Axisymmetric contact problem for an elastic layer and an elastic foundation

This paper is concerned with the axisymmetric problem of an elastic layer lying on a semi-infinite foundation. The layer is pressed against the foundation by a uniform clamping pressure applied over its entire surface and a uniform vertical body force due to the effect of gravity. In addition, an axisymmetric vertical line load is applied to the layer. It is assumed that the contact between the layer and the foundation is frictionless and that only compresive normal tractions can be transmitted through the interface. The contact along the interface will be continuous if the value of the line load is less than a critical value. However, interface separation takes place if it exceeds this critical value. The problem is formulated and solved for the cases of tensile and compressive line loads. Numerical results for contact stress distributions are given for different material combinations.     

Axisymmetric contact problem for an elastic layer on a rigid foundation with a cylindrical hole

A solution of an axisymmetric contact problem for an isotropie homogeneous, elastic layer bonded to the rigid foundation with a cylindrical hole, is obtained. The effect of the weight of the layer is also taken into account. The problem is reduced to the solution of two simultaneous Fredholm integral equations of the second kind whose iterative solution is obtained for $\text{h > max (a, b)}$ where $\text{h}$ is the layer thickness and $\text{a}$ and $\text{b}$ are respectively the radius of contact of the axisymmetrie punch and the radius of the cylindrical hole. Simple expressions are obtained for the total pressure on the punch, the distribution of normal pressure under the punch, the shape of the deformed surface outside the punch and the normal displacement in the cylindrical hole.     

Axisymmetric contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the elastostatic contact problem of a semi-infinite cylinder compressed against a layer lying on a rigid foundation. It is assumed that all the contacting surfaces are frictionless and that only compressive normal tractions can be transmitted through the interfaces. Upon loading the contact along the layer-foundation interface shrinks to a circular area whose radius is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of the contact area between the layer and the foundation, and the stress intensity factor round the edge of the cylinder are calculated for various material pairs.     

Axisymmetric contact and crack problems for an initially stressed neo-Hookean elastic layer

A solution of the contact problem for an initially stressed neo-Hookean infinite layer is obtained. The layer is assumed to be bonded to a rigid foundation. The punch is taken to be axisymmetric and in particular, the conical and cylindrical shapes of the punch are considered in detail. The expression for the total load applied to the punch to maintain a given displacement is obtained. We have also considered the penny-shaped crack in the layer under an initial stress. The crack is taken to lie in the central plane of the layer. The surfaces of the layer are considered to be either stress free or fixed. Expressions for the stress intensity factors are obtained. Numerical values of the physical quantities are exhibited graphically.     

Surface loads on granular material underlain by a rough rigid base

A general theoretical analysis with numerical evaluation for the complete distribution of stress within a granular layer underlain by a rough rigid base, and the surface displacements of the layer beneath the footing is presented. The cases of a point load, a uniformly loaded circular footing acting at the surface of the layer but transmitting three usual types of contact pressure, have been considered. It is considered that the material is elastic, cross-anisotropic in which the ratio $\text{E}\text{G}\text{> 2(1 + μ)}$. The effect of the material parameter $\text{E}\text{G}$ on stresses and displacements in granular materials has been presented in a non-dimensional form.     

Interface separation for an elastic layer loaded by a rigid stamp

The frictionless contact problem for an infinite elastic layer lying on a horizontal rigid plane is considered. The external load is applied to the layer through a rigid stamp. It is assumed that the layer is subjected to uniform vertical body forces due to the effect of gravity. For values of the resultant compressive force P acting on the stamp vertically which are less than a critical value P_cr the continuous contact along the layer-substrate interface is maintained. However, if tensile tractions are not allowed on the interface, for P >P_cr the layer separates from the interface along a certain finite region. The problem is formulated and solved for both cases and numerical results for P_cr, separation initiation distance, contact stresses, distances determining the separation area, and the vertical displacement in the separation zone are given.     

Rolling contact of a rigid cylinder over a smooth elastic or viscoelastic layer

Summary The rolling contact of a rigid cylinder over a smooth elastic or viscoelastic layer is investigated. Steady rolling is assumed and inertia terms are neglected so slow rolling in some sense is implied. The boundary-value problem is formulated for a finite spectrum viscoelastic model and illustrated by a standard linear solid. A singular integral equation is derived for the contact pressure, whose kernel is approximated to high accuracy by a logarithm multiplied by a polynomial plus a second polynomial. The latter integral equation is solved approximately and accuracy of solution verified. A geometrical parameter defining the ratio of layer thickness to cylinder arises, and valid solutions within the framework of infinitesimal deformation gradient are obtained down to =10^–3, classified as a thin layer.

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Der Rollkontakt eines starren Zylinders auf einer glatten elastischen oder viskoelastischen Schicht

Zusammenfassung Der Rollkontakt eines starren Zylinders auf einer glatten elastischen oder viskoelastischen Schicht wird untersucht. Vorausgesetzt wird stationäres Rollen und, da Beschleunigungsglieder vernachlässigt werden, langsames Rollen. Das Randwertproblem wird für ein viskoelastisches Modell mit endlichem Spektrum formuliert und diskutiert für das lineare Standardmodell. Für den Kontaktdruck wird eine singuläre Integralgleichung hergeleitet, deren Kern mit großer Genauigkeit durch die Summe eines mit einem Polynom multiplizierten Logarithmus und einem zweiten Polynom genähert wird. Diese Integralgleichung wird näherungsweise gelöst und die Genauigkeit der Lösung bestimmt. Ein geometrischer Parameter , das Verhältnis der Schichtdicke zum Zylinderradius, tritt in den Gleichungen auf, und im Rahmen infinitesimaler Deformationsgradienten gültige Lösungen werden von =10^–3 aufwärts ermittelt.

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With 4 Figures     

Influence of a concentrated force on the contact of an elastic body with a rigid base with depression

We study the contact interaction of an elastic half plane with a rigid base containing a smooth depression on the boundary. The bodies are kept in contact by the action of a uniform external pressure and a concentrated force applied to an elastic half plane above the center of the depression. The presence of the depression in the rigid base results in the appearance of an intercontact gap between the bodies whose length depends on the load. The contact problem is reduced to a singular integral equation for the height of the gap. This equation is solved analytically. The influence of the force on the length and height of the gap is analyzed. It is shown that, for some magnitudes of the force and distances from the point of its application to the interface of the bodies, the bodies may be in contact even in the center of the depression. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 6, pp. 27–31, November–December, 1999.     

The axisymmetric double contact problem for a frictionless elastic layer indented by an elastic cylinder

This paper is concerned with the smooth receding contact between an elastic layer and a half space when the layer is compressed by a frictionless semi-infinite elastic cylinder. Upon loading, the contact along the layer-subspace interface shrinks to a circular area, radius of which is unknown. The analysis leads to a system of singular integral equations of the second kind. The integral equations are solved numerically and the contact pressures, extent of contact and the stress intensity factor round the edge of the cylinder are calculated for various material combinations.     

Static response of a rigid strip bonded to an elastic layer

Summary The problem of a rigid, bonded, rectangular indenter acting on a plane, transversely isotropic elastic layer is solved. The layer is fixed to a rigid foundation and the cases of applied moment and horizontal displacement are considered. In the first case the moment per unit rotation and horizontal displacement are calculated, while in the latter case the horizontal load per unit displacement and the associated torque are found. From the two problems influence coefficients are calculated and their values are given for different materials and layer thicknesses.

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Das statische Verhalten eines mit einer elastischen Schicht verschweißten starren Streifens

Zusammenfassung Gelöst wird das Problem eines mit einer auf einer starren Unterlage befestigten orthotropen elastischen Schicht verschweißten starren Streifens. Betrachtet werden die Fälle eines auf den Streifen wirkenden äußeren Momentes und einer reinen Horizontalverschiebung des Streifens. Im ersten Fall werden das für eine bestimmte Verdrehung nötige Moment und die zugehörige Horizontalverschiebung ermittelt, im zweiten Fall die Horizontalkraft und das Moment pro Einheitsverschiebung. Einflußzahlen werden berechnet und ihre Werte für verschiedene Werkstoffe und Schichtdicken angegeben.

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With 1 Figure     

Plane contact problem of interaction between a rigid heat-conducting wedge and an elastic layer under nonstationary friction heat release

We consider a solution to the new plane problem of contact interaction with an elastic layer of a rigid, heat-conducting cross-sectional die in the form of a bounded or unbounded wedge. Using the apparatus of integral transformations, we have obtained an exact solution of the heat-conduction equations for the die and the layer, which permits reducing the problem stated to a system of integral equations whose structure is determined by the kind of thermophysical conditions on the interaction surface. This enables us to make the mathematical statement of the problem more realistic in terms of the distribution of thermoelastic stresses and adequately estimate the influence of thermal fields on the value of the contact pressure and the character of its distribution. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 4, pp. 755–766, July–August, 2008.     

The indentation of an elastic layer by a rigid stamp under conditions of complete adhesion

Summary In this paper the indentation of an elastic layer by a rigid stamp is treated under conditions of complete adhesion beneath the stamp, where the ratio of the half-width of the contact region and the thickness of the layer is assumed to be small. The cases of a flat stamp and a polynomial one are considered successively and two applications are treated.

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Die Einprägung einer elastischen Schicht von einem starren Stempel bei vollständiger Befestingung under conditions of complete adhesion

Zusammenfassung Die vorliegende Arbeit befaßt sich mit dem Eindringen eines starren Stempels in eine elastische Schicht für den Fall, daß die Unterseite des Stempels auf der Schicht befestigt ist.Dabei wird angenommen, daß die halbe Breite des Berührungsgebietes klein ist gegenüber der Dicke der Schicht.Zwei Sonderfälle werden näher untersucht: der flache Stempel und der polynomförmige Stempel. Zwei Anwendungen werden behandelt.

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With 6 Figures     

Tangential contact problem for a transversely isotropic elastic layer bonded to an elastic foundation

A system consisting of an elastic layer made of a transversely isotropic material bonded to an elastic half-space made of a different transversely isotropic material is considered. An arbitrary tangential displacement is prescribed over a domain S of the layer, while the rest of the layer’s surface is stress-free. The tangential contact problem consists of finding a complete field of stresses and displacements in this system. The generalized-images method developed by the author is used to get an elementary solution to the problem. It is also shown that an integral transform can be interpreted as a sum of generalized images. The case of a circular domain of contact is considered in detail. The results are valid for the case of isotropy as well.     

The impact of a rigid sphere with an elastic layer of finite thickness

Summary The problem of the impact of a rigid sphere with an elastic isotropic layer is considered in the initial stage of dynamic interaction. The initial stage is characterized by the fact that the velocity of the displacement of the intersection points of the sphere with the upper boundary of the layer is larger than the velocity of longitudinal waves, hence the free surface normal to the contact domain with the body is undisturbed. The method of successive approximations as well as the ray method, according to which the solution behind the fronts of incident and reflected waves is constructed in terms of power series (ray expansions), are used as methods of solution. In the problem under consideration, we used one-term ray expansions whereby the main characteristics of the shock interaction have been obtained, and the possibility of localized damage of the material of the layer at the points lying along the central ray has been examined.     

Axisymmetric graph model of an elastic solid

The method of a deformable solid analysis based on graph theory principles is used in mechanics to construct discrete models that allow a numerical simulation of the solid displacement, strain, and stress fields. It provides the alternative opportunity to derive solid equations by transformation of the generalized coordinate systems introduced by the graph structural objects (paths, cut-sets, cycles, chords).Published in Problemy Prochnosti, No. 6, pp. 83–103, November–December, 1996.