Three-dimensional free vibration analysis of four-parameter continuous grading fiber reinforced cylindrical panels resting on Pasternak foundations

This research investigates three-dimensional free vibration analysis of four-parameter continuous grading fiber reinforced (CGFR) cylindrical panels resting on Pasternak foundations by using generalized power-law distribution. The functionally graded orthotropic panel is simply supported at the edges, and it is assumed to have an arbitrary variation of matrix volume fraction in the radial direction. A four-parameter power-law distribution presented in literature is proposed. Symmetric and asymmetric volume fraction profiles are presented. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are solved by generalized differential quadrature method, and natural frequency is obtained. The fast rate of convergence of the method is demonstrated, and to validate the results, comparisons are made with the available solutions for functionally graded isotropic shells with/without elastic foundations. The effect of the elastic foundation stiffness parameters and various geometrical parameters on the vibration behavior of the CGFR cylindrical panels is investigated. This work mainly contributes to illustrate the influence of the four parameters of power-law distributions on the vibration behavior of functionally graded orthotropic cylindrical panels resting on elastic foundation. This paper is also supposed to present useful results for continuous grading of matrix volume fraction in the thickness direction of a cylindrical panel on elastic foundation and comparison with similar discrete laminated composite cylindrical panel.     

Calculating the flows of a viscoplastic medium in tubes using the local variation method

This article considers the problem of the motion of a visco-plastic medium in tubes and channels. The results of [1, 2] are used, which present the variational formulation and the qualitative analysis of this problem. The method of local variations suggested in [3] is used for the numerical solution of the variational problem. A more detailed presentation of the algorithm of this method in application to boundary-value and variational problems is given in [4]. Results of calculations of certain concrete problems on an electronic computer are presented.The author wishes to express his sincere gratitude to F. L. Chernous'ko for the problem formulation and helpful counsel, and G. I. Barenblatt and S. S. Grigoryan for useful discussions.     

A novel 2-D six-parameter power-law distribution for three-dimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation

This paper is motivated by the lack of studies in the technical literature concerning to the three-dimensional vibration analysis of bi-directional FG rectangular plates resting on two-parameter elastic foundations. The formulations are based on the three-dimensional elasticity theory. The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. This paper presents a novel 2-D six-parameter power-law distribution for ceramic volume fraction of 2-D FGM that gives designers a powerful tool for flexible designing of structures under multi-functional requirements. Various material profiles along the thickness and in the in-plane directions are illustrated using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori-Tanaka scheme. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. Some new results for natural frequencies of the plates are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the natural frequency than conventional 1-D FGM.     

Choice of a self-similar solution in boundary-layer theory

If the speed of the outer flow at the edge of the boundary layer does not depend on the time and is specified in the form of a power-law function of the longitudinal coordinate, then a self-similar solution of the boundary-layer equations can be found by integrating a third-order ordinary differential equation (see [1–3]). When the exponent of the power in the outerflow velocity distribution is negative, a self-similar solution satisfying the equations and the usually posed boundary conditions is not uniquely determinable [4], A similar result was obtained in [5] for flows of a conducting fluid in a magnetic field. In the present paper we study the behavior of non-self-similar perturbations of a self-similar solution, enabling us to provide a basis for the choice of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–46, July–August, 1974.     

Method of generalized similitude in free convection problems with arbitrary distribution of the temperature or heat flux on a vertical wall

The self-similar problem of free convection near a heated vertical plate was solved for the first time in [1] for the simplest case of a constant wall temperature. In [2], Yang proved the existence of a self-similar solution to the problem of free convection for vertical plates and cylinders on the surfaces of which the temperature has a power-law distribution. In [3], Yang's solution was generalized to the case of free convection near a slender figure of revolution, but also only in the self-similar case of a power-law distribution of the temperature on the wall. In [4], this problem was solved in an extended nonsimilar formulation but by an artificial and not general method similar to Gertler's, the convergence of the approximations being slow. The present paper contains the solution to the problem of free convection near a vertical plate with arbitrary distribution of the temperature or heat flux on its surface. Rigorous application of the method of generalized similitude [5] leads in this case to universal equations that present insuperable computational difficulties, which forces one to use a simplified but fairly general method to solve this class of problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 167–170, May–June, 1980.I thank L. G. Loitsyanskii and E. M. Smirnov for discussing the results and for valuable comments.     

Motion of a strong shock front in a two-dimensional gas layer with moving internal boundary

We examine the problem of planar one-dimensional motion of a strong shock wave with moving internal boundary in which the initial position of the front, its intensity, the mass of the gas involved in the motion, and the energy contained in this gas are known. The problem is not self-similar and its exact solution, which involves working with partial differential equations, presents serious difficulties. In the following we determine the law of shock-front motion in this problem via the method of [1], which makes it possible to find a system of ordinary differential equations for the problem. The method is based on an initial specification of the power-law coupling between the dimensionless Lagrangian and Eulerian variables and replacement of the energy equation by this coupling and the energy integral. The solution is sought in the first approximation.     

Note on the Heat Transfer of Flows over a Stretching Wall in Porous Media: Exact Solutions

In this note, heat transfer over a stretching sheet with mass transfer in a porous medium is revisited. Analytical solutions are presented for two cases including a prescribed power-law wall temperature case and a prescribed power-law wall heat flux case. The solutions are expressed by the Kummer’s function. Closed-form solutions are found and presented for some special parameters. The solutions might offer more insights of the heat transfer characteristics compared with the numerical solutions.     

Mass transfer effects on the non-Newtonian fluids past a vertical plate embedded in a porous medium with non-uniform surface heat flux

 The present study is devoted to investigate the influences of mass transfer on buoyancy induced flow over vertical flat plate embedded in a non-Newtonian fluid saturated porous medium. The Ostwald–de Waele power-law model is used to characterize the non-Newtonian fluid behavior. Similarity solution for the transformed governing equations is obtained with prescribed variable surface heat flux. Numerical results for the details of the velocity, temperature and concentration profiles are shown on graphs. Excess surface temperature as well as concentration gradient at the wall associated with heat flux distributions, which are entered in tables, have been presented for different values of the power-law index n, buoyancy ration B and the exponent λ as well as Lewis number Le. Received on 26 April 2000     

Mass Transfer on a Delta Wing in a Hypersonic Flow

The effect of mass transfer on the parameters of the laminar boundary layer flow on a cold delta wing in a hypersonic viscous perfect-gas flow is investigated for the case of a strong viscous-inviscid interaction regime. The influence of the mass transfer intensity on the supercritical-subcritical transition coordinate and the local and total aerodynamic parameters of a delta wing with a power-law cross-section is numerically determined.     

Steady flow of a power-law non-Newtonian fluid across an unconfined square cylinder

A two-dimensional flow of a non-Newtonian power-law fluid directed normally to a horizontal cylinder with a square cross section is considered in the present paper. The problem is investigated numerically with a finite volume method by using the commercial code Ansys Fluent with a very large computational domain so that the flow could be considered unbounded. The investigation covers the power-law index from 0.1 to 2.0 and the Reynolds number range from 0.001 to 45.000. It is found that the drag coefficient for low Reynolds numbers and low power-law index (n ≤ 0.5) obeys the relationship C_D = A/Re. An equation for the quantity A as a function of the power-law index is derived. The drag coefficient becomes almost independent of the power-law index at high Reynolds numbers and the wake length changes nonlinearly with the Reynolds number and power-law index.     

Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities

The free thermal vibration of functionally graded material (FGM) cylindrical shells containing porosities is investigated. Both even distribution and uneven distribution are taken into account. In addition, three thermal load types, i.e., uniform temperature rise (UTR), nonlinear temperature rise (NLTR), and linear temperature rise (LTR), are researched to explore their effects on the vibration characteristics of porous FGM cylindrical shells. A modified power-law formulation is used to describe the material properties of FGM shells in the thickness direction. Love’s shell theory is used to formulate the strain-displacement equations, and the Rayleigh-Ritz method is utilized to calculate the natural frequencies of the system. The results show that the natural frequencies are affected by the porosity volume fraction, constituent volume fraction, and thermal load. Moreover, the natural frequencies obtained from the LTR have insignificant differences compared with those from the NLTR. Due to the calculation complexity of the NLTR, we propose that it is reasonable to replace it by its linear counterpart for the analysis of thin porous FGM cylindrical shells. The present results are verified in comparison with the published ones in the literature.     

Dynamic modeling of soil–tool interaction: An overview from a fluid flow perspective

The study of tillage tool interaction centers on soil failure patterns and development of force prediction models for design optimization. The force-deformation relationships used in models developed to date have been considering soil as a rigid solid or elasto-plastic medium. Most of the models are based on quasi-static soil failure patterns. In recent years, efforts have been made to improve the conventional analytical and experimental models by numerical approaches. This paper aims at reviewing the existing methods of tillage tool modeling and exploring the use of computational fluid dynamics to deal with unresolved aspects of soil dynamics in tillage. The discussion also focuses on soil rheological behaviour for its visco-plastic nature and its mass deformation due to machine interaction which may be analyzed as a Bingham plastic material using a fluid flow approach. Preliminary results on visco-plastic soil deformation patterns and failure front advancement are very encouraging. For a tool operating speed of 5.5 m s⁻¹, the soil failure front was observed to be about 100-mm forward of the tool.     

Detwinning of High-Purity Zirconium: <Emphasis Type="Italic">In-Situ</Emphasis> Neutron Diffraction Experiments

Twinning is an important deformation mode in hexagonal metals to accommodate deformation along the c-axis. It differs from slip in that it accommodates shear by means of crystallographic reorientation of domains within the grain. Such reorientation has been shown to be reversible (detwinning) in magnesium alloy aggregates. In this paper we perform in-situ neutron diffraction reversal experiments on high-purity Zr at room temperature and liquid nitrogen temperature, and follow the evolution of twin fraction. The experiments were motivated by previous studies done on clock-rolled Zr, subjected to deformation history changes (direction and temperature), in the quasi-static regime, for temperatures ranging from 76 K to 450 K. We demonstrate here for the first time that detwinning of { 10[`1] 2 } á 10[`1][`1] ñ\left\{ {10\overline 1 2} \right\}\left\langle {10\overline 1 \overline 1 } \right\rangle tensile twins is favored over the activation of a different twin variant in grains of high-purity polycrystalline Zr. A visco-plastic self-consistent (VPSC) model developed previously, which includes combined slip and twin deformation, was used here to simulate the reversal behavior of the material and to interpret the experimental results in terms of slip and twinning activities.     

Effects of shear-thinning/thickening nature on the stability of Couette-like flow with uniform crossflow

The linear stability of wall-injected pressure- driven Couette-like flow in power-law fluids is studied. Previous study on this kind of flow for Newtonian fluids by Nicoud and Angilella [Phys. Rev. E 56, 3000 （1997）] was extended to power-law fluids to understand the effects of shear-thinning/thickening nature on the flow stability. A related expression between the critical crossflow Reynolds number for Newtonian fluids and that for power-law fluids is obtained as the streamwise Reynolds number is large enough based on numerical computations, and verified theoretically in the case of a limiting condition with the power-law index.     

The Run-Off Condition for Rimming Flow of a Power-Law Fluid

The rimming flow of a power-law fluid in the inner surface of a horizontal rotating cylinder is investigated. Exploiting the fact that the liquid layer is thin, the simplest lubrication theory is applied. The generalized run-off condition for the steady-state flow of the power-law liquid is derived. In the bounds implied by this condition, film thickness admits a continuous solution. In the supercritical case when the mass of non-Newtonian liquid exceeds a certain value or the speed of rotation is less than an indicated limit, a discontinuous solution is possible and a hydraulic jump may occur in the steady-state regime. The location and height of the hydraulic jump for the power-law liquid is determined. Received 8 February 2001 and accepted 19 June 2001     

Free vibrations of four-parameter functionally graded parabolic panels and shells of revolution

Basing on the First-order Shear Deformation Theory (FSDT), this paper focuses on the dynamic behaviour of moderately thick functionally graded parabolic panels and shells of revolution. A generalization of the power-law distribution presented in literature is proposed. Two different four-parameter power-law distributions are considered for the ceramic volume fraction. Some symmetric and asymmetric material profiles through the functionally graded shell thickness are illustrated by varying the four parameters of power-law distributions. The governing equations of motion are expressed as functions of five kinematic parameters. For the discretization of the system equations the Generalized Differential Quadrature (GDQ) method has been used. Numerical results concerning four types of parabolic shell structures illustrate the influence of the parameters of the power-law distribution on the mechanical behaviour of shell structures considered.     

Wall effects on the transportation of a cylindrical particle in power-law fluids

The present work deals with the numerical calculation of the Stokes-type drag undergone by a cylindrical particle perpendicularly to its axis in a power-law fluid. In unbounded medium, as all data are not available yet, we provide a numerical solution for the pseudoplastic fluid. Indeed, the Stokes-type solution exists because the Stokes’ paradox does not take place anymore. We show a high sensitivity of the solution to the confinement, and the appearance of the inertia in the proximity of the Newtonian case, where the Stokes’ paradox takes place. For unbounded medium, avoiding these traps, we show that the drag is zero for Newtonian and dilatant fluids. But in the bounded one, the Stokes-type regime is recovered for Newtonian and dilatant fluids. We give also a physical explanation of this effect which is due to the reduction of the hydrodynamic screen length, for pseudoplastic fluids. Once the solution of the unbounded medium has been obtained, we give a solution for the confined medium numerically and asymptotically. We also highlight the consequence of the confinement and the backflow on the settling velocity of a fiber perpendicularly to its axis in a slit. Using the dynamic mesh technique, we give the actual transportation velocity in a power-law “Poiseuille flow”, versus the confinement parameter and the fluidity index, induced by the hydrodynamic interactions.     

Self-similar solutions of non-Newtonian fluid boundary layer in MHD

The self-similar solutions of the boundary layer for a non-Newtonian fluid in MHD were considered in [1, 2] for a power-law velocity distribution along the outer edge of the layer and constant electrical conductivity through the entire flow. However, the MHD flows of many conducting media, which are solutions or molten metals, cannot be described by the MHD equations for non-Newtonian fluids.The self-similar solutions of the boundary layer for a non-Newtonian fluid without account for interaction with the electromagnetic field were studied in [3].In the following we present the self-similar solutions for the boundary layer of pseudoplastic and dilatant fluids with account for the interaction with an electromagnetic field for the case of a power-law velocity distribution along the outer edge of the layer, when the conductivity of the fluid is constant throughout the flow and the magnetic Reynolds number is small.Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 2, No. 6, pp. 77–82, 1967The author wishes to thank S. V. Fal'kovich for his interest in this study.     

Flows induced by power-law stretching surface motion modulated by transverse or orthogonal surface shear

Boundary-layer solutions to Banks' problem for the flow induced by power-law stretching of a plate are obtained for two generalizations that include arbitrary transverse plate shearing motion. In one extension an arbitrary transverse shearing motion is the product of the power-law stretching. In the other extension the streamwise stretching coordinate is added to an arbitrary transverse shearing and together raised to the power of stretching. In addition we find that Banks' power law stretching may be accompanied by orthogonal power-law shear. In all cases, the original boundary-value problem of Banks [1] is recovered. Results are illustrated with velocity profiles both at the plate and at fixed height in the fluid above the plate.     

Conservation of the resultant force vector in the plane of the angle of attack for slender star-shaped bodies in supersonic flows

At high supersonic flight speeds bodies with a star-shaped transverse and power-law longitudinal contour are optimal from the standpoint of wave drag [1–3]. In most of the subsequent experimental [4–6] and theoretical [6–9] studies only conical star-shaped bodies have been considered. For these bodies in certain flow regimes ascent of the Ferri point has been noted [10]. In [11] the boundary-value problem for elongated star-shaped bodies with a power-law longitudinal contour was solved for the case of supersonic flow. The present paper deals with the flow past these bodies at an angle of attack. It is found that for arbitrary star-shaped bodies with any longitudinal (in particular, conical) profile the aerodynamic forces can be reduced to a wave drag and a lift force, the lateral force on these bodies being equal to zero for any position of the transverse contour.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 135–141, November–December, 1989.