Showing papers in "Acta Mechanica in 1979"

On the analysis of rotation and stress rate in deforming bodies

Abstract: When a solid element experience large deformations, the components of stress will, in general, vary as a result of material rotation. These changes occur even in the absence of additional strain, and need to be accounted for in formulating constitutive laws that involve the rate of change of stress. In this paper the correction terms are extended to the case when material axes become strongly skewed. An expression for the rate of material rotation as an explicit function of vorticity, rate of deformation and stretch is derived. It is then shown that the rate of change of stress depends on the rate of material rotation. As an example, expressions for material rotation and stress are derived for a hypoelastic material undergoing uniform, rectilinear, shear. The shear stress is compared with a solution that neglects skewing of the axes, and it is found that, for the example, skewing may be neglected for strains less than 0.4. Finally, the use of these relations in numerical calculations involving finite deformation is discussed.

On plane waves in generalized thermoelasticity

Abstract: The propagation and stability of harmonically time-dependent thermoelastic plane waves of assigned frequency in the generalized thermoelasticity theory of Green and Lindsay are considered. The results are contrasted with those of the theory of Lord and Shulman. In general there is no a priori correspondence between the two theories. However, after introducing some assumptions on the constitutive coefficients α and α*, which appear in the theory of Green and Lindsay, all the corresponding results of the theory of Lord and Shulman are recovered.

Bifurcation analysis of the triaxial test on sand samples

Abstract: A bifurcation analysis of a strain hardening dilatant sand sample in the triaxial test is carried out. The analysis shows that the triaxial test yields only then the limiting soil properties if 1) the sample is compact enough and 2) if the confining pressure does not exceed a critical value depending on the soil anisotropy and the slenderness of the sample.

One dimensional dynamic electromechanical constitutive relations of ferroelectric materials

Abstract: We present a macroscopic theory for the dynamic response of a poled polycrystalline ferroelectric material describing its coupled electromechanical interactions. The treatment is restricted to an idealized material representing the simplest system capable of displaying ferroelectricity; it includes changes in both the magnitude of the electric dipoles and the orientation of the domains. Coupling between the electrical and mechanical properties of the material is considered and the constitutive equations are linearized to illustrate the resulting dynamic response.

On electromagneto-thermoelastic plane waves

Abstract: The propagation of harmonically time-dependent electromagneto-thermoelastic plane waves of assigned frequency in an unbounded, homogeneous, isotropic, elastic, thermally and electrically conducting medium is considered. The theory of thermoelasticity recently proposed by Green and Lindsay is used to account for the interactions between the elastic and thermal fields. The results pertaining to phase velocity and attenuation coefficient of various types of waves are compared with those of Nayfeh and Nemat-Nasser who have dealt with a theory of thermoelasticity having a thermal relaxation time.

A mathematical analysis of nonlinear waves in a fluid filled visco-elastic tube

Abstract: Our investigations in this paper are centred around the mathematical analysis of a “modal wave” problem. We have considered the axisymmetric flow of an inviscid liquid in a thinwalled viscoelastic tube under certain simplifying assumptions. We have first derived the propagation space equations in the long wave limit and also given a general procedure to derive these equations for arbitrary wave length, when the flow is irrotational. We have used the method of operators of multiple scales to derive the nonlinear Schrodinger equation governing the modulation of periodic waves and we have elaborated on the “long modulated waves” and the “modulated long waves”. We have also examined the existence and stability of Stokes waves in this system. This is followed by a discussion of the progressive wave solutions of the long wave equations. One of the most important results of our paper is that the propagation space equations are no longer partial differential equations but they are in terms of pseudo-differential operators.

The V-notched elastic half-plane problem

Abstract: The problem of a V-notched isotropic elastic half-plane under generalized plane stress or plane strain conditions can be reduced, by using the complex variable technique, to a complex Cauchy type singular integral equation along one of the V-notch edges. This equation can be numerically solved by use of the Gauss-Jacobi method and reduction to a system of linear equations. The values of the stress intensity factorKI at the V-notch tip were evaluated for some notch angles in the case of pure tension and the results obtained are in accordance with the available results in the case of a V-notched finite isotropic plane elastic medium. The difficulties faced in evaluatingKI are investigated and a discussion on them is made. The method is also applicable even when the V-notch edges are curvilinear and their loading arbitrary.

Analysis of mixed-mode isochromatic crack-tip fringe patterns

Abstract: Westergaard stress function representations of the stress field around the tip of a stationary crack subjected to mixed-mode loading are employed to develop a general relationship between the isochromatic fringe orderN, its position parametersr and θ and the general stress field expressed in terms of several parameters, the stress intensitiesK 1,K 2, a far field stress parameter α, and higher order term parametersβ 1 andβ 2. These parameters are varied and isochromatic crack tip fringe patterns are constructed for a set of classified combinations. Since the size, shape, orientation, and other particular features of the fringe patterns depend strongly on the combination of parameters chosen they can be used to classify crack tip fringe patterns. The introduction of fringe loop diagrams not only facilitates the classification procedure and enables a classification without plotting fringe patterns, but it also permits qualitative and quantitative evaluation of the isochromatic patterns. Isochromatics which classify fourtyeight different states of stress are investigated and the majority of them have been illustrated.

Numerical solution of the thermal instability of a micropolar fluid layer between rigid boundaries

Abstract: In this paper, we consider the convective instability of a heat conducting micropolar fluid layer between rigid boundaries. The eigenvalue problem governing the perturbations is solved, numerically, using finite difference method and Wilkinson's iteration technique. The heat induced by microrotation leads to the onset of instability not only due to adverse temperature gradient but also for positive temperature gradient. In the case of rigid boundaries, the critical Rayleigh number is seen to be higher than that of free boundaries. Here we notice that the cell pattern for positive temperature gradient are highly elongated when compared with that of negative temperature gradient.

Steady convection in a horizontal fluid layer

Abstract: Steady Rayleigh-Benard-convection in a rectangular box which is heated from below and cooled from above has been investigated. The onset of convection was calculated using linear Boussinesq equations and a Galerkin method. These calculations have demonstrated the stabilising effect of vertical boundaries. A special differential interferometer served to visualize the flow configurations in gases as well as in silicone oil. The density profiles were measured by means of a laser-differential interferometer with a moving mirror, evaluated quantitatively and compared with previously published results. The test fluids silicone oil (Pr=1780) and nitrogen (Pr=0.71) were chosen to demonstrate the influence of the Pr number both on the onset of convection and on the steady flow pattern at large Ra numbers.

Couette flow of a dusty gas

Abstract: Couette flow of a dusty gas between two parallel infinite plates has been studied for impulsive start as well as for uniformly accelerated start of one of the plates. The problem has been solved with the help of Laplace Transform technique. It is found that the dust velocity in the case of accelerated start of the plate is less than the fluid velocity, for moderate value of the relaxation time of the dust particles and they become equal when the dust particles become very fine. It is observed that the magnitude of the shear stress is larger when the plate starts with uniform acceleration than when it is impulsively started to move with uniform velocity.

Thin-walled beams as directed curves

Abstract: A non-linear theory of thin-walled elastic beams of open cross section is formulated in terms of a space curve endowed with several director fields, one for each of the plates composing the beam. Physically meaningful constraints among the directors are introduced and a technique for the algebraic elimination of the corresponding Lagrange multipliers from the field equations is discussed. Compatibility conditions for a minimal set of strain measures are presented.

Über die Konvexität von Fließkörpern isotroper und anisotroper Stoffe

Abstract: In der vorliegenden Arbeit wird die Frage der Konvexitat von Flieskorpern zunachst allgemein diskutiert. Zur Nachprufung der Konvexitat wird eine Konvexitatsbedingung bei Anisotropie aufgestellt, aus der man die Gultigkeitsbereiche von Ansatzfreiwerten ermitteln kann. Isotropie ist als Sonderfall enthalten. Einige spezielle Fliesbedingungen isotroper und anisotroper Stoffe werden auf ihre Konvexitat hin untersucht und dargestellt.

Effect of injection on the flow of second order fluid in the inlet region of a channel

Abstract: In this paper the effect of injection and elasticity of the liquid on the flow pattern of a second order fluid in the inlet region of a porous channel has been studied. The pressure gradient has been calculated by an energy integral applied over the entire flow field (including the inviscid flow region) rather than by simply applying Euler's equation of inviscid motion to the flow outside the boundary layer. The effects of suction and non-Newtonian parameter on the growth of boundary layer thickness in the inlet region, core velocity and pressure distribution in the inlet region have been studied.

Free vibration of rectangular beams of arbitrary depth

Abstract: The state space approach is extended to the two dimensional elastodynamic problems The formulation is in a form particularly amenable to consistent reduction to obtain approximate theories of any desired order Free vibration of rectangular beams of arbitrary depth is investigated using this approach The method does not involve the concept of the shear coefficientk It takes into account the vertical normal stress and the transverse shear stress The frequency values are calculated using the Timoshenko beam theory and the present analysis for different values of Poisson's ratio and they are in good agreement Four cases of beams with different end conditions are consideredDie Zustandsraum-Technik wird auf zweidimensionale elastodynamische Probleme ausgedehnt Die Formulierung ist besonders geeignet fur die Aufstellung von Naherungstheorien beliebigen Grades Freie Schwingungen von Rechteckbalken beliebiger Hohe wurden mit Hilfe dieser Technik untersucht Das Verfahren umgeht den Begriff des Schubbeiwertsk Es berucksichtigt die senkrechte Normalbeanspruchung und die Querkraft Die Frequenzwerte werden mit Hilfe der Balkentheorie von Timoshenko und der vorliegenden Analyse berechnet, und zwar fur verschiedene Werte der Querdehnzahl Die berechneten Werte befinden sich in guter Ubereinstimmung Vier Falle von Balken mit verschiedenen Endbedingungen werden untersucht

On temperature and heat source distributions in sliding contact problems

Abstract: The heat conduction problem of sliding contact as found in a tool work-piece interaction is analyzed by assuming the presence of a thin boundary layer between the tool wear-flat and the work-piece. Within the layer heat is produced at a constant rate per unit volume of the layer material as a result of large frictional forces. The layer material itself constitutes in the case of cutting of a brittle material a very fine powder whose thermal properties may differ significantly from those of the tool and the work-piece. The presence of the boundary layer permits thus the tool and the work-piece surface temperature distributions to be totally different Furthermore, the evolution in time of these distributions is determined by the heat conduction characteristics of the entire system under consideration and consisting of tool, layer and work-piece. The mathematical formulation of the problem results in a mixed boundary value problem which can be recast into a pair of coupled integral equations of the Fredholm type for the unknown heat source distributions over the tool and work-piece wear-flat area. The resulting tool wear-flat surface temperature distribution is shown to depend on the boundary layer thickness and thermal properties to a significant extent. The heat source distribution obtained for a set of given parameter combinations may serve as a prime input quantity in e.g. a finite element program designed to calculate the entire tool temperature distribution as a function of time.

On a gradient method in nonconservative mechanics

Abstract: The paper is concerned with the analysis of nonconservative dynamical systems under the supposition that Hamilton's momentum vector can be expressed as the gradient of a scalar function which depends on the generalized coordinates and time. All components of the momentum vector are obtained as the solutions of a system of partial differential equations called the basic system. It is shown that the solution is known if a complete solution of the basic system is available. As an illustration of the theory, several examples of practical interest are solved.

Flow and heat transfer due to small torsional oscillations of a disk about a constant mean

Abstract: The paper deals with the study of flow and heat transfer in a viscous fluid from a disk performing small rotating oscillations about a constant mean. Separate solutions for low and high frequency ranges are developed. In the low frequency range the solution is obtained by Karman-Pohlhausen method. It is found that near the disk the oscillating components of radial and tangential velocities lag behind the disk oscillations. This tendency to phase lag is more than compensated as we move away from the disk. The oscillating components of the radial stress and the rate of heat transfer at the disk always lag behind, while the tangential stress has a phase lead. For very high frequencies the velocity field is of “shear-wave” type, predicting a phase lead of 45° in the tangential stress at the disk.

The second fundamental crack problem and the rigid line inclusion problem in plane elasticity

Abstract: The second fundamental problem of plane isotropic elasticity for a cracked infinite medium, that is the problem in which the displacement derivatives are given along the crack edges, as well as the closely related rigid line inclusion problem in plane elasticity are treated by using the method of complex potentials. Without restrictive assumptions on geometry and boundary conditions, these problems are reduced to a complex Cauchy type singular integral equation along the crack or the inclusion, the numerical solution of which can easily be obtained by using the Lobatto-Chebyshev method.

Oscillatory Flow past a Porous Bed

Abstract: Taking the most general form of Darcy law, flow past and against a porous bed is analysed when the free stream oscillates with or without a non-zero mean. Boundary layer characteristics within the porous bed are derived.

Plastic deformation in form of strain trajectories of constant curvature —Theory and comparison with experimental results

Abstract: Based on the constitutive equations for plastic materials of any deformation history presented in previous papers of the author analytical relations are given for the case of plastic deformation in form of constant curvature strain trajectories. Magnitude and direction of the stress vector are determined for various radii of curvature and then compared with the experimental results described in [4]. Finally, a comparison is made between the theory applied here and the analytical method proposed in [5].

Dispersion of solutes in combined free and forced convective flow through a channel

Abstract: Dispersion of a solute in combined free and forced convective laminar flow through a parallel plate channel has been considered, when there is a uniform axial temperature variation along the channel walls. It is observed that the effective Taylor diffusion coefficient increases with increase in Grashof number and also the interesting fact that it is same for both heating and cooling of the plates.

On the Growth and Decay of Acceleration Waves in Transient Gas Flows With Vibrational Relaxation

Abstract: The problem of breakdown of acceleration waves in the neighbourhood of the leading frozen characteristics in transient gas flows with vibrational relaxation has been studied. The critical timetc is determined when the breakdown of the wave will occur at the cusp of the envelope of the intersectying forward characteristics. It is concluded that there exists a critical amplitude of the wave such that all compressive waves with an initial amplitude greater than the critical one will break down and a shock-type discontinuity will be formed due to non-linear steepening, while an initial amplitude less than the critical one will result in a decay of the wave. It is also concluded that the point of breakdown moves outward along the leading characteristics as the relaxation time decreases and that there is no breakdown, if the gas has a sufficiently small, but not zero, relaxation time.

On the ubonded contact between elastic and elastic-rigid media

Abstract: According to the published analyses of unbonded contact problems in which a rigid or elastic sphere, or circular cylinder, presses against an elastic half-space, the compressive contact stress tends to zero at the separation contour. Some authors assume this as an a priori condition for solving contact problems of this type. One purpose of the present paper is to show, by means of three examples, that this assumption is not valid in general, and that its validity depends not only on the geometry of the indentor but also upon the used elastic response equations. The other purpose is to demonstrate the utility of variational calculus with variable matching points for the solution of a class of contact problems.