Showing papers in "Acta Mechanica in 1969"

An attempt to describe the behavior of metals under cyclic loads using a more general workhardening model

Abstract: In many problems of plastic deformation, when the prescribed loads or displacement do not increase in proportion but vary in a complex manner, for instance alternating between prescribed limits, more general work-hardening models are needed in order to describe the plastic behavior. One of such models, previously proposed by the author, is applied here in order to discuss superposition of fixed and alternating loads. In particular, the case of steady tension and alternating torsion is considered in detail and the rate of axial strain accumulation is computed. There is a qualitative aggreement between theoretical prediction and experimental data.

Eine Cosserat-Theorie des plastischen Flie\ens

Abstract: In geeigneter Ubertragung der Ideen von E. undF. Cosserat (1896) auf die Plastizitatstheorie werden Drehgeschwindigkeiten der infinitessimalen Werkstoffelemente (Korner) betrachtet, die unabhangig von den durch das klassische Geschwindigkeitsfeld induzierten Rotationen sind. Die Differenz beider sowie die Relativdrehung benachbarter Korner sind zusatzliche Verformungsgro\en. Dementsprechend wird die Statik durch asymmetrische Schubspannungen und Momentenspannungen vervollstandigt. Fur ein angemessenes Grundgleichungssystem sind mindestens zwei Flie\kriterien, beide im Zusammenspiel als plastische Potentiale, und die zugehorige Flie\regel erforderlich. Davon werden drei spezielle Formen als Verallgemeinerungen desTrescaschen sowie desHuber-Levy-Misesschen Stoffgesetzes aufgestellt und angewendet (a) auf die Torsion des Kreiszylinders — wobei in einem Fall der experimentell beobachtete isotropePoynting-Effekt (Langenanderung) herauskommt — und (b) auf die allgemeine ebene Formanderung. Hier erweist sich nur die verkurzte Theorie (keine antimetrischen Schubspannungen) als genugend einfach: Alle klassischen Gleitlinienlosungen bleiben gultig, und die zusatzlichenCosserat-Gro\en kann man auf geometrischem Wege ermitteln.

Large deflections of heated circular plates

Abstract: Large deflections of heated circular plates with or without a concentric circular hole and with different boundary conditions and temperature distributions are analyzed in the light ofBerger's analysis. The subject is treated in a simple and unified manner. Numerical results are shown in graphical form.

Second Order Effects in the Theory of Plasticity

Abstract: A simple yield condition for the elastic-plastic medium is developed which admits a stress-strain-increment relation including the observed second-order extension associated with pure torsion and the resulting dissimilarity between theMohr circles of stress and strain-increments. This yield condition is then used as the loading function of a strain-hardening medium in conjunction with a rule of kinematic hardening.

On fading memory in materials of evolutionary type

Abstract: It is shown that a material whose thermodynamical state is described by internal variables α i which are governed by equations of evolution of the form $$\dot \alpha _i = f_i \left( {u_j ,\alpha _k } \right)$$ , where theu j represent the “external” variables (deformation, temperature, temperature gradient) possesses the property of fading memory (as postulated byColeman) if thef i are differentiable and if the matrix of their partial derivatives with respect to the α k is negative definite.

Contact problems of a rectangular block on an elastic layer of finite thickness. Part I: The thin layer

Abstract: We consider a layer of finite thickness loaded in plane strain by a stamp with a straight horizontal base, which is smooth and rigid. The stamp is pressed vertically into the layer and is slightly rotated by an external moment load subsequently. Two cases are considered successively: the lower side of the layer is attached to a rigid base and, secondly, the lower side is allowed to slide without friction along the base. The paper is divided into two parts. The first part deals with an asymptotic investigation of a thin layer and the second one refers to a thick one.

On the elastic stability of thick columns

Abstract: Stability of a rectangular strip of isotropic highly elastic incompressible material is analyzed using the theory ofGreen, Rivlin andShield. The strip is subject to a uniaxial compression in planes of plane strain. Both the asymmetric and symmetric forms of buckling are studied and the results found in full agreement with the corresponding results obtained from the theory ofBiot.

Störwellenlängen-Messung bei Längswirbeln in laminaren Grenzschichten an konkav gekrümmten Wänden

Abstract: Bei der plotzlichen Abbremsung eines um seine Achse rotierenden flussigkeitsgefullten Zylinders treten in der sich bildenden Wandgrenzschicht instationareGortler-Taylor-Wirbel auf. Diese werden sichtbar gemacht und ihre Storwellenlangen im ersten Moment ihres Erscheinens gemessen. Es zeigt sich, das ihre Grose nur abhangt von einerReynolds-ZahlRe a =ω·R a 2/v, die den Zustand der Rotation des Zylinders vor dem Abbremsen kennzeichnet. Eine theoretische Abschatzung der Grose der Storwellenlangen stimmt mit den Mesergebnissen gut uberein.

The solution of mixed boundary value problems that involve time-dependent boundary regions for viscoelastic materials with one relaxation function

Abstract: A homogeneous and isotropic linear viscoelastic material, whose relaxation function in shear is a constant multiple of its relaxation function in isotropic compression, occupies a region which does not change with time. It is shown that, if all the regions where stress (displacement) boundary conditions are prescribed are either monotone increasing or monotone decreasing for all time, the solution of a viscoelastic problem may be reduced to the solution of a one-parameter family of adjunct elastic problems. This result is generalized to give a tie between the solution of boundary value problems for certain anisotropic and inhomogeneous viscoelastic bodies, with one relaxation function, and solutions of anisotropic, inhomogeneous elastic boundary value problems.

Wave propagation in inhomogeneous elastic media, solution in terms of Bessel functions

Abstract: Wave propagation in an inhomogeneous elastic solid, when the elastic parameters of the solid depend on one space co-ordinate only, is considered. The stress and displacement components are assumed to depend on this same space co-ordinate and time alone. For the situations considered here the motion is governed by the wave equation with variable wave speed. The values of the elastic parameters which correspond to the solution of this equation in terms ofBessel functions are determined and some particular problems are considered.

The convexity of the stability boundary of symmetric structural systems

Abstract: The elastic stability of a discrete conservative structural system with independent loading parameters is studied Attention is restricted to systems which satisfy certain analytical symmetry conditions and which lose their initial stability at a point of bifurcation A theorem, concerning the convexity of the stability boundary of such systems, is established

Schallnahe Strömungen mit Wärmezufuhr

Abstract: Es werden die Grundgleichungen fur die schallnahen Profilumstromungen mit Warmezufuhr hergeleitet. Man erhalt eine nichtlineare Differentialgleichung vom gemischten Typ, bei der die Warmezufuhr in Form eines Quellgliedes auftritt. Eine Naherungslosung dieser Gleichung wird fur verschiedene Grosen der zugefuhrten Warme untersucht. Hierbei spielt die sogenannte kritische Warmezufuhr eine wichtige Rolle. Generell zeigt sich, das eine Warmezufuhr im Uberschallteil des Stromfeldes zu einer Widerstandsreduktion fuhrt.

Zur Berechnung der Eigenspannungen bei einem durch endliche Biegung verformten inkompressiblen Plattenstreifen

Abstract: Die Eigenspannungen eines inkompressiblen Plattenstreifens, der elastisch-plastisch durch endliche Biegung verformt wird, werden ermittelt. Es zeigt sich, das die sich ergebenden Eigenspannungen in hinreichender Genauigkeit auch mit Hilfe eines Superpositionsprinzips ermittelt werden konnen, indem man dem am Ende der Belastung sich ergebenden Spannungszustand die Spannungen einer elastischen Biegung mit entgegengesetztems gleich grosen resultierenden Moment uberlagert.

Stress analysis of an infinite plate containing an elastic rectangular inclusion

Abstract: A solution is presented for determining the stresses in an infinite elastic plate containing an elastic rectangular inclusion subjected to a uniform stress field. The bond stresses are obtained by treating the plate and the inclusion separately and satisfying continuity of displacements at discrete points rather than continuously. An example of a square elastic inclusion is treated in detail. Comparison is made between the solution obtained by point-matching techniques and the exact solution for a rigid square inclusion. Excellent agreement is observed. The practical motivation behind this research was the estimation of the bond stresses induced between the fibers and matrix of multifiber composite sheets.

Stresses in a thick plate containing an eccentric spherical inclusion or cavity Part I

Abstract: The paper presents an analytic solution for the stresses in a thick plate of infinite size containing a spherical inclusion or cavity eccentrically located between the surfaces. The plate is subject to a stress system symmetrical about the axis of revolution of the plate while the surfaces are stress-free. In the inclusion problem, the inclusion is assumed to be rigid and no slip occurs between the plate, and inclusion. In the cavity problem, the surface of cavity is assumed to be stress-free. The solutions of both problems are illustrated by numerical examples for the cases of all-round tension and circular bending.

A contribution to the theory of viscous damping of stratified wave flows

Abstract: A boundary layer method is described for the calculation of viscous damping coefficients associated with small amplitude progressive waves propagating in non-homogeneous fluid. The analysis is formulated essentially in terms of the perturbation pressures and it is suggested that this procedure might be of use in a variety of wave damping calculations.

Buckling of viscoelastic maxwell plates under random temperature

Abstract: Considered is the problem of thermal buckling of a square plate, simply supported along the edges, of a linear incompressible viscoelasticMaxwell material, and exposed to a temperature field varying randomly with time and across the width of the plate.

Creep buckling of simply supported moderately thin circular cylindrical shells

Abstract: The creep deformations of a circular cylindrical shell are calculated under the action of uniform axial compression. The edges of the shell of finite length are simply supported and the deformations are assumed to be axially symmetric. Such axisymmetric deformations have been observed to occur when the shell is moderately thin-walled. The solution is obtained in the form of trigonometric series for the stresses and displacements.

Approximate solutions for the structure of shock waves

Abstract: Approximate solutions of the B-G-K model equation are obtained for the structure of a plane shock, using various moment methods and a least squares technique. Comparison with available exact solution shows that while none of the methods is uniformly satisfactory, some of them can provide accurate values for the density slope shock thickness delta n . A detailed error analysis provides explanations for this result. An asymptotic analysis of delta n for largeMach numbers shows that it scales with theMaxwell mean free path on the hot side of the shock, and that their ratio is relatively insensitive to the viscosity law for the gas.

On the derivation of the constitutive equations of ideal plasticity

Abstract: TheLevy-Mises equations for an ideal plastic material are derived from a general implicit relation between the stress and rate of deformation tensors. This derivation provides an alternative to the usuala priori postulates of a yield criterion and flow rule.

Non-symmetrical plastic collapse modes of a symmetrically loaded circular cylinder

Abstract: Plastic collapse modes of a thick-walled homogeneous cylinder under circularly-symmetrical loading (internal and external normal pressures) are analyzed. Ideal plasticity,Huber-Mises-Hencky yield condition, incompressibility and plane strain conditions are assumed. The possibility of existence of non-symmetrical plastic collapse modes corresponding to circularly-symmetrical stress distribution is found. General non-symmetrical solution is presented in the form ofFourier series and illustrated by some numerical examples.

A uniqueness theorem in the linear theory of elasticity with microstructure

Abstract: It is shown that in theMindlin theory of elastic materials with microstructure, uniqueness holds for the traction, the displacement, and a wide variety of mixed problems, provided the potential energy function is homogeneous of order two.

Die Anwendung der Monge-Gleichung auf Ausbreitungsvorgänge in Gasen

Abstract: Fur das im Rahmen der Naherung exakte Entwicklungsverfahren des zweiten Verfassers wurden die Voraussetzungen untersucht und die Grundgleichungen sowie weitere Beziehungen in allgemeiner Form abgeleitet.

Thermal waves in an elastic highly stretched cylindrical bar

Abstract: The propagation of thermoelastic waves in a cylindrical bar of incompressible material and circular cross-section is investigated assuming that the bar, kept at a uniform constant temperature, undergoes a finite initial stretch, has an infinite length and the superposed disturbance is axially symmetric and of small amplitude. Three governing equations for the infinitesimal radial and longitudinal displacements as well as for the secondary pressure function and the secondary transient temperature field are derived. The remaining field equation is supplied by the incompressibility condition. The system of coupled second order partial differential equations involving three independent variables is solved by the method ofFrobenius assuming a propagation of simple harmonic waves. Three approximations to the phase velocity are found that in the isothermal case coincide with the known results both of the classical and the finite deformation theory.

Transient heat transfer for flow over a rotating disk

Abstract: The development of the thermal boundary layer over a rotating disk, following a step change in temperature of the disk, is discussed. The unexpected non-monotonic behaviour of the response time as a function of thePrandtl number is examined by considering the asymptotic form of the solution of the governing equation for large time. The results and conjectures which are to be found in earlier papers are, respectively, extended and verified.

Transient Energy Transfer in Hot and Cold Layers of Radiating Gas during Expansion

Abstract: Transient energy transfer in the hot and the surrounding cold layers of gray radiating gas during expansion has been studied. The gases have been assumed to be perfect and in local thermodynamic equilibrium. The heating of the gas by a diffuse and a collimated component of radiation flux incident on the boundaries from some external source has been considered and the dependence of the physical and radiative properties of the gas on the temperature has also been accounted for. The governing equations have been transformed toLagrangian coordiantes and then solved simultaneously by a finite difference method. The distributions of temperature, pressure and velocity are presented in hot as well as cold gas layers as a function of time for a variety of parameters and initial conditions of physical interest.

Minimal design of sandwich axisymmetric cylindrical shells obeying Mises' criterion

Abstract: The solution is presented of the problem of minimum volume design of the circular cylindrical sandwich shell obeying theMises yield criterion and carrying loads transverse to the lateral surface and symmetric about the axis of the cylinder. Detailed conditions are established, which are necessary and sufficient for the determination of the minimal stress both bounded and unbounded. In the case of one-directional loads the minimal and optimal designs corresponding to the above minimal stresses are specified up to the point from which only routine calculations are required. It is shown that for such loads the characteristic features of the minimal and optimal designs are independent of the peculiarities of the loading. All six possible combinations of edge support are considered. The results are compared with those of the only previous investigation of the same structure but underTresca's criterion. Some significant discrepancies between the results of the two papers are pointed out.

Berechnung von Überschallströmungen um schlanke Körper mit Hilfe bewegter Singularitäten

Abstract: In dieser Arbeit werden Ausstrahlungslosungen der instationaren Wellengleichung fur ein ruhendes Storzentrum angegeben, die die Anfangsbedingung erfullen und langs einer Achse singular werden. Durch Uberlagerung solcher, langs einer Kurve verteilter Losungen entsteht das Potential einer bewegten Singularitat. Dabei kann die zur Erfullung der Randbedingung notwendige Symmetrie der bewegten Singularitat durch Einschaltung einerLorentz-Transformation ebenso erreicht werden, wie die Erfullung der Anfangsbedingung an der richtigen, zur Anstromrichtung symmetrischenMach-Front. Bekannte Ergebnisse werden bestatigt und fur beschleunigte und pendelnde Korper neue hinzugefugt.

Steady temperature fields in second order fluids

Abstract: It was shown earlier that if a viscometric flow of an incompressible simple fluid takes place between two surfaces held at constant but differing temperatures, and if the temperature gradient exists in the direction of the velocity gradient only, two conductivity coefficients determine the heat flow and the temperature field. In this article, after the zeroth, first and second order asymptotic expansions for the heat flux vector are developed under a fading memory assumption, the temperature fields in a few viscometric flows are determined und the above condition on the temperature gradient. It is found that a reduction in heat transfer can occur under these conditions, which suggests that the present theory may be used to explain the observations in turbulent flows of polymer additives in water.