Showing papers in "Acta Mechanica in 2000"

Transient wave propagation in a one-dimensional poroelastic column

Abstract: Biot's theory of porous media governs the wave propagation in a porous, elastic solid infiltrated with fluid. In this theory, a second compressional wave, known as the slow wave, has been identified. In this paper, Biot's theory is applied to a one-dimensional continuum. Despite the simplicity of the geometry, an exact solution of the full model, and a detailed analysis of the phenomenon, so far have not been achieved. In the present approach, an analytical solution in the Laplace transform domain is obtained showing clearly two compressional waves. For the special case of an inviscid fluid, a closed form exact solution in time domain is obtained using an analytical inverse Laplace transform. For the general case of a viscous fluid, solution in time domain is evaluated using the Convolution Quadrature Method of Lubich. Of all the inverse methods previously investigated, it seems that only the method of Lubich is efficies and stable enough to handle the highly transient cases such as impact and step loadings. Using properties of three widely different real materials, the wave propagating behavior, in terms of stress, pore pressure, displacement, and flux, are examined. Of most interest is the identification of second compressional wave and its sensitivity of material parameters.

Lie symmetries and conserved quantities of constrained mechanical systems

Abstract: The Lie symmetries and conserved quantities of constrained mechanical systems are studied. Using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the systems are established. The structure equation and the form of conserved quantities are obtained. We find the corresponding conserved quantity from a known Lie symmetry, that is a direct problem of the Lie symmetries. And then, the inverse problem of the Lie symmetries-finding the corresponding Lie symmetry from a known conserved quantity-is studied. Finally, the relation between the Lie symmetry and the Noether symmetry is given.

A Reissner-Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect

Abstract: This paper is concerned with flexural vibrations of composite plates, where piezoelastic layers are used to generate distributed actuation or to perform distributed sensing of strains in the plate. Special emphasis is given to the coupling between mechanical, electrical and thermal fields due to the direct piezoelectric effect and the pyroelectric effect. Moderately thick plates are considered, where the influence of shear and rotatory inertia is taken into account according to the kinematic approximations introduced by Mindlin. An equivalent single-layer theory is thus derived for the composite plates. It is shown that coupling can be taken into account by means of effective stiffness parameters and an effective thermal loading. Polygonal plates with simply supported edges are treated in some detail, where quasi-static thermal bending as well as free, forced and actuated vibrations are studied.

Eshelby's stress tensors in finite elastoplasticity

Abstract: This work examines critically the role that the Eshelby (energy-momentum) tensor or its degenerate form, the Mandel stress, should logically play as the driving force in an invariant formulation of the thermomechanics of finite-strain elasto-plasticity. Here the stress measure of which Mandel advocated the use in elastoplasticity, is shown to coincide, up to a sign, with the quasi-static Eshelby stress tensor expressed in the elastically released intermediate configuration. The various “constitutive” representations for the plastic rate are then discussed in terms of various thermodynamically conjugate pairs of “forces” and “velocities” for anisotropic materials.

The onset of Darcy-Benard convection in an inclined layer heated from below

Abstract: We present an, account of the linear instability of Darcy-Boussinesq convection in a uniform, unstably stratified porous layer at arbitrary inclinations α from the horizontal. A full numerical solution of the linearized disturbance equations is given and the detailed graphical results used to motivate various asymptotic analyses. A careful study shows that at large Rayleigh numbers two-dimensional instability can only arise when α≤31.30°. However it is also demonstrated that the maximum inclination below which this instability may be possible is the slightly greater value of 31.49° which corresponds to a critical Rayleigh number of 104.30.

Some reflections on the Mori-Tanaka and Ponte Castañeda-Willis methods with randomly oriented ellipsoidal inclusions

Abstract: For a two-phase isotropic composite consisting of an isotropic matrix and oriented isotropic ellipsoidal inclusions, Mori-Tanaka's (MT) [6] method and the more recent Ponte Castaneda-Willis (PCW) [1] method are perhaps the only two methods that deliver explicit results for its effective moduli. An attractive feature of the MT method is that it always stays within the Hashin-Shtrikman [3] bounds, while the novel part of the PCW approach is that it has a well defined microstructure. In this paper, we made a comparative study on these two models, for both elasticity and their applications to plasticity. Over the entire range of inclusion shapes, the PCW estimates are found to be consistently stiffer than the MT estimates. An investigation of the possibility of a PCW microstructure for the MT model indicates that the MT moduli could be found from the PCW formulation, but this would require a spatial distribution that is identical to the oriented inclusion shape. Such a requirement implies that the underlying two-point joint probability density function is not symmetric, and thus it is not permissible. One is led to conclude that, unlike the aligned case, the MT model cannot be realized from the PCW microstructure with randomly oriented inclusions.

Effects of porous boundaries on peristaltic transport through a porous medium

Abstract: Peristaltic pumping by a sinusoidal traveling wave in the porous walls of a twodimensional channel filled with a viscous incompressible fluid through a porous medium is investigated theoretically and graphically. It has been considered that the fluid is entering the flow region through one plate at the same rate as it is leaving through the other plate. A perturbation solution is obtained, which satisfies the momentum equation for the case in which the amplitude ratio is small. It has been noticed that the mean axial velocity and the reversal flow increase by increasing the permeability parameterW. The mean axial velocity and the reversal flow decrease by increasingV until at the upper quarter of the channel it increases by increasingW. Also, the fluid motion is nonsymmetric. Numerical results are reported for various values of the physical parameters of interest.

Modal coupling in one-dimensional electromechanical structured continua

Abstract: A passive continuously distributed control of mechanical vibrations is proposed. The piezoelectric actuators are interconnected by a linear electric transmission line. We introduce coupling and internal resonance criteria to determine the optimal choices for electric parameters. These criteria can be found decomposing the differential operator appearing in the linear evolution equations according to a partition of the state vector into mechanical and electrical parts. The results we find allow for the design of an experimental set up.

Bending of square aluminium extrusions with aluminium foam filler

Abstract: An experimental programme consisting of 24 tests was carried out to study the three-point-bending behavior of square AA6060 aluminium extrusions filled with aluminium foam under quasi-static loading conditions. The outer cross section width and span of the beams were kept constant at 80 mm and 800 mm, respectively. The main parameters investigated were the foam density, the extrusion wall strength and the extrusion wall thickness. The experiments showed that the foam filler significantly altered the local deformation pattern of the beams. Simple design formulae were developed in order to predict the load bearing capacity of the foam filled beams.

The inplane elastic properties of circular cell and elliptical cell honeycombs

Abstract: The inplane elastic properties of perfectly circular and elliptic cell honeycombs are derived through an analytical method and validated numerically. In the case of perfectly circular cell hexagonally packed honeycomb, the inplane elastic properties are shown to be isotropic. However, a departure from circularity of the cells leading to cell ellipticity results in the inplane properties becoming orthotropic. The orthotropic elastic constants are also derived analytically and validated numerically.

Application of higher-order tensor theory for formulating enhanced continuum models

Abstract: In this paper attention is focussed on the derivation of higher-order isotropic tensors and their application in the formulation of enhanced continuum models. A mathematical theory will be discussed which relates formal orthogonal invariant polynomial functions to isotropic tensors. Using this theory, the second-order to the sixth-order isotropic tensor will be derived. When the tensor order increases, the derivation procedure clearly reveals a repeatable character. Thereafter, an example will be given of how the higher-order isotropic tensors can be used in the formulation of an enhanced continuum model. It will be demonstrated that symmetry conditions significantly reduce the number of material parameters.

Phenomenological nonlocal approaches based on implicit gradient-enhanced damage

Abstract: The present paper focusses on five phenomenological approaches in gradient-enhanced damage, several of which have been proposed in the literature to simulate material degradation. These different gradient-damage based nonlocal models are examined with respect to their ability to describe crack initiation and crack propagation. The models are applied to identical mechanical benchmark tests, where the material damage evolution law is taken as good as possible equal for each model. Interesting differences between the different models arise, and it is shown that care must be taken in the interpretation and application of these models. One-dimensional results cannot be extrapolated in a straightforward fashion to two dimensions, and the physical relevance of some results is in some cases debatable. Furthermore, it is shown that the response of some models is strongly influenced by small differences in the applied damage evolution law. A discussion is made on the use of two different types of such evolution laws, which are frequently applied in the literature.

Aerodynamic forces and flow structures of an airfoil in some unsteady motions at small Reynolds number

Abstract: The aerodynamic forces and flow structures of a NACA 0012 airfoil in some unsteady motions at small Reynolds number (Re=100) are studied by numerically solving the Navier-Stokes equations. These motions include airfoil acceleration and deceleration from one translational speed to another and rapidly pitching up in constant freestream (equivalent to pitching up during translational motion at constant speed). It is shown that at small Reynolds number (Re=100), when the airfoil is performing fast acceleration or deceleration from one speed to another, a large aerodynamic force can be generated during and for a time period after the acceleration or deceleration; a large aerodynamic force can also be generated when the airfoil is performing a fast pitching motion in a constant freestream. In these fast unsteady motions, an airfoil in low Re flow can produce a large aerodynamic force as effective as in large Re flow, or the effect of unsteady motion dominates the Reynolds number effect. During the fast unsteady motion of the airfoil, new layers of strong vorticity are formed near the upper and lower surfaces of the airfoil under the previously existing thick vorticity layers, and it is the generation and motion of the new vorticity layers that is mainly responsible for the generation of the large aerodynamic force; the large-scale structure and movement of the newly produced vorticity layers are similar to that of high Re flow.

The effects of in-plane core stiffness on the wrinkling behavior of thick sandwiches

Abstract: This contribution presents a refined analytical solution for the wrinkling of sandwich plates with isotropic face layers and thick orthotropic cores, taking into accountin-plane deformations of the core. A single explicit equation for the critical wrinkling load in an asymptotic sense in derived. The results have been verified extensively by a numerical model [1] and show that, when dealing with highly orthotropic cores (e.g., honeycombs), the wrinkling loads and deformation patterns can strongly depend on the in-plane stiffness of the core. This new theoretical finding which is of considerable practical importance is the main motivation for this paper. Classical wrinkling formulae [2], [3], [4] can lead to significant errors when used in connection with highly orthotropic cores.

On singular features of acoustic wave propagation in weakly dissipative anisotropic thermoviscoelasticity

Abstract: The joint effect of viscosity and heat conductivity on the propagation of plane bulk acoustic waves is studied in the low-loss approximation. Weak dissipation has significant impact on wave properties in the neighborhood of those acoustic axes (directions of phase-speed degeneracy occurring in the absence of dissipation) which are not parallel to symmetry axes and which therefore split into pairs of degenerate complex wave vectors not lying in symmetry planes. The bifurcation nature of the wave branches in the neighboring domain and the drastically changing polarization features are analyzed. Wave solutions containing linear spatial dependence of the pre-exponential amplitude admitted along pairwise directions of wave-vector degeneracy are studied.

Edge crack problem in a semi-infinite FGM plate with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading

Abstract: The temperature distribution in structural elements in practical cases usually changes in two or three directions. Based on such facts, aiming at more effectiveness, a functionally graded material (FGM), whose properties change in two or three directions, is introduced, that investigated here called bi-directional FGM. The current study aims at the formulation, solution and investigation of a semiinfinite edge cracked FGM plate problem with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading. The solution of the boundary value problem that one obtains from the mathematical formulation of the current crack problem under thermal loading reduces to an integral equation with a generalized Cauchy kernel. This integral equation contains many two-dimensional double strongly singular integrals, which can be solved numerically. In order to separate the singular terms and overcome the divergence of the integrals an asymptotic analysis for the singular parts in the obtained integral equation was carried out. Also, the exact solution for many singular integrals is obtained. The obtained numerical results are used in the representation of the thermal stress intensity factor versus the thermal/mechanical nonhomogeneous parameters. The numerical results show that it is possible to reduce and control the thermal stress intensity factor.

Heat transfer characteristics of a non-isothermal surface moving parallel to a free stream

Abstract: Heat transfer from a surface in motion relative to a quiescent or moving fluid occurs in many manufacturing processes such as hot rolling, continuous casting, extrusion, and drawing. In this study, an analysis has been carried out to predict thermal transport occurring in the boundary layer on a non-isothermal flat surface that moves in the same direction of the flowing surrounding fluid. The surface temperature is assumed to have a power-law variation,T w (x)=T ∞ +Ax m . The effect of various governing parameters, such as Prandtl number Pr, wall temperature exponentm, free stream velocityu∞, and the normalized velocity difference|u w −u ∞ |/u r , whereu r is the largest ofu∞ andu w , on the temperature profiles and the Nusselt number are clearly illustrated. For the same wall temperature exponent, Prandtl number, and normalized velocity difference, a higher value of Nusselt number results fromu w >u∞ than fromu w

Some comments on objective rates of symmetric Eulerian tensors with application to Eulerian strain rates

Abstract: In recent years the role of a convenient objective rate of objective quantities has been passionately discussed. There is a large number of well-justified formulations, e.g., [8], [13], [16]. For an overview of some selected derivatives see, e.g., [21]. However, unreliable results obtained in specific computations [11] complicate the right choice. Moreover, from a physical point of view there exist some additional requirements on time derivatives besides the principle of objectivity [5]. In this paper we try to show that there is a need for using corotational rates. For that purpose we give different approaches. In an application to the aforementioned facts we prove that only the Hencky strain [6] can have an objective corotational rate. We do that by identifying the objective strain rate and the deformation rate. Moreover, the spin involved in this rate is the logarithmic spin as defined in [23].

Exact solutions of non-Newtonian fluid flows with prescribed vorticity

Abstract: The equations of motion of a non-Newtonian second-grade fluid flow are highly nonlinear partial differential equations. For this reason, there exists only a limited number of exact solutions. Due to the complexity of the equations, inverse methods described by Nemenyi [1] have become attractive in the study of non-Newtonian fluids. In these methods, certain physical or geometrical properties of the flow field are assumed a priori. Lin and Tobak [2] studied steady plane viscous incompressible flows for a chosen vorticity function by decomposing the nonlinear fourth-order partial differential equation in the streamfunction. This excellent approach yielded two second-order linear partial differential equations in the streamfunction. Hui [3] used this approach to study unsteady plane viscous incompressible flows. During the past decade, there has been substantial interest in flows of viscoelastic liquids due to the occurrence of these flows in industrial processes. In this paper, we study the steady and unsteady incompressible viscous non-Newtonian second-grade fluid flows in which the vorticity is proportional to the streamfunction perturbed by a uniform stream. The solutions obtained are exact solutions and represent various non-parallel flows of second-grade fluids. The plan of this paper is as follows: In Sect. 2, the equations of motion of an unsteady plane incompressible second-grade fluid are given, and the vorticity function is assumed to be∇ 2 ψ=A(ψ−Ux−BUy 2). In Sect. 3, solutions for the steady flow are obtained. In Sect. 4, solutions for unsteady flow are obtained.

Contact zone models for an interface crack in a piezoelectric material

Abstract: A plane strain problem for an interface crack along the fixed edge of a piezoelectric semi-infinite space is examined. Electrically conducting and electrically insulated crack surfaces are considered. By using Fourier transforms the systems of singular integral equations are formulated for both cases. It was found that in the second case for the most commonly used piezoelectric materials instead of oscillating singularity the real singularity of general power type occurs. The dependence of this singularity on the piezoelectric parameters has been investigated. The contact zone model is considered as an alternative one for the case of the oscillating singularity, and the way this model can be used for the investigation of interface cracks in finite size piezoelectrics is suggested.

Axisymmetric dynamics of composite spherical shells with active piezoelectric/composite stiffeners

Abstract: The paper presents a theoretical formulation for spherical shells reinforced by meridional and circumferential stiffeners Active damping of the shell is introduced through control action of piezoelectric coupled pairs bonded to the meridional stiffeners The induced loads can include radial pressure and a thermal field that are independent of the circumferential coordinate Neglecting local deformations between adjacent meridional stiffeners, the response of the shell will be axisymmetric The analysis employs the Donnell-Mushtari-Vlasov version of Love's theory of shells together with a smeared stiffeners technique The paper also considers a particular case of shell mounted piezoelectic coupled pairs without conventional stiffeners A closed form solution is derived for spherical panels without conventional stiffeners within the range of the meridional coordinate between 75° and 90° using a version of the Geckeler approximation

Effect of uniform blowing/suction on MHD-natural convection over a horizontal cylinder: UWT or UHF

Abstract: A laminar boundary-layer analysis is used to investigate the flow and heat transfer characteristics of MHD-natural convection over a horizontal cylinder under the effect of uniform blowing/suction. The surface of the horizontal cylinder is maintained at a uniform wall temperature (UWT) or a uniform heat flux (UHF). The nonsimilar governing equations are obtained by using a suitable transformation and solved by the Keller box method. Numerical results for the dimensionless velocity profiles, the dimensionless temperature profiles, the skin friction coefficient, and the Nusselt number are presented for various values of the dimensionless coordinate ξ, the magnetic parameterM, the Prandtl number Pr, and the blowing/suction parameterf w . Increasing the Prandtl number Pr or the magnetic parameterM and decreasing the blowing/suction parameterf w decreases the skin friction coefficient. The Nusselt number increases (decreases) with an increase inf w or Pr (ξ orM).

Piezoelectric vibrations of composite beams with interlayer slip

Abstract: Actuating piezoelectric effects in two-layer beams with interlayer slip are described in detail, and special attention is given to the identification of the piezoelectric actuation as eigenstrains. It is demonstrated that piezoelectrically induced strains conveniently can be interpreted as eigenstrains acting in a background composite beam without piezoelectric actuators. The analogy between the piezoelectric effect and that of thermal strains is utilized in the present paper, where a layer-wise first-order flexural theory is applied to two-layer beams with various boundary conditions. The layers are assumed to be made of piezoelectric materials. Bernoulli-Euler hypothesis is assumed to hold for each layer separately, and a linear constitutive relation between the horizontal slip and the interlaminar shear force is considered. The governing sixth-order initial-boundary value problem is solved by separating the dynamic response in a quasistatic and in a complementary dynamic portion. The quasistatic solution that may also contain singularities or discontinuities due to sudden load changes is determinded in a closed form. The remaining complementary dynamic part is nonsingular and can be approximated by a truncated modal series of accelerated convergence. The proposed procedure is illustrated for piezoelectrically induced flexural deformations, where the forcing function is the piezoelectric curvature.

On the dynamic behavior of honeycomb based composite solids

Abstract: The aim of this contribution is twofold. First, a dispersive model of periodic composite solids made of an isotropic matrix reinforced by a hexagonal system of slender fibres or by a honeycomb-like slender skeleton is formulated. Second, this model is applied to the analysis of vibration and wave propagation problems in the above honeycomb based composites. Contrary to the known homogenized models the main feature of the proposed model is that it describes the effect of cell size on the overall dynamic behavior of a composite solid. It is shown that on the macro-level the response of honeycomb based composites is isotropic. It is also proved that there exist dispersive dilatational-type and shear-type waves, which can propagate in these composites. Simple formulae for lower and higher free vibration frequencies are derived, and the existence of certain restrictons imposed on the physically allowable wave propagation speeds is shown.

Momentum and heat transfer from a continuous moving surface to a power-law fluid

Abstract: Momentum and heat transfer from a continuous moving surface with an arbitrary surface velocity distribution and uniform surface temperature in a power-law fluid have been considered. Using a coordinate transformation, the boundary layer equations are reduced to a simple form. Modified Merk's series method has been used for momentum equation and universal function approach for energy equation. The resulting equations have been integrated numerically by using fourth-order Runge-Kutta method and method of continuation. Two types of plate velocity distributions are considered: (i) surface velocity proportional to positive power of distance from the slot, (ii) linearly stretched velocity distribution with nonzero slot velocity. It is found that the displacement thickness is much thicker for pseudoplastic fluids than for Newtonian and dilatant fluids for both cases. The local Nusselt number, obtained by the universal function method, has been compared with non-similar results. The results are in good agreement.

Three-dimensional analysis of an antiparallel piezoelectric bimorph

Abstract: Three-dimensional electromechanical responses of a piezoelectric bimorph are studied. The bimorph is antiparallel in the sense that it consists of two identical, plate-like piezoelectric elements with opposite poling directions. Both the top and bottom surfaces of the bimorph are fully covered with negligibly thin conductive electrodes. By introducing a small parameter and using the transfer matrix method it is shown that a three-dimensional solution of the problem can be readily constructed, provided the solution to a set of two-dimensional equations very similar to those in the classic plate theory is obtainable. The three-dimensional solution satisfies all the field equations as well as the boundary conditions on the major surfaces and at the interface between the two piezoelectric plates. In many special cases, the electric edge condition can be fulfilled point by point, and thus the solution is exact in Saint-Venant's sense. The formulation and new analytical results for a strip-shaped cantilever bimorph under the action of applied voltage and end moment are presented.

Effects of throughflow and internal heat generation on the onset of convection in a fluid layer

Abstract: The throughflow and internal heat generation effects on the onset of convection in an infinite horizontal fluid layer are investigated. The boundaries are considered to be rigid (however permeable) and perfectly conducting. The resulting eigenvalue problem is solved by using the Galerkin method, and the effects of various parameters in the stability results are analyzed. The results indicate that the stability of the system is significantly affected by both throughflow and internal heat generation in the fluid layer. The Prandtl number comes into play due to the presence of throughflow and it has a profound effect on the stability of the system. It is found that, in the presence of internal heating, throughflow in one direction supresses convection while throughflow in the other direction encourages it.

Fluid flow and heat transfer analysis of Couette flow in a composite duct

Abstract: This paper deals with a fully developed Couette flow through a composite channel, partially filled with a clear fluid and partially with a fluid-saturated porous medium. The flow occurs because of a moving plate, which bounds the clear fluid region. The porous region is bounded by a fixed plate. Assuming that the moving plate is adiabatic, and that the fixed plate is subject to a uniform heat flux, a boundary layer solution for the velocity and temperature fields is obtained.

Laminar boundary-layer separation over a circular cylinder in uniform shear flow

Abstract: By appealing to the classical boundary-layer theory, the present paper investigates the effect of the freestream shear on the separation of the laminar boundary layer around a circular cylinder. It is shown that on the side of the cylinder with faster freestream velocity the location of the separation point (point of vanishing wall shear) is virtually unaffected by the freestream shear, while on the other side of the cylinder a critical shear rate is observed. Below this critical value, separation occurs typically at the rear surface of the cylinder and is found to shift towards the downstream direction with increasing freestream shear. Above the critical shear rate, the boundary layer separates from the windward side of the cylinder. Further increase of the freestream shear then causes the separation point to move towards the upstream direction. The present findings may have important implication on the issue regarding to the orientation of the lift force exerting on the cylinder.

Natural convection flow of a viscous fluid about a truncated cone with temperature dependent viscosity

Abstract: The two-dimensional natural convection flow of a viscous incompressible fluid with temperature dependent viscosity about a truncated cone is considered. The governing equations for the flow are obtained by using suitable transformations and solved by using an implicit finite difference method. Perturbation solutions are obtained near the leading edge and in the downstream regime, and the results are obtained in terms of the local skin friction coefficient and the local Nusselt number. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The dimensionless velocity profiles and viscosity distribution are also presented graphically for various values of the viscosity variation parameter, ɛ, for small values of the Prandtl number appropriate for liquid metals.