Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

A solution of the E-ε model for nearly homogeneous turbulence with a mean shear

An analytical solution of the E-ε model for the downstream evolution of a stationary and nearly homogeneous turbulent shear flow is presented. In case that the turbulent time scale has adjusted itself to the time scale imposed by the shear, an asymptotic solution can be derived from the full solution, which shows that both E and ε increase downstream exponentially. By comparing this asymptotic solution with experimental data a value for the unknown constant c _lε , in the ε-equation, is derived. Moreover, we find an expression for the downstream development of the variance of a scalar, which is also compared with experimental data. The analytical solution shows that a homogeneous shear flow with a uniform velocity gradient can only be obtained if the shear is sufficiently small. In the experiments this condition is not always satisfied. A discussion is given on how a nearly homogeneous shear flow can be obtained over a limited downstream interval by changing the initial conditions in E and ε, and a comparison is made with experimental data. Finally it is shown that better transverse homogeneity can be obtained by taking an exponential velocity profile instead of a linear profile.     

Effect of novel synthetic jet on wake vortex shedding modes of a circular cylinder

A novel actuator signal achieved by changing the ratio of the suction duty cycle to the blowing duty cycle is adopted to enhance the control effect of the synthetic jet for the flow around a circular cylinder. The suction duty cycle factor k defined as the ratio between the time duration of the suction cycle and the blowing cycle and the equivalent momentum coefficient C_μ are introduced as the determining parameters. The synthetic jet is positioned at the rear stagnation point in order to introduce symmetric perturbations upon the flow field. The proper orthogonal decomposition (POD) technique is applied for the analysis of the spanwise vorticity field. Increasing the suction duty cycle factor, the momentum coefficient is enhanced, and thus a stronger and larger scale synthetic jet vortex pair with a higher convection velocity is generated. The synthetic jet vortex pair interacts with the spanwise vorticity shear layers behind both sides of the cylinder, resulting in the variations of the wake vortex shedding modes at Re=950: for k=0.25, C_μ=0.148, vortex synchronization at the subharmonic excitation frequency with antisymmetric shedding mode; for 0.50≤k≤1.00, 0.213≤C_μ≤0.378, vortex synchronization at the excitation frequency with the symmetric or antisymmetric shedding modes; for 2.00≤k≤4.00, 0.850≤C_μ≤2.362, vortex synchronization at the excitation frequency with symmetric shedding mode. Hence, the control effect of the synthetic jet upon the wake vortex of a circular cylinder can be enhanced by increasing the suction duty cycle factor so as to increase the momentum coefficient. This is also validated at a higher Reynolds number Re=1600.     

Nanofluids thermal behavior analysis using a new dispersion model along with single-phase

A numerical study has been performed to analyze nanofluids convective heat transfer. Laminar α-Al₂O₃-water nanofluid flows in an entrance region of a horizontal circular tube with constant surface temperature. Numerical analysis has been carried out using two different single-phase models (homogenous and dispersion) and two-phase models (Eulerian–Lagrangian and mixture). A new model is developed to consider the nanoparticles dispersion. The transport equations for the tube with constant surface temperature were solved numerically using a control volume approach. The effects of nanoparticles volume fraction (0.5, 1 %) and Reynolds number (650 ≤ Re ≤ 2300) on nanofluid convective heat transfer coefficient were studied. The results are compared with the experimental data and it is shown that the homogenous single-phase model is underestimated and the mixture model is overestimated. Although the Eulerian–Lagrangian model gives a reasonable prediction for the thermal behavior of nanofluids, the dispersion single-phase model gives more accurate prediction despite its simplicity.     

Entrance flow of a bingham fluid in a tube

The numerical solution of the entrance flow in a tube has been obtained for a Bingham fluid. The numerical procedure used is that of Patankar and Spalding [1]. The accuracy of the numerical results is demonstrated by comparing the fully-developed velocity profiles with analytical exact solutions. The results of the entrance flow in a tube for the case of a zero yield stress are compared with the entrance flow solution for a Newtonian fluid. Detailed results are presented for a wide range of yield numbers (=τ _y D/ūμ).     

Analytical solutions for finite cylinders compressed between platens and the strain effect on the valence-band structure of Si1−xGex alloy

This paper presents an analytical solution for inhomogeneous strain and stress distributions within finite circular cylinders of Si_1−xGe_x alloy under compression test with end friction. The method follows Lekhnitskii’s stress function approach, but a new expression for the stress function is proposed so that all of the governing equations and boundary conditions are satisfied exactly. Numerical results show that the axial, radial, circumferential and shear strains are all inhomogeneous within finite cylinders, and local strain concentrations near two end surfaces were usually developed as long as friction exists between end surfaces and loading platens. Moreover, by using envelope-function method, the effect of strain on the valence-band structure of Si_1−xGe_x alloy is also studied. It was found that strain can induce band splitting, alteration of the shape of constant energy surfaces of the heavy-hole and the light-hole bands of Si_1−xGe_x alloy.     

On one skew-symmetric problem of electroelasticity for inhomogeneous cylinder of finite length

In the present paper, static bending problem of the electroelasticity for an inhomogeneous cylinder of finite length with sliding fixed end-supports is investigated. The given boundary value problem is reduced to a system of 12 k (k = 1, 2, …) integro-differential equations. Expressions for the components characterizing the state of stress for the inhomogeneous cylinder are presented. Based on the developed analytical algorithm, extensive numerical investigations associated with the stress analysis of an inhomogeneous piezoceramic cylinder have been conducted. The results of these investigations are illustrated graphically, demonstrating the stress distributions in piezoceramic circular and elliptical cylinders with inclusions of various geometries.     

Reconstruction of the three mechanical material constants of a lossy fluid-like cylinder from low-frequency scattered acoustic fieldsReconstruction des trois constants mécaniques matériels d'un cylindre fluide dissipatif à partir de champs acoustiques basses fréquences

The inverse medium problem for a circular cylindrical domain is studied using low-frequency acoustic waves as the probe radiation. To second order in k₀a (k₀ the wavenumber in the host medium, a the radius of the cylinder), only the first three terms (i.e., of orders 0, ?1 and +1) in the partial wave representation of the scattered field are non-vanishing. This enables the scattered field to be expressed algebraically in terms of the unknown material constants, i.e., the density ρ₁, and the real and imaginary parts of complex compressibility κ₁ of the cylinder. It is shown that these relations can be inverted to yield explicit, decoupled expressions for ρ₁ and κ₁ in terms of the totality of the far-zone scattered field. These expressions furnish accurate estimations of the material parameters provided the probe frequency is low and the radius of the cylinder is known very precisely. To cite this article: T. Scotti, A. Wirgin, C. R. Mecanique 332 (2004).     

Application of the γ-Reθ Transition Model to Simulations of the Flow Past a Circular Cylinder

Based on the finite volume method, the flow past a two-dimensional circular cylinder at a critical Reynolds number (Re = 8.5 × 10⁵) was simulated using the Navier-Stokes equations and the γ-Re_θ transition model coupled with the SST k ? ω turbulence model (hereinafter abbreviated as γ-Re_θ model). Considering the effect of free-stream turbulence intensity decay, the SST k ? ω turbulence model was modified according to the ambient source term method proposed by Spalart and Rumsey, and then the modified SST k ? ω turbulence model is coupled with the γ-Re_θ transition model (hereinafter abbreviated as γ-Re_θ-SR model). The flow past a circular cylinder at different inlet turbulence intensities were simulated by the γ-Re_θ-SR model. At last, the flow past a circular cylinder at subcritical, critical and supercritical Reynolds numbers were each simulated by the γ-Re_θ-SR model, and the three flow states were analyzed. It was found that compared with the SST k ? ω turbulence model, the γ-Re_θ model could simulate the transition of laminar to turbulent, resulting in better consistency with experimental result. Compared with the γ-Re_θ model, for relatively high inlet turbulence intensities, the γ-Re_θ-SR model could better simulate the flow past a circular cylinder; however the improvement almost diminished for relatively low inlet turbulence intensities The γ-Re_θ-SR model could well simulate the flow past a circular cylinder at subcritical, critical and supercritical Reynolds numbers.     

Combined convection and wall conduction effects in laminar pipe flow: numerical predictions and experimental validation under uniform wall heating

Laminar combined convection in horizontal circular ducts is investigated both numerically and experimentally, under uniform wall heating. A series of experiments for the heating of water in a long horizontal copper tube are simulated numerically in order to assess the reliability of the theoretical results. Peripheral and axial wall conduction effects, inherently present in the experiments, are accounted for in the numerical model. The cross validation of experimental and numerical data allows significant conclusions to be reached on conjugate conduction and convection with buoyancy effects in horizontal duct flows. Buoyancy is considered for values of the modified Rayleigh number,Ra _qo , up to 5·10⁶; the forced convection contribution is considered for two values of the entry Reynolds number,Re _o=500 and 1000.     

Analytical and experimental evaluations of the influence of fracture surface roughness,its sliding actions and the associated plasticity on fatigue crack propagation

An investigation of fatigue crack propagation in rectangular AM60B magnesium alloy plates containing an inclined through crack is presented in this paper. The behavior of fatigue crack growth in the alloy is influenced by the fracture surface roughness. Therefore, in the present investigation, a new model is developed for estimating the magnitude of the frictional stress intensity factor, k_f, arising from the mismatch of fracture surface roughness during in-plane shear. Based on the concept of k_f, the rate of fatigue crack propagation, db/dN, is postulated to be a function of the effective stress intensity factor range, Δk_eff. Subsequently, the proposed model is applied to predict crack growth due to fatigue loads. Experiments for verifying the theoretical predictions were also conducted. The results obtained are compared with those predicted using other employed mixed mode fracture criteria and the experimental data.     

The Facet method: A hierarchical multilevel modelling scheme for anisotropic convex plastic potentials

An entirely new analytical expression describing plastic anisotropy is presented. It is designed to be used in combination with multilevel models. It makes use of the theory of dual plastic potentials, which is shortly revisited. An analysis of convexity is presented. Note that the new method is not optimal when not used in combination with a multilevel model; other methods are better suited for identification on the basis of mechanical test data. Compared with already existing methods which work with multilevel models, the new method has the following advantages: (i) it is automatically convex anywhere in the six-dimensional stress or strain rate space; (ii) it can be used for materials with a stress differential effect, such as hcp metals or pre-strained cubic materials; (iii) its identification procedure is such that not only the Taylor theory, but also more advanced theories, such as the Alamel-model or self-consistent models, can be used to identify the parameters; (iv) an analytical expression of the plastic potential can be obtained in both strain rate and stress space, which is an important advantage when implementing the model in finite element codes for metal forming. Equipotential surfaces in strain rate space and corresponding yield loci obtained by the new method for four materials (one ferrite single crystal, one aluminium alloy and two types of steel) are presented and discussed.     

The center of a system of non-parallel forces

From a discrete system F of applied forces given by a collection of vectors F_k applied to corresponding points P_k, a new system QF can be obtained through a rotation by Q of all F_k without changing P_k. In this note we examine invariant properties of F under arbitrary rotations. We also examine invariant properties of the family QF when all rotations share a fixed axis, giving a coordinate-free approach to the results of Kolosov (1927) .     

Thermal-hydraulic performance of oval tubes in a cross-flow of air

The thermal-hydraulic performance of five oval tubes is experimentally investigated and compared with that for a circular tube in a cross-flow of air. The range of Reynolds numbers Re_D is approximately between 1,000 and 11,000. The nominal axis ratios R (major axis/minor axis) for three of the investigated oval tubes are 2, 3, and 4. Two other configurations of oval tubes are also tested, an oval tube R=3 with two wires soldered on its upper and lower top positions, and a cut-oval tube. The performance of the tubes is corrected for the effects of area blockage and turbulence intensity. The measurement results show that the mean Nusselt numbers Nu_D for the oval tubes are close to that for the circular tube for Re_D<4,000. For a higher Re_D, the Nu_D for the oval tubes is lower than that for the circular tube and it decreases with the increase in the axis ratio R. The drag coefficients C_d for the tubes are measured and the combined thermal-hydraulic performance is indicated by the ratio Nu_D/C_d, which shows a better combined performance for the oval tubes.     

Studies of dual and tandem rigid wheel performance in sand: by G. D. Swanson

Soil parameters c, ?, k_c, k_? and n were determined by use of a Penetro-Shear apparatus. This device utilizes the rotating and penetrating motion of a circular plate. The performance of a full scale M113 track is predicted using soil parameters calculated from the Penetro-Shear Apparatus data and compared to experimental determination of the drawbar pull-weight ratio vs.per cent of track slip curves.     

Contact of rough surfaces: A simple model for elasticity, conductivity and cross-property connections

Two rough plates with multiple contacts of diverse microgeometries are pressed against one another. Two types of contacts can be distinguished: Hertzian ones and “welded” areas. We find that for circular contacts (1) the two types produce the same effect on the incremental stiffness of the interface and on the effective conductivity across it if their contact areas are the same; (2) for both contact types, the compliance and the conductivity are controlled by the same microstructural parameter, where S_k is the kth contact area; (3) the explicit cross-property connection is established that gives the (incremental) stiffness in terms of the conductivity; it holds for an arbitrary mixture of Hertzian and welded contacts and does not require any knowledge of their microgeometries. Whereas the two types of contacts produce the same effect on the incremental stiffness, the Hertzian contacts cause non-linearities (the incremental stiffness increases with loading). The non-linearity is controlled by the microstructural parameter that is sensitive to contrasts of curvatures of the contacting parts. Model predictions are generally in good agreement with experimental data on conductivities of rough metal surfaces and stiffnesses of rough rock surfaces.     

A Summary of New Predictive High Frequency Thermo-Vibrational Models in Porous Media

In this paper, we consider the effect of mechanical vibration on the onset of convection in porous media. The porous medium is saturated either by a pure fluid or by a binary mixture. The importance of a transport model on stability diagrams is presented and discussed. The stability threshold for the Darcy–Brinkman case in the Ra _Tc -R and k _c -R diagrams is presented (where Ra _Tc , k _c and R are the critical Rayleigh number, the critical wave number and the vibration parameters, respectively). It is shown that there is a significant deviation from the Darcy model. In the thermo-solutal case with the Soret effect, the influence of vibration on the reduction of multi-cellular convection is emphasized. A new analytical relation for obtaining the threshold of mono-cellular convection is derived. This relation shows how the separation factor Ψ is related to the controlling parameters of the problem, Ψ = f (R, ε*, Le), when the wave number k → 0. The importance of vibrational parameter definition is highlighted and it is shown how, by using a proper definition for vibrational parameter, we may obtain compact relationship. It is also shown how this result may be used to increase component separation.     

Analytical Solutions to Gas Flow Problems in Low Permeability Porous Media

In most of conventional porous media the flow of gas is basically controlled by the permeability and the contribution of gas flow due to gas diffusion is ignored. The diffusion effect may have significant impact on gas flow behavior, especially in low permeability porous media. In this study, a dual mechanism based on Darcy flow as well as diffusion is presented for the gas flow in homogeneous porous media. Then, a novel form of pseudo pressure function was defined. This study presents a set of novel analytical solutions developed for analyzing steady-state and transient gas flow through porous media including effective diffusion. The analytical solutions are obtained using the real gas pseudo pressure function that incorporates the effective diffusion. Furthermore, the conventional assumption was used for linearizing the gas flow equation. As application examples, the new analytical solutions have been used to design new laboratory and field testing method to determine the porous media parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs. Then, permeability (k) and effective diffusion coefficient (D _e) was determined; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.     

Deficiency of Approximate Interblock Conductivities for Simulation of Horizontal Unsaturated Flow

Determination of appropriate horizontal interblock conductivities approximations (k _app) is a critical step for numerical solution of unsaturated flow by finite difference method. This study characterized the horizontal effective interblock conductivity (k _H) and the classical k _app by contour map. The deficiencies of the k _app were determined by both statistical method and wetting front prediction. Mathematical expressions of the k _H were redeveloped for three constitutive hydraulic functions. A large database including ten different k _app formulas, three hydraulic functions, and a large range of soil texture is used to determine the deficiencies of the k _app for simulation of horizontal unsaturated flow. Some k _app may be either higher or lower than the k _H since they depend on both the expression forms of the k _app and the specific hydraulic functions. For the soil described by the Gardner function, the horizontal wetting front generated from the k _H is almost equivalent to the analytical solution, but those predicted from the k _app with certain errors. For the soils described by the Brooks–Corey function and the van Genuchten function, the best k _app for approximating the k _H were also determined, but with conversions for the soils described by the van Genuchten function. For any two k _app that hold compatible medians, the k _app (>k _H) better tracks the wetting front and makes mass conservation than the k _app (<k _H) does. The ranking list of k _app with respect to the quality of approximation k _H is ultimately determined from the analysis of mass balance of the simulation based on the large database. Ten k _app can be divided into four clusters. The logarithm mean and the arithmetic mean calculated from saturation can produce the best simulation results with relative mass balance error averagely less than 2%. Simulations conducted with harmonic mean calculated from hydraulic conductivity are mostly associated with the largest mass balance errors. Such a mean can therefore not be used to simulate horizontal water front advance, whatever the soil type and whatever the model of hydrodynamic function. It is also suggested that use of the classical k _app in a numerical process may generate improper results, thus application of the k _H is convincingly recommended on the prediction of horizontal wetting front.     

Three-dimensional analysis of piezocomposite plates with arbitrary geometry and boundary conditions

In this paper, an accurate series solution in conjunction with an energy formulation for the treatment of piezocomposite plates with arbitrary geometry and aspect ratio, under both electrical and mechanical loadings are proposed. A remedy for dealing with nonhomogeneous boundary conditions is also presented. Through introduction of amending polynomials of order p_k for the kth layer, the accuracy and convergence rate are dramatically improved. These polynomials ensure continuity of the generalized displacement fields across the interfaces, while their derivatives can have the required discontinuities up to a desired order. Moreover, depending on the nature of the physical problem under consideration, incorporation of the appropriate functions result in greater convergence rate and precision of the solution.