The problem of a slender die

The paper deals with the contact behaviour of a slender die indenting an elastic half-space. It is shown that the problem of determining the pressure on the elastic half-space may be reduced (with an error exponentially small relative to the elongation) to a single-variable integral equation, whose solution is commonly represented by an asymptotic series in a small parameter. It was shown for a die of oval form that, depending on the type of contact region, either an increase or decrease in the force acting from the elastic half-space on the die upon approaching the end-points of the die are possible.     

Impact of a plane die on an elastic orthotropic half-plane

To study the process of impact of a rigid body on the surface of an elastic body made of a composite material, we consider a nonstationary dynamic contact problem about the impact of a plane rigid die on an elastic orthotropic half-plane. The problem is reduced to solving an integral equation of the first kind for the Laplace transform of the contact stresses under the die base. An approximate solution of the integral equation is constructed with the use of a special approximation to the symbol of the kernel of the integral equation in the complex plane. The inverse Laplace transform of the solution results in determining the scalar contact stress field on the die base, the force exerted by the die on the elastic medium, and the vertical displacement field of the free surface of the orthotropic medium out side the die. The solutions thus obtained permit studying specific features of the process of die penetration into an orthotropic medium and the strain properties of the medium.     

Contact problems on soft and rigid coatings of an elastic half-plane

The solutions of contact problems on a soft or rigid coating of an elastic half-plane are of great practical interest. Accordingly, the present paper is divided into two parts: in the first part, we consider the problem of interaction between a rigid biquadratic die and an elastic half-plane through a thin soft coating; in the second part, we consider the problem of interaction between a rigid plane die and an elastic half-plane through a thin rigid coating. We derive integral equations for the problems under study and construct their approximate solutions by a regular asymptotic method. Earlier, the question of studying such problems was posed, for example, in [1–3]. Here we use these results to a large extent.     

Biegung metallischer Kreisplatten mit elastischer Unterlage

Übersicht Vorgestellt wird die Analyse rotationssymmetrischer Platten, welche auf einer elastischen Bettung ruhen. Die Platten bestehen aus elastisch/perfekt-plastischem Werkstoff, für den Tresca's Fließ bedingung und die assoziierte Fließregel gelten. Vorgeschlagen wird eine Methode zur analytischen Beschreibung des Verformungsvorgangs unter einer statischen Last, die monoton anwächst. Lösungsbeispiele werden angegeben.

---

Bending of metallic circular plates resting on an elastic subgrade

Summary Rotationally symmetric plates resting on an elastic (Winkler) subgrade are studied. They are made of an elastic, perfectly plastic material obeying Tresca's yield condition and associated flow rule. The proposed method gives closed form solutions for monotonically loaded sandwich type plate. Examples for clamped and free edge plates are given.

     

Constitutive Model Calibration for Powder Compaction Using Instrumented Die Testing

Calibration procedures for constitutive models for powder compaction are presented. A practical calibration method based on a die compaction experiment is presented. A newly developed apparatus consisting of a die instrumented with radial stress sensors is described. The paper proposes two contributions to account for errors present in instrumented die testing, which are due to 1) elastic compliance of the testing frame, influencing the measurement of axial strain and 2) the presence on non-homogeneous stress state in the test specimen. It is shown that system compliance is important for generating an accurate stress-strain curve for compression. The effect of different compliance correction methods is evaluated with regard to the accuracy of models predicting pressing forces. The system compliance becomes more significant during unloading in the die; this information is used to determine the elastic properties. A new compliance correction method is introduced following a detailed analysis of the forces and deformations of different parts of the loading frame. In instrumented die compaction the axial and radial stresses are measured at fixed locations and the specimen is subject to non-homogeneous stresses and strains due to the effect of friction between the powder and die wall. Starting from the Janssen-Walker method of differential slices a method to account for non-homogeneous stress and strain is developed.     

Design of Inhomogeneous Plates on an Elastic Half-Space

An applied model is developed for designing an inhomogeneous plate on an elastic half-space. The plate is discretized by the finite-element method and the half-space by the boundary-element method. The model allows us to evaluate the stress–strain state of the plate immediately under a die, whose dimensions are comparable with the plate's cross section. The plate may be inhomogeneous across the thickness and in plan. Separation of the plate from the elastic base is possible.     

Zur Formulierung von Stoffgesetzen der Plastomechanik im Dehnungsraum nach Ilyushins Postulat

Übersicht Stoffgesetze der Plastizitätstheorie, die im Dehnungsraum formuliert sind, vergleichen die Referenzmit der aktuellen Konfiguration, aber auch mit einer Zwischenlage, in welcher der betrachtete materielle Punkt spannungsfrei ist. Durch geeignete Kombinationen von rechten Cauchy-Green-Tensoren und ihren Inversen kann man insbesondere ein elastisch isotropes Verhalten beschreiben. Ausgehend vom zugehörigen elastischen Potential und von einer gegebenen Konsistenzbedingung, die an die Stelle des Fließkriteriums im Spannungsraum tritt, erzeugt Ilyushins Postulat eine Fließregel, die zu einer sehr praktiblen und expliziten Form vereinfacht werden kann, wenn man kleine elastische Deformationen voraussetzt.

---

Summary Constitutive equations of the plasticity theory formulated in terms of strains, compare the reference and the actual configuration, but also with an intermediate one in which the considered material point is free of stresses. Suitable combinations of right Cauchy-Green-Tensors and their inverses make possible to describe especially an isotropic elastic behaviour. On the bases of an appropriate elastic potential and of a given condition of consistency which is used instead of a yield-condition expressed in terms of stresses, Ilyushin's postulate induces a flow rule which can be simplified to very practicable and explicite forms if small elastic deformations are assumed.

---

---

Gewidmet Prof. Th. Lehmann, Ruhr-Universität Bochum, zum 60. Geburtstag     

Torsion des unendlichen Halbraumes durch einen kreisförmigen Stempel

Übersicht Es wird ein numerisches Berechnungsverfahren für die Einleitung des Torsionsmomentes im unendlichen Halbraum durch einen kreisringförmigen Stempel, der fest mit dem Halbraum verbunden ist, entwickelt. Das gemischte Randwertproblem wird mit Hilfe geeigneter Singularitäten auf eine Fredholmsche Integralgleichung zurückgeführt und numerisch gelöst. Die Lösung der Integralgleichung gibt direkt die gesuchten Schubspannungen in der Kontaktfläche an. An Testreohnungen wird die Brauchbarkeit der Methode aufgezeigt.

---

Torsion of an elastic half-space by a flat annular stamp

Summary The problem considered here is that of an elastic isotropic half-space twisted by the rotation of a flat annular stamp to which it is joined. Using singularities the mixed boundary value problem is reduced by means of Betti's reciprocical theorem into a Fredholm integral equation for the unknown shearing stresses in the contact region. Some numerical results are presented and compared with published ones.

     

On the stretching and swelling of an elastic liquid extruded from a capillary die

A theoretical model for calculating the profile of a polymer jet extruded from a circular die and stretched by an axial force is elaborated. The model exploits the idea of matching the solutions for the polymer flow inside and outside the capillary by equating the elastic energy flux at the die outlet. The rheological equations used to obtain the two solutions have been previously tested in many situations. Comparison of the results of computations made according to the theoretical model with those of some special experiments revealed satisfactory agreement.     

Die Ausbreitung einer elastischen Kugelwelle in einem unendlichen Körper mit transversaler Isotropie

Übersicht In einem unendlichen elastischen Körper mit sphärisch transversaler Isotropie wird die Oberfläche einer um den Ursprung geschlagenen Hohlkugel mit konstantem Druck belastet. Die die Wellenausbreitung beschreibende Differentialgleichung wird der Laplace-Transformation unterworfen. Die Bildlösung baut sich aus modifizierten Besselfunktionen zweiter Art auf, deren Zeiger von den elastischen Konstanten abhängt. Ist die Formänderungsenergie des Körpers eine positiv definite quadratische Form, so gelten für die elastischen Konstanten einschränkende Ungleichungen. Sind diese erfüllt, so ist die Bildlösung in der positiven Halbebene und auf der imaginären Achse mit Ausnahme des Ursprungs regulär. Mit Hilfe des Umkehrintegrals der Laplace-Transformation kann die Lösung im Oberraum berechnet werden. Diese Lösung wird numerisch ausgewertet für spezielle Werte der elastischen Konstanten.

---

Summary In an infinite elastic body possessing spherical transversal isotropy the surface of a sphere around the origin is subjected to constant pressure. Laplace's transformation is applied to the equation for the radial displacements. The solution in the subdomain is constructed with aid of modified Bessel functions of the second kind, the order of which depends on the elastic stiffnesses. If the strain energy is positive definite, the elastic stiffnesses are submitted to certain inequalities. Therefore the solution in the subdomain possesses no singularities in the positive half of the complex domain and on the imaginary axis with exception of the origin. By use of the inversion theorem of Laplace's transformation the solution in the time domain is derived. For a special choice of the elastic stiffnesses numerial results are given.

     

The buckling load of an elastically supported cantilevered column with continuously varying cross section and distributed axial load

Summary The solution for the buckling load of an elastic supported cantilevered column with continuously varying cross section and distributed axial load is developed. Boundary conditions are formulated for fixed and elastic support. The critical loads are tabled for an usual variation in cross section and axial load distribution as function of the ratio between respectively rigidities and axial forces at fixed and free end, the variation of the rigidity and axial forces given by their degrees of the polynomial variation and rotation stiffness of the elastic support.

---

Übersicht Es wird die Knicklast für eine elastisch gelagerte, freitragende Säule mit stetig veränderlichem Querschnitt und stetig verteilter Axiallast berechnet. Die Randbedingungen sind für feste sowie für elastische Lagerung formuliert worden. Für einen übliczen Bereich von Querschnitten und Axiallasten werden die kritischen Lasten in Tabellen angegeben. Dabei wurde das Verhältnis zwischen der Steifigkeit und den Axiallasten am festen und freien Ende variiert.

     

Photoelastic stress analysis of a V-shaped die for plastic work

The stress state in the V-shaped die for plastic compression was investigated by using a photoelastic stress analysis in which an Araldite in a glassy elastic state and a softened celluloid were used as model materials for the die and work specimen, respectively. It was found that the direction of the frictional shear stress is reversed at a certain point on the die surface. Because the frictional shear stress of the die mainly depends on the flow speed of the work material, the popular assumption that the coefficient of friction is a constant over the die surface such as in the case of Coulomb friction appears unrealistic.     

The dynamic indentation of an elastic half-space by a rigid punch

The two-dimensional contact problem between a rigid die and an elastic half-space is considered. A numerical method of solution is proposed which involves an iterative process which is continued until the correct solution is obtained according to certain criteria. The method is general enough and can handle punches of arbitrary shape as well as time-dependent indentation velocities. The treatment is unified for subsonic, transonic and supersonic indentations. The numerical procedure is checked with analytical results which are known in several special cases and good agreement is obtained. Results are presented for the smooth as well as frictional indentation by a wedge-shaped die and for a smooth parabolic punch.     

Zur Theorie elastoplastischer Schalentragwerke

Übersicht Für dünne Schalentragwerke wird eine allgemeine Theorie zur Berechnung elastoplastischer Formänderungen bei kleinen Verzerrungen angegeben. Als Grundlagen dienen dazu für elastische Formänderungen das Hookesche Gesetz und für plastische Formänderungen eine verallgemeinerte v. Misessche Fließbedingung, die die Möglichkeit isotroper oder anisotroper Verfestigung noch offen läßt, verbunden mit einem Formänderungsgesetz, das die Inkremente von Verzerrungen und Spannungen miteinander verknüpft. Im Bereich elastoplastischer Formänderungen gilt eine Addition elastischer und plastischer Verzerrungsgeschwindigkeitsanteile; ein Entlastungsvorgang wird durch ein differentielles elastisches Verhalten beschrieben.

---

Summary A general theory of elastoplastic deformations of thin shells is developed considering small strains. The basic equations are (i) Hooke's law in the elastic range and (ii) a generalized v. Mises criterion, which is able to describe hardening in an isotropic as well as in an anisotropic way, together with a deformation law connecting strain rates and stress rates in the plastic range. The range of elastoplastic deformations is defined by an addition of elastic and plastic strain rates. Unloading occurs in a differentiated elastic way.

     

Elastic fields of a dislocation loop in a two-phase material

The purpose of this study is to set forth the mathematical framework for an efficient treatment of a plane dislocation loop in a two-phase material, and to carry the elastostatic part of the analysis far enough, so that the results are immediately tractable for applications to solid state. The two-phase material is idealized as two isotropic elastic half-spaces with perfect adhesion, while the loop is placed in a plane parallel to the interface. It is shown that the elastic fields can be obtained by differentiating two types of integrals and, thus, are readily evaluated for any shape of the loop for which the integrals are available from potential theory. Explicit results are given for circular prismatic and glide loops.

---

Zusammenfassung Mit der vorliegenden Arbeit wird die Absicht verfolgt, den mathematischen Rahmen zu einer leistungsfähigen Behandlung von ebenen Versetzungsschleifen in einem Zweiphasenmaterial zu erweitern und den elastostatischen Teil der Berechnung so weit fortzuführen, daß die Resultate für die Anwendung in der Festkörperphysik unmittelbar zur Verfügung stehen. Das Zweiphasen-Material wird idealisiert durch zwei isotrope elastische Halbräume mit perfekter Adhäsion, während die Versetzungsschleife in einer Ebene parallel zur Trennfläche angenommen wird. Es wird gezeigt, daß die elastischen Felder durch Differentiation zweier Integraltypen und damit für jede beliebige Form der Versetzungsschleife erhalten werden können, für die die Integrale sich aus der Potentialtheorie ergeben. Explizite Resultate werden für kreisförmige prismatische und gleitende Schleifen angegeben.

     

Some research directions in finite elasticity theory

Summary A number of topics in finite elasticity theory which appear to lend themselves to further development were briefly discussed. These include (i) the effect of kinematic constraints which are exactly, or approximately, satisfied; (ii) the mechanics of elastic membranes; (iii) the applicability of results in finite elasticity theory to problems involving stress relaxing materials; (iv) the development of necessary and sufficient conditions for material stability of isotropic elastic materials; (v) the conditions for bifurcation solutions to exist in deformed elastic bodies.

---

Zusammenfassung Es werden einige Themen der nichtlinearen Elastizitätstheorie, die sich für weitere Entwicklungen anbieten, kurz besprochen. Darunter sind (i) die Wirkung kinematischer Zwangsbedingungen, die exakt oder näherungsweise erfüllt werden; (ii) die Mechanik elastischer Membranen; (iii) die Anwendbarkeit von Ergebnissen in der nichtlinearen Elastizitätstheorie auf Probleme, die Materialien mit Spannungsrelaxation einschließen; (iv) die Entwicklung notwendiger und hinreichender Bedingungen für materielle Stabilität isotroper elastischer Materialien; (v) die Bedingungen für die Existenz von Verzweigungslösungen in deformierten elastischen Körpern.

---

---

With 1 figure     

Bemerkungen zu dem Einfluß der Massenverteilung bei nichtkonservativen Stabilitätsproblemen elastischer Stäbe

Übersicht Ein unten eingespannter, oben frei beweglicher elastischer Stab stehe unter der Einwirkung einer richtungstreuen und einer mitgehenden Kraft. Es wird untersucht, wie sich die Massenverteilung und die Reihenfolge der Belastung auf die kritische Belastung auswirken.

---

Summary The influence of the mass distribution and of the sequence of the loading is investigated for an elastic column. The column is assumed to be clamped at the base and free at the upper end. The load consists of one force acting in a fixed direction and of another force which maintains its orientation with respect to the deforming column.

     

Contact electroelasticity problem for a nonplane elliptical die

The stress-strain state of an elastic piezoceramic halfspace under the action of a rigid elliptical die with a convex base is considered, in the case where the symmetry axis of the body is in the direction of the preliminary-polarization field. The contact area is within the isotropy plane. Attention is confined to the case of die translation. Expressions are obtained for the semimajor axis and eccentricity of the contact area and the translational points of the die base under different electric boundary conditions. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 17–26, June, 1999.     

Rollreibung mit axialem Schub

übersicht Das Problem des Schräglaufes eines elastischen Kreiszylinders auf der Oberfläche eines elastischen Halbraumes wird unter der Voraussetzung, daß das Coulomb'sche Reibungsgesetz für jedes Flächenelement des Berührgebietes gilt, im Rahmen der Elastizitätstheorie erster Ordnung untersucht. Nach Herleitung der dieses Problem beherrschenden nichtlinearen singulären Integralgleichungen werden für gleiche Materialkonstanten der beiden Körper Methoden entwickelt, die eine numerische Lösung ermöglichen. Die Ergebnisse sind in Schaubildern dargestellt, die die gesamte Lösungsmannigfaltigkeit für die spezielle Querzahl v = 0,3 (Stahl auf Stahl) umfassen.

---

Summary The problem of an elastic cylinder of circular cross-section moving obliquely on the surface of an elastic half-space is treated by means of an elasticity theory of first order, assuming that Coulomb's law of friction is valid for each element of the contact area. After the derivation of the nonlinear singular integral equations governing the above problem, methods for a numerical solution for the case of equal material constants are presented. For a Poisson's ratio of 0.3 (steel on steel) the complete range of possible solutions is given and shown in graphs.

     

Slow nonstationary vertical motions of a die on the surface of an elastic half-space

We consider the dynamic contact problem on vertical motions of an absolutely rigid body on an elastic half-space. We assume that the contact region does not vary during the motion and there is no friction under the die bottom. We construct an approximate solution of the problem under the assumption that the variation in the contact pressure under the die bottom on the time interval in which the Rayleigh wave runs the distance equal to the contact area diameter is small. Computational formulas are obtained for the cases of circular and elliptic dies.