Showing papers in "Acta Mechanica in 1982"

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Abstract: A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator δ into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.

Laminarescent, relaminarizing and retransitional flows

Abstract: This report examines in detail all accelerated turbulent boundary layers and subcritical pipe or channel flows undergoing relaminarization and possible retransition, with a view to evolving a broad picture in regard to the status of experiments in these flows, the trustworthiness or shortcomings of the data, the sources of difficulties peculiar to these flows, etc. With the hindsight so acquired, a discussion is provided of the directions in which future work would most usefully supplement the existing data.

Boundary layer flow of a dusty fluid over a semi-infinite flat plate

Abstract: Both the drag force due to slip and the transverse force due to slip-shear have been considered in boundary layer equations. The solution has been found in a power series of non-dimensionalx, x being the distance in the down-stream direction. Solutions for high slip region and small slip region characterised byx≪1 andx≫1 respectively, have been found separately. In the high slip region the effect of increase in concentration parameter α of the dust particles is to increase the magnitude of the longitudinal fluid phase velocityu. Also the magnitude of the longitudinal particle slip velocityup-u is becoming maximum on the plate and decreasing along the plate withx. The transverse particle velocityvp is independent of α but it is directly proportional to β, the transverse force coefficient. An interesting result is thatvp is assuming small positive value on the plate. The transverse force has taken an important role in migration of particles away from the plate. In the small slip region the flow of dust particles is mainly governed by the fluid-phase. The effect of α on the flow field in this region is to decrease the boundary layer thickness. In this region the particles are having some tendency to accumulate near the plate. Lastly, it has been found that the shearing stress, skinfriction and the dimensionless drag-coefficient on the plate increase with increase of α.

On a proposal for a continuum with microstructure

Abstract: A recently proposed model for a continuum with microstructure is further substantiated by identifying the microstructure with dislocations. In particular, the continuum is viewed as a superimposed state composed of a perfect lattice state, an immobile dislocation state, and a mobile dislocation state. It is assumed that each state evolves continuously in space-time and transitions from one state to another take place spontaneously according to the balance laws of effective mass and momentum. When the constitutive equations are subjected to the requirements of invariance, familiar statements from dislocation dynamics are deduced. When plastic strain and yield are identified in terms of the parameters characterizing the dislocation states, familiar flow rules and yield surfaces are produced. The capability of the model to predict not only Tresca and Von-Mises plastic behavior but also phenomena such as kinematic hardening, different responses in tension and compression, latent hardening, and the Bauschinger effect, is shown. Finally, the appropriateness of our equations to model creep, cyclic plasticity, and fatigue, is illustrated.

On the theory of stress-assisted diffusion, II

Abstract: Similarity and general steady-state solutions of a recently developed strees-assisted diffusion theory are derived. General transient solutions are obtained analytically for certain classes of stress distributions. For general stress distributions a perturbation method is employed to produce transient solutions. Under appropriate conditions the derived solutions are reduced to previous formulae that have unsystematically appeared in the literature. Examples of interesting crack problems involving stress singularities as well as the elimination of singularities are considered. An equilibrium solution is utilized together with a straightforward physical argument to produce rationally two empirical formulae previously proposed in the literature to model embrittlement and stress corrosion cracking phenomena. This solution is further used to model embrittlement and stress corrosion cracking data more successfully than previous attempts. An appendix on certain preliminary elastodiffusive fracture criteria is given.

Photoelastic Investigation of Dynamic Load Transfer in Granular Media

Abstract: Dynamic photoelasticity in conjunction with high-speed photography is utilized to study impact wave propagation and dynamic load transfer in granular soil. Series of sequentially recorded isochromatic fringe patterns provide full field information of the dynamic event. Experimental results show that the wave propagation process is governed by a contact wave which increases the density of contacts in propagation direction. The study of dynamic load transfer along specific representative load chains provides basic information about wave velocities, contact duration, and directional stability necessary for a rigorous analysis of the general complex dynamic wave-soil interaction problem.

Flüssigkeitsschwingungen in Kegelbehälterformen

Abstract: Es werden fur Kegelbehalterformen Naherungsformeln fur die naturlichen Frequenzen einer Flussigkeit mit freier Oberflache bestimmt. Die Naherungswerte ergeben fur Kegelbehalterformen mit kleinem Offnungswinkel, bei denen die freie Flussigkeitsoberflache durch eine Koordinatenflache angenahert werden kann, gute Ergebnisse.

Free convection in the laminar boundary layer flow of a thermomicropolar fluid past a vertical flat plate with suction/injection

Abstract: The effect of suction/injection in the laminar free convection flow of a thermomicropolar fluid past a nonuniformly heated vertical flat plate has been considered. The conditions under which similarity exists have been examined. The resulting system of non-linear ordinary differential equations has been solved numerically after transforming the infinite domain of boundary layer coordinate into a finite domain. The effects of variation of the boundary condition parameter and suction/injection parameter on the velocity, microrotation and temperature fields and the heat transfer coefficient have been studied graphically. The skin-friction parameter and the gradient of microrotation on the wall have been tabulated. It is found that there is significant increase in velocity, skin-friction and the heat transfer coefficient with the decreasing concentration of microelements.

Flssigkeitsschwingungen in Kegelbehlterformen

Abstract: ZusammenfassungEs werden für Kegelbehälterformen Näherungsformeln für die natürlichen Frequenzen einer Flüssigkeit mit freier Oberfläche bestimmt. Die Näherungswerte ergeben für Kegelbehälterformen mit kleinem Öffnungswinkel, bei denen die freie Flüssigkeitsoberfläche durch eine Koordinatenfläche angenähert werden kann, gute Ergebnisse.Auf diese Art kann man das Flüssigkeitsverhalten in vielen für die Praxis wichtigen Behälterformen angenähert bestimmen. Experimentelle Ergebnisse zeigen gute Übereinstimmung mit der Theorie.SummaryApproximate expressions are obtained for natural frequencies of an incompressible liquid with a free surface in an arbitrary container of conical geometry. These approximations yield good results if the apex angle of the cone is small, such that the spherical coordinate surface presents a good approximation to the plane free liquid surface.This way a great many of container shapes of practical importance may be treated. Experimental results exhibit good agreement with those of theory.

A continuum theory for two phase media

Abstract: A continuum theory of a two phase solid-fluid media is formulated. The basic balance laws for the solid phase as well as for the fluid phase are presented. Based on thermodynamical consideration a set of constitutive equations are derived and the basic equations of motions of the distributed solid and fluid continua are obtained and discussed. It is shown that the theory contains as its special cases, Mohr-Coulomb criterion of limiting equilibrium of granular materials, Saffman theory of dusty gas, as well as Darcy's law of flow through porous media. It is then concluded that the present theory covers the full spectrum of two phase solid-fluid media from low porosity granular media with Darcy's law of fluid motion to low and high concentration two phase flows such as dusty gas and blood flow.

Yield functions that account for the effects of initial and subsequent plastic anisotropy

Abstract: A descriptive initial yield function is presented from an examination of experimentally determined yield loci, component plastic strain paths and Lode's parameters that indicate either a severe textural anisotropy in a material or a slight departure from the von Mises condition. The transition from the initial function to one that describes a subsequent yield surface which translates with the stress vector is developed and compared with experimental results. Observations on the Bauschinger, Swift and hardening effects in subsequent yield loci, defined at the limit of proportionality, are adequately represented through Ziegler's translation law when modified for non-linear work hardening. It is shown that the marked distortion, particularly apparent, in the presence of shear stress, may be represented by considering separately the translation for each quadrant of the initial yield locus.

Projective invariance of shaky structures

Abstract: A framework consisting of rigid rods which are connected in freely moveable knots, in general is stable if the number of knots is sufficiently large. In exceptional cases, however, the rodwork may allow an infinitesimal deformation. Due to a theorem of Liebmann, this apparently metric property of existing shakiness in fact is a projective one, as it does not vanish if the structure is transformed by an affine or projective collineation. The paper presents a new analytic proof of this remarkable phenomenon. The developments are applicable also to polyhedra with rigid plates and to closed chains of rigid links.

Buoyancy and surface-tension driven instabilities in presence of negative Rayleigh and Marangoni numbers

Abstract: Benard-Marangoni instabilities are theoretically discussed: emphasis is placed on the role of negative Rayleigh and Marangoni numbers. Marginal, supercritical and subcritical instabilities are respectively examined. The first part is concerned with the response of an unbounded fluid layer with respect to small disturbances. A variational principle describing marginal stability is proposed. Rayleigh-Ritz method is used to obtain approximate solutions for the critical Rayleigh and Marangoni numbers. In a second part, corrections to the linear theory, by including weak nonlinearities, are introduced. The amplitude of the supercritical temperature and velocity fields are calculated in the framework of Stuart's shape approximation. Finally, the possibility of subcritical instability with respect to disturbances of arbitrary amplitude is investigated by the method of energy.

Hamilton's principle and the calculus of variations

Abstract: In the recent literature of the Calculus of variations, mathematical proofs have been presented for what the writers claim to be a more precise statement of Hamilton's Principle for conservative systems. Nothing is said about Hamilton's Principle for nonconservative systems. According to these writers, the action integral, the variation of which is Hamilton's Principle for conservative systems, is a minimum for discrete systems for small time intervals only and is never a minimum for continuous systems. The proof of this more precise statement is based in the sufficiency theorems of the Calculus of Variations. In this paper, two contradictions to the statement are demonstrated — one for a discrete system and one for a continuous system.

Comparison of one-dimensional solutions with Fabri theory for ejectors

Abstract: Ejector performance for constant area or constant pressure mixing is theoretically examined using a one-dimensional approach as well as the inviscid, theory of Fabri. The variety of solutions generated by the one-dimensional approach is shown to be severely restricted by the Fabri theory. This restriction produces solution regions that are not allowed, as well as regions where the back pressure alters the secondary flow rate. Operation with a slightly supersonic secondary flow is examined, but may not be realized due to shock wave formation. Operation at a high supersonic value for the secondary flow provides appreciably poorer ejector performance than is attainable when this flow is subsonic.

On the elasto-plastic stress concentration at a circular hole in an anisotropic sheet

Abstract: The classical problem of determining the stress concentration factor at a circular hole embedded in an infinite sheet subjected to remote uniform tension is investigated. A finite strain elasto-plastic deformation theory based on Hill's new anisotropic flow theory [7] is used. It is shown that the governing field equations can be reduced to a single first order differential equation from which the stress concentration factor is obtained by a standard numerical method. The solution covers the entire elasto-plastic range and is valid for any strain hardening function. Comparison with experimental results, for a few materials, shows good agreement. With a pure power hardening law and within the framework of small strain plasticity, our results agree with those obtained from a more general solution discovered by Budiansky [8].

Hall effects on free and forced convective flow in a rotating channel

Abstract: The influence of Hall currents on the free and forced convective flow of a viscous rotating fluid between two horizontal plates is analysed. An exact solution for the velocity and temperature have been obtained. the effects of Hall currents on the velocity, the temperature and shear stress are discussed analytically and graphically.

Solution of plane anisotropic elastostatical boundary value problems by singular integral equations

Abstract: Basis for the presented method is the knowledge of certain fundamental solutions for theinfinite anisotropic medium. By superimposing these singular solutions in a suitable fashion, the given boundary value problem can be formulated as a tensorial integral equation (singularity method).

Kinematics of a multi-dimensional shock of arbitrary strength in an ideal gas

Abstract: It has been shown that the kinematics of a shock front in an ideal gas with constant specific heat can be completely described by a first order nonlinear partial differential equation (called here — shock manifold equation or SME) which reduces to the characteristic partial differential equation as the shock strength tends to zero. The condition for the existence of a nontrivial solution of the jump relations across the shock turns out to be the Prandtl relation. Continuing the functions representing the state on the either side of the shock to the other side as infinitely differentiable functions and embedding the shock in a one parameter family of surfaces, it has been shown that the Prandtl relation can be treated as a required shock manifold equation for a function Φ, where Φ=0 is the shock surface. We also show that there are other forms of the SME and prove an important result that they are equivalent. Shock rays are defined to be the characteristic curves of a SME and it has been shown that when the flow on either side of the shock is at rest, the shock rays are orthogonal to the successive positions of the shock surface. Certain results have been derived for a weak shock, in which case the complete history of the curved shock can be determined for a class of problems.

The propagation of Love waves in water-saturated soil underlain by a heterogeneous elastic medium

Abstract: The propagation of Love waves in water-saturated soil overlying a non-homogeneous elastic medium has been investigated in the present paper. The solution of the problem is evaluated with the help of Fourier transform technique. It has been found that the velocity of body wave depends on the direction of propagation, and the velocity of Love wave depends on the porosity of the layer and the non-homogeneity character of the medium as well. Further, the effect of porosity and heterogeneity has been shown graphically.

Steady two dimensional incompressible laminar visco-elastic flow in a converging or diverging channel with suction and injection

Abstract: The solution for a viscous incompressible fluid in convergent and divergent channels can be expressed exactly in terms of elliptical functions [1]. The discussion of the exact solution is rather complicated. Rosenhead [2] has studied the steady two dimensional radial flow of an incompressible viscous fluid between two impermeable inclined plane walls. I f the l~eynolds number is large and if there is suction or blowing, a t the walls whose magnitude is inversely proportional to the distance along the wall from the origin of the channel, a solution of the laminar boundary layer equation can be obtained [B]. Terrill [4] obtained the solution for slow flow through the channel for the ease of suction at one wall and equal blowing at the other wall. There seems to be some error in his solution. The aim of this paper is to s tudy Terrill 's problem replacing viscous fluid b y Walters ' B ' viseo-elastic fluid.

Rapid couette flow of cohesionless granular materials

Abstract: Presented is an analysis on the Couette flow of cohesionless granular materials between two co-axial rotating cylinders. The constitutive equations employed have been postulated on the basis of available experimental and theoretical results which take into account the particle collisions as well as dynamic pressures induced by the trace of the unsemble phase average of the square of flow fluctuations. These constitutive equations loosely resemble the Reiner-Rivlin fluid behavior, and predict normal stress effects.

The influence of biaxiality of loading on the form of caustics in cracked plates

Abstract: Summary The optical method of reflected caustics was applied up-to-now to problems of cracked plates under uniaxial loading. Only the problem of the biaxial tension of the plate has been considered for the particular case where the crack is transverse to the longitudinal axis of the plate which coincided with the loading axis. In this paper the influence of a biaxial loading of the plate on the form and orientation of the caustic was studied in connection with the orientation of the crack. New modified relations were given for the evaluation of the complex stress intensity factor K = K I -- iKri in terms of the angle ~v of the angular displacement of the caustic axis. For the accurate evaluation of K I and K H nomograms of correction factors 6vmax, 6x max and dx rain were given in terms of the angle of inclination of the crack O9 = (90 -- fl) and the biaxiality factor k. Experimental evidence with PMMA internally cracked plates corroborated the results of theory.

Effect of porous lining on the MHD flow between two concentric rotating cylinders under the influence of a uniform magnetic field

Abstract: In this paper we have studied the viscous incompressible and slightly conducting fluid flow between two concentric rotating cylinders with non-erodable and non-conducting porous lining on the inner wall of the outer cylinder under the influence of a uniform radial magnetic field of the formB r =A/r (Hughes and Young). We have solved the equations using the method given in Hughes and Young. We have evaluated the velocity and temperature distributions, shearing stress and the rates of heat transfer coefficients. We have investigated the effects of the magnetic parameter, the non-dimensional measure of the thickness of the porous lining, the ratio of the velocities of the cylinders, the slip parameter, the porosity parameter, the Biot number and the productP r E on the physical quantities evaluated.

Saint-Venant's principle for a micropolar helical body

Abstract: The analog of Toupin's version of Saint-Venant's principle is proved for an isotropic, linear elastic micropolar body of arbitrary length and of uniform cross-section which in the unstressed state is helical. That is, when such a body is loaded by self-equilibrated stresses and couple stresses at one end only, we show that the elastic strain energy stored in the part of the body beyond a certain distance from the loaded end, decreases exponentially with the distance.

On the Theory of Diffusion in Linear Viscoelastic Media

Abstract: A recently developed viscoelastodiffusive theory is further examined in terms of its mathematical foundations and obvious generalizations. A derivation of the basic equations under quite general assumptions is provided and an equivalence between a “hereditary” version and a “differential operator” version of the theory is established. Both relaxation and creep types of viscoelastodiffusive behavior is considered. The concepts of chemical potential, fading memory, and a correspondence principle are discussed. The general development is concluded with an illustrative example.

Non-unique solutions of the boundary layer equations for the flow near the equator of a rotating sphere in a rotating fluid

Abstract: In this paper it is shown, using a numerical technique, that axially-symmetric solutions of the boundary layer equations which describe the rotating flow near the equator of a rotating sphere are not unique. In certain regimes it is found that at least three possible solutions are possible. When the sphere and fluid rotate with almost the same angular velocity it is shown that the approach to solid body rotation is a non linear process.

A mathematical model of post-instability in fluid mechanics

Abstract: Mathematical models of continua are based on the assumption that the functions describing their states can be differentiated as many times as necessary. This artificial mathematical limitation follows neither from the principles of mechanics nor from the definition of a continuum. The price paid for such a mathematical convenience is instability (in the class of smooth functions) of the solutions to the corresponding governing equations. This instability can appear in the form of failure of hyperbolicity [1], [2], [3], [4] or in the form of “cascade instability” (stretching of vorticity in an inviscid fluid or instability of the Navier-Stokes equations [5], [6], [7]). All the cases are characterized by an unlimited decrease in the scale of the motions, in the course of which the derivatives of the corresponding functions tend to infinity, although the functions themselves remain finite. In other words, the solution tends to “go out” from the class of differentiable functions. Such an instability shows that the corresponding state of a continuum cannot be properly simulated by smooth functions. Hence, the original model must be corrected by giving up the requirement about differentiability and enlarging the class of functions. In this article postinstability in fluids is simulated by giving up the principle of impenetrability and introducing multivalued fields of velocities. The fundamental properties of enlarged model of fluid with application to turbulence are discussed.

On the concept of stress-strain relations in plasticity

Abstract: In some cases the plastic stress-strain relations based on the so-called normality rule lead to an unsatisfactory agreement with experimental results. This is particularly true in bifurcation problems or more generally in problems with non-proportional loading paths. Within the frame of a phenomenological theory of non-isothermic large deformations it is shown how plastic stress-strain relations can be modified in order to improve the agreement with the real material behaviour.