - function relating _iso ^* to (f _iso ^* tod) - function relating _kin ^* to (f _kin ^* tod) - (l) parameter to describe the full extent of plastic loading up to the present, giving the arc length of plastic strain (stress relaxation) trajectory; see (H10) - function relating the direction for image-point translation of the loading surface to various other tensorial directions associated with the current state; see (H5). With 6 Figures     

The contributions of microrotation of lubricant molecules in a thin film journal bearing

Summary Characteristics of a journal bearing were computed for thin film lubrication accounting for microrotation of the lubricant molecules using both the half-Sommerfeld and Reynolds boundary conditions. Although the Reynolds boundary conditions produced higher pressure and loads, the effects of microrotation studied by both schemes showed similar trends. Primary characteristics that effect the contributions of microrotation to the load carrying capacity of the journal bearing were identified. These characteristics were varied and their effects on the load capacity of the journal bearing are shown.Nomenclature a circumference of the shell, 2R - a ₁,a ₂,a ₃ constants - c radial difference between the shaft and shell, [R-r] - c ₁,c ₂,c ₃,c ₄ solving partial differential equations constants - e eccentricity - F ^* body force per unit mass - G substitute function for integration - h film thickness - j microinertia constant of the fluid - K ₁,K ₂ defined after Eq. (2.27) - l material length, - L ^* body couple - m - N coupling number, - p pressure - p ₀ ambient pressure - Q fluid flux flow through the cavity cross section - r shaft radius - R shell radius - R _h modified Reynold number, - R _l Reynolds number, - t time - u, v velocity components inx-andy-direction, respectively - u ₀ velocity of the shaft surface - V velocity vector - W load carrying capacity - W load component resulting from pressure parallel to the line of centers - W load component resulting from pressure perpendicular to the line of centers - x, y, z Cartesian coordinates - , , , micropolar viscosity coefficients - ₁, ₂ parameters of boundary conditions for the microrotation vector at the shell and shaft, respectively - the deviate angle of the load direction from the line of centers - e/c - , Newtonian viscosity coefficients - microrotation velocity vector - microrotation velocity component in thez-direction - angular velocity of the shaft - ^* thermodynamic pressure - mass density of the lubricant fluid - polar angle around the journal bearing - ^* angle which satisfies Reynold's B.C.     

On a stability theory of non-selfadjoint mechanical systems

Summary The concept of the Hamiltonian functional is generalized in such a way that a bilinear functional results, which plays the role of the Hamiltonian for non-selfadjoint systems. For this generalized Hamiltonian the condition leads to the so called hybrid Galerkin's equations, and the condition , to the load-frequency reationship. This relationship can be interpreted as a surface in the load-frequency space, the projection of which on the load-planes yields the stability boundaries, i.e. the buckling loads.

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Zu einer Stabilitätstheorie nicht-selbstadjungierter mechanischer Systeme

Zusammenfassung Der Begriff des Hamiltonschen Funktionals wird in solcher Weise verallgemeinert, daß ein bilineares Funktional bei nicht-selbstadjungierten Systemen an seine Stelle tritt. Für dieses verallgemeinerte Hamiltonsche Funktional führt die Bedingung auf die sogenannten hybriden Galerkinschen Gleichungen und die Bedingung auf die Last-Frequenz-Funktion. Diese Funktion kann im Last-Frequenz-Raum als eine Fläche aufgefaßt werden, deren Projektion auf die Last-Ebenen die Stabilitätsgrenzen und damit die Knicklasten liefert.

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Dedicated to Professor Kurt Magnus in honor of his sixtieth birthday.     

Laminar thermal boundary layer on a continuous accelerated sheet extruded in an ambient fluid

Summary Exact boundary layer similarity solutions are developed for flow, friction and heat transfer on a continuously accelerated sheet extruded in an ambient fluid of a lower temperature.Melt-spinning, polymer and glass industries and the cooling of extruded metallic plates are practical applications of this problem.Results for skin-friction and heat-transfer coefficients are given. Larger acceleration is accompanied by larger skin-friction and heat-transfer coefficients. Rapid cooling of the sheet is accompanied by a larger Nusselt number.Nomenclature sheet width - c dimensionless constant - c _f local skin friction coefficient - F dimensionless transformed stream function - G dimensionless transformed temperature - local heat transfer coefficient - fluid thermal conductivity - length of deformation zone - m exponent of surface speed variation - q exponent of surface temperature variation - T dimensionless temperature - sheet surface temperature - solidification temperature - ambient temperature - sheet thickness - u velocity component along the sheet - u _s sheet surface velocity - wind up velocity - v velocity component normal to the sheet - x dimensionless coordinate along the sheet - y dimensionless coordinate normal to the sheet - Nu Nusselt number, - Pr Prandtl number, - Re Reynolds number, - =Re^–0.5 - dimensionless similarity coordinate - dynamic viscosity - kinematic viscosity - fluid mass density - sheet mass density - wall shear stress - dimensionless stream function With 3 Figures     

A simple quadrature method for computing laminar boundary layers

Summary An approximate method is developed for calculating two-dimensional steady laminar boundary layers. Simple quadrature formulae have been obtained to determine energy thickness and a shape factor which subsequently determines the position of the point of separation; the only requirement being knowledge of the free-stream velocity. The relations between the shape factor and the other important characteristics of the boundary layer are also given.The method is extended to compressible flow assuming that the surface is adiabatic, that the viscosity is proportional to temperature, and that the Prandtl number is unity. Calculations and comparison of the results with other known solutions show that the quadrature method is not only simple to use, but also that it supplies closer approximation than do most other approximate methods.

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Eine einfache Quadraturmethode zur Berechnung laminarer Grenzschichten

Zusammenfassung Es wird ein Verfahren zur Berechnung zweidimensionaler stationärer laminarer Grenzschichten entwickelt. Einfache Quadraturformeln, die nur die Kenntnis der Geschwindigkeitsverteilung der Außenströmung erfordern, dienen der Berechnung der Energieverlustdicke und eines Geschwindigkeitsformparameters. Letzterer bestimmt die Lage des Ablösungspunktes. Zwischen dem Formparameter und anderen wichtigen Größen der Grenzschicht bestehen einfache Zusammenhänge.Bei Anwendung der Methode auf kompressible Strömungen wird angenommen, daß sich die Wand wärmeisoliert (adiabat) verhält, die Zähigkeit proportional der Temperatur ist und die Prandtl-Zahl den Wert Eins besitzt. Durchgeführte Rechnungen zeigen nicht nur die einfache Handhabung des Quadratur-Verfahrens, sondern durch Vergleich mit bekannten Ergebnissen auch eine bessere Übereinstimmung als die meisten anderen Näherungsmethoden.

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List of Symbols a speed of sound - c constant - c _p specific heat at constant pressure - G integral of the free-stream velocity - H modified shape factor - known shape factors - k, m, n, p exponents - l reference length - Mach number - p pressure - Prandtl number - R gas constant - Reynolds number with reference to energy thickness - Reynolds number for a reference state - T temperature - u velocity component in thex-direction - x, y coordinate axes parallel and perpendicular to the surface - x _S point of separation - function appearing in shearing stress - function appearing in energy dissipation ( value for the flat plate) - ₁, ₂, ₃ displacement, momentum and energy thickness, respectively - function appearing in dissipation integral - viscosity - ratio of specific heats, =1.4 (air) - thermal conductivity - kinematic viscosity - density - _w shearing stress at the wall With 3 FiguresDedicated to Professor Kurt Magnus in honor of his sixtieth birthday.     

Free convection from a heated vertical cylinder embedded in a fluid-saturated porous medium

Summary We consider the free convection boundary layer flow induced by a heated vertical cylinder which is embedded in a fluid-saturated porous medium. The surface of the cylinder is maintained at a temperature whose value above the ambient temperature of the surrounding fluid varies as then ^th power of the distance from the leading edge. Asymptotic analyses and numerical calculations are presented for the governing nonsimilar boundary layer equations and it is shown that, whenn<1, the asymptotic flowfield far from the leading edge of the cylinder takes on a multiple-layer structure. However, forn>1, only a simple single layer is present far downstream, but a multiple layer structure exists close to the cylinder leading edge. We have shown that the fully numerical and asymptotic calculations are in stisfactory agreement, especially for exponentsn close to zero. Comparisons of the present numerical solutions obtained using the Keller-box method with previous numerical solutions using local methods are also given.List of symbols a radius - scaled streamfunctions - f ₀,f ₁,f ₂ inner zone streamfunctions whenn<1 - leading order streamfunctions inn>1, 1 asymptotic solution - F ₀,F ₁ outer zone streamfunctions whenn<1 - G large parameter satisfyingG=X ² lnG - g gravitational acceleration - K permeability of the porous medium - n exponent in prescribed temperature law - r radial co-ordinate - r rescaled radial co-ordinate - R Darcy-Rayleigh number - T temperature of convective fluid - T _w temperature of cylinder at leading edge - T ambient temperature of fluid - u velocity in axial direction - v velocity in azimuthal direction - w velocity in radial direction - x axial co-ordinate - x escaled axial co-ordinate - X dimensionless axial co-ordinate - thermal diffusivity of the saturated medium - coefficient of thermal expansion - constant in the boundary conditions forF ₀ - dimensionless radial co-ordinate - co-ordinate for the outer zone in then<1 solution - scaled radial co-ordinates - scaled fluid temperature - similarity variable for then=1 problem - nondimensionalisation constant (Eq. (9)) - viscosity of fluid - scaled axial co-ordinates - density of fluid - co-ordinate for the inner zone in then<1 solution - azimuthal co-ordinate - similarity variables for then>1 problem - streamfunction     

Asymptotic analysis of wave propagation in a finite viscoplastic bar

Summary Axially symmetric wave propagation in a finite extent cylindrical sample, whose viscoplastic behaviour is described by the Bodner-Partom model, involves variables which can be scaled so as to bring out a small parameter characteristic of the problem at hand. Asymptotic solutions can be derived from power series expansions of the variables with respect to the small parameter.In the one-dimensional case, a finite difference solution is shown to agree with the zero order term of the asymptotic expansion which is closely related to the Hopkinson pressure bar simplified theory.Then it is shown in a special case that retaining the sole first term of each asymptotic expansion amounts to eliminating the needlessly intricate fluctuations of the exact solution due to successive wave reflections at the specimen's both ends, while keeping the essentials of the response and simplifying the numerical work.Afterwards, the relative orders of magnitude of the various stress components with respect to the small parameter are estimated in the more general framework of the two-dimensional problem, with a view to re-deriving the so-called inertia correction; the latter relies on an appropriate definition of the mean stress experienced by the sample and uses the first two terms of the asymptotic expansions.This study suggests the applicability of the method in a more general scope, taking into account in a consistent way other phenomena (friction and lateral motion of the adjacent bars, for instance) or more refined constitutive equations.Notation a=h/R aspect ratio of specimen - A specimen cross-section area - A ₁ bar cross-section area - c elastic wave speed in specimen - c ₁ elastic wave speed in bars - E specimen Young's modulus - E ₁ bar Young's modulus - h specimen length - H Z/E - r radial coordinate - R specimen radius - s _ij stress deviator - t time - T incident signal duration - u _i physical displacement in specimen - U _i dimensionless displacement in specimen - V ₀,V ₁ dimensionless velocities on end faces - w _p plastic work - W _p dimensionless plastic work - x axial coordinate - Z Bodner-Partom hardening parameter - small parameter - zero order longitudinal extension rate - strain rate tensor - dimensionless axial coordinate - dimensionless axial coordinate - polar angle - Poisson's ratio - specimen density - _ij stress tensor - _ij dimensionless stress tensor - equivalent stress - dimensionless time - dimensionless viscoplastic strain rate in axial direction     

Heat flow model of rapid solidification with nonhomogeneous thermal contact

A heat flow model is presented of the solidification process of a thin melt layer on a heat conducting substrate. The model is based on the two-dimensional heat conduction equation, which was solved numerically. The effect of coexisting regions of good and bad thermal contact between foil and substrate is considered. The numerical results for thermal parameters of the Al-Cu eutectic alloy show considerable deviations from one-dimensional solidification models. Except for drastic differences in the magnitude of the solidification rate near the foil-substrate interface, the solidification direction deviates from being perpendicular to the substrate and large lateral temperature gradients occur. Interruption of the thermal contact may lead to back-melting effects. A new quantity, the effective diffusion length, is introduced which allows some conclusions to be drawn concerning the behaviour of the frozen microstructure during subsequent cooling.Nomenclature _i ,a _i Thermal diffusivity _i = _i /c _{i i} ,a _i = _i / ₁ - c _i Specific heat capacity - d Foil thickness - D Solid state diffusion coefficient - e_x, e_z Unit vectors - H Latent heat of fusion - h ,h Foil-substrate heat transfer coefficients - i Index: 1, melt; 2, solidified foil; 3, substrate - _i ,k _i Thermal conductivityk _i = _i / ₁ - n Normal unit vector - Nu ,Nu Nusselt numbers for regions of badNu(x,) and good thermal contact, respectivelyNu =h Nu _d / ₁,,Nu(x, )=h(x,)d/ ₁ - R Universal gas constant - , s Position of the liquid-solid interface ¯s/d=s=s _xe_x+s _ze_z - Local solidification rate /d = s =s _xe_x +s _ze_z - t Real time - T _i Temperature field - T ₀ Ambient temperature - T _f Melting temperature - u _i Dimensionless temperature fieldu _i (x, z,)=T _i (x,z,)/T _f - u ₀ Dimensionless ambient temperatureu ₀=T ₀/T _f - _i Local cooling rate within the foil _i = du _i /d - W Stefan numberW=H/c ₁ T _f - ,x Cartesian coordinate parallel to the foil-substrate interfacex= /d - ₀,x ₀ Lateral extension of foil sectionx ₀= ₀/d - ₁,x ₁ Lateral contact lengthx ₁= ₁/d - ,z Cartesian coordinate perpendicular to the foil-substrate interfacez= /d - ₀,z ₀ Substrate thicknessz ₀= ₀/d - E Activation energy of diffusion - T Initial superheat of the melt - u Dimensionless initial superheat u=T/T _f - (x) Step function - _eff Dimensionless effective diffusion length - _i Mass density - Dimensionless time=t ₁/d ² - _f, _f(x, z) Total and local dimensionless freezing time, respectively     

Thermomechanic behavior of a polymer upon its formation in a crystallization process: Theoretical principles,hypotheses, and mathematical model

A model of thermomechanic behavior of a polymer upon its formation in a crystallization process is proposed. Based on methods of nonequilibrium thermodynamics governing relationships are obtained which make it possible to establish the dependence of the final degree of crystallicity of the material on the history of the crystallization process and to explain the mechanism of formation of the remanent stresses in a polymer article.Notation u translation vector - v velocity vector - acceleration vector - absolute temperature - density - c specific heat capacity - deformation tensor - strain tensor - specific enthropy - U ^* internal energy - z specific free enthalpy - _i internal parameters of state - t time - q heat flux vector - matrix of heat conduction coefficients - W ^* energy dissipation - F vector of mass forces - the 4th rank tensor of elastic pliabilities - matrix of heat expansion coefficients - tensor of contribution of structural variations to deformation - function of equilibrium value ^* - p mean pressure - deviator of the tensor of deformations - spherical part of the deformation tensor - deviator of the tensor of stresses - K volume modulus - unity tensor - Q enthalpy of the crystallization process - Q _eq enthalpy of the equilibrium crystallization process - _g glass transition temperature - ^*() the curve obtained in the equilibrium crystallization process - f final degree of crystallicity Institute of Mechanics of Continuous Media of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Institute of Technical Chemistry of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 3, pp. 479–485, May–June, 1995.