Completely hierarchical p-version curved shell element for laminated composite plates and shells

This paper presents a new completely hierarchical three dimensional curved shell finite element formulation for linear static analysis of laminated composite plates and shells. The element displacement approximation can be of arbitrary polynomial orders p , p and p in the , , and directions thereby permitting strains of at least (p –1), (p –1) and (p –1) order. The element approximation functions as well as the nodal variables are hierarchical. The element formulation ensures C ⁰ continuity. The lamina properties are incorporated by numerically integrating the element stiffness matrix for each lamina. The formulation has no restriction on either the number of laminas or the layup pattern of the laminas. The geometry of the laminated shell element is described by the coordinates of the nodes lying on the middle surface of the element and the lamina thicknesses at each node. The element formulation is equally effective for very thin as well as very thick laminated plates and curved shells. The results obtained from the present formulation are compared with those available in the literature as well as available analytical solutions.     

Nonlinear bending of unsymmetric angle-ply plates with edges elastically restrained against rotation

Summary Based on the von Kármán-type plate theory a solution is formulated for the large deflection of an unsymmetrically laminated angle-ply rectangular plate under transverse and inplane loads. Each pair of opposite edges is assumed to be elastically restrained against rotation to the same degree. The transverse deflection and force function in governing equations are expanded into generalize double Fourier series such that all required boundary conditions are exactly satisfied. Numerical results for maximum deflection, bending moment and inplane force in the plate are graphically presented for various aspect ratios, types of transverse load, high-modulus composite materials, numbers of layers, angles of orientation and boundary conditions. Present results for simple boundary conditions are in good agreement with previous data.Notations plate stiffnesses defined in [4], [9] - a, b, h length, width and thickness of plate - a _i ,b _i plate constants defined in [4] - E _L ,E _T ,G _LT ,v _itLT principal elastic constants of orthotropic material - F nondimensional force function defined in Eqs. (3) - F _mn ,Q _mn ,W _mn Fourier constants defined in Eqs. (9), (10) and (21) - H _i constant given by Eq. (19) - N ,M ,M nondimensional moments defined in Eqs. (6) - N ,N ,N nondimensional inplane forces defined in Eqs. (6) - nondimensional inplane load defined in Eqs. (8) - Q nondimensional transverse load defined in Eqs. (3) - q,q _o intensities of general and uniform transverse loads - R _m ,S _n ,X _m ,Y _n orthogonal functions of or defined by Eqs. (11) and (12) - r inplane load ratio defined in Eqs. (8) - x, y, z rectangular cartesian coordinates - W, w nondimensional deflection,w/h, and deflection - _m , _m , _m constant coefficients given by Eqs. (13)–(15) - , nondimensional coordinates defined in Eqs. (3) - aspect ratio (a/b) - ₁, ₂ rotational edge-restraint coefficients With 6 Figures     

Uniform suction/blowing effect on forced convection about a wedge: Uniform heat flux

Summary The effect of constant suction/blowing on steady two-dimensional laminar forced flow about a uniform heat flux wedge is numerically analyzed. The nonlinear boundary-layer equations were transformed and the resulting differential equations were solved by an implicit finite difference scheme (Keller box method). Numerical results for the velocity distribution, the temperature distribution, the local skin friction coefficient and the local Nusselt number are presented for various values of Prandtl number Pr, pressure gradient parameterm and suction/blowing parameter . In general, it has been found that the local skin friction coeffcient and the local Nusselt number increase owing to suction of fluid. This trend reversed for blowing of fluid. In addition to, as the blowing effect is strong enough, i.e. –0.65, the flow separation only occurred in the case ofm=0.0.Notation C Constant defined in Eq. (4.2) - C _f Local skin friction coefficient, 2v(u/y) _y=0/U ² - f Dimensionless stream function defined in Eq. (5.3) - g Gravitational acceleration - h Local heat transfer coefficient - k Thermal conductivity - m Pressure gradient parameter, /(2–) - Nu _x Local Nusselt number,hx/k - Pr Prandtl number,v/ - q _w Wall heat flux - Re _x Local Reynolds number,U x/v - T Temperature - T _w Wall temperature - T Temperature of ambient fluid - u Velocity component in thex-direction - U Potential flow velocity,Cx ^m - v Velocity component in they-direction - V _w Surface mass transfer - x Coordinate along the wedge surface - y Coordinate normal to the wedge surface - Thermal diffusivity - Angle factor of the wedge - Pseudosimilarity variable defined in Eq. (5.2) - Suction/blowing parameter defined in Eq. (5.1) - Total angle of the wedge - Dimensionless temperature defined in Eq. (5.4) - Kinematic viscosity - Stream function     

Underpinning of a simple blunt flaw fracture initiation relation

With the cohesive process zone representation of the micro-mechanistic processes that are associated with fracture as a basis, the author is involved in a wide-ranging research programme, the objective being to extend the fracture mechanics methodology for sharp cracks to blunt flaws, so as to take credit for the blunt flaw geometry. In earlier work, a Mode I fracture initiation relation has been derived, subject to the restriction that the process zone size s is small compared with the flaw depth (length) and any characteristic dimension other than the flaw root radius . The relation gives the critical elastic flaw-tip peak stress _pcr, and has been derived using a two-extremes procedure, whereby the separate, and indeed exact, solutions for small and large s/ values are blended together to give an all-embracing relation that is valid for all s/ values. _pcr is expressed in terms of the process zone material parameters and geometrical parameters but, for a wide range of flaw geometry parameters, _pcr essentially depends on only one geometrical parameter . This paper provides underpinning for the general thrust of the two-extremes procedure by appealing to exact results for the complete spectrum of s/ values from analyses of appropriate Mode III models. Results obtained by applying the two-extremes procedure are shown to be in very good agreement with the exact results.     

Polymorphism in Sr3Fe2O7−x

The compound Sr₃Fe₂O_7–x , with variable iron valence, was investigated by X-ray powder techniques, both at room and at high temperatures. If the material is examined in massive form, a single phase called -Sr₃Fe₂O_7–x appears as previously reported in the literature. This -phase is tetragonal and exhibits the lattice parameters: a=3.874 and c=40.314 Å. Two other phases, called and -Sr₃Fe₂O_7–x , respectively, can be obtained on heating the finely powdered material when laid on a flat platinum support. The form is stable up to 1275° C, while the form is revealed only above 1275° C and changes always into -Sr₃Fe₂O_7–x when quenched. Both and phases are tetragonal, with a=4.001 and c= 58.251 for the form and a=4.013, c=57.092 Å for the form. The transition involves a true phase equilibrium, while the transformation is possible only by means of a suitable mechanical treatment of the material.     

AC electrical conductivity of electron beam evaporated Cu-GeO2 thin cermet films

AC electrical properties of 410 nm think 30 at.wt% Cu-70 at.wt% GeO₂ thin films are reported for the frequency range 10⁴ to 10⁶ Hz and temperature range 150 to 425 K. The loss tangent (tan ) and the dielectric loss (/₀) are found to show striking minima around a cut-off frequency 10⁵ Hz. In the lower frequency range (10⁵ Hz), ₁() ^s T ⁿ is obeyed with s (0 to 0.51) increasing as a function of temperature and n (0.10 to 0.14) showing a very weak temperature dependence. In the higher frequency region (10⁵ Hz), ₁() and /₀ increase sharply leading to the quadratic behavior of ₁() with s equal to 2. These processes are discussed by analyzing an equivalent circuit which shows that at lower frequencies, the effects of series resistance in leads and contacts can be neglected, while at higher frequencies such effect give rise to spurious ² dependance for the conductance. A weakly activated AC conductivity and a frequency exponent s that increases with increasing temperature suggest that the low frequency behavior originates from carrier migration by tunneling process.     

Velocity of second sound and superfluid density near the superfluid transition in3He-4He mixtures

Using superleak condenser transducers, the velocity of second soundU ₂ has been measured near the superfluid transition temperature T in³He-⁴He mixtures with molar concentrationsX of³He of 0.0, 0.038, 0.122, 0.297, and 0.440. We have obtained the superfluid density _s/ fromU ₂ on the basis of linearized two-fluid hydrodynamics. The results for _s/ are consistent with those obtained from the oscillating disk method, as expected from two-fluid hydrodynamics. The value of _s/ at eachX could be expressed by a single power law, _s/=k, where =1-T/R, with the experimental uncertainty. It is found that the exponent is independent of concentration forX0.44 within the experimental uncertainty. This concentration independence of is in agreement with the universality concept. From the conclusion that the values of are universal forX0.44, the concentration dependence of the superfluid component _s is expressed by an empirical equation _s(X, )=²_s(0, ). It is found that corresponds to the volume fraction of⁴He in the superfluid³He-⁴He mixture. The value of is in agreement with that obtained from the measurement of the molar volume by others.This paper is based on a thesis submitted to Tokyo University of Education in partial fulfillment of the requirements for the Ph.D. degree.     

Singular Band Behavior of the Extended Emery Model for the Superconducting Cuprates

Mean field slave-boson approximation is performed on the extended Emery model for the CuO₂ conducting plane. The model is parameterized by Cu–O charge transfer energy _pd , copper–oxygen overlap t ₀, oxygen–oxygen overlap t', and Coulomb interaction U on the copper site taken as infinite. Special emphasis is placed on the role of t in the renormalization trends of the effective band parameters _pf and t, replacing _pd and t ₀, at small doping . It is shown that small, negative t expands the range of stability of the metallic phase, changing, in the second order of the perturbation theory, the nature of the metal–insulator transition point. In the nonperturbative limit, t modifies strongly the renormalization of _pf , making it saturate at the value of 4t. Finite doping suppresses the insulating state approximately symmetrically with respect to its sign. The regime _pf 4t fits very well the ARPES spectra of Y123, Bi2212, and LSCO and also explans, in the latter case, the evolution of the FS with doping accompanied by the spectral weight-transfer from the oxygen to the resonant band.     

Fatigue crack growth behaviour of tungsten carbide-cobalt hardmetals

Fatigue crack propagation studies have been carried out on a range of WC-Co hardmetals of varying cobalt content and grain size using a constant-stress intensity factor double torsion test specimen geometry. Results have confirmed the marked influence of mean stress (throughK _max), which is interpreted in terms of static modes of fracture occurring in conjunction with a true fatigue process, the existence of which can be rationalized through the absence of any frequency effect. Dramatic increases in fatigue crack growth rate are found asK _max approaches that value of stress intensity factor ( 0.9K_IC) for which static crack growth under monotonic load (or static fatigue) occurs in these materials. Lower crack growth rates, however, produce fractographic features indistinguishable from those resulting from fast fracture. These observations, and the important effect of increasing mean free path of the cobalt binder in reducing fatigue crack growth rate, can reasonably be explained through a consideration of the mechanism of fatigue crack advance through ligament rupture of the cobalt binder at the tip of a propagating crack.     

Coherent Brillouin Spectroscopy in Solid Parahydrogen

We applied coherent Brillouin spectroscopy to solid parahydrogen, and measured the Brillouin spectra of longitudinal acoustic modes at 5.6K. It was found that the linewidth of these spectra is 1.5MHz. From the observed Brillouin shift and the crystal orientation, the elastic stiffness was determined as C ₁₁=0.355±0.016GPa and C ₃₃=0.432±0.022GPa.     

Shear viscosity of liquid4He and3He-4He mixtures,especially near the superfluid transition

High-resolution measurements of are reported for liquid⁴He and³He-⁴He mixtures at saturated vapor pressures between 1.2 and 4.2 K with particular emphasis on the superfluid transition. Here is the mass density, the shear viscosity, and in the superfluid phase both and are the contributions from the normal component of the fluid ( _n and _n ). The experiments were performed with a torsional oscillator operating at 151 Hz. The mole fraction X of³He in the mixtures ranged from 0.03 to 0.65. New data for the total density and data for _n by various authors led to the calculation of . For⁴He, the results for are compared with published ones, both in the normal and superfluid phases, and also with predictions in the normal phase both over a broad range and close to T. The behavior of and of in mixtures if presented. The sloped/dT near T and its change at the superfluid transition are found to decrease with increasing³He concentration. Measurements at one temperature of versus pressure indicate a decreasing dependence of on molar volume asX(³He) increases. Comparison of at T, the minimum of _n in the superfluid phase and the temperature of this minimum is made with previous measurements. Thermal conductivity measurements in the mixtures, carried out simultaneously with those of , revealed no difference in the recorded superfluid transition, contrary to earlier work. In the appendices, we present data from new measurements of the total density for the same mixtures used in viscosity experiments. Furthermore, we discuss the data for _n determined for⁴He and for³He-⁴He mixtures, and which are used in the analysis of the data.     

hp-version finite elements for the space-time domain

A bilinear formulation of elasto-dynamics is offered which includes, as a special case, Hamilton's law of varying action. However, the more general bilinear formulation has several advantages over Hamilton's law. First, it admits a larger class of initial-value and boundary-value problems. Second, in its variational form, it offers physical insight into the so-called trailing terms of Hamilton's law. Third, numerical applications (i.e., finite elements in time) can be proven to be convergent under correct application of the bilinear formulation, whereas they can be demonstrated to diverge for specific problems under Hamilton's law. Fourth, the bilinear formulation offers automatic convergence of the natural velocity end conditions; while these must be constrained in present applications of Hamilton's law. Fifth, the bilinear formulation can be implemented in terms of a Larange multiplier that gives an order of magnitude improvement in the convergence of velocity. This implies that, in this form, the method is a hybrid finite-element approach.List of symbols b arbitrary constant - A _i, A _i vector of integrals, i = 0, j - A() linear operator on - () Hamilton's form of A - B (u, ) bilinear operator u, - B (u, ) Hamilton's form of B - B _i,j , B _ij , B _ij matrix of integrals - C constant, N/m - c number of floating-point operations per coef. evaluation - f, f(x) force per unit length, N/m - F, F ₀, F _L forces, N - J number of functions in series for û - k spring rate per unit length, N/m² - K spring rate, N/m - K _max maximum value of K - L _a Lagrangian, non-dimensional - L length of beam, m - m mass per unit length, kg/m - M mass, kg - M _max maximum value of M - n number of functions in series for - N number of elements in domain - p momentum density, kg/sec - P, P ₀, P _T momentum, kg-m/sec - q _i generalized coordinates - r _j coefficients of _j - t time, sec - t ₀, t ₁ limits of action integral, Hamilton's law - T end of time period, sec - u solution for displacement, m - û approximation to u, m - u ₀ initial value for u, m - test function, m - limited class of , m - x spatial coordinate, m - flapping angle, rad - Lock number - time increment, sec - Lagrange multiplier - longitudinal stiffness EA, N (Eqs. 1–18) - advance ratio of rotor (Eqs. 33–34 and figures) - _i , _r polynomial functions - non-dimensional time, azimuth angle - () variation of ( ) - W virtual work - ( ) d ( )/dx - ( .) d ( )/dt - (*) d/d - [ ] matrix - { } column vector - row vector     

Hydraulic resistance of rotating tubes

Parametric relations have been obtained, based on experimental data, for the hydraulic resistance coefficient of rotating tubes, allowing for the action of mass forces on the fluid flow.Notation d tube internal diameter - F mass force referred to unit volume - F^* mass force applied to a fluid particle - F excess mass force (difference between the mass forces of two characteristic points of the system) - K similarity parameter - j difference between the inertia accelerations at two points of the system - l characteristic dimension - M moment of momentum of a fluid particle - n number of revolutions of the tube in unit time - r radial coordinate, tube radius - Re Reynolds number, based on the mean mass velocity of the fluid motion - Re_circ Reynolds number based on the circular velocity of fluid motion - S similarity parameter - u circular velocity of the inner surface of the tube - v volume of a fluid particle - w mean mass velocity of the fluid - v kinematic viscosity - ₀ - coefficient of hydraulic resistance of a rotating tube - ₀ same for a fixed tube - fluid density - angular velocity of rotation of the tube     

Elastodynamic parameters for dynamic interface fracture mechanics

Summary Dynamic extension of cracks running along curvilinear interfaces of brittle bimaterials subjected to mechanical crack surface loads and superimposed thermal strains acting along the ligament is considered. This paper especially addresses the provision and discussion of elastodynamic interface parameters in order to assess quantitatively the bimaterial fracture in view of the governing physical features: applied mechanical and thermal strain loading, existence of an interface, crack-tip velocity and curvature of the interface contour. By utilizing the linear theory of thermoelasticity and adopting Stroh's method of generalized complex potentials, from the corresponding boundary and continuity conditions vectorial Hilbert problems are derived. It is shown that the parameters of the eigenvalues and of the eigenvectors of the Hilbert problems can be interpreted as elastodynamic interface mechanics parameters reading (,v _p, _Hf , _Hf ). Generalized Dundurs parameters of dynamics (, ) and consequently an associated generalized Dundurs diagram of dynamics are proposed. While the aforementioned elastodynamic interface parameters (, ,v _p, _Hf , _Hf ) do not assume the interface to be damaged, interfaces with running interface cracks generally cause two additional interface parameters, denoted as bimaterial constants (, _Hf ), where the latter is specific to the curvature of the interface in conjunction with the velocity of the interface crack. However, the bimaterial constants (, _Hf ) can be traced back to interface parameters for an uncracked bimaterial, namely to (, _Hf ).     

Submillimeter surface impedance and relaxation: A case study of quasi-two-dimensional YBa2Cu3Ox

Surface impedance measurements in the normal and superconducting state are an excellent method to study the conduction electron dynamics in metals. This holds especially in the relaxation range, i.e., for distances traveled in one r.f. periods= _F/(_f is the Fermi velocity) being smaller or of the order of the penetration depth and mean free pathl. For materials with _F<-10⁷ cm/sec the relaxation range is easily accessible forf>0.1 THz. Then, in the normal state, relaxation defines the surface impedance with a penetration depth approaching the London penetration depth _L, andR _0l/2 as surface resistance allowing a measure of _L and relaxation time (T, ). In the superconducting state the photon interaction scales with _F/_L=l/ ( _f is the dimension of Cooper pairs for l) and causes at low frequencies an absorption rate growing with, which is decreasing with _F/l. The rate increase proportional to turns to a decrease above 0.1 THz, being accompanied by a decrease of with frequency which is stronger for large and small _F/l. These characteristic dependences allow a measurement of material parameters, anisotropy, and dynamics of electrons. To evaluate the consequences of theâ, b, and-direction anisotropy, the integral kernels for normal and superconducting surface impedances in its nonintegrated, angle-dependent form are presented, analyzed, and compared with impedance measurements above 0.1 THz of YBa₂Cu₃O_x.     

Flow of viscoelastic fluids under high shearing stresses

Some characteristics of viscoelastic fluid flow under high shearing stresses are analyzed here.Notation _w tangential shearing stress at a channel wall - h half-height of a channel - L channel length - =w/¯w flow velocity, dimensionless - ¯w mean flow velocity at a given section - fluidity of viscoelastic fluid - P_xx-P_yy first difference of normal stresses, N/m² - =y/h distance from channel wall, dimensionless Translated from Inzhenerno-Fizicheskii Zhurnal, vol. 20, No. 5, pp. 918–920, May, 1971.     

Computation of particle settling speed and orientation distribution in suspensions of prolate spheroids

A numerical technique for the dynamical simulation of three-dimensional rigid particles in a Newtonian fluid is presented. The key idea is to satisfy the no-slip boundary condition on the particle surface by a localized force-density distribution in an otherwise force-free suspending fluid. The technique is used to model the sedimentation of prolate spheroids of aspect ratio b/a=5 at Reynolds number 03. For a periodic lattice of single spheroids, the ideas of Hasimoto are extended to obtain an estimate for the finite-size correction to the sedimentation velocity. For a system of several spheroids in periodic arrangement, a maximum of the settling speed is found at the effective volume fraction (b/a)²04, where is the solid-volume fraction. The occurence of a maximum of the settling speed is partially explained by the competition of two effects: (i) a change in the orientation distribution of the prolate spheroids whose major axes shift from a mostly horizontal orientation (corresponding to small sedimentation speeds) at small to a more uniform orientation at larger , and (ii) a monotonic decrease of the the settling speed with increasing solid-volume fraction similar to that predicted by the Richardson–Zaki law (1–)⁵⁵ for suspensions of spheres.     

Universality and the scaling laws in the superfluid phase transition of3He-4He mixtures

From the second-sound velocityU ₂ near the superfluid transition point, the superfluid densities in³He-⁴He mixtures, _s (X) and _s (), were deduced along the paths of constant³He concentrationX and of constant chemical potential difference of³He and⁴He. The following critical exponents of _s are determined: (a) =XX for _s (X) in the(X, T) plane,(b) X for _s (X) in the(, T) plane, and(c) for _s () in the(, T) plane. It is found that and _X change by about 4–6% relative to with increasing³He concentration up toX=0.4 and by 8–10% up toX=0.53. It seems that, belowX=0.53, universality hold for . Values of have been found to be in good agreement with the critical exponent of _s in pure⁴He under constant pressure. The values of and _X forX0.53 are also found to be consistent with the scaling relations in the (,T) plane of³He-⁴He mixture.Work performed in part while at the Electrotechnical Laboratory.     

Specific Heat of LaMnO3 + δ at 50 K ≤ T ≤ 160 K

We have developed a shell model, which includes the long-range coulomb, van der Waals interaction, and the short-range Hafemeister–Flygare repulsive interaction operative up to second neighbor atom to study the cohesive and thermal properties of LaMnO_{3 +} . The results on cohesive energy obtained by us are in good agreement with that of calculated value by DeSouza et. al. (R. A. DeSouza, M. S. Islam, and E. I. Tiffee, J. Mater. Chem. 9, 1621 (1999)). In addition, we have also calculated molecular force constant (f), compressibility (), restrahlen frequency (_o), Debye temperature (_D), and the low temperature specific heat at 50 K T 160 K. Our results on Debye temperature and specific heat for the temperature range 50 K T 160 K are closer to the recently measured data with an automated quasi-adiabatic pulse technique.     

A flat jet in a granular bed of finite height

The distribution of gas flows in the vicinity of the jet is discussed and the conditions of disruption of the static equilibrium of the bed, the formation and growth of a cavity, and the jet breakthrough of the bed are investigated qualitatively.Notation a, b functions calculated in [11] - C, C constants in (7) - F derivative of the complex potential - f function in (6) - G function defined in (19) - H dimensionless height of bed - h height of cavity - k coefficient introduced in (15) - p, po pressure inside bed and in cavity - p dimensionless pressure drop - Q, q dimensional and dimensionless jet flow rates - q₁, q₂ critical values - T dimensionless height of cavity - T₀, T₁ T₁, T₂ characteristic values of T - u,v filtration velocities - u, u_* initial filtration velocity in the bed and minimum fluidization velocity - u_o velocity scale introduced in (14) - u * velocity scale introduced in (14) - u* velocity of fictitious flow defined in (15) - U complex velocity - Z=X+iY, z=x+iy dimensionless coordinates - z=x+iy dimensional coordinates - coefficient of hydraulic resistance - parameter from (5) - specific weight of particles' material - porosity - =+i coordinates in the plane obtained from z=x+iy as a result a of conformai transformation - _m value of giving a minimum of the function G - f complex and real flow potentials - angle of internal friction - stream function - angle of inclination of boundaries of the region of plastic flow to the vertical Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 5, pp. 804–812, November, 1979.