Showing papers in "Acta Mechanica in 1977"

Theoretical derivation of Darcy's law

Abstract: Darcy's law for anisotropic porous media is derived from the Navier-Stokes equation by using a formal averaging procedure. Particular emphasis is placed upon the proof that the permeability tensor is symmetric. In addition, it is shown that there is a one-to-one relationship between the local and macroscopic velocity fields. This leads to the interesting phenomenological observation that the local velocity vector at any given point must always lie either on a fixed line or in a fixed plane. All of this holds true for an incompressible homogeneous Newtonian fluid moving slowly through a rigid porous medium with uniform porosity under isothermal and steady state conditions. The question whether Darcy's law is applicable under nonsteady or compressible flow conditions, or when the medium has nonuniform porosity, is also discussed. Finally, it is shown that the Hagen-Poiseuille equation, as well as the expression describing Couette flow between parallel plates, can be derived from the equations presented in this work and may thus be viewed as special cases of Darcy's law.

On yielding of oriented solids

Abstract: Yielding and failure of oriented solids is studied within the framework of tensor representations. Transversely isotropic materials are considered. Using the information available regarding the tensor generators and the set of independent stress and mixed stress-material orientation invariants the general form of constitutive relation for incipient plastic motion of transversely isotropic solid is given and the yield condition is discussed. Specific forms of yield criteria for cohesive materials as well as for materials with internal friction are developed and compared with experimental information. A novel approach to flow and failure of oriented solid is thus explained on example of stratified material, starting with experimental motivation, passing through theoretical development and terminating on comparisons with experiments.

The foundations of thermodynamics, its basic postulates and implications. A review of modern thermodynamics

Abstract: This article reviews modern theories of irreversible thermodynamics. It is known that the second law of thermodynamics is not a unique statement defined by precise rules. On the contrary, there are various versions of the second law and, likewise, also various degrees of generality to which these are exploited. All these laws express some notion of irreversibility and the implications drawn from them necessarily differ from each other. In this article we discuss these versions. They are motivated from the balance law of entropy. It is shown how the Clausius-Duhem theory, the entropy free thermodynamics of Meixner and the theory of Muller naturally follow from such a balance law. The approaches of irreversible thermodynamics and that of rational thermodynamics are compared using a simple heat conducting fluid. Muller's version of the second law, which appears to be the most general form of it, is discussed in detail. It is shown that whereever it has been applied already, its implications are farther reaching than other theories permit. Finally we discuss some criticism that has been raised against one or the other theory. — Physical arguments are emphasized and on the mathematical side the reader need only be familiar with basic calculus.

An experimental investigation concerning yield surfaces and loading surfaces

Abstract: This paper presents the results of experiments concerning the motion of the yield and the loading surfaces and the interrelation between these two motions. It is shown that the yield surface tends to become tangent to the loading surface, and that the plastic strain vector is normal to the yield surface.

Array of Periodic Curvilinear Cracks in an Infinite Isotropic Medium

Abstract: The problem of an array of periodic curvilinear cracks in an infinite isotropic medium under conditions of generalized plane stress or plane strain is reduced, by using the method of complex potentials of Muskhelishvili [1], to a complex Cauchy-type singular integral equation on one of the cracks, which can be further numerically solved by reduction to a system of two real Cauchytype singular integral equations and application of the Lobatto-Chebyshev method of numerical solution of such equations. Applications to the cases of arrays of straight or arc-shaped cracks are also given.

Kinematische Wellen@@@Kinematic waves

Abstract: ZusammenfassungDas Auftreten kinematischer Wellen ist an das Bestehen eines funktionellen Zusammenhanges zwischen den Feldgrößen gebunden. Ein interessantes Anwendungsgebiet ihrer Theorie stellen Strömungsvorgänge dar, bei denen sich diese Relation dadurch ergibt, daß die Einflüsse von Reibungskräften und Schwerekräften dominieren, während Trägheitseffekte global nur von untergeordneter Bedeutung sind. Als typische Beispiele dafür werden turbulente und laminare Filmströmungen, Ausbreitungsvorgänge im Geschiebe von Flüssen sowie Sedimentationsvorgänge in Zweiphasenströmungen behandelt.SummaryThe occurrence of kinematic waves is characterized by the existence of a functional relationship between the dependent flow-variables. Interesting flow problems which can be handled by their theory are phenomena where this relation expresses the fact that frictional forces and gravity forces dominate while inertia effects are neglegible. As typical examples turbulent and laminar flows of thin sheets of fluid, wave propagation phenomena in erodible beds and sedimentation processes in two-phase-flows are considered.

Effect of wall conductances on hydromagnetic flow and heat transfer in a rotating channel

Abstract: Wall conductance effects on the hydromagnetic flow and heat transfer between two parallel plates in a rotating frame of reference has been studied when the liquid is permeated by a transverse magnetic field. An exact solution of the governing equations has been obtained. It is found that the velocity, current density and the temperature depend only on the sum of the wall conductances φ1 + φ2 = φ but magnetic field depends on the individual values of φ1 and φ2, where φ1 and φ2 are respectively the wall conductance ratios of the upper and lower walls.

A discrete probabilistic model for mechanical response of a granular medium

Abstract: A study has been made of the compressibility and force transmission in a granular material, modeled as a two-dimensional random packing of like spheres in elastic contact. The packing geometry is represented by a stochastic planar graph, with the nodes of the graph taken as centers of the spheres and the branches, as contacts between adjacent spheres. The stochastic graph is replaced by a lattice each of whose branches has a random stiffness modulus assigned to it. The corresponding set of stiffness moduli is considered a two-dimensional random process characterized by the porosity, average coordination number and internal-angle distribution of the packing. For mechanical response calculations, the lattice is treated as an elastic structure with branch stiffness in the form of the Hertz contact law, and analyzed by the node method.

Electromagnetic wave propagation in non-linear dielectric media

Abstract: A modification of Bergman's integral operator method is applied to a hodograph system descriptive of linearly polarised plane electromagnetic pulse propagation in non-linear dielectrics Series solutions of the system are thereby generated corresponding to general D-E and B-H relations For certain non-linear ‘constitutive laws’ the governing wave propagation equations are reduced to a hyperbolic canonical form associated with the classical wave equation and closed form integration is thereby achieved The availability of arbitrary parameters in these laws allows approximation to non-linear dielectric laws derived empirically Moreover, it is noted how a generalised version of Weinstein's Correspondence Principle may be utilised to generate iteratively more general non-linear constitutive laws for which reduction to canonical form may be obtained

Increase of the first natural frequency and buckling load of plates by optimal fields of initial stresses

Abstract: Departing from the common way of optimization of vibrating structures by an optimal mass distribution this paper deals with the problem of maximizing the fundamental frequency of structures by optimizing fields of initial stresses without varying the given appropriate shape of the structure. Thin elastic circular and rectangular plates are considered, which may be loaded by external inplane forces, and optimal initial membrane stress fields are calculated, which produce values of the first natural frequency of the free bending vibrations as high as possible. As a constraint, the strain energy caused by the field of initial stresses in the resting, externally unloaded plate is given. The computation leads to a max-min-problem solved numerically with the help of gradient methods. The optimal fields of initial stresses of the buckling plates are calculated as extreme cases in the same manner.

Random vibrations of orthotropic plates clamped or simply supported all round

Abstract: An approximate method is presented for determining the probabilistic response of rectangular orthotropic plates clamped all round. For solving the stochastic boundary value problem, the probabilistically given and sought functions are expressed in terms of series of approximate modes of vibration, which satisfy the boundary conditions but not the field equation. Galerkin's procedure then yields a set of linear equations for the cross-spectral densities of the displacements. The cross-spectral density of the external pressure is taken to be a product of longitudinal and transverse correlation coefficients which depend on frequency and separation distance. When the approximate method presented here is applied to cases capable of closed solutions (i.e. plates having a pair of opposite edges simply supported), the result coincides with that obtained by the classical normal-mode approach.

Gaseous diffusion in a stressed-thermoelastic solid. Part I: The thermomechanical formulation

Abstract: A phenomenological theory is proposed for the diffusion of a dilute solution of a gas in a thermoelastic solid. It is assumed that the thermomechanical behavior of the solid is unaffected by the presence of the gas. On the other hand the presence of the solid is recognized by the gas by letting certain thermomechanical variables of the solid to enter into the constitutive equations of the gas. The constitutive functionals of the gas are restricted by the principles of continuum physics. These principles are currently referred to as equipresence, material objectivity, entropy production inequality, as well as the balance equations of mass, linear and angular momentum and internal energy. By this approach, axiomatic statements on the existence of equations of state are avoided and the classical results of linear irreversible thermodynamics are obtained by further specialization of the proposed theory.

On quasistatic problems of cracks in a non-homogeneous elastic layer

Abstract: Simple solutions are given for plane and anti-plane problems of cracks in strips when the strip is made of two or more different elastic media. Cases considered are when time harmonic displacements are applied to the strip sides with a stationary crack, or when fixed displacements are applied to the strip sides and the crack moves uniformly. These results generalise in some respects results given recently by Matcynski [1], [2], [3].

Nonlinear viscoelasticity and relaxation phenomena of polymer solids

Abstract: In the light of a three-chain model of statistical network theories of rubberlike elastic models, it is assumed that the free energy function of incompressible viscoelastic polymer solids is a separable, symmetric function of the principle stretch ratios and the hidden thermodynamic coordinates along the same directions. This assumption leads to a characterization of those viscoelastic polymer solids which exhibit the property of factorizability between the time and strain functions.

Gaseous diffusion in a stressed-thermoelastic solid. Part II: Thermodynamic structure and transport theory

Abstract: In part I of the present work a constitutive theory is established for a gas diffusing, as a dilute solution, in a linear thermoelastic solid. The theory is compatible with the laws of thermomechanics i.e., mass, momentum and energy balance, as well as with the principles of positive entropy production and material objectivity. The assumptions of small concentrations of the gas and the absence of viscous effects simplify the analysis considerably. This allows a thermomechanical model for the solid to be assumed, independent of the presence of the gas. The constitutive model adopted for the solid is the linear uncoupled theory of thermoelasticity. In this level of generality a transport theory is developed for the gas by using the conservation laws of mass, linear momentum and internal energy. Some uncommon identities for the thermodynamic quantities of the gas are also established, based on the derived functional dependence for the internal energy, entropy, free energy and stress tensor of the gas. Thus a Gibb's type equation is obtained. In addition a comparison with classical irreversible thermodynamics, previous intuitive theories and experiments is made.

On the steady motion of jets with elliptical sections

Abstract: In 1960 Taylor discussed the motion of a jet of inviscid fluid issuing from an elongated orifice in a flat sheet, which transforms itself into a jet which is first flattened in the direction of the long axis of the orifice and then subsequently in the perpendicular direction. Such a motion is re-considered here using a system of dynamical equations obtained from a one-dimensional model.

On uniqueness criteria and minimum principles for crystalline solids at finite strain

Abstract: New uniqueness criteria and minimum principles for quasi-static, rate-type boundary value problems in crystalline solids are derived. The analysis is based upon a general constitutive theory of finite elastic-plastic deformation encompassing the distinct mechanisms of lattice straining and crystallographic slip. Locally sufficient uniqueness criteria and a related minimum principle permitting the independent variation of velocity and plastic shear rates are established for the case when the two dominant principal stresses are everywhere tensile.

Zur Modifikation des Spannungsdeviators

Abstract: In der vorliegenden Arbeit wird der Einflus der Kompressibilitat auf das mechanische Verhalten isotroper Stoffe eingehend untersucht. Dazu benutzt man einen modifizierten Spannungsdeviator, der zur Regulierung der plastischen Kompressibilitat einen Ansatzfreiwertm enthalt und furm=0 in den Spannungstensor ubergeht, wahrend furm=1 der ubliche Deviator anfallt. Im Hinblick auf die charakteristische Gleichung des modifizierten Deviators kann dieser Ansatzfreiwert nur reelle Werte annehmen. Als Anwendungsbeispiel wird ein quadratischer Ansatz fur das plastische Potential gemacht, der bei Inkompressibilitat (m=1) das Misessche Potential als Sonderfall enthalt. Fur dieses Beispiel wird gezeigt, das auch aus Konvexitatsgrunden und im Hinblick auf eine mechanisch sinnvolle Volumendilatation der Ansatzfreiwert reell sein mus. Fur imaginarem-Werte erhalt man konkave Flieskorper und ein falsches Vorzeichen fur die Volumendilatation.

On the application of the energy method to the stability problem of nonconservative autonomous and nonautonomous systems

Abstract: The energy approach is extended to cover the stability problem of nonconservative mechanical systems. The eigenvalue curve is obtained by the condition that a certain matrix be singular, and flutter loads follow from the requirement that the derivative of the determinant of this matrix with respect to the frequency of the motion be zero. In the case that the nonconservative forces of the system are being allowed to tend toward zero, all conditions of this general theory change into the well known conditions of the classical theory of stability of conservative systems. The here presented theory allows also for a post-buckling analysis and for the inclusion of nonautonomous systems.

Stability of magnetogasdynamic shear flow

Abstract: Some general results on the stability of a compressible parallel shear flow permeated by an aligned magnetic field are derived. It is shown that the complex wave speed of an unstable wave lies in a semi-circle in the upper half plane with the range of the shear flow as diameter. Further both compressibility and magnetic field are found to be stabilizing. It is also shown that in a flow near a wall, two acoustic waves carry energy away from the wall. A semi-circle theorem for a non-planar compressible shear flow is also deduced.

Dynamic expansion of a compressible hyperelastic spherical shell

Abstract: This paper is concerned with the finite spherically symmetric motion of a compressible hyperelastic spherical shell, subjected to a spatially uniform step funtion application of pressure at its inner surface. A method, given in a previous paper [1], for the determination of the field of characteristics, for expansion of a spherical cavity in an unbounded solid is adapted to consider the spherical shell problem. Results are presented graphically for a particular strain energy function and are compared with results obtained for an incompressible material and from linear elasticity theory.

On the thermodynamics of nonlinear single integral representations for thermoviscoelastic materials with applications to one-dimensional wave propagation

Abstract: Thermodynamic theory is used to develop single integral constitutive relations for the nonlinear thermoviscoelastic response to arbitrary stress and temperature histories; the thermomechanically coupled energy equation is also obtained. The thermorheologically simple material, modified superposition and the isotropic stress power law are discussed in detail. A modified Fourier heat conduction law is employed to ensure that the propagation of thermal disturbances takes place at a finite velocity. Using the nonlinear thermoviscoelastic stress power law along with the linearized energy equation and modified Fourier law, one-dimensional wave front solutions are obtained.

On a lower bound theorem for the buckling load of elastic beams subjected to nonconservative, compressive follower loads

Abstract: For beams loaded withuniformly distributed follower forces, there exists a lower bound theorem, which allows to evaluate the actual buckling load approximately by means of the buckling load of the same beam loaded correspondingly with conservative unidirectional forces of the same magnitude. In this paper it is shown that the theorem remains valid for monotonically decreasing or at least not increasing follower forces. Moreover, a modified lower bound theorem can be formulated for follower loads which monotonically increase over a part or even over the whole length of the beam. In that case, a “lower bound curve” has to be constructed using the eigenvalue curve of the conservatively loaded beam. The intersection of the lower bound curve with the vertical axis yields a lower bound for the buckling load of the nonconservatively loaded beam.

Hall effects on hydromagnetic flow over an impulsively started porous plate

Abstract: An exact solution has been obtained for the problem of Hall effects on the hydromagnetic flow due to an impulsive start of a porous flat plate. It is observed that in the initial stages there is no inertial oscillation while at large time the steady state is reached through inertial oscillations. The time to attain the steady state increases with increase in Hall parameter.

Elastic-plastic analysis of a flat ring subject to internal pressure

Abstract: An exact elastic-plastic solution for the stresses in a flat ring subject to pressure is obtained on the basis ofJ2 deformation theory together with a modified Ramberg-Osgood law. The results are assessed on the basis of Budiansky's criterion for the acceptability ofJ2 deformation theory. By using exact elastic-plastic stresses, the radial enlargement and the permanent increase in thickness of the ring at the hole are obtained. Upon release of the pressure, residual stresses are produced around the hole.