Showing papers in "Acta Mechanica in 1995"

The dynamics of avalanches of granular materials from initiation to runout. Part II. Experiments

Abstract: This paper describes a model to predict the flow of an initially stationary mass of cohesionsless granular material down a rough curved bed and checks it against laboratory experiments that were conducted with two different kinds of granular materials that are released from rest and travel in a chute consisting of a straight inclined section, a curved segment that is followed by a straight horizontal segment. This work is of interest in connection with the motion of landslides, rockfalls and ice and dense flow snow avalanches. Experiments were performed with two different granular materials, nearly spherical glass beads of 3 mm nominal diameter, Vestolen particles (a light plastic material) of lense type shape and 4 mm nominal diameter and 2,5 mm height. Piles of finite masses of these granular materials with various initial shapes and weight were released from rest in a 100 mm wide chute with the mentioned bent profile. The basal surface consisted of smooth PVC, but was in other experiments also coated with drawing paper and with sandpaper. The granular masses under motion were photographed and partly video filmed and thus the geometry of the avalanche was recorded as a function of position and time. For the two granular materials and for the three bed linings the angle of repose and the bed friction angle were determined. The experimental technique with which the laboratory avalanches were run are described in detail as is the reliability of the generated data. We present and use the depth-averaged field equations of balance of mass and linear momentum as presented by Savage and Hutter [28]. These are partial differential equations for the depth averaged streamwise velocity and the distribution of the avalanche depth and involve two phenomenological parameters, the internal angle of friction, o, and a bed friction angle, δ, both as constitutive properties of Coulomb-type behaviour. We present the model but do not derive its equations. The numerical integration scheme for these equations is a Lagrangian finite difference scheme used earlier by Savage and Hutter [27],[28]. We present this scheme for completeness but do not discuss its peculiarities. Comparison of the theoretical results with experiments is commenced by discussing the implementation of the initial conditions. Observations indicate that with the onset of the motion a dilatation is involved that should be accomodated for in the definition of the initial conditions. Early studies of the temporal evolution of the trailing and leading edges of the granular avalanche indicate that their computed counterparts react sensitively to variations in the bed friction angle but not to those of the internal angle of friction. Furthermore, a weak velocity dependence of the bed friction angle, δ, is also scen to have a small, but negligible influence on these variables. We finally compare the experimental results with computational findings for many combinations of the masses of the granular materials and bed linings. It is found that the experimental results and the theoretical predictions agree satisfactorily. They thus validate the simple model equations that were proposed in Savage and Hutter [28].

Exact solutions for some simple flows of an Oldroyd-B fluid

Abstract: We present two simple but elegant solutions for the flow of an Oldroyd-B fluid. First, we consider the flow past an infinite porous plate and find that the problem admits an asymptotically decaying solution in the case of suction at the plate, and that in the case of blowing it admits no such solution. Second, we study the longitudinal and torsional oscillations of an infinitely long rod of finite radius. The solutions are found in terms of Bessel functions.

Harmonic differential quadrature method and applications to analysis of structural components

Abstract: A harmonic differential quadrature (HDQ) method with application to the analysis of buckling and free vibration of beams and rectangular plates is presented A new approach is proposed for the determination of the weighting coefficients for differential quadrature It is found that the HDQ method is more efficient than the ordinary differential quadrature (DQ) method, especially for higher order frequencies and for buckling loads of rectangular plates under a wide range of aspect ratios Also, some shortcomings existing in theDQ method are removed

An exact solution of the Navier-Stokes equations for magnetohydrodynamic flow

Abstract: Viscous flow past a stretching sheet in the presence of a uniform magnetic field is considered. An exact similarity solution for velocity and pressure of the two-dimensional Navier-Stokes equations is presented, which is formally valid for all Reynolds numbers. The solution for the velocity field turns out to be the identical solution derived earlier by Pavlov [1] within the framework of high-Reynolds-number boundary layer theory, in which the pressure distribution cannot be determined.

The effect of temperature-dependent viscosity on free-forced convective laminar boundary layer flow past a vertical isothermal flat plate

Abstract: The effect of a temperature-dependent viscosity on an incompressible fluid in steady laminar free-forced convective boundary layer flow over an isothermal vertical semi-infinite flat plate is studied. The local similarity solution is used to transform the system of partial differential equations, describing the problem under consideration, into a boundary value problem of coupled ordinary differential equations and an efficient numerical technique is implemented to solve the reduced system. Numerical calculations are carried out, for various values of the dimensionless parameters of the problem, which include a Prandtl number, a mixed convection parameter and a viscosity/temperature parameter. The results are presented graphically and the conclusion is drawn that the flow field and other quantities of physical interest are significantly influenced by these parameters. In particular, it is concluded that when the viscosity of a working fluid is sensitive to the variation of temperature, care must be taken to include this effect, otherwise considerable error can result in the heat transfer processes. In the literature, such care is not always evident.

Hall effects on magnetohydrodynamic boundary layer flow over a continuous moving flat plate

Abstract: Hall effects on magnetohydrodynamic boundary layer flow over a continuous moving semiinfinite flat plate in a viscous incompressible electrically conducting fluid are investigated when the liquid is permeated by a uniform transverse magnetic field on the assumption of a small magnetic Reynolds number. The transformed governing equations have been solved numerically using a difference-differential scheme in combination with an iterative method in solving the resulting ordinary differential equations. The velocity profiles and skin friction on the plate are computed and discussed in detail for various values of the Hall and magnetic parameters. The present analysis is more general than any previous investigation.

Plane surface suddenly set in motion in a non-Newtonian fluid

Abstract: The flow field of a fluid being called “the third order fluid” or the “fluid of grade three” is considered for a non-Newtonian flow in the vicinity of a plane wall, suddenly set in motion. The velocity field of the flow is described by a fourth order, non-linear partial differential equation. The solution of this differential equation shows that for short time a strong non-Newtonian effect is present in the velocity field. However, for long time the velocity field becomes a Newtonian one.

The existence of Taylor vortices and wide-gap instabilities in spherical Couette flow

Abstract: The flow of a viscous incompressible fluid in the gap between two concentric spheres was investigated for the case where only the inner sphere rotates and the outer one is stationary. Flow visualization studies were carried out for a wide range of Reynolds numbers (Re≦2·105) and aspect ratios (0.08≦β≦0.5) to determine the instabilities during the laminar-turbulent transition and the corresponding critical Reynolds numbers as a function of the aspect ratio. It was found that the laminar basic flow loses its stability at the stability threshold in different ways. The instabilities occurring depend strongly on the aspect ratio and the initial conditions. For small and medium aspect ratios (0.08≦β≦0.25), experiments were carried out as a function of the Reynolds number to determine the regions of existence for basic flow, Taylor vortex flow, supercritical basic flow and furthermore, to give the best fit for the maximum number of pairs of Taylor vortices as a function of aspect ratio. For wide gaps (0.33≦β≦0.5), however, Taylor vortices could not be detected. The first instability manifests itself as a break of the spatial symmetry and non-axisymmetric secondary waves with spiral arms appear depending on the Reynolds number. For β=0.33, secondary waves with an azimuthal wave numbern=six, five and four were found, while in the gap with an aspect ratio of β=0.5 secondary waves withn=five, four and three spiral arms exist. Frequencies of these secondary waves were measured, the corresponding critical Reynolds numbers and the transition Reynolds numbers during the transition to turbulence were found. The flow modes occurring at the poles look very similar to those found in the flow between two rotating disks. Effects of non-uniqueness and hysteresis were determined as a function of the acceleration rate.

A heat-flux dependent theory of thermoelasticity with voids

Abstract: A thermoelasticity theory for elastic materials with voids is formulated. This theory includes the heat-flux among the constitutive variables and assumes an evolution equation for the heat-flux. A class of mixed initial boundary value problems is defined and the uniqueness is established.

An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate

Abstract: This paper concerns an asymptotic analysis of the dispersion relation for wave propagation in a pre-stressed incompressible elastic plate. Asymptotic expansions for the wave speed as a function of wavenumber and pre-stress are obtained. These expansions have important potenatial applications to many dynamic problems such as impact problems. It is shown that in the large wavenumber limit the wave speed of the fundamental modes of both symmetric and anti-symmetric motions tends to the associated Rayleigh surface wave speed, on the other hand, the wave speeds of all the harmonics tend to a single limit which is the corresponding body wave speed. It is also shown that, whereas the fundamental modes are very sensitive to changes in the underlying pre-stress, the harmonics are little affected by such changes, espcially in the small and large wavenumber limits.

The nonuniqueness of the MHD flow of a viscoelastic fluid past a stretching sheet

Abstract: Two closed-form solutions were found for the boundary layer equations of the title problem. Discussions are made to trace among them the physically realistic solution.

Nonclassical thermal effects in Stokes' second problem

Abstract: The MCF model is used to study the nonclassical heat conduction effects in Stoke's second problem. The structure of the waves and the influence of the thermal relaxation time on the temperature and velocity fields are investigated. The displacement thickness, skin friction and the rate of the heat transfer at the plate are determined.

Reduction of mesh sensitivity in continuum damage mechanics

Abstract: Continuum damage theories can be applied to simulate the failure behaviour of engineering constructions. In the constitutive equations of the material a damage parameter is incorporated. A damage criterion and a damage evolution law are postulated and quantified based on experimental data. The elaboration of the mathematical formulation is performed by common finite element techniques. Without special precautions the numerical results appear to be unacceptably dependent on the measure of the spatial discretization. It is shown that a simple but effective procedure leads to the conservation of objectivity.

On the role of strain gradients in adiabatic shear banding

Abstract: The effect of higher order strain gradients on adiabatic shear banding is investigated by considering the simple shearing of a heat conducting thermoviscoplastic material with a gradient-dependent flow stress. The competition between the gradient-dependent plastic flow and heat conduction and their influence on the shear band width and structure are examined. Two internal length scales, i.e., the deformation internal length and the thermal internal length, are incorporated into the linear stability analysis, which shows that the band width size scales either with the square root of the strain gradient coefficient (in the absence of heat conduction) or the thermal conductivity (in the absence of strain gradients), respectively. The numerical computation for the nonlinear problem reveals that the “diffusive” effect of the strain gradient is much stronger than that of the heat conduction and dictates the constitutive response of the material in the postlocalization regime, and shows that the deformation length scale is much larger than the termal length scale. The band width predicted by the gradient theory agrees reasonably well with the experimental observations found in the literature.

Vibration of pretwisted cantilever trapezoidal symmetric laminates

Abstract: A first known investigation on the vibratory characteristics of pretwisted composite symmetric laminates with trapezoidal planform is presented. A governing eigenvalue equation is derived based on the Ritz minimization procedure. This formulation shows that bending and stretching effects of these symmetric laminates are coupled by the presence of twisting curvature. For the solution method, a set of orthogonally generated shape functions is employed to approximate the transverse and in-plane displacements. These functions are generated through a proposed recurrence formula. During the orthogonalization process, a basic function is introduced to ensure the satisfaction of the essential boundary conditions. This proposed method is applied to determine the vibration response of the titled problem. The effects of angles of twist and laminated layers upon the vibration frequencies are examined. Convergence tests for selected examples are presented in which the accuracy of the results is established. It has been found that an increase in the angle of twist also strengthens the torsional stiffness of the laminated plates and therefore results in higher twisting frequencies. The present solutions, where possible, are verified by comparison with data of simplified examples that are available in the literature.

Exact analysis for three-dimensional free vibrations of cross-ply cylindrical and doubly-curved laminates

Abstract: The free vibrations of simply-supported cross-ply cylindrical and doubly-curved laminates are investigated. The three-dimensional equations of motion, expressed in terms of displacements, are reduced to a system of coupled ordinary differential equations, which are then solved using the power series method. Numerical results are presented for cylindrical, spherical and saddle-shape laminates having a various number of layers.

Constitutive modelling for granular media using an anisotropic distortional yield model

Abstract: It has been attempted here to model the mechanical response of granular media through a combination of analytical, numerical and experimental techniques. Based on the experimental evidence obtained from a series of triaxial laboratory experiments on specimens made of glass beads, the model attempts to simulate the movement of yield surfaces and the stress-strain relationships. It is found that the yield surface distorts in the direction of loading in a manner analogous to that of metals. An anisotropic distortional yield model is formulated in order to describe the experimental behavior exhibited by these granular media. From the experimental yield surfaces the parameters of the model have been evaluated and a hardening rule, based on the Phillips rule, has been determined. Associativity on the π-plane has been observed experimentally. Using these concepts the constitutive formulation has been presented and the stress-strain curves have been generated. From the comparisons of these curves we observe a good correlation between the model and the experimental observations.

Thermodynamics of fluid-saturated porous media with a phase change

Abstract: In the present paper some aspects of the thermodynamics of a fluid-saturated porous solid, taking into account the phase change of liquid into gas, are discussed. Based on the principles of the thermodynamics of irreversible processes, restrictions for the constitutive equations of the constituents are developed, in particular for the phase transition. The porous body is assumed to be incompressible and elastically deformed, and the pore space is filled with an immiscible liquid-gas mixture (moisture). The theory presented here provides some background that can be found suitable, e.g., for drying aspects.

Elastic moduli of heterogeneous solids with ellipsoidal inclusions and elliptic cracks

Abstract: The influence of ellipsoidal inclusions and elliptic cracks on the overall effective moduli of a two-phase composite and of a cracked body, respectively, is investigated by means of Mori-Tanaka's theory for three types of inclusion and four types of crack arrangements: monotonically aligned, 2-D randomly oriented (two kinds for cracks), and 3-D randomly oriented. The effective moduli of the composite in the aligned case are known to coincide with Willis' orthotropic lower (or upper) bounds with a two-point ellipsoidal correlation function if the matrix is the softer (or harder) phase. With 2-D randomly oriented inclusions, the effective moduli are examined under Willis' transversely isotropic bounds with a two-point spheroidal correlation function, and it is found that, as the cross-sectional aspect ratio of the ellipsoidal inclusions flattens from circular shape to disc-shape, the two effective shear moduli and the plane-strain bulk modulus all lie on or within the bounds. The effective bulk and shear moduli of an isotropic composite containing randomly oriented ellipsoidal inclusions also fall on or within Hashin-Shtrikman's bounds as the shape of the ellipsoids changes. The obtained moduli are then extended to a cracked body containing elliptic cracks, which are generated by compressing the thickness of ellipsoidal voids to zero. It is found that only selected components of the effective moduli are dependent upon the crack density parameter η. Their dependence on η and the crack shape γ are explicitly established.

Forchheimer and Brinkman extended Darcy flow model on natural convection in a vertical cylindrical porous annulus

Abstract: In the present numerical work, the Forchheimer and Brinkman extended Darcy flow model is used for studying the natural convection heat transfer in a vertical cylindrical porous annulus. Forchheimer inertial and Brinkman viscous terms have been characterized by two non-dimensional numbers. The present description renders the formulation suitable for vertical annuli as well as for rectangular cavity. Numerical results obtained by SAR scheme indicate, that Brinkman viscous terms lead to a higher decrease in the average Nusselt number compared to the Forchheimer inertial terms. Numerical results obtained with the present non-Darcy flow model are in good agreement with the available experimental results of a high permeability porous medium for which results obtained with the Darcy flow model show considerable deviation.

Influence of large deflections on the dynamic stability of nonlinear viscoelastic plates

Abstract: The dynamic stability analysis of nonlinear viscoelastic plates is presented. The problem is formulated within the large deflections theory for isotropic plates, and the Leaderman representation of nonlinear viscoelasticity for the material behavior. The influence of the various parameters on the stability/instability possible situation is investigated within the concept of the Lyapunov exponents. In addition, it is shown that in some cases the system has a chaotic behavior.

The concept of physical metric in thermodynamics

Abstract: In this paper we introduce a new geometric object, the physical metric,g, for the measurement of distance between material particles in the reference configuration. We then treatg as an internal variable and thereby obtain hereditary constitutive equations for dissipative solids under conditions of large deformation. We demonstrate thatg is in fact a measure of non-affine deformation and introduce the concept of the hypercontinuum which allows the existence of multiple velocities in a material neighborhood, thereby leading to a deformation field with multiple metrics.

Effects of Hall currents and Coriolis force on Hartmann flow under general wall conditions

Abstract: The influence of Hall currents and rotation on a generalized Hartmann flow and heat transfer is investigated. The channel is rotating with a constant angular velocity around an axis perpendicular to the walls in a uniform transverse magnetic field. The walls may have the same thickness, electrical and thermal conductivity as well as Hall parameter or different ones. The flow may be driven either by a pressure gradient or by motion of one of the walls. Exact solutions are derived for the velocity, magnetic field, viscous stress, current density, temperature distribution, yield components, electric current components, mean temperature as well as Nusselt numbers. Representative numerical results are presented in diagrams and effects of different parameters are discussed.

On the interaction of solitary waves of flexure in elastic rods

Abstract: A numerical study is made to determine whether the partial differential equations governing planar, twist-free motions in a theory of the dynamics of elastic rods are completely integrable in the sense of soliton theory. The theory in which the equations arise is due to Kirchhoff and Clebsch and is complete to within an error of order two in an appropriate dimensionless measure of thickness and strain. A recently developed energy-preserving finite-difference scheme is employed to determine the consequences of the interaction of solitary traveling waves, which, in the present twist-free case, are loops traveling at constant speed. It is found that the change induced in such a loop-wave upon collision with another is more than a shift in phase.

Vibration of stiffened skew Mindlin plates

Abstract: This paper presents a formulation for the free vibration analysis of skew Mindlin plates with intermediate parallel stiffeners attached in two directions The Mindlin theory is used to account for the effects of transverse shear deformations and rotary inertia of the plate while the Engesser theory associated with the consideration of torsion is employed for stiffeners Based on these two theories, the energy functionals for the plate system have been derived To obtain the vibration frequencies, these energy functionals are minimized with the shape functions assumed in a set of two-dimensional mathematically complete polynomials This procedure has been implemented numerically to compute the vibration solutions Convergence studies have been performed to verify the accuracy of this method Sets of first known results have been presented for several stiffened plate structures

Heterogeneity and implicated surface effects : statistical, fractal formulation and relevant analytical solution

Abstract: The aim of this paper is to examine the implications of material heterogeneity on brittle material response and on relevant surface effects. Statistical and fractal concepts are used for this purpose. In the statistical formulation the displacement gradients of the micro-medium are considered to be random fields characterized by stationary exponential or Gaussian auto-covariance and by the relevant correlation length or scale of fluctuation. Through Taylor series expansion around the mean of the random field, an important analogy is found between the statistical formulation and the micro-structural theory, originally introduced by Mindlin, where higher order gradients of deformation appear in the constitutive equations. The analogy is valid only when fluctuations are small, so that some simplifications are allowed. It is found that the so-called internal length appearing in the micro-structural formulation is analogous to the correlation length in the statistical one. In the statistical approach there are no extra boundary conditions in the formulation, as is the case when higher order gradients are introduced. However, what is know as “conditioning” of the field at the boundaries effects its behavior near/on them. The statistical approach can provide further information in the form of higher order moments not captured by the gradient theory. Material/structure response is strongly dependent on the aforementioned scale. Its effect is most pronounced near the boundaries of a structure where its role on surface related phenomena is paramount. In order to study heterogeneity at a hierarchy of scales, i.e. absence of characteristic length, complex disorderly system, fractal concepts and relevant power decay laws are considered. The formulation introduces the fractal dimension of the heterogeneous displacement gradient of the micro-medium, a length describing the overall size of the structure, and the lower cutoff of the scaling law. The physical interpretation of the lower cutoff is the lower limit of applicability of the power scaling law. Mathematically it is important since in this case the fractal can be “followed” in the spatial domain. Similarly to the statistical case, an analogy between the fractal formulation and gradient theories is identified. No extra boundary conditions appear in the fractal formulation. However, there are still open questions with respect to the behavior of a fractal after conditioning, as is the case on boundaries. The analytical solution of a relevant surface instability problem for the gradient, statistical, and fractal formulation is presented. The solution was obtained through symbolic computations by computer because the analytical work is tedious and error prone. The analytical solution provides significant insight into the problem of heterogeneity and skin effects in brittle materials, internal length estimation, and the role of fractal scaling properties. Finally, the concepts introduced herein are discussed with respect to experimental information and numerical implementation.

Dynamic response of cross-ply laminated circular cylindrical shells with various boundary conditions

Abstract: Closed-form solutions of the dynamic response of cross-ply laminated circular cylindrical shells are developed for arbitrary boundary conditions and under arbitrary loadings. The equations of motion of the classical, first-order and third-order theories are converted into a single-order system of equations by using state variables. To solve for the dynamic response, the biorthogonality conditions of principle modes of the original and adjoint eigenfunctions are used to decouple the state space equation. The study reveals that the disagreement between shear deformation theories in much less than the disagreement between them and the classical theory.

Free vibration of cantilevered cylinders: effects of cross-sections and cavities

Abstract: This study investigates the influence of a cavity on the natural frequencies and mode shapes of a homogeneous, isotropic, elastic solid cylinder. Cavities of different shapes and sizes are often found in engineering applications to provide access, reduce weight and to cut cost in production. As a result, detailed quantitative solutions to this class of problem will be of great interest to engineering practitioners. In this study, a highly efficient and accurate numerical algorithm is proposed to examine the vibrations of elastic solid cylinders with a deep cavity. The technique uses the exact theory of three-dimensional elasticity in conjunction with the Ritz form of minimum energy principle to derive the governing eigenvalue equation. Within the context of three-dimensional elasticity, displacement functions composed of a set of twodimensional lateral surface functions and one-dimensional longitudinal functions are defined for each displacement component. The orthogonality inherent in these functions has resulted in well form eigen value matrices which can be manipulated and solved with ease. The technique yields upper-bound natural frequencies and three-dimensional deformed mode shapes for a wide range of elastic hollow cylinders with an arbitrary cross section. Most of the results presented herein are believed to be new to the existing literature.

The ray method for solving boundary problems of wave dynamics for bodies having curvilinear anisotropy

Abstract: The ray method for solving boundary problems involving the propagation of surfaces of strong discontinuity in curvilinear anisotropic media is developed. The method employs the solution behind the surface of strong discontinuity which is constructed in terms of power series (ray series). Shock subjections to boundaries of cavities in transversely isotropic bodies having cylindrical, toroidal and spherical anisotropies are investigated. It is demonstrated that in some cases the uniform validity of truncated ray series is not impaired over domains of the wave motion existence. For their improvement, the authors suggest a new method, the method of forward-area regularization, which allows one to extend the truncated ray series over the whole region of the wave solution existence.

A mathematical model of suspension with saltation

Abstract: A theoretical approach to the treatment of wind erosion data, particularly from a wind tunnel, is presented. Considerations are given to the utilisation of a real data set in validation of the model, data that will be presented in a forthcoming paper. Following this, the physics of particle suspension, saltation and the turbulent boundary layer are examined. Two different mathematical models evolve: one considers only suspension, another evokes Bagnold's observation that eroding material merely shifts the velocity profile and the effect of the airborne material on the effective density of the air parcel. These produce a final, relatively simple expression that credibly fits the data of Gerety and Slingerland. A critique of the approach reveals it to be an adequate expression of the known mechanisms of suspension and saltation. Derived algebraic forms for integrated collectors show several of the same “logarithmic power” dependences. Importantly, the results show little influence of saltation itself on the profile. It appears that the saltation process is responsible for a feedback such that the eddy diffusion process for particle movement is effectively enhanced. The combination of an appropriate correction of the pitot data (following Scott and Carter) and a complete mass balance has removed the “kink” from the velocity profile and also the need to consider the saltation process itself in the particle mass balance.