Showing papers in "Acta Mechanica in 1981"

Nonsteady flow through porous media in the presence of a threshold gradient

Abstract: In this paper the threshold gradient effect in the nonsteady flow through a porous media is investigated and its role on the pressure and flow rate distributions in a flow system is evaluated. The system considered in this investigation is a linear oil reservoir. Two situations of practical interest, i.e. reservoir depleted at a constant pressure and at a constant flow rate, are considered. The solutions in a closed form for this flow system are obtained. A comparison between the pressure distributions obtained analytically, numerically and by the integral method is also presented. In order to illustrate the effect that threshold gradient would have on the flow behavior, several numerical examples are given. The results derived from this investigation show that the flow behavior in the presence of a threshold gradient depends strongly on two dimensionless parameters. These parameters are directly related to the threshold gradient, the physical properties of the porous medium and the flow parameters at the outface flow.

The impulsive motion of a flat plate in a viscoelastic fluid

Abstract: In this note the flow near a wall suddenly set in motion has been studied for a particular class of non-Newtonian viscoelastic fluids. For the description of such a fluid one has used the l~ivlin-Ericksen constitutive equation. Only the first three material constants have been taken into consideration. Because it is not possible in this cas9 to obtain similarity solutions a series expansion with respect to a non-similarity parameter will he given. Finally the velocity distribution with its separate functions will be shown.

The infinitesimal group of the Navier-Stokes equations

Abstract: The Navier-Stokes equations for an incompressible viscous fluid admit time translation, time dependent change of the pressure origin, a scale change, rotation of axes, and time dependent spatial translation. No other transformations appear if dependence on derivatives is allowed.

Unsteady hydromagnetic laminar flow of a conducting dusty fluid between two parallel plates started impulsively from rest

Abstract: The unsteady laminar motion of an electrically conducting, viscous and incompressible dusty fluid between two infinitely extended non-conducting parallel plates under a uniform transverse magnetic field, fixed relative to the fluid has been considered. The lower and the upper plate are started impulsively from rest and thereafter move with different but uniform velocities. The velocity fields for the conducting dusty fluid and non-conducting dust particle have been obtained in terms of three non-dimensional parametersl (concentration), σ (relaxation time parameter) andM (Hartmann number). The expressions for the discharge per unit breadth of the plate and the skin-friction at the lower plate are calculated. It is observed, from numerical calculations, that as the Hartmann number increases velocities of the dusty gas and dust particle increase when both the plates are in motion (velocity of the upper plate being equal to, greater than and less than that of the lower plate in the same direction) and decrease in case of Couette motion.

Closed-Form Solutions for Finite Length Crack Moving in a Strip Under Anti-Plane Shear Stress

Abstract: In this paper exact expressions for the anti-plane dynamic stress distributions around finite length cracks propagating with constant velocity in infinitely long finite width strips are determined. Two cases of practical importance are investigated. Firstly, the lateral boundaries of the strip are clamped and displaced in equal and opposite directions, to produce anti-plane shear resulting in a tearing motion along the leading edge of the crack and, secondly, the lateral boundaries of the strip are subjected to shearing stresses. Employing Fourier transforms the solution of each problem is reduced to solving a pair of dual integral equations. Closed-form solutions of these integral equations are obtained leading to exact expressions for the stress intensity factors. Numerical results are presented in graphical form.

Application of the Green's function method to thin elastic polygonal plates

Abstract: Recently, the Green's function method has been applied successfully to problems of plane elasticity, using influence functions of some finite basic domain of simple geometrical shape, which contains the given one as a subdomain. The result of this formulation is a pair of integral equations, which have to be defined only along that part of the boundary not coinciding with the border of the basic domain. A rather general formulation for the solution of bending of plates of arbitrary convex planform and loading is presented, where, for the sake of brevity, plates of polygonal shape are considered. The polygonal plate is embedded in a rectangular domain, thereby applying coincidence of boundaries as far as possible. Those boundary conditions in the actual problem, which are not already satisfied, lead to a pair of coupled integral equations for a density function vector with components to be interpreted as line loads and moments distributed in the basic domain along the actual boundary. Thus, the kernel is the corresponding Green's matrix. Hence, having solved the integral equations, deflections and stresses in the actual problem are explicitly known. Solution of the integral equations is generally achieved by a numerical procedure. The method is tested in example problems by considering a trapezoidal plate under various boundary conditions.

Transient hydromagnetic convective flow in a rotating channel with porous boundaries

Abstract: The paper studies the combined free and forced convection in a rotating, viscous, incompressible fluid confined between two parallel porous plates. Assuming that the temperature varies linearly along the plates and the pressure gradient maintained uniform over the planes parallel to the plates, the velocity, temperature and the stresses are calculated analytically. Their behaviours for different values of the parameters viz., the Hartmann number, the Grashoff number and the suction parameter etc., are discussed graphically giving out the interplay between the various forces.

Determination of the polar equilibrium properties of the ferroelectric ceramic PZT 65/35

Abstract: The mechanical, piezoelectric and dielectric properties of ferroelectric ceramics can be quite complicated. These materials exhibit both instantaneous and transient responses depending on various physical mechanisms such as dipole dynamics and domain switching. Hence the constitutive relations for the stress and the electric displacement which have been formulated are quite complicated and their characterization can be rather intricate. These constitutive relations must also describe the equilibrium responses of the ceramics when the transient processes have come to rest. In the paper, we show that knowledge of the sound speeds of a virgin and a poled specimen of a ferroelectric ceramic and certain associated properties of the hysteresis and butterfly loops permits the characterization of its equilibrium material properties in the polar direction consistent with an explicit version of the constitutive relations for the stress and the electric displacement.

Analytical approximations to the solution of the Blasius equation

Abstract: Direct integration of the classical Blasius equation yields a weighted integral condition for the flow. The accuracy of the solution is improved dramatically when the series is rewritten to avoid the region of divergence. The first term of this new series approximates the stress accurately to 0(10{-4}).

One dimensional polar responses of the electrooptic ceramic PLZT 7/65/35 due to domain switching

Abstract: The constitutive relations and rate laws proposed by Chen and Peercy [1] have been successfully implemented in describing the one dimensional polar mechanical and dielectric responses of the electrooptic ceramic PLZT 7/65/35 to a slowly varying cyclic electric field. In particular, we examine the consequences of domain switching and determine the results associated with the butterfly and hysteresis loops. The agreement between the numerical and the experimental results is quite good.

Flüssigkeitsschwingungen mit freier Oberfläche in keilförmigen Behältern

Abstract: Es werden die naturlichen Frequenzen einer reibungsfreien und inkompressiblen Flussigkeit mit freier Oberflache in keilformigen Behaltern mit kleinem Offnungswinkel bestimmt. Druckverteilung, Oberflachenauslenkung, Flussigkeitskrafte und Momente konnen aus dem Geschwindigkeitspotential bestimmt werden. Fur erzwungene Schwingungen der Anregung inx- undz-Richtung sowie fur Rotationsanregung ϕ des Behalters konnen Krafte und Momente der Flussigkeit abgeleitet werden.

Axisymmetric and non-axisymmetric convection in a cylindrical container

Abstract: Benard convection within a cylindrical box heated from below was investigated experimentally using interferometric measuring devices and three-dimensional evaluation techniques. Nonaxisymmetric longitudinal convection rolls have been observed atquasistatic heating rates and for diameter-to-height ratios of 6 and 6.8. The quantitatively evaluated density profiles show that the flow pattern is three-dimensional.

Dynamic and incremental variational inequality principles, differential inclusions and their applications to co-existent phases problems

Abstract: The aim of the present paper is the derivation of some classes of variational principles in form of variational inequalities (unilateral problems), with respect to material laws and boundary conditions including strong physical nonlinearities and discontinuities describing the co-existence, of many phases, for instance Masson-Savart effects, sliding and adhesive friction phenomena etc. To treat the appearing nondifferentiable “potentials” and inequality subsidiary conditions, the “subdifferentiability”, a concept of convex analysis, and the related notion of “superpotential” is used. Some classes of unilateral dynamic problems are further discussed and the respective variational inequalities and differential inclusions are derived for several combinations of material laws and boundary conditions. For the respective static problems the propositions of minimum potential and complementary energy are proved and some uniqueness and existence questions are discussed. Finally some applications illustrate the theory.

A new method for the numerical solution of singular integral equations appearing in crack and other elasticity problems

Abstract: The solution of many elasticity problems and particularly crack problems can be reduced to the solution of a singular integral equation with a Cauchy type kernel (or a system of such equations). In this paper a new method of numerical solution of singular integral equations with Cauchy type kernels is proposed, based on the replacement of the unknown function by another function. This method presents some advantages over the previous methods under appropriate conditions. An application of the method to a crack problem in plane elasticity is also made.

The response equations for a cross-tie track

Abstract: Early analyses of the lateral response of cross-tie railroad tracks are based on the assumption that the rail-tie structure responds like a beam in bending. Because of the difficulties encountered in determining the lateral bending rigidity of a cross-tie track, a more recent approach is to model the rail-tie structure as a beam in bending that is also subjected to continuous resistance moments, transmitted to the rails by the fasteners. Also this approach exhibits a number of shortcomings. To date there are no generally accepted equations for the lateral response of the the cross-tie track. The main purpose of the present paper is to derive such equations. Since the rail-tie structure consists of a repeated pattern of identical units, the corresponding difference equations are derived first. Then, by a limiting process, in which the tie spaces tend to zero, the difference equations are reduced to differential equations. This approach yields equations with well-defined coefficients, in terms of the geometrical and mechanical parameters of the track structure. Difference and differential equations are derived for the track response in the lateral, as well as the vertical, plane. The derived equations are then discussed and compared with those suggested by other investigations.

Integral equation method for the study of two dimensional stokes flow

Abstract: An integral equation method applicable to two-dimensional Stokes flows for incompressible Newtonian isoviscous fluids is presented here. This new formulation only involves the physical quantities usually defined on the boundary in fluid mechanics. Through a discretization of boundary integral equations, we prove the numerical efficiency of our method by solving three problems.

Constitutive properties of dry sand observable in the triaxial test

Abstract: The so-called “rigid-granular” model for dry sand is tested here, within the frame of a bifurcation analysis of continuous and discontinuous deformation modes occurring in the triaxial test. For describing incipient shear parallel to the geometric axes of the sample and at 45° to them, two shear moduli μ and\(\mathop \mu \limits^* \) must be introduced. For obtaining predictions conformable to the experimental evidence, μ and\(\mathop \mu \limits^* \) must be proportional to the governing stress level and their ratio\(\mathop \mu \limits^* /\mu \) must be less then 1 (anisotropy). The analysis shows then that: 1) in the compression test the diffuse mode (bulging) occurs in the hardening regime whereas the localized modes (rigid wedges, shear bands) occur in the softening regime, and 2) in the extension test localized necking is always occurring in the hardening regime.

Time-dependent oscillating systems with damping, slowly varying parameters, and delay

Abstract: Mechanical and physical oscillating systems with slowly varying parameters, delay, and large damping, modeled by certain nonlinear ordinary and partial differential equations are considered. The time-dependent terms in these eqiations may not be periodic in time. The study is based on an extension of the asymptotic method of Krylow-Bogoliubov-Mitropolskii.

The unsteady boundary-layer development on a rotating disc in counter-rotating flow

Abstract: The boundary-layer growth, consequent on the impulsive counter-rotation of a disc from a state of solid-body rotation with the ambient fluid, is investigated numerically. It is found that the results presented are consistent with the existence of a singularity at a finite time, as first suggested by Bodonyi & Stewartson [1], and that the first terms of their asymptotic expansion do satisfactorily describe the behaviour in the vicinity of the singularity. However, our results also convincingly indicate that the next terms in the asymptotic expansion are of a larger order than that postulated previously.

Unsteady mixed convection from an isothermal circular cylinder

Abstract: Combined unsteady convection from an isothermal horizontal cylinder in a stream flowing vertically upwards has been investigated. Numerical solutions of the unsteady boundary-layer equations have been obtained at any station along the cylinder using the series truncation method. Solutions which are valid near the front and near stagnation points have been obtained using standard finite-difference methods. A series solution in powers of time has been obtained with which the numerical solutions has been checked.

Failure of a composite with a rigid fiber inclusion

Abstract: The failure behavior of a composite plate consisting of a matrix reinforced by a rectilinear rigid fiber inclusion and subjected to a uniaxial uniform stress at any angle to the axis of the fiber is studied. The critical stress of fracture and the characteristic angle which the fracture path forms with the fiber are determined by using two different approaches. According to the first approach it is assumed that fracture initiation is controlled by the maximum circumferential stress, while the second approach is based on the concept of the minimum strain energy in the vicinity of the end of the fiber. The critical stress and the characteristic angle of fracture are determined using both these approaches. A satisfactory agreement between these two theories was found. Finally, useful results concerning the dependence of the critical stress of fracture and the corresponding angle on the fiber orientation and Poisson's ratio of the material of the matrix were derived.

Removal of the zero eigenvalues of integral operators in elastostatic boundary value problems

Abstract: Discretization of various integral equations of the second kind in elastostatic boundary value problems leads to singular or almost singular systems of algebraic equations, i.e. the determinants of the coefficient matrices vanish or are extremely small. The reason for this lies in the fact that the integral operators have isolated zero eigenvalues. In this paper a method of removing the zero eigenvalues is developed which is based mainly on mechanical aspects.

Wave propagation in a fluid-filled elastic tube

Abstract: In a recent paper [1] the present authors (T.B.M. and J.B.H.) studied dispersive wave motions in a tethered, fluid-filled elastomer tube. There the radial inertia of the fluid was taken into account by employing an approximation similar to that proposed by Love [2] for analysis of wave propagation in bars and a simple bending theory of shells was employed for the tube wall. Here, by solving the fluid equations exactly we determine conditions under which the Love approximation is valid. We then extend our previous results to include the effect of shear deformation of the tube wall and analyze this extended theory to ascertain the relative importance of including shear in fluid-filled tube models designed for biological applications.

Mass transfer effects on the flow past an accelerated infinite vertical plate with variable heat-transfer

Abstract: An analysis of the mass transfer in a unsteady two-dimensional flow past an accelerated infinite vertical plate has been carried out under the following conditions: (1) The heat-transfer is varied at the plate (2) the non-dimensional timet is small and (3) Prandtl and Schmidt number is equal to one

Weak discontinuities in a dissociating gas

Abstract: The effects of non-equilibrium dissociation and that of the wave front curvature on the propagation of weak discontinuities headed by wave fronts of arbitrary shape and their consequent formation into shock waves are examined It is found that all compressive waves, except in one special case of converging waves, grow without bound only if the magnitude of the initial discontinuity associated with the wave exceeds a critical value It is shown that, in this special case, the stabilizing influence of curvature for converging waves is not strong enough to overcome the instabilities associated with the gasdynamic phenomenon involved

Unsteady pressure waves and shock waves in elastic tubes containing bubbly air-water mixtures

Abstract: The flow of a two-phase fluid through elastic tubes is more complex than that of a single phase fluid The mathematical model is based on an one-dimensional approach to the flow of a liquid-gas mixture The one-dimensional equations for transient two-phase flow through elastic tubes are a system of nonlinear hyperbolic partial differential equations if the bubbles and the liquid particles move with the same velocity Included in the model are the effects of wall elasticity, compressibility of the gas and the liquid, the surface tension and the variable area change The propagation of finite pressure waves and shock waves in a liquid containing gas bubbles has been investigated The results show a differently strong influence of the parameters on the wave propagation speed and on the shock wave relations

Constitutive relations of metal crystals at arbitrary strain

Abstract: At any generic state, the tangent moduli and compliances of a metal crystal are derived in terms of its elastic moduli and compliances, and its physical slip system hardening modulih ij. The structure ofh ij is explored in conjunction with a mixed hardening law. It is found that the latent hardening moduli (h ij,i≠j) are related to the active hardening moduli (h ij,i+j) through the latent hardening coefficients, and that each active hardening modulus is composed of the selfhardening, single slip modulush and the latent structural-change hardening modulih ij ′ . The theory is supplemented with some suggested functions forh andh ij ′ , suitable for metal forming analysis. The derived constitutive relations are finally applied to calculate the tensile stress-strain relations of aluminum and zinc crystals under finite strains.

On the Numerical Evaluation of Two- and Three-Dimensional Cauchy Principal-Value Integrals

Abstract: In the present paper two techniques are introduced for the numerical evaluation of either two-, or three-dimensional Cauchy principal-value integrals defined either over a shallow Lyapunov shell, or a closed Lyapunov domain respectively. A Gaussian cubature formula was established for the evaluation of these integrals which makes use of the concept of the finite-part integrals. Several numerical and theoretical applications are given to illustrate the results.