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     

Nonlinear axisymmetric response of orthotropic thin truncated conical and spherical caps

Summary An approximate analytical solution of the large deflection axisymmetric response of polar orthotropic thin truncated conical and spherical shallow caps is presented. Donnell type equations are employed. The deflection is approximated by a one term mode shape satisfying the boundary conditions. The Galerkin's method is used to get the governing equation for the deflection at the hole. Nonlinear free vibration response and the response under uniformly distributed static and step function loads are obtained. The effect of various parameters is investigated.Notations A, A ^* Inward and outward amplitudes - a, b, h Base radius, inner radius and thickness of the cap - D M h ³/[12(–v ² )] - E ,E Young's moduli - H ^*,H Apex height, dimensionless apex heght:H ^*/h - N _, Stress resultants - p ^1/2 - q Uniformly distributed load - Q,Q₀ Dimensionless load: , dimensionless step load - Q, Q ₀ Dimensionless load: , step load - t, Time, dimensionless time: t - T _A Ratio of nonlinear periodT for inward amplitudeA and the linear periodT _L - w ^* Normal displacement at middle surface - w Dimensionless displacement:w ^*/h - ₁ Linear parameter of static response - Orthotropic Parameter:E /E - Mass density - ₂,₃ Quadratic and cubic nonlinearity parameters - b/a - v ,v Poisson's ratios - Dimensionless radius:r/a - ^*, Stress function, dimensionless stress function: - ₀ ^* ,₀ Linnear frequency, dimensionless frequency: With 7 Figures     

Integral-equation representations of flow elastoplasticity derived from rate-equation models

Summary Two integral-equation representations are presented in this paper, based on the exact integrations of the conventional rate-equation model of associativeJ ₂ flow elastoplasticity with combined-isotropic-kinematic hardening-softening. Among them the strain-controlled integral-equation representation has two new naturally defined material functionsY(Z) andU(Z) of the normalized active workZ, which plays the role of intrinsic time. One of the immediate benefits derivable from the new representations is, owing to the explicit unfolding of the highly nonlinear path-dependence between stress and strain without a detour to the evolutions of internal state variables, their adaptability for direct calculations without any iteration. Indeed, it is itself a constructive algorithm. It is shown that at a realistic level of precision, the strain-controlled integral-equation representation saves 99% or more of the CPU time compared with the widely used elastic predictor-radial return algorithm of the rate-equation representation.List of symbols e _ij ,e _ij ^e ,e _ij ^p strain deviator, elastic strain deviator, plastic strain deviator - effective strain - ^p effective plastic strain - e ₁,e ₂,e ₃ principal strain deviator,e ₃=–e ₁–e ₂ - e _tan,e _rad tangential strain increment, radial strain increment - E Young's modulus, assumed to be constant - f yield function in stress space - F yield function in strain space - G shear modulus, assumed to be constant - G(Z ₁,Z ₂) shear relaxation function of elastoplasticity - h( ^p ),k( ^p ) material functions of plasticity for the stress-space rate-equation representation - material functions of plasticity for the strain-space rate-equation representation - I ₂ second invariant of the deviatoric strain tensor - J ₂ second invariant of the deviatoric stress tensor - J(z ₁,z ₂) shear creep function of elastoplasticity - K bulk modulus, assumed to be constant - p dummy variable of integration in place of the effective plastic strain - r _ij active stress - R _ij active strain - effective active-stress, i.e. times Euclidean length of active stress - effective active-strain, i.e. times Euclidean length of active strain - S _ij ,S _ij ^e ,S _ij ^r stress deviator, elastic stress deviator, stress relaxation - effective stress - effective stress relaxation - S ₁,S ₂,S ₃ principal stress deviator,S ₃=–S ₁–S ₂ - t, , , time - t ₀ zero-value time - t _u latest unloading time - y(z), u(z) material functions of plasticity for the stress-controlled integral-equation representation - Y(Z), U(Z) material functions of plasticity for the strain-controlled integral-equation representation - z normalized active complementary-work - material functions defined for use in convertingh( ^p ) andk( ^p ) toy(z) andu(z) - Z normalized active work - material functions defined for use in convertingh( ^p ) andk( ^p ) toY(Z) andU(Z) - _ij back stress - A _ij back strain - _ij , _ij ^e , _ij ^p strain, elastic strain, plastic strain - _y (initial) yield strain, _y =h(0)/2G - Poisson's ratio assumed to be constant - _ij , _ij ^e , _ij ^r stress, elastic stress, stress relaxation - _y (initial) yield stress, yield strength, _y =h(0)     

Equilibrium equations for plane slender bars undergoing large shear deformations

Summary Equilibrium equations for plane slender bars are derived using the principles of Continuum Solid Mechanics. Large shear deformations are accounted for. The equations are derived in an explicit form, yielding upon introduction of a constitutive law (not given here) a system of differential equations to be integrated on the undeformed length. Within certain limitations, the equations, derived in an exact form, incorporate expressions for finite strain. The stress distribution along a deformed section is studied.

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Gleichgewichtsbedingungen ebener schlanker Stäbe bei großen Schubverformungen

Zusammenfassung Gleichgewichtsbedingungen ebener schlanker Stäbe werden unter Verwendung der Sätze der Kontinuumstechnik hergeleitet. Große Schubverformungen werden berücksichtigt. Die Gleichungen werden in expliziter Form angegeben, sie führen mit einem hier nicht angegebenen Werkstoffgesetz auf ein System von Differentialgleichungen. Mit gewissen Einschränkungen schließen diese in exakter Form angegebenen Gleichungen die für endliche Verformungen ein. Die Spannungsverteilung in einem verformten Abschnitt wird untersucht.

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Notation position vector of material point in undeformed state - position vector of material point in deformed state - displacement vector of material point - u displacement component inx-direction - w displacement component inz-direction - covariant base vector in deformed state (i=1,2) - G _ij metric tensor in deformed state - g _ij metric tensor in undeformed state - slope of geometrical axis (elastic line) - shear deformation angle - rotation of plane section of bar - K curvature of geometrical axis - curvature of geometrical axis due to bending only - unit tangent vector to geometrical axis - unit normal vector to geometrical axis - _(ij) Lagrangian strain tensor - ^ij stress tensor - ^(ij) physical stress tensor - N normal force (parallel to geometrical axis) acting on plane section - S shear force (parallel to deformed section) acting on plane section - M bending moment - H horizontal external force acting on plane section - V vertical external force acting on plane section - vector of conservative body forces uniformly distributed along geometrical axis - f _x component of inx-direction - f _z component of inz-direction - unit normal vector to plane section - F _i force component on distorted plane section - F _(i) physical force component on distorted plane section - F _ij component ofF ⁱ inj-direction - inclination angle ofF ¹¹ toF ¹ (see Fig. 8) With 8 Figures     

On the number of term orders

By an admissible order on a finite subsetQ of ⁿ we mean the restriction toQ of a linear order on ⁿ compatible with the group structure of ⁿ and such that ⁿ is contained in the positive cone of the order. We first derive upper and lower bounds on the number of admissible orders on a given setQ in terms of the dimensionn and the cardinality ofQ. Better estimates are possible if the setQ enjoys symmetry properties and in the case whereQ is a discrete hyperbox of the form In the latter case, we also give asymptotic results as d _k for fixedn. We finally present algorithms which compute the set of admissible orders that extend a given binary relation onQ and their number. The algorithms are useful in connection with the construction of universal Gröbner bases.AMS Classification: primary 06F20 secondary 06-04, 11N25     

The activation energy for superplastic flow in Al-6Cu-0.4Zr

A study of the activation energies for superplastic flow in an essentially single phase Al-6Cu-0.4Zr alloy has been made in the temperature range 430 to 490° C. Straight line Arrhenius plots for bothQ and were obtained in Regions I, II and III. In all cases the ratio corresponded well to the average strain rate sensitivity as determined by both change rate testing and from the slope of the In versus In curves. Values ofQ of 35.2, 19.0 and 20.1 kcal mol^–1 were obtained in Regions I, II and III respectively. These values were expected to be close to the true activation energies, and corresponded to the measured lattice and predicted grain boundary diffusion activation energies. These energies, together with microstructural observations made on deformed material, were used to identify possible deformation mechanisms.     

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     

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     

Incremental finite element equations for thermomechanical response of elastomers: Effect of boundary conditions including contact

Summary The present investigation concerns the solution of nonlinear finite element equations by Newton iteration, for which the Jacobian matrix plays a central role. In earlier investigations [1], [2], a compact expression for the Jacobian matrix was derived for incremental finite element equations governing coupled thermomechanical response of near-incompressible elastomers. A fully Lagrangian formulation was adopted, with three important restrictions: (a) the traction and heat flux vectors were referred to theundeformed coordinates; (b) Fourier's law for heat conduction was expressed in terms of theundeformed coordinates; and (c) variable contact was not considered. In contrast, in the current investigation, the boundary conditions and Fourier's law of heat conduction are referred to thedeformed coordinates, and variablethermomechanical contact is modeled. A thermohyperelastic constitutive equation introduced by the authors [3] is used and is specialized to provide a thermomechanical, near-incompressible counterpart of the two-term Mooney-Rivlin model. The Jacobian matrix is now augmented with several terms which are derived in compact form using Kronecker product notation. Calculations are presented on a confined rubber O-ring seal submitted to force and heat.List of symbols A contact area - A _i - A _MM coefficient matrix for foundation model - a _TM coefficient vector for foundation model - a _MT coefficient vector for foundation model - a _TT coefficient scalar for foundation model - B _t,B _q matrices related to boundary terms due to large deformations - B _{c t} ,B _{c q} matrices related to boundary terms due to variable thermomechanical contact - B _{f t} ,B _{f q} matrices related to boundary terms due to nonlinear foundation model - B _{c MM} submatrix inB _{c t} - B _{c TM} submatrix inB _{c q} - B _{c TT} submatrix inB _{c q} - B _{f MM} submatrix inB _{f t} - B _{f TM} submatrix inB _{c q} - B _{f TT} submatrix inB _{f q} - B _{t MM} submatrix inB _t - B _{q TM} submatrix inB _q - B _{q TT} submatrix inB _q - C Cauchy-Green strain tensor - C ₁,C ₂ constants in strain energy density functions for the elastomer - c vec (C) - c ₂ vec (C ²) - c _e specific heat at constant strain - _e - c _hi parameters in contact heat conductance model - D _nl stiffness matrix due to geometric nonlinearity - D _T isothermal tangent modulus matrix - D _{T e} modulus matrix at constantT ande - D _T tangent modulus matrix at constantT and - d g _i nodal vectors related to prescribed traction and heat flux - d g _{c q} nodal vector related to thermal contact - d g _{c t} nodal vector related to contact traction - d g _{f q} nodal vector related to heat flux - d g _{f q 0} nodal vector related to heat flux - d g _{f q 0} nodal vector related to heat flux - d g _{f t} nodal vector related to traction - d g _{f t 0} nodal vector related to prescribed traction - d g _{f q 0} nodal vector related to prescribed traction - d g _{q 0} nodal vector related to heat flux - d g _{q 0} nodal vector related to heat flux - d g _{t 0} nodal vector related to traction - d g _{t 0} nodal vector related to traction - d(n ^TT} q) prescribed heat flux increment - d tt} prescribed traction increment - e vec() - e _d vec( _d ) - e _r shift parameter in elastic foundation model - f, f(T) thermal expansion function, = - F deformation gradient tensor - f _c nodal vector from the contact traction - g gap function - g _i nodal vectors related to mechanical and thermal loads - g _M nodal vector related to mechanical load - g _{i T} nodal vectors related to thermal terms - h time step - h _n - I _i invariants ofC - I ₉ 9×9 identity tensor - I 3×3 identity tensor - i vectorial counterpart ofI: vec(I) - J Jacobian matrix for Newton iteration - J determinant ofF - k thermal conductivity - k _H high stiffness in elastic foundation model - k _L low stiffness in elastic foundation model - K(g) stiffness function for elastic foundation model - K tangent stiffness matrix - K _MM tangent stiffness submatrix - K _MT tangent stiffness submatrix - K _MP tangent stiffness submatrix - K _PP tangent stiffness submatrix - K _PT tangent stiffness submatrix - K _TT tangent stiffness submatrix - M ₁ strain-displacement matrix - M ₂ strain-displacement matrix - m unit vector normal to target surface - N interpolation matrix - n vector normal to current surface - n ₀ vector normal to undeformed surface - n _i - p (true) pressure - Q heat rate across contact surface - q heat flux vector referred to the deformed configuration - q ₀ heat flux vector referred to the undeformed configuration - qq} prescribed heat flux - r residual vector in combined equilibrium equation - r _M residual vector from mechanical equilibrium - r _T residual vector from thermal equilibrium - r residual vector from near-incompressibility constraint - R matrix of heat conduction in domain - S surface in current configuration - s vec () - S ₀ surface in undeformed configuration - S _c candidate contact surface in current configuration - S _{c 0} candidate contact surface in undeformed configuration - S _{f M} surface corresponding to nonlinear foundation in current configuration - S _{f M 0} surface corresponding to nonlinear foundation in undeformed configuration - S _{f T} surface corresponding to nonlinear foundation in current configuration - S _{f M 0} surface corresponding to nonlinear foundation in undeformed configuration - S _T prescribed temperature boundary surface in current configuration - S _{T 0} prescribed temperature boundary surface in undeformed configuration - S _t prescribed traction boundary surface in current configuration - S _{t 0} prescribed traction boundary surface in undeformed configuration - S _q prescribed heat flux boundary surface in current configuration - S _{q 0} prescribed heat flux boundary surface in undeformed configuration - S _u prescribed displacement boundary surface in current configuration - S _{u 0} prescribed displacement boundary surface in undeformed configuration - T current temperature - T ₀ reference temperature - T _r temperature of rigid foundation - t time - t _n contact traction normal to contact surface - t _{n, n} solution value oft _n at thenth load step - t _ti components of tangential contact traction vector - t traction referred to current configuration - tt} prescribed traction - u displacement vector - v combined vector of nodal parameters - V ₀ volume in undeformed configuration - V volume in deformed configuration - w - x position vector in deformed configuration - X position vector in undeformed configuration - y possible contact point in the target surface - volumetric thermal expansion coefficient - _i parameters in metal-metal thermal contact models - _hi coefficients in thermal contact model - _k coefficient in elastic foundation model - interpolation matrix for strain field - _T interpolation matrix for thermal gradient: ₀ T - - vector of nodal displacements - Lagrangian strain tensor - _d deviatoric portion of Lagrangian strain tensor - interpolation function for - entropy density - vector of nodal temperatures - þ isothermal bulk modulus - surface area factor - interpolation function forT - temperature-adjusted pressure, - mass density in the deformed configuration - ₀ mass density in the undeformed configuration - 2nd Piola-Kirchhoff stress tensor - Helmholtz free energy density function - _M Helmholtz free energy density function - ₀ Helmholtz free energy density function - _i - _i - _ij - _ij - nodal vector for pressure field - near-incompressibility constraint function - the target surface equation: (y)=0 - (·) variational operator - vec(·) vectorization operator - symbol for Kronecker product of two tensors - tr(·) trace of a tensor - det(·) determinant of a tensor - divergence operator with respect to current configuration - ₀ divergence operator with respect to undeformed configuration - · the norm of vector     

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)     

Schwache Auftriebseffekte in laminaren,vertikalen, runden Freistrahlen

Zusammenfassung Der auftriebslose laminare, runde Freistrahl beliebigen Öffnungswinkels kann durch eine exakte Lösung der vollständigen Navier-Stokes-Gleichungen beschrieben werden. Wirkt die Schwerkraft parallel zur Strahlrichtung, so stellen sich auf Grund der von einer Energiequelle erzeugten Temperaturunterschiede in der Nähe des Strahlzentrums schwache Auftriebseffekte ein. Dieses reguläre Störproblem kann mit einem additiven Störansatz bewältigt werden. Der Entwicklungsparameter ist proportional zum Quadrat des Abstandes vom Ursprung, lediglich bei der Druckstörung tritt noch ein logarithmischer Term hinzu. Die Störgleichungen werden numerisch gelöst. Es zeigt sich, daß die Breite der Geschwindigkeitsstörung von der Strahlbreite des auftriebslosen Strahls praktisch unabhängig ist. Die Störungen steigen mit Werten der Prandtlzahl Pr und des Reibungsparametersk, die beide unabhängig voneinander im Intervall [0,01; 20] variiert werden. Die Ergebnisse werden anhand graphischer Darstellungen diskutiert.

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Weak buoyancy effects in laminar, vertical, round jets

Summary It is possible to describe the non buoyant, laminar, round jet with arbitrary angle of aperture by an exact solution of the complete Navier-Stokes-equations. If the gravity force is parallel to the jet axis, small buoyancy effects arise due to temperature differences caused by an energy source located in the center of the jet. In this regular perturbation problem all functions are expanded in terms of a parameter which is proportional to the square of the distance from the origin, but in the expansion of the pressure an additional logarithmic term arises. The equations for the perturbation quantities are solved numerically. It appears that the spatial extension of the perturbation of the radial velocity component is almost independenl of the width of the non-buoyant jet. The magnetude of the perturbations increases witt increasing Prandtl number Pr and increasing friction parameterk. Both parameters arh varied independently in the interval [0,01; 20]. Discussion of the results is based on severae diagrams.

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Bezeichnungen a Temperaturleitfähigkeit - c _p spez. Wärme bei konstantem Druck - c _T konstanter Faktor der Temperaturfunktion - D Differentialoperator - F, f Stromfunktion - g Temperaturfunktion - g _s Schwerebeschleunigung - k Reibungskonstante - K Druckkonstante - L Differentialoperator - L _b Bezugslänge - M ₀ Quellstärke der Impulsquelle - reduzierte Quellstärke der Impulsquelle - O, o Landau-Symbole - p Druck - kinematische Druckdifferenz - P _b Bezugsgröße für den Druck - Pr Prandtl-Zahl - Q Quellstärke der Energiequelle - reduzierte Quellstärke der Energiequelle - r, , sphärische Polarkoordinaten - Re Reynolds-Zahl - U _b Bezugsgeschwindigkeit - v Geschwindigkeit - x transformierte Variable - Temperaturdifferenz - kinematische Zähigkeit - Impulsstromtensor - _ij Komponenten des Impulsstromtensors - Dichte - _ij Schubspannungskomponenten Indizes 1 auftriebsloser Strahl - 2 Störgrößen - r r-Richtung - -Richtung - * dimensionsbehaftete Größen Mit 12 AbbildungenTeil der Dissertation (TU Wien 1978, Referenten Prof. Dr. W. Schneider, Prof. Dr. A. Kluwick).     

Group codes on certain algebraic curves with many rational points

We construct a series of algebraic geometric codes using a class of curves which have many rational points. We obtain codes of lengthq ² over _q , whereq = 2q ₀ ² andq ₀ = 2 ⁿ , such that dimension + minimal distance q ² + 1 – q ₀ (q – 1). The codes are ideals in the group algebra _q [S], whereS is a Sylow-2-subgroup of orderq ² of the Suzuki-group of orderq ² (q ² + 1)(q – 1). The curves used for construction have in relation to their genera the maximal number of GF _q -rational points. This maximal number is determined by the explicit formulas of Weil and is effectively smaller than the Hasse—Weil bound.Supported by Deutsche Forschungsgemeinschaft while visiting Essen University     

Free vibration of rectangular beams of arbitrary depth

Summary The state space approach is extended to the two dimensional elastodynamic problems. The formulation is in a form particularly amenable to consistent reduction to obtain approximate theories of any desired order. Free vibration of rectangular beams of arbitrary depth is investigated using this approach. The method does not involve the concept of the shear coefficientk. It takes into account the vertical normal stress and the transverse shear stress. The frequency values are calculated using the Timoshenko beam theory and the present analysis for different values of Poisson's ratio and they are in good agreement. Four cases of beams with different end conditions are considered.

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Freie Schwingungen rechteckiger Balken beliebiger Höhe

Zusammenfassung Die Zustandsraum-Technik wird auf zweidimensionale elastodynamische Probleme ausgedehnt. Die Formulierung ist besonders geeignet für die Aufstellung von Näherungstheorien beliebigen Grades. Freie Schwingungen von Rechteckbalken beliebiger Höhe wurden mit Hilfe dieser Technik untersucht. Das Verfahren umgeht den Begriff des Schubbeiwertsk. Es berücksichtigt die senkrechte Normalbeanspruchung und die Querkraft. Die Frequenzwerte werden mit Hilfe der Balkentheorie von Timoshenko und der vorliegenden Analyse berechnet, und zwar für verschiedene Werte der Querdehnzahl. Die berechneten Werte befinden sich in guter Übereinstimmung. Vier Fälle von Balken mit verschiedenen Endbedingungen werden untersucht.

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Notation 2h depth of beam - k Timoshenko shear constant - L length of the beam - n mode number - u, v displacement inx, y directions - A area of cross section - A _n coefficient in series representation - E modulus of elasticity - G modulus of rigidity - I moment of inertia aboutz-axis - mass density - Poisson's ratio - r - r×n - _{x y} direct stresses - _xy shear stress - eigenvalue of square matrix - frequency of harmonic vibration - eigenvalue= - frequency parameter= - ^* frequency parameter=× With 1 FigureOn leave from M. A. College of Technology, Bhopal, 462007, India     

Glassy behavior of low-temperature energy relaxation in YBa2Cu3O7 and YBa2Cu3O6

Measuring the power release after rapid cooling a YBa₂Cu₃O₇ sample (m=42.85 g, T_c=91 K) from the equilibrium temperature T₁ (2.35 KT₁15.1 K) to T₀=1.5 K, we observed a time dependence typical of a glass: is proportional to t^–1. The results allow us to determine the linear term of the heat capacity (0.8 mJ/mole · K²) due to the two-level systems. While the low-temperature heat capacity anomaly noticeably decreases, the power release is essentially unchanged after oxygen reduction of the sample.     

Application of the singularity method for the formulation of plane elastostatical boundary value problems as integral equations

Summary A method for the formulation of elastostatical boundary value problems as integral equations is presented, the basic idea of which consists of superimposing in a suitable fashion singular solutions for the infinite medium. Since mechanical aspects play an important role in the concept of the method, all quantities in the equations can be interpreted physically. The applicability of the method is illustrated by examples of the geometrical and statical boundary value problem ofplane elastostatics for which 32 different formulations as integral equations are established.The second aim of the paper consists of revealing an analogy between the most important notions of the singularity method, viz. between state variables and singularities. The analogy is manifested by certain symmetries of influence functions, and enables the systematical representation of the basic relations and their interpretation within a larger context.

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Anwendung der Singularitätenmethode zur Formulierung von Randwertproblemen der ebenen Elastostatik als Integralgleichungen

Zusammenfassung Es wird eine Methode zur Formulierung von Randwertproblemen der Elastostatik als Integralgleichungen beschrieben, deren Grundgedanke darin besteht, singuläre Lösungen für das unendliche Medium in geeigneter Weise zu überlagern. Da beim Aufstellen der Gleichungen mechanische Gesichtspunkte im Vordergrund stehen, lassen sich alle auftretenden mathematischen Größen physikalisch deuten. Die Anwendbarkeit der Methode wird anhand des geometrischen und des statischen Randwertproblems derebenen Elastostatik erklärt. Es ergeben sich dabei 32 verschiedene Formulierungen der Probleme als Integralgleichungen.Weiterhin wird in dem Aufsatz eine Analogie zwischen den wichtigsten Begriffen der Singularitätenmethode, den Zustandsgrößen und den Singularitäten, aufgedeckt. Die Analogie macht sich durch gewisse Symmetrien der Einflußfunktionen bemerkbar und erlaubt es, die grundlegenden Beziehungen systematisch darzustellen und in einen größeren Zusammenhang einzuordnen.

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Nomenclature

state variables; torso state variables Airy's stress function - u _i displacement vector - _i stress function vector - U _ij distortion tensor - _ij stress tensor - U _i distortion vector - {ei33-3} stress vector - N collective denotation for state variables singularities; torso singularities v wedge dislocation - C _i edge dislocation - R _i single force - G _i Rieder's singularity - F _i Massonnet's singularity - c _i generating vector ofc _ij - r _i generating vector ofr _ij - g _i generating vector ofg _ij - f _i generating vector off _ij - c _ij dipole ofC _i - r _ij dipole ofR _i - g _ij dipole ofG _i - f _ij dipole ofF _i - M collective denotation for singularities - m _i dipole of a singularityM - d moment of a force dipole - b dilation intensity of a force dipole influence functions (NM) collective denotation for influence functions further influence functions: see Chapters 4.1, 4.2 - [NM] collective denotation for proportionality factors of non integral terms corresponding to (NM). further proportionality factors see Chapter 8 geometry x _i , radius vector of the field and source point respectively - ,q _i vector and unit vector respectively in the direction of the connecting line between the field and the source point - distance between the field and the source point - S curve on the infinite plane congruent to the boundary of the elastic body - S ¹,S ² sections ofS - curve equidistant fromS - s, arc length of a field and a source point respectively onS - arc length of a field point on - n_i, normal vector ofS at the field and the source point respectively - curvature ofS further denotations _i , vector of differentiation with respect to field and source point co-ordinates respectively - _ij identity tensor - e _ij permutation tensore ₁₁=e ₂₂=0e ₁₂=–e ₂₁=1 - m Poisson's ratio - G shear modulus - Cauchy principal value - b _i arbitrary constant vector With 10 Figures     

Experimente zum Strömungswiderstand in gekrümmten,rotierenden Kanälen mit quadratischem Querschnitt

Zusammenfassung Bei ausgebildeter Strömung wurde der Druckverlust in rotierenden, gekrümmten Kanalstrecken experimentell ermittelt. Speziell interssierte der Einfluß der Krümmung auf den Reibungswiderstand und im besonderen der der Rotation, die sowohl in Richtung der Durchflußgeschwindigkeit als auch entgegengesetzt dazu erfolgte. Die strömungsspezifischen Parameter wie Reynolds-und Taylor-Zahl erfuhren eine Änderung im Bereich 3·10²<Re<2·10⁵ und 0Ta<10⁵. Das Krümmungsverhältnis variierte zwischen 0,0496a/r _m 0,337. Aufgezeigt wird die Möglichkeit einer universellen Darstellung des Widerstandsbeiwertes für den nicht rotierenden Kanal bei einerseits laminarer und anderseits bei turbulenter Bewegung. Im rotierenden Kanal beobachtet man beträchtliche Druckverlusterhöhungen, deren Größe von der Rotationsrichtung abhängt. Bei großen Taylor-Zahlen wurde eine Unabhängigkeit des Strömungswiderstandes vom Krümmungsverhältnis festgestellt.

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Experimental investigations on the friction resistance in rotating curved channels with square cross-section

Summary For fully developed flow the pressure loss in rotating curved channels was investigated experimentally. The influence of the curvature on the friction coefficient and particularly on the rotation, which was realized in the direction of the through flow as well as in the opposite direction was of special interest. The specific parameters of the flow field, like the Reynolds-number and the Taylor-number were varied within the range of 3·10²<Re<2·10⁵ and 0Ta<10⁵. The curvature ratio were varied within 0,0496a/r _m 0,337. The possibility of an universal presentation of the friction coefficient for the nonrotating channel is shown for laminar and turbulent through flow. In the rotating channel a considerable pressure loss increase can be observed. The magnitude depends on the direction of rotation. In case of great Taylor-numbers the curvature ratio was found to be independent of the friction resistance.

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Bezeichnungen a Kantenlänge - mittlere Geschwindigkeit - l Kanallänge - r _m Krümmungsradius - Volumenstrom - _p Druckdifferenz - Widerstandsbeiwert - =57/Re Widerstandsbeiwert des geraden Kanals - dynamische Viskosität - Dichte - Winkelgeschwindigkeit - Dean-Zahl - Reynolds-Zahl - Taylor-Zahl Mit 6 Abbildungen     

Correlation between magnetic hysteresis and magnetic relaxation in YBaCuO single crystals

The dependence of the magnetic momentm obtained from the hysteresis loops on the speed of the magnetic field sweep =dH _ext/dt is explained on the basis of Anderson's interpretation of the magnetic flux creep. In addition, a phenomenological model is suggested which predicts a linear dependence ofm on ln with the slope m/ ln , numerically equal to the relaxation rate m/ ln(t) from the usual magnetic relaxation. Such linear relations betweenm and ln were observed experimentally in single crystals of YBaCuO. Preliminary experiments on the complementary time dependent relaxation ofm after a simulated step change ofH _ext gave mostly relaxation rates close to the predicted values. The model here presented also enables one to compare the critical state in the superconductor at a field sweep rate with the critical state at some timet _eff after a step change ofH _ext. The values of analyzed in our experiments actually correspond to the critical state at timest _eff between0.04 and4 sec after an imaginary large step change ofH _ext.     

Effects of phase fluctuation on a mixture of rotating superfluids

It is shown that if the root-mean-square of the gradient of the phase fluctuation of either of the components exceeds the corresponding inverse of the coherence length or if the chemical potential exceeds ₁ ⁰ or ₂ ⁰ , where is the volume integrals of the interaction function between the components, and ₁ ⁰ , ₂ ⁰ are the densities of the two components, the mixture of two rotating superfluids has an instability.     

The laminar stagnation-point flow against a solidifying moving shell

Summary This paper considers the two-dimensional laminar stagnation-point flow due to a jet impinging onto a solidifying moving boundary. The flow is of interest in connection with the horizontal belt strip casting process. An exact solution to the Navier-Stokes equations is found that is shown to depend on a single ordinary differential equation. The solution is useful in the study of morphological and hydrodynamic instabilities within the impingement region. Solutions for the steady-state shape of the initial stages as well as the asymptotic behavior of the solidifying interface are also discussed in a perturbative manner.Nomenclature A suction velocity in boundary layer variables - a jet width [m] - c specific heat of the solid metal [J/m³K] - h Newtonian heat transfer coefficient [W/m²K] - k velocity gradient in units ofU/a - m dS ^*/dX ^* local inclination of the solidifying phase - S ^* (L)/L average slope of the solidifying phase - S ^* local thickness of the solidified phase [m] - S, S local thickness of the solidified phase in units ofL and , resp. - T absolute temperature [K] - T _f fusion temperature of metal [K] - T ₀ temperature of cooling water [K] - U jet velocity [m/s] - V belt velocity [m/s] - +i complex velocity potential in units ofUa - x coordinate tangential to the solidifying interface in units ofa - X ^* coordinate tangential to the belt [m] - X, X coordinates tangential to the belt in units ofL and , resp. - y coordinate orthogonal to the solidifying interface in units ofa - Y ^* coordinate orthogonal to the belt [m] - Y, Y coordinates orthogonal to the belt in units ofL and , resp. - z x+iy complex coordinates in units ofa - unit vector along the belt - unit vector orthogonal to the belt - local unit normal vector to the solidifying interface - h _f latent heat of fusion of metal [J/m³] - thermal diffusivity of solid metal [m²/s] - belt velocity in units ofU - { _n }, { _n } asymptotic sequences of the outer and inner expansion, resp. - m suction velocity outer variables - velocity potential in units ofUa - jet inclination relative to the local solidifying interface - coordinate orthogonal to the solidifying interface in units of - x c thermal conductivity of solid metal [W/mK] - displacement thickness in units of - v kimematic viscosity of liquid metal [m ²/s] - arctan (dS ^*/dX ^*) local angle of inclination of the solidifying interface - =(T–T ₀)/(T _f –T ₀) dimensionless temperature - perturbation parameter - coordinate tangential to the solidifying interface in units ofa/k - stream function in units ofUa - magnified stream function valid within the boundary layer - solidification constant Dimensionless parameter P _eL VS ^* (L)/ Peclet number - Q h/(cV) Heat flux number - R Ua/v Reynolds number - St Stefan number     

Developing combined convection of non-Newtonian fluids in an eccentric annulus

Summary Laminar combined convection of non-Newtonian fluids in vertical eccentric annuli, in which the inner and outer walls are held at different constant temperatures is considered and a new economical method of solution for the three-dimensional flow in the annulus is developed. Assuming that the ratio of the radial to the vertical scale, , is small, as occurs frequently in many industrial applications, then the governing equations can be simplified by expanding all the variables in terms of . This simplification gives rise to the presence of a dominant cross-stream plane in which all the physical quantities change more rapidly than in the vertical direction. The solution trechnique consists of marching in the vertical streamwise direction using a finite-difference scheme and solving the resulting equations at each streamwise step by a novel technique incorporating the Finite Element Method. The process is continued until the velocity, pressure and temperature fields are fully developed, and results are presented for a range of the governing non-dimensional parameters, namely the Grashof, Prandtl, Reynolds and Bingham numbers.List of symbols Bn Bingham number, - d ^* difference between the radii of the outer and inner cylinders,r _o ^*–r_i ^* - e ^* distance between the axes of the inner and outer cylinders - e eccentricity,e ^*/d^* - F ^* external force acting on the fluid - g ^* acceleration due to gravity - g ^* gravitational vector, (0,0,g ^*) - Gr Grashof number, _m *² g**(T ₀*–T _e*)d*³/ _m *² - K ^* consistency of the fluid - L ^* height of the cylinders of the annulus - n flow behaviour index - p ^* dimensional pressure - P dimensionless pressure gradient - Pr Prandtl number, _m */ _m ** - r _i ^* radius of the inner cylinder of the annulus - r _o ^* radius of the outer cylinder of the annulus - r _T wall temperature difference ratio,(T _i ^*–T_e ^*)/(T_o ^*–T_e ^*) - Re Reynolds number, _m d*w _m */ _m * - T dimensionless temperature of the fluid,(T ^*–T_e ^*)/(T_o ^*–T_e ^*) - T _dif ^* temperature difference between the walls of the annulus - T _e ^* temperature at the fluid at the entrance of the annulus - T _i ^* temperature at the inner cylinder of the annulus - T _o ^* temperature at the outer cylinder of the annulus - u dimensionless transverse velocity in thex direction,u ^*/(w_m ^*) - U dimensionless transverse velocity in the annulus,Reu - u ^* fluid velocity vector, (u ^*, v^*, w^*) - v dimensionless transverse velocity in they direction,v ^*/(w_m ^*) - V dimensionless transverse velocity in the annulus,Rev - w dimensionless vertical velocity,w ^*/w_m ^* - w _m scaling used to non-dimensionalise the vertical velocity - x dimensionless transverse coordinate,x ^*/d^* - y dimensionless transverse coordinate,y ^*/d^* - z dimensionless vertical coordinate,z ^*/L^* - Z dimensionless vertical coordinate,z/Re - Z _r dimensionless distance in the vertical direction where the final wall temperatures are attained,Z _r ^*/L^* - * dimensional molecular thermal diffusivity - * coefficient of thermal expansion, - dimensional rate of strain tensor - dimensionless ratio of the length scales in the annulus,d ^*/L^* - * dimensional apparent non-Newtonian viscosity - _m * mean viscosity, - * dimensional fluid density - _m * dimensional reference fluid density - * dimensional stress tensor - yield stress