Die Herleitung von Schalengleichungen über ein erweitertes Variationsprinzip bei Berücksichtigung der Querschubverzerrungen

Ohne ZusammenfassungBezeichnungen L Bezugsgrößen für dimensionslose Koordinaten - L charakteristische Schalenabmessung - t Schalendicke - Schalenparameter - körperfeste, krummlinige, dimensionslose Koordinaten der Schalenmittelfläche - Dimensionslose Koordinate in Richtung der Schalennormalen - i, j,...=1,2,3 Indizierung des dreidimensionalen Euklidischen Raumes - ,,...=1,2 Indizierung des zweidimensionalen Riemannschen Raumes - (...), Partielle Differentiation nach der Koordinate - (...), Kovariante Differentiation für Tensorkomponenten des zweidimensionalen Raumes nach der Koordinate - (...)| Kovariante Differentiation für Tensorkomponenten des dreidimensionalen Raumes nach der Koordinate - Variationssymbol - a ,a ₃ Basisvektoren der Schalenmittelfläche - V Verschiebungsvektor - U ,U ₃ Verschiebungskomponenten des Schalenraumes - v ,w,w ,W Verschiebungskomponenten der Schalenmittelfläche - Verhältnis der Metriktensoren des Schalenraumes und der Schalenmittelfläche - _ik Verzerrungstensor des Raumes - _{(, )}, Symmetrische Verzerrungstensoren der Schalenmittelfläche - _{[, ]} Antimetrischer Term des Verzerrungsmaßes - ^, Spannungstensor - n ,m ,q Tensorkomponenten der Schnittgrößenvektoren - p ,p,c Tensorielle Lastkomponenten     

Superconducting Gap in the Multi-band System MgB2

Based on the angle-resolved photoemission spectra of single crystals, it is demonstrated that a couple of bands cross the Fermi energy in MgB₂, which is in good agreement with band theory. The superconducting gap in this multiband system is carefully examined by Raman scattering spectroscopy with various polarizations. It has been revealed that the large gap (24k _B T _c) that is typical for a clean limit s-wave superconductor is restricted to the -bands, while the gap on the -bands is much smaller (21.1k _B T _c) and strongly affected by the impurity scattering, which gives a dirty limit behavior. This unusual two-gap behavior might be caused by the lack of interband scattering due to special separation of the - and -bands, as predicted by Mazin et al.     

Broadband Frequency Study of the Zero Sound Attenuation Near the Quantum Limit in Normal Liquid 3He Close to the Superfluid Transition

The zero sound attenuation, ₀(,T, P), of normal liquid ³ He has been studied over a broad range of frequency (/2 = 8 – 50 MHz). Data has been collected at a constant temperature (T 1.1 mK) which is just above the superfluid transition temperature, T _c , when the liquid is near a pressure, P, of 1 bar. The results are compared to Landau's prediction in the quantum limit, k _B T k _B T _F , where ₀(,T,P) = (P) T ²[l + (/2k _B T)²]. Deviations from Landau's prediction are compared to the results of other workers and are discussed with respect to additional (unidentified) extrinsic background effects and (possible) intrinsic scattering mechanisms due to fluctuations in the liquid.     

Theoretical analysis of the basic parameters of ultrasonic flowmeters

Conclusions A comparison of the final results of the two methods shows that the analytical expressions for the interchannel phase difference and repetition frequency difference F, and in the case of pulsed circuits also of the pulse propagation time difference , differ in the two methods by their corrections B and B.Errors in using the analytical expressions obtained by the approximate method do not exceed 1% when the hydrodynamic correction B for b_n0.1 is applied. If this value is taken as the basic error, the application of the approximate method is justifiable for pipe diameters exceeding 200 mm with piezoelectric element radii of r10 mm, and for pipe diameters exceeding 100 mm with element radii of r5 mm. Therefore, the application of the approximate method is limited in the main to the range of pipes with large and to a certain extent medium diameters. For the greater part of pipes with medium diameters and those with small diameters, it is necessary to use the analytical expressions obtained by the rigorous method. The use of these equations is also required for pipes with large diameters in designing high-precision flowmeters.     

Flow of a generalized second grade fluid between heated plates

Summary We examine the fully developed flow of a generalized fluid of second grade between heated parallel plates, due to a pressure gradient along the plate. The constant coefficient of shear viscosity of a fluid of second grade is replaced by a shear dependent viscosity with an exponentm. If the normal stress coefficients are set equal to zero, this model reduces to the standard power-law model. We obtain the solution for the case when the temperature changes only in the direction normal to the plates for the two most commonly used viscosity models, i.e. (i) when the viscosity does not depend on temperature, and (ii) when the viscosity is an exponentially decaying function of temperature.

List of symbols

Alphanumeric A ₁,A ₂ Kinematical tensor - b Body force - C Dimensionless parameter related to the pressure gradient - h Separation between the plates - L Velocity gradient - m Power-law index - M Constant appearing in the Reynolds viscosity model - p Pressure field - Modified pressure field - q Heat flux vector - r Radiant heating - T Cauchy's stress tensor - l Unit tensor - v Velocity vector - V Characteristic velocity - x Axis along the plate - y Axis perpendicular to the plate Greek ₁, ₂ Normal stress coefficient - Specific internal energy - Dimensionless parameter related to the viscous dissipation - Conservative body force field - Specific entropy - Thermal conductivity - Coefficient of viscosity - ₀ Reference viscosity - Second invariant of the stretching tensor - Temperature - ₁ Temperature of the lower plate - ₂ Temperature of the upper plate - Density - Specific Helmholtz free energy Operators div Divergence - grad Gradient - tr Trace     

Magnetic field dependence of zero-sound attenuation close to the pair-breaking edge in3He-B due toJ=1−,J z =±1 collective modes

Our previous theory yielded for the Zeeman splitting of the imaginaryJ=1 collective mode in³He-B the result =2+0.25J _z ( is the effective Larmor frequency). In this paper we take into account the downward shift of the pair-breaking edge from 2 to 2₂– (₂ and ₁ are the longitudinal and transverse gap parameters). This leads to a complex Landé factor: the frequencies of theJ _z =±1 components become =2+0.39J _z , and the linewidths of these resonances become finite: =0.18. The coupling amplitudes of theJ _z =±1 components to density are found to be proportional to gap distortion, (₁–₂/(/)². Our results for the ultrasonic attenuation due to theJ _z =±1,J=1 modes are capable of explaining the field dependence of the attenuation close to the pair-breaking edge as observed by Dobbs, Saunders, et al. The observed peak is caused by theJ _z =–1 component: its height increases due to gap distortion as the field is increased, and the peak shifts downward in temperature and its width increases with the field due to the complex Landé factor. TheJ _z =+1 component gives rise to a corresponding dip relative to the continuum attenuation.     

Theory of RF SQUIDs Operating in the Presence of Large Thermal Fluctuations

A comprehensive analytical theory is presented for non-hysteretic RF SQUIDs operating in the adiabatic mode in the presence of large thermal fluctuations. When 1 ( = 2LI_c/₀ is the hysteresis parameter, L is the SQUID inductance, I_c is the critical current of the Josephson junction, and ₀ is the flux quantum) the theory is applicable also for RF SQUIDs operating in the non-adiabatic mode. In contrast to previous theories in which the noise is treated perturbatively and which therefore are applicable only if the product 1 ( = 2k_BT/ ₀ I_c is the noise parameter, k_B is the Boltzmann constant, and T is the absolute temperature)—the case of small thermal fluctuations—the present theory is valid for around unity or higher. In the limit 0 the theory reproduces the results of small thermal fluctuations theories. It has been found that in the presence of large thermal fluctuations the screening current in the SQUID inductance is suppressed by a factor that increases with increasing . Taking into account this new basic fact, all SQUID characteristics (output signal, transfer function, noise spectral density and energy sensitivity) have been recalculated and a good agreement with experimental data has been obtained. It has been also found that RF SQUIDs can be operated with substantially higher values of the inductance and of the noise parameter than DC SQUIDs. These two aspects, which are of particular importance at liquid nitrogen temperature, make high T_c RF SQUIDs very attractive.     

On the measurement and physical meaning of the cleavage fracture stress in steel

Elastic-plastic two-dimensional (2D) and three-dimensional (3D) finite element models (FEM) are used to analyze the stress distributions ahead of notches of four-point bending (4PB) and three-point bending (3PB) specimens with various sizes of a C-Mn steel. By accurately measuring the location of the cleavage initiation sites, the local cleavage fracture stress _f and the macroscopic cleavage fracture stress _F is accurately measured. The _f and _F measured by 2D FEM are higher than that by 3D FEM. _f values are lower than the _F, and the _f values could be predicted by _f=(0.8––1.0)_F. With increasing specimen sizes (W,B and a) and specimen widths (B) and changing loading methods (4PB and 3PB), the fracture load P _f changes considerably, but the _F and _f remain nearly constant. The stable lower boundary _F and _f values could be obtained by using notched specimens with sizes larger than the Griffiths–Owen specimen. The local cleavage fracture stress _f could be accurately used in the analysis of fracture micromechanism, and to characterize intrinsic toughness of steel. The macroscopic cleavage fracture stress _F is suggested to be a potential engineering parameter which can be used to assess fracture toughness of steel and to design engineering structure.     

Thermal diffusivity of inhomogeneous systems

The possibility of analyzing the nonsteady temperature fields of inhomogeneous systems using the quasi-homogeneous-body model is investigated.Notation t, t_I, t_i temperature of quasi-homogeneous body inhomogeneous system, and i-th component of system - a, , c thermal diffusivity and conductivity and volume specific heat of quasi-homogeneous body - a_{i i}, c_i same quantities for the i-th component - q heat flux - S, V system surface and volume - x, y coordinates - macrodimension of system - dimensionless temperature Fo=a/² - Bi=/ Fourier and Biot numbers - N number of plates - =h/ ratio of micro- and macrodimensions - _V, volumeaveraged and mean-square error of dimensionless-temperature determination - time - m_i i-th component concentration Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 126–133, July, 1980.     

Analytical Theory of DC SQUIDS Operating in the Presence of Thermal Fluctuations

A comprehensive analytical theory of symmetric DC SQUIDs is presented taking into account the effects of thermal fluctuations. The SQUID has a reduced inductance < 1/ where = 2LI_c/₀, L is the loop inductance, ₀ is the flux quantum, and I_c is the critical current of the identical Josephson junctions which are assumed to be overdamped. The analysis, based on the two dimensional Fokker–Planck equation, has been successfully performed in first order approximation with considered a small parameter. All important SQUID characteristics (circulating current, current-voltage curves, transfer function, and energy sensitivity) are obtained. In the limit 1( = 2k_BT/I_c0 is the noise parameter, k_B is the Boltzmann constant, and T is the absolute temperature) the theory reproduces the results of numerical simulations performed for the case of small thermal fluctuations. It was found that for < 1 the SQUID energy sensitivity is optimum when is higher than 1/, i.e., outside the range for which the present analysis is valid. However, for 1 the energy sensitivity has a minimum at L = L_F , where L_F = ( ₀ /2) ²/k_B , and therefore, in this case, the optimal reduced DC SQUID inductance is _opt = 1/, i.e., within the range for which the present analysis is valid. In contrast to the case of an RF SQUID, for a DC SQUID the transfer function decreases not only with increasing L/L_F but also with increasing (as 1/). As a consequence, the energy sensitivity of a DC SQUID with < 1/ degrades more rapidly (as ⁴ ) with the increase of than that of an RF SQUID does (as ² ).     

Determination of the thermophysical characteristics of alternatively freezing and thawing dispersed media by the method of solving inverse problems of heat conduction

The article explains an algorithm for determining the thermophysical characteristics of dispersed media with phase transitions based on the method of solving inverse problems of heat conduction.Notation r space coordinate - time - T temperature of the specimen - T₀ initial temperature - c_i, c_w, c_sk specific heat of ice, water, and of the organic-mineral skeleton, respectively - c_f, c_m, _f, _m specific heat and thermal conductivity in the frozen and melted zones, respectively - c effective heat capacity - thermal conductivity - p density - ₀, _sb bound and strongly bound moisture, respectively - (T) amount of nonfrozen water - R radius of the cylinder - q() heat flux - I functional - u₁(), U₂() measured temperatures of the specimen at the points r = 0 and r = R, respectively, at the instant - ₁, ₂ degree of confidence of the supplementary information - final instant of time - a, b, k, _s positive constants - L specific heat of melting - N number of grid nodes over space - n number of grid nodes over time - h grid step over space - grid step over time - solution of the conjugate system - s number of iteration Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 2, pp. 292–297, August, 1980.     

On the analysis of tunneling data for A15 superconductors

We examine the applicability of the standard McMillan inversion of Eliashberg's equations for superconductors with a nonconstant electronic density of statesN(). We do this usign simple models forN() and a realistically shapedN() taken from recent band structure work for Nb₃Sn. It turns out that peak structure inN() near _F may lead to gross errors in the derived Eliashberg function ²F() when the energy dependence ofN() is omitted in the inversion procedure. For Nb₃Sn, this leads to a 40% overestimate of when ²F() is evaluated via the standard McMillan program.     

A superconducting vibrating reed applied to flux-line pinning. I. Theory

A theoretical treatment is given of a superconducting reed clamped at one end and performing flexural vibrations in a homogeneous longitudinal magnetic fieldB _a. When the flux lines are rigidly pinned the reed behaves like an ideal diamagnet whose bending distorts the external field. This generates a magnetic restoring force (line tension) B _a ² which is independent of the reed thicknessd, whereas the mechanical restoring force (stiffness) is d ³. Therefore, the resonance frequency /2 of a thin superconducting reed increases drastically when a fieldB _a is applied, or for a givenB _a, when the reed is cooled below its critical temperatureT _c. With decreasing pinning strength (characterized by Labusch's parameter ) the resonance frequency decreases, ²²– _pin ² where _pin ^{2 –1}, and an attenuation _v ^–2 occurs due to the viscous motion of flux lines. For larger vibration amplitudes an additional, amplitude-dependent damping _h ^–3 occurs due to the hysteretic losses caused by elastic instabilities during flux motion.On leave from Centro Atómico, Bariloche, Argentina.     

Specific heat near the lambda point in 4He and 3He-4He mixtures: Test of universality of the critical exponent and the amplitude ratio,and observation of the critical-tricritical crossover effect

The specific heat under saturated vapor pressure of pure ⁴He and of six ³He-⁴He mixtures up to X = 0.545 was measured in the temperature range 3 × 10^–6 <¦T-T ¦ <10^–2 K. The critical exponents and along the path = are independent of X up to X = 0.545, where (= ₃ – ₄) is the difference between chemical potentials. If we take account of higher order terms, the exponent (= ) and the amplitude ratio A /A are independent of X up to X = 0.545. The values of and A /A are –0.023 and 1.090, respectively. The critical-tricritical crossover effect was observed for X = 0.545 and the boundary of crossover region closest to the critical region was at /T = (1–2) × 10^–4, where is the distance ¦T – T ¦ along the path = . This value is in good agreement with the estimated value by Riedel et al. But, remarkably, in the case of X = 0.439 this effect was not observed.     

A numerical method for flexible airfoils in incompressible inviscid flow

Summary The paper deals with the aerodynamic analysis of flexible airfoils, based on a quasi-lattice vortex method (QVLM). The analysis is formulated in matrix form and leads, as in other similar studies, to a linear algebraic system when the angle of attack is nonzero, and to an eigenvalue problem when the incidence angle is zero. The aerodynamic characteristic curvesC _L -,C _m - are presented. Finally, the airfoil shapes for several values of the tension coefficient and angles of attack are drawn. The results obtained with the present method are in good agreement with those reported in previous studies and evidentiate the flexibility of the QVLM as applied to flexible airfoils.Notation A aerodynamic matrix, defined in QVL method, (8) - B matrix, see Eq. (18) - c chord of airfoil - C matrix defined asAB - C _L lift coefficient, 2L/V ² c) - C _p moment coefficient, 2M/(V ² c ²) - C _p pressure coefficient,C _p =2p/(V ² ) - C _T tension coefficient, 2T/(V ² c) - D matrix, see Eq. (11) - I unit matrix - l curvilinear length of the flexible airfoil - N number of collocation points on the airfoil shape - q dynamic pressure, V ² /2 - T tension force in the sail - V freestream velocity - w downwash - x nondimensional coordinate,x/c - X _i control points, Eq. (9) - X _max dimensionless position of the maximum camber - Y _k source points, Eq. (9) - z coordinate normal tox axis - Z nondimensional coordinate,z/c - Z _s camber equation in dimensionless form,z _s /c - incidence with respect to the upstream flow velocity - column vector of the local curvatures {₁, ₂,..., _N } ^T - nondimensional membrane excess ratio - eigenvalue of the problem (23) - _k zeroes of the Chebyshev polynomia of the first kind, 1kN - column vector of the local slopes, {₀, ₁, ₂,..., _N } ^T - column vector, {₁, ₂,..., _N } ^T - ₀ slope at airfoil leading edge     

Thermal conductivity of hydrocarbons in the naphthene group under high pressures

The thermal conductivity of hydrocarbons in the naphthene group has been experimentally determined. An equation is now proposed for calculating the thermal conductivity over the given temperature and pressure ranges.Notation thermal conductivity - ₂₀ and ₃₀ values of the thermal conductivity at 20 and 30°C, respectively - t₀,P₀ thermal conductivity at t₀, p₀ - _t p thermal conductivity at temperature t and under pressure P - change in thermal conductivity - P pressure - P_melt melting pressure - P₀ atmospheric pressure - t₀ 20°C temperature - T, t temperature - T_cr critical temperature - temperature coefficient of thermal conductivity - ₂₀ temperature coefficient of density - density - ₂₀ density at 20°C - _cr critical density - M molar mass - =T/T_cr referred temperature - v specific volume - v₀ specific volume at 20°C - v change in specific volume - _{3 0} a coefficient - B (t) a function of the temperature - S a quadratic functional - W_i, weight of the i-th experimental point - _i error of the i-th experimental value of thermal conductivity - B _{y, =0.6} value of B (t) at T = 0.6T_cr - B = B (t)/B_{, =0.6} referred value of coefficient B (t) Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 3, pp. 491–499, September, 1981.     

Interface shapes in a torsionally oscillating,layered medium of viscoelastic liquids

Summary The free surface motion of a layered medium of liquids in a gravitationally stable configuration, resting on top of a layer of mercury, driven by a torsionally oscillating, cylindrical outer wall is investigated. The non-linear problem in the unknown physical domain is expressed as a series of linear problems in the rest state by means of a domain perturbation method. The flow variables and the stress are expanded into series in terms of the amplitude of the oscillation of the cylinder. The shapes in the mean of the interfaces between layers and the flow field are determined up to second order in the perturbation parameter, the amplitude of the oscillation.Nomenclature Density - Modified pressure field - Amplitude of the oscillation - Frequency of the oscillation - Interfacial value of the surface tension - Dynamic viscosity - , , Material functions - Complex viscosity - Stream function - Position vector at timet= - ₁, ₂ The first two Rivlin-Ericksen constants - Quadratic shear relaxation modulus - ,t Time - u Velocity vector - u,v,w Velocity components - S Extra stress tensor - h Interface elevation - D Stretching tensor - G Strain history tensor - A ₁ The first Rivlin-Ericksen tensor - J Mean curvature - p Pressure - t Unit tangent vector - n Unit normal vector - G Shear relaxation modulus - X Position vector in the rest stateD ₀ - r, ,z Rest state coordinates - x Position vector in the physical spaceD - R, ,Z Physical space coordinates - r ₀ Radius of the oscillating cylinder - e _r ,e ,e _z Physical basis vectors inD ₀ - e _R ,e ,e _Z Physical basis vectors inD - Indicates the jump in the enclosed quantity across an interface With 1 FigurePresented at the Xth Canadian Congress of Applied Mechanics, The University of Western Ontario, London, Ontario, Canada, June 2–7, 1985.     

Heat transfer in a “pseudoturbulent” bed of dispersed material

A two-phase model is proposed for the steady heat exchange between a surface and a pseudoturbulent bed of dispersed material. Expressions are obtained for the temperature fields of the gaseous and solid phases.Notation _g effective thermal conductivity of gaseous phase - _s effective thermal conductivity of the mixed solid phase - porosity - _m molecular thermal conductivity - d particle diameter - temperature of dispersed bed at a large distance from heat source - , g gas temperature - _p particle temperature - _w wall temperature - x current coordinate in the direction perpendicular to the wall - l bed thickness - q heat flux - coefficient of heat exchange between wall and pseudoturbulent bed of dispersed material - ^* coefficient of interphase heat exchange - _g=_g/_w dimensionless gas temperature - _p = _p/_w dimensionless particle temperature - Y = x/d dimensionless coordinate - L =l/d dimensionless bed thickness - A_h dimensionless coefficient of interphase heat exchange - Nu_g = d/_s Nusselt number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 3, pp. 465–469, September, 1981.     

Stability of the nonequilibrium state of a superconductor irradiated by an electromagnetic field

Irradiation of a superconductor by an electromagnetic field with a frequency ₀ larger than twice the energy gap (order parameter) decreases the order parameter . We obtain the quasiparticle distribution function n and the dependence of the order parameter on the power of the electromagnetic field P by solving numerically the kinetic equations for n and in the steady state. We take ₀/₀ = 2.1, 8, and 20, where ₀ is the equilibrium value of the order parameter at T = 0 K. In the examples considered the dependence of on the pumping power P becomes double-valued above a critical power. We allow phonons also to be out of thermal equilibrium. To discuss the stability of the steady state thus obtained, we derive kinetic equations for small deviations of the quasiparticle distribution function and the order parameter from the steady state n and by means of the nonequilibrium Green's function theory. Assuming n, exp(i kr – t), (k) is computed using n _s/, where n _sis the steadystate quasiparticle distribution function for arbitrary . It is concluded in general that the steady state on the upper branch ( > _{c 2}) is stable with respect to both spatially homogeneous and inhomogeneous fluctuations, and the lower branch ( > _{c 2}) is unstable; _{c 2}is the value where the upper and lower branches of (P) coalesce.This work is partly supported by a Grant-in-Aid for Special Project Research from the Ministry of Education, Science and Culture of Japan.     

Viscoelastic and rheological behavior of concentrated colloidal suspensions

Molecular approaches are discussed to the density (), viscoeleastic (), and rheological () behavior of the viscosity(,,) of concentrated colloidal suspensions with 0.3 < < 0.6, where, is the volume fraction, the applied frequency, and ; the shear rate. These theories are based on the calculation of the pair distribution functionP ₂(r,,), wherer is the relative position of a pair of colloidal particles. The linear viscoelastic behavior(,,=0) follows from an equation forP ₂(r,,) derived from the Smoluchowski equation for small, generalized to large by introducing the spatial ordering and (cage) diffusion typical for concentrated suspensions. The rheological behavior(,,=0) follows from an equation forP ₂(r,) of a dense hard-sphere fluid derived from the Liouville equation. This leads to a hard-sphere viscosity^hs(,) which yields the colloidal one(,) by the scaling relation(,) ₀=^hs(,) _B, where ₀ is the solvent viscosity. _B is the dilute hard-sphere (Boltzmann ) viscosity and the's are appropriately scaled,(,) and(,) agree well with experiment. A unified theore for(,,) is clearly needed and pursued.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994. Boulder, Colorado, U.S.A.