Showing papers in "Acta Mechanica in 2001"

Magnetohydrodynamic stagnation-point flow towards a stretching sheet

Abstract: Steady two-dimensional stagnation-point flow of an incompressible viscous electrically conducting fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that the velocity at a point decreases/increases with increase in the magnetic field when the free stream velocity is less/greater than the stretching velocity. Temperature distribution in the flow is obtained when the surface is held at a constant temperature.

Unsteady three-dimensional MHD-boundary-layer flow due to the impulsive motion of a stretching surface

Abstract: The development of velocity and temperature fields of an incompressible viscous electrically conducting fluid, caused by an impulsive stretching of the surface in two lateral directions and by suddenly increasing the surface temperature from that of the surrounding fluid, is studied. The partial differential equations governing the unsteady laminar boundary-layer flow are solved numerically using an implicit finite difference scheme. For some particular cases, closed form solutions are obtained, and for large values of the independent variable asymptotic solutions are found. The surface shear stresses inx-andy-directions and the surface heat transfer increase with the magnetic field and the stretching ratio, and there is a smooth transition from the short-time solution to the long-time solution.

First-order chemical reaction on flow past an impulsively started vertical plate with uniform heat and mass flux

Abstract: A finite-difference solution of the transient natural convection flow of an incompressible viscous fluid past an impulsively started semi-infinite plate with uniform heat and mass flux is presented here, taking into account the homogeneous chemical reaction of first order. The velocity profiles are compared with the available theoretical solution and are found to be in good agreement. The steady-state velocity, temperature and concentration profiles are shown graphically. It is observed that due to the presence of first order chemical reaction the velocity decreases with increasing values of the chemical reaction parameter. The local as well as average skin-friction, Nusselt number and Sherwood number are shown graphically.

Micro-mechanical modelling of granular material. Part 1: Derivation of a second-gradient micro-polar constitutive theory

Abstract: This contribution is one in a series of two papers. In the current paper a constitutive law is developed that includes the micro-structural effects by particle displacement as well as particle rotation. Both degrees of freedom can be related to corresponding macroscopic kinematic continuum variables, where the resulting gradients of displacement are selected up to the fourth-order and the gradients of rotation up to the third order. The elastic micro-structural properties for an individual particle are used to derive the macro-level behavior for a fabric of equal-sized spherical particles, leading to a second-gradient micro-polar formulation. In this model, all coefficients are expressed in terms of particle stiffness and particle structure. It is shown that the second-gradient micro-polar model can be reduced to simpler forms, such as the classic linear elastic model, the second-gradient model and the Cosserat model. In the accompanying paper these reduced forms are treated in more detail by analyzing the corresponding dispersion relations for plane body wave propagation.

Experimental and numerical analyses of bending of foam-filled sections

Abstract: Numerical simulations and experiments are conducted to study the bending crush behavior of thin-walled columns filled with closed-cell aluminum foam. A nonlinear dynamic finite element code was used to simulate quasi-static three point bending experiments. The aluminum foam filler provides a higher bending resistance by retarding inward fold formation at the compression flange Moreover, the presence of the foam filler changes the crushing mode from a single stationary fold to a multiple propagating fold. The progressive crush prevents the drop in load carrying capacity due to sectional collapse. Henceforth, the aluminum foam filling is very attractive to avoid global failure for a component which undergoes combined bending and axial crushing. This phenomenon is captured from both experiment and numerical simulation. It was found that partially foam-filled beams also still offer, high bending resistance, and the concept of the effective foam length is developed. Potential applications of foam-filled sections for crashworthy structures are suggested.

On a class of exact solutions of the equations of motion of a second grade fluid

Abstract: By means of Fourier sine transform, the velocity field corresponding to a flow of a suddenly moved flat plate in a second grade fluid is determined. The adequate solution for the Rayleigh-Stokes problem for the edge is also presented.

Scale and boundary conditions effects in elastic properties of random composites

Abstract: We study elastic anti-plane responses of unidirectional fiber-matrix composites. The fibers are of circular cylinder shape, aligned in the axial direction, and arranged randomly, with no overlap, in the transverse plane. We assume that both fibers and matrix are linear elastic and isotropic. In particular, we focus on the effects of scale of observation and boundary conditions on the overall anti-plane (axial shear) elastic moduli. We conduct this analysis numerically, using a two-dimensional square spring net-work, at the mesoscale level. More specifically, we consider finite “windows of observation”, which we increase in size. We subject these regions to several different boundary conditions: displacement-controlled, traction-controlled, periodic, and mixed (combination of any of the first three) to evaluate the mesoscale moduli. The first two boundary conditions give us scale-dependent bounds on the anti-plane elastic moduli. For each boundary condition case we consider many realizations of the random composite to obtain statistics. In this parametric study we cover a very wide range of stiffness ratios ranging from composites with very soft inclusions (approximating holes) to those with very stiff inclusions (approaching rigid fibers), all at several volume fractions.

A continuum model for granular materials: Considering dilatancy and the Mohr-Coulomb criterion

Abstract: In this paper we will explore the consequences of the Mohr-Coulomb criterion on the constitutive equation proposed by Rajagopal and Massoudi [1]. This contunuum model which is based on the earlier works of Cowin [2] has also the ability to predict the dilatancy effect which is related to the normal stress effects. At the same time, if a proper representation is given to some of the material parameters, this model would also comply with the Mohr-Coulomb criterion. We also present, as a special case, an exact solution for the case of simple shear flows.

Numerical studies on the identification of the material parameters of Rivlin's hyperelasticity using tension-torsion tests

Abstract: This paper deals with the identification of material parameters of elasticity relations based on Rivlin's hyperelasticity for incompressible material response, where the free energy evolves as a polynomial in the first and second invariant of the right Cauchy-Green tensor. This elasticity relation has the advantage of incorporating the material parameters linearily. The numerical studies are applied to tension, torsion and combined tension-torsion tests with cylindrical carbon black-filled rubber specimens represented in Haupt and Sedlan [1] and [2]. In the identification process the analytical solution of the resulting boundary value problem leads to a linear least square solution. In this article attention is focused on the numerical solution of several models proposed in the literature and their behavior for both a large and a small number of test data.

Thermal buckling of cross-ply laminated composite beams

Abstract: Thermal buckling of thick, moderately thick and thin cross-ply laminated beams subjected to uniform temperature distribution are analyzed. Exact analytical solutions of refined beam theories are developed to obtain the critical buckling temperature of cross-ply beams with various boundary conditions. The state space concept in conjunction with Jordan canonical form will be used to solve exactly the governing equations of the thermal buckling problems. The effects of length to thickness ratio, modulus ratio, thermal expansion coefficients ratio, various boundary conditions and number of layers on the critical buckling temperature are investigated.

Unsteady convetion flow of micropolar fluids past a vertical porous plate embedded in a porous medium

Abstract: The unsteady two-dimensional laminar flow of a viscous incompressible micropolar fluid past a semi-infinite porous plate embebbed in a porous medium is studied. The free stream velocity follows an exponentially increasing or decreasing small perturbation law. The porous surface absorbs the micropolar fluid with a time varying suction velocity which has a small amplitude. The effects of material parameters on the velocity and temperature fields across the boundary layer are investigated. Numerical results of velocity distribution of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Results show that for the case of a surface cooling by natural convection the skin friction on the porous plate shows an increasing nature up to the critical value of ciscosity ratio. And the surface heat transfer tends to decrease slightly by increasing the magnitude of suction velocity with a given permeablity parameter, and given Prandtl number. However, for a surface heating case, the surface skin friction shows an opposite nature as compared with a surface cooling case.

Free convection from a vertical permeable circular cone with non-uniform surface temperature

Abstract: Laminar free convection from a vertical permeable circular cone maintained at nonuniform surface temperature is considered. Non-similar solutions for boundary-layer equations are found to exist when the surface temperature follows the power law variations with the distance measured from the leading edge. The numerical solutions of the transformed non-similar boundary-layer equations are obtained by using three methods, namely, (i) a finite difference method, (ii) a series solution method, and (iii) an asymptotic solution method. Solutions are obtained in terms of skin friction, heat transfer, velocity profile and temperature profile for smaller values of Prandtl number and temperature gradient are displayed in both tabular and graphical forms. Finite difference solutions are compared with the solutions obtained by perturbation and asymptotic techniques and found to be in excellent agreement.

Aerodynamic properties of a wing performing unsteady rotational motions at low Reynolds number

Abstract: The aerodynamic forces and flow structures of a wing of relatively small aspect ratio in some unsteady rotational motions at low Reynolds number (Re=100) are studied by numerically solving the Navier-Stokes equations. These motions include a wing in constant-speed rotation after a fast start, wing accelerating and decelerating from one rotational speed to another, and wing rapidly pitching-up in constant speed rotation. When a wing performs a constant-speed rotation at small Reynolds number after started from rest at large angle of attack (α=35°), a large lift coefficient can be maintained. The mechanism for the large lift coefficient is that for a rotating wing: the variation of the relative velocity along the wing-span causes a pressure gradient and hence a spanwise flow which can prevent the dynamic stall vortex from shedding. When a wing is rapidly accelerating or decelerating from one rotational speed to another, or rapidly pitching-up during constant speed rotation, even if the aspect ratio of the wing is small and the flow Reynolds number is low, a large aerodynamic force can be obtained. During these rapid unsteady motions, new layers of strong vorticity are formed near the wing surfaces in very short time, resulting in a large time rate of change of the fluid impulse which is responsible for the generation of the large aerodynamic force.

Micro-mechanical modelling of granular material. Part 2: Plane wave propagation in infinite media

Abstract: This paper discusses the propagation of plane body waves through a second-gradient micropolar elastic continuum. In an accompanying paper, this macroscopic constitutive law has been derived from the micro-level particle characteristics, which are the inter-particle stiffness, the particle size and the package density. As a result of incorporating the micro-scale effects, the body waves propagate in a dispersive manner, where dispersion becomes more prominent when the wavelength of the generated body waves reaches the order of magnitude of the particle size. After successively deriving the equations of motion and the dispersion relations for plane body wave propagation, the compressional wave properties for the second-gradient micro-polar model are compared to those for the Born-Karman lattice structure. Furthermore, distinguished features of the second-gradient micro-polar model are exhibited by comparing the dispersion relations of the coupled propagation of the shear wave and the micro-rotational wave with those of more simple constitutive models. The paper ends with a parameter study, where the effect by the translational particle contact stiffness and the rotational particle contact stiffness is examined.

Transient thermal stress analysis of multilayered hollow cylinder

Abstract: This paper deals with the transient response of one-dimensional axisymmetric quasistatic coupled thermoelastic problems. Laplace transform and finite difference methods are used to analyze the problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal stress distribution in a transient state. Moreover, the computational procedures established in this article can solve the generalized thermoelasticity problem for a multilayered hollow cylinder with orthotropic material properties.

Layout optimization of trusses using improved GA methodologies

Abstract: The main concern of the paper is the simultaneous treatment of size, shape and topology variables in the optimum design of space trusses. As compared to only size optimization, this is a challenging, more difficult and complex problem. The paper discusses a solution algorithm which is based on the use of GAs. Two new methodologies, “annealing perturbation” and “adaptive reduction of the design space”, are introduced in conjunction with GAs, bringing additional increase in computational efficeiency. Some common problems in handling shape and topology design considerations are eliminated, which in turn provides a large and a flexible design environment. A numerical problem is presented to test the performance of the proposed methodologies and to compare the results with those existing in the literature. Furthermore, the paper studies a second problem designed to observe the efficiency of GAs in a considerably large and complex design space.

Fully developed flow of a modified second grade fluid with temperature dependent viscosity

Abstract: In this paper we will study the fully developed flow of a modified (and sometimes referred to as the generalized) second grade fluid down an inclined plane. The reasons for using such a model for the flow of non-Newtonian fluids are (i) the capability of predicting the normal stress differences and (ii) allowing for the possibility of shear dependent viscosity. The boundary value problem is solved numerically, and the special case of constant viscosity amends itself an exact solution (as previously reported in the literature) which serves as a test case to check the accuracy of our numerical scheme. The velocity and temperature profiles are obtained for various dimensionless numbers, for the case where the viscosity is also a function of temperature.

Control of the transient thermoelastic displacement of a functionally graded rectangular plate bonded to a piezoelectric plate due to nonuniform heating

Abstract: In this study, the theoretical analysis of a control of the transient thermoelastic displacement is developed for a functionally graded rectangular plate bonded to a piezoelectric plate due to nonuniform heat supply. Assuming that the functionally graded plate has nonhomogeneous thermal and mechanical material properties in the thickness direction, the three-dimensional temperature in a transient state and the three-dimensional transient thermal stresses of a simply supported plate for a functionally graded material are analyzed by introducing the theory of laminated composites as a theoretical approximation. By using the solution for a functionally graded plate and the exact solution for a piezoelectric plate of crystal class mm2, the theoretical analysis of three-dimensional transient piezothermoelasticity is developed for a simply supported combined plate. the analysis of a piezothermoelastic problem leads to an appropriate electric potential applied to the piezoelectric plate which suppresses the induced thermoelastic displacement in the thickness direction at the midpoint on the free surface of the functionally graded plate. As an example, numerical calculations are carried out for a functionally graded rectangular plate made of zirconium oxide and titanium alloy, bonded to a piezoelectric plate of a cadmium selenide solid. Some numerical results when the transient thermoelastic displacements are controlled are shown in figures.

Heat and mass transfer effects on flow past an impulsively started vertical plate

Abstract: An exact solution to the problem of lfow past an impulsively started infinite vertical plate in the presence of uniform heat and mass flux at the plate is presented by the Laplace-transform technique. The velocity, the temperature and the concentration profiles are shown graphically. The rate of heat transfer, the skin-friction, and the Sherwood number are also shown on graphs. The effect of different parameters like Grashof number, mass Grashof number, Prandtl number, and Schmidt number are discussed.

Closed and approximate analytical solutions for rectangular Mindlin plates

Abstract: Basing on the Nadai-Levy and the Vlasov-Kantorovich methods closed and approximate analytical solutions of Mindlin's plate equations in the case of rectangular plates are discussed. For elastic, homogeneous and isotropic plates three unknowns of the governing two-dimensional boundary value problem are formulated as series of products of functions depending on a single coordinate. Specifying the functions for one of the in-plane coordinate directions the governing partial differential equations for a special type of boundary conditions and the principle of virtual displacements for the general case yield a set of ordinary differential equations. The analytical solution of these equations provides expressions for functions depending on the other in-plane coordinate. For plates with simply supported edges for one of the coordinate directions and for arbitrary homogeneous boundary conditions for the other one the Nadai-Levy method provides a closed or exact solution in the sense that the infinite series for displacements and stress resultants can be truncated to obtain any desired accuracy. In the general case of nonsimply supported edges the iterative Vlasov-Kantorovich method yields an approximate analytical solution. Both methods are nonsensitive to a reduction of the thickness with respect to accuracy and represent the boundary layer solutions in terms of exponential functions. Applications to rectangular plates with various types of boundary conditions are presented.

MHD boundary-layer flow of a non-Newtonian fluid over a continuously moving surface with a parallel free stream

Abstract: The MHD flow and heat transfer of a non-Newtonian power-law fluid over a continuously moving surface with a parallel free stream have been investigated. The partial differential equations governing the non-similar flow have been solved numerically using an implicit finite-difference scheme. The skin friction and heat-transfer coefficients increase with the magnetic parameter, and they are more for the pseudoplastic fluid than for the dilatant fluid. The heat-transfer coefficient increases significantly with the Prandtl number. The gradient of the velocity at the surface is negative when the wall velocity is greater than the free stream velocity, and it is positive when the wall velocity is less than the free stream velocity.

Interaction of peristaltic flow with pulsatile magneto-fluid through a porous medium

Abstract: The interaction of purely periodic mean flow with a peristaltic induced flow is investigated within the framework of a two-dimensional analogue. The mathematical model considers a viscous incompressible fluid under the effect of a transverse magnetic field through a porous medium between infinite parallel walls on which a sinusoidal traveling wave is imposed. A perturbation solution to the complete set of Navier-Stokes equations is found for the case in which the frequency of the traveling wave and that of the imposed pressure gradient are equal. The ratio of the traveling wave amplitude to channel width is assumed to be small. For this case a first-order steady flow is found to exist, as contrasted to a second-order effect in the absence of the imposed periodic pressure gradient. The effect of the magnetic parameter, permeability parameter and the various parameters included in the problem are discussed numerically.

Natural convection in an annulus between two rotating vertical cylinders

Abstract: A numerical study is conducted to understand the effect of rotation on the axisymmetric flow driven by buoyancy in an annular cavity formed by two concentric vertical cylinders which rotate about their axis with different angular velocities. The inner and outer side walls are maintained isothermally at temperature θ c and θ h , respectively, while the horizontal top and bottom walls are adiabatic. The vorticity-stream function form of the Navier-Stokes equations and the energy equation have been solved by modified Alternating Direction Implicit method and Successive Line Over Relaxation method. Numerical results are obtained for a wide range of the Grashof number, Gr, nondimensional rotational speeds Ω i , Ω o of inner and outer cylinders and for different values of the Prandtl number Pr. The effects of the aspect ratio,A, on the heat transfer and flow patterns are obtained forA=1 and 2. The numerical results show that when the outer cylinder alone is rotating and the Grashof number is moderate, the outward bound flow is confined to a thin region along the bottom surface while the return flow covers a major portion of the cavity. For a given inner or outer cylinder rotation the temperature field is almost independent of the flow in the annulus for fluids with low Prandtl number, while it depends strongly for high Prandtl number fluids. At a high Grashof number, with moderate rotational speeds, the dominant flow in the annulus is driven by thermal convection, and hence an increase in the heat transfer rate occurs. In the case of unit aspect ratio, the flow pattern is unicellular for the rotation of the cylinders in the same direction, and when they rotate in the opposite direction two or more counter rotating cells separated by a stagnation surface are formed. The rate of heat transfer at the hot cylinder is suppressed when its speed of rotation is higher than that of the cooler cylinder. The computed heat transfer and flow patterns are compared with the available results of a nonrotating cylindrical annulus, and good agreement is found.

Unsteady MHD flow due to non-coaxial rotations of a porous disk and a fluid at infinity

Abstract: An exact solution of the unsteady three-dimensional Navier-Stokes equations is derived for the case of flow due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field. An analytical solution of the problem is established by the method of Laplace transform, and the velocity field is presented in terms of the tabulated functions. It is found that the boundary layer thickness in the cases of suction/blowing decreases with the increase in the magnetic parameter.

Symmetry of turbulent boundary-layer flows: Investigation of different eddy viscosity models

Abstract: The symmetrical properties of the turbulent boundary-layer flows and other turbulent flows are studied utilizing the Lie group theory technique. The self-similar forms of the indepedent variables and the solution function for the turbulent boundary layer flows with three different models of the turbulent (eddy) viscosity are obtained. Proceeding from this analysis, a simple numerical method for computation of turbulent flows is developed.

A new solution branch of the Falkner-Skan equation

Abstract: We present here results for a new solution branch of the Falkner-Skan equation with parameter β. It is found that there are two turning points on this new branch which results in two solutions of the problem for 37.844<β<∞, three solutions for β=37.844, four solutions for 14.533<β<37.844, three solutions for β=14.533, and two solutions for 1<β<14.533. This solution branch is found to end singularly at β=1; its structure is analytically investigated and the principal characteristics described. The spatial stability of such solutions is also commented on.

Power law and log law velocity profiles in turbulent boundary-layer flow: equivalent relations at large Reynolds numbers

Abstract: The open equations of a turbulent boundary layer subjected to a pressure gradient analysed for classical two layers (inner wall and outer wake), while matched in the overlap region of MAX through the Millikan-Kolmogorov hypothesis leads to an open functional equation, and its classical solution for the velocity distribution is the log. region. It is shown here that the same open functional equation also predicts a power law velocity distribution and a power law skin friction in the overlap region. The uniformly valid solution for the composite wall power law and wake velocity profile is obtained. The connection between the power law and the classical log. law solutions of the open functional equation is analyzed. At large Reynolds number, the power law solutions reduce to the classical log. law solutions, and the equivalence predicts a certain relationship between the constants in power and log. laws. The results are compared with the experimental data.

Thermally induced vibrations of cross-ply laminated shallow shells

Abstract: An exact analytical solution of the dynamic response of cross-ply laminated shallow shells subject to rapid heating is presented. The classical theory (based on Love-Kirchhoff assumption), involving three coupled partial differential equations, is used. The solution is applicable to shells whose parallel edges are simply supported and the remaining ones are clamped. A generalized modal approach is used to obtain the solution. The equations of motion are converted into a single-order system of equations by using state variables. The biorthogonality conditions of principal modes of the original and adjoint eigenfunctions are used to decouple the state space equation. Histories of deflection of graphite-reinforced aluminum shell panels are presented through numerical examples.

Buckling and free vibration of elastic plates using simple and mixed shear deformation theories

Abstract: Simple and mixed shear deformable theories accounting for the transverse shear, in the sense of Reissner-Mindlin's thick plate theory, are employed in the construction of variational statements for rectangular flat plates. Analytical solutions for natural frequencies and buckling loads of anisotropic plates under various boundary conditions are developed. A variety of simply-supported, clamped and free boundary conditions is considered, and comparisons with the existing literature are made.

Dynamic stress intensity factor of a functionally graded material under antiplane shear loading

Abstract: The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.