The contact problem of a rigid stamp with friction on a functionally graded magneto-electro-elastic half-plane

We consider in this study the frictional sliding contact problem between a functionally graded magneto-electro-elastic material and a perfectly conducting rigid punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the unknowns are the normal contact stress, the electric displacement and the magnetic induction. These integral equations are then solved numerically to obtain the distributions of the normal contact stress, electric displacement and magnetic induction at the surface of the graded medium. The main objective of this paper is to study the effect of the non-homogeneity parameter, the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat and circular punch profiles.     

An elastic contact problem for a half-space indented by a flat annular rigid stamp

We consider an elastic contact problem for a half-space indented by a flat annular rigid stamp, which is a three-part mixed boundary value problem. If the distribution of pressure on the contact region is assumed, the contact problem will be reduced to the solution of an infinite set of simultaneous equations. Numerical results are also illustrated for the distributions of the displacements and stresses in the half-space.     

An elastic contact problem for a neo-hookean half-space indented by a annular rigid stamp of arbitrary profiles

The present paper deals with an elastic contact problem for a neo-Hookean half-space indented by a annular rigid stamp of arbitrary profile, which constitutes a three part mixed boundary value problem. The problem is reduced into triple integral equations which are further reduced into an infinite set of simultaneous equations. Numerical results are illustrated graphically for the distributions of the normal displacement and stress in the case of parabolic concave punch for different values of a/b at $\text{z = 0}$     

Frictional contact of flexible and rigid bodies

 This paper is about planar frictional contact problems of both flexible and rigid bodies. For the flexible case a nonlinear finite element formulation is presented, which is based on a modified Coulomb friction law. Stick-slip motion is incorporated into the formulation through a radial return mapping scheme. Linearly interpolating four node elements and three node contact elements are utilized for the finite element discretization. The corresponding tangent stiffness matrices and residual vectors of the equations of motion are presented. In the rigid body case the contact problem is divided into impact and continual contact, which are mathematically described by linear complementarity problems. The impact in normal direction is modeled by a modified Poisson hypothesis, which is adapted to allow multiple impacts. The formulation of the tangential impact is grounded on Coulombs law of friction. The normal contact forces of the continual contact are such that colliding bodies are prevented from penetration and the corresponding tangential forces are expressed by Coulombs law of friction. Examples and comparisions between the different methods are presented. Received: 10 January 2001     

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.     

Thermoelastic contact between a functionally graded elastic cylindrical punch and a half-space involving frictional heating

Plane contact of a cylindrical punch and a half-space is considered in this article. Both the punch and the half-space are assumed to be functionally graded and thermally heat conducting. Frictional heat generation inside the contact region due to sliding of the punch over the half-space surface is taken into account. It is assumed that heat exchange between the free boundary of the half-space and the surrounding medium obeys Newton’s law, while the conditions for ideal thermal contact exists in the region in which the solids interact. Elastic parameters, thermal expansion coefficients and coefficients of thermal conductivities of the punch and the half-space are assumed to vary along the normal to the plane of contact. With the help of Fourier integral transforms, the problem is reduced to a system of three singular integral equations. The effects of non-homogeneity parameters of the graded materials on various subjects of interest are discussed and shown graphically.     

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     

Frictional contact between 3D beams

 The paper deals with frictional contact between 3D beams with rectangular cross-sections. In the analysis large displacements and small strains are allowed. Hence the cross-section behaves like a rigid body undergoing displacement and rotations. Contact between the beams is assumed to be point-wise between their edges. For the friction the simple non-associated Coulomb law is adopted. The analogy to rigid plasticity is used where stick and slip conditions are considered and backward Euler scheme is applied whenever the yield condition is violated. Contact constraints for normal and tangential part are introduced using the active set strategy and the penalty method. A consistent linearisation of both contact contributions is derived and expressed in suitable matrix form, easy to use in FEM approximations. Several numerical examples including the comparison to the full 3D-solid analysis depict the efficiency of the presented approach. Received 5 May 2001     

The dynamic contact with perfect adhesion and frictional slip between a rigid indentor and an elastic half-space

Summary The problem of the dynamic indentation in loading of an elastic half-space by a rigid axisymmetric punch when a finite friction in the contact region is assumed to exist is considered. The problem involves the determination of a timedependent unknown contact area which is composed of an adhesive circular region whose time-dependent radius is a priori unknown, surrounded by a frictional annulus. A numerical procedure based on an iterative process is developed, which is continued until the complete solution is obtained. This solution is determined by the elastodynamic equations of motion, the moving mixed boundary conditions, the requirement that the contact normal stresses are compressive and that no interpenetration occurs outside the contact region. Furthermore, the radius of separation between the locked and slip regions is determined from the requirement that the current shear stresses at the adhesive region cannot exceed the product of the coefficient of friction and the normal stresses, and that within the slip annulus Coulomb law holds. The method is illustrated for a paraboloid indentor and the effect of friction on the elastic field is exhibited.

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Der dynamische Kontakt bei perfekter adhäsion und Reibungsschlupf zwischen einem starren Stempel und einem elastischen Halbraum

Zusammenfassung Das Problem der dynamischen Beanspruchung eines elastischen Halbraumes bei Belastung durch einen starren, achsensymmetrischen Stempel mit begrenzter Reibung in der Kontaktzone, wird betrachtet. Das Problem betrifft die Bestimmung einer zeitabhängigen unbekannten Kontaktzone, bestehend aus einem Kreisbereich mit Haftung, dessen zeitabhängiger Radius apriori unbekannt ist und einem umgebenden Kreisbereich, wo Reibung auftritt. Ein numerisches Verfahren basierend auf einer Iterationsmethode wird entwickelt, das bis zum Erhalt der kompletten Lösung fortgesetzt wird. Die Lösung ist bestimmt durch die elastodynamischen Bewegungsgleichungen, die bewegten gemischten Randbedingungen, der Bedingung, daß die Kontaktnormalspannungen Druckspannungen sind und keine gegenseitige Durchdringung außerhalb der Kontaktzone auftritt. Weiters wird der Radius der Separation zwischen der Haft- und Rutschzone durch die Forderung festgelegt, daß die auftretende Schubspannung in der Haftzone nicht das Produkt aus Reibungskoeffizient und Normalspannung erreichen kann und während der Schlupfabnahme das Coulombsche Reibungsgestz gilt. Die Methode wird für einen parabolischen Stempel angewendet und der Effekt der Reibung auf das elastische Feld aufgezeigt.

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With 3 Figures     

Frictional contact analysis of functionally graded materials with Lagrange finite block method

Based on the one‐dimensional differential matrix derived from the Lagrange series expansion, the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials. In this paper, the formulation of the Lagrange finite block method with boundary type in the strong form is presented and applied to non‐conforming contact problems for the functionally graded materials subjected to either static or dynamic loads. The first order partial differential matrices are only needed both in the governing equations and in the Neumann boundary condition. By introducing the mapping technique, a block of quadratic type is transformed from the Cartesian coordinate of global system to the normalized coordinate with eight seeds. Time dependent partial differential equations are analyzed in the Laplace transformed domain and the Durbin's inversion method is applied to determine all the physical values in the time domain. Conforming and non‐conforming contacts are investigated by using the iterative algorithm with full load technique. Illustrative numerical examples are given and comparisons have been made with analytical solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Creating self-organized submicrometer contact instability patterns in soft elastic bilayers with a topographically patterned stamp

The surface of a thin elastic bilayer becomes spontaneously unstable when it is brought in proximity to another rigid contactor. The instability patterns, which are random and isotropic, exhibit a dominant lateral length scale of instability λ, which linearly scales with the bilayer thickness (h) as: λ = R(F)h. It is known that for an elastic bilayer, R(F) exhibits a nonlinear dependence on the ratios of individual film thicknesses (H) and shear moduli (M) of the two constituent layers, and can have values as low as 0.5 under specific conditions. This is in contrast to a near constant value of R(F) ≈ 3 for a single layer elastic film. (1) These isotropic contact instability patterns in a bilayer can be ordered, aligned and modulated using a topographically patterned stamp. The precise morphology of the aligned structures depends on commensuration between λ and the stamp periodicity (λ(P)), and on the intersurface separation distance. A variety of patterns, like an array of circular holes, double periodic channels, etc., in addition to a positive and a negative replica of the stamp pattern, can be engineered with a simple stamp having 1D grating structure. A lower value of R(F) in a bilayer allows generating patterns with sub 500 nm lateral resolution, which is impossible to create by elastic contact lithography (ECL) of a single layer film due to strong surface tension effects in ultrathin films. Thus, control of elastic instability in a bilayer with a patterned stamp represents a flexible soft lithography tool allowing modulation of length scales, morphology, and order.     

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     

Frictionless elastic contact problem for a curved rigid punch of arbitrary shape

Summary A new method is proposed for the analysis of elastic contact problems for a curved punch of non-elliptic planform under the action of a normal force. The punch base is assumed to be a quadratic surface. The method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier. Some general relationships are established between the applied force and the punch settlement. Specific formulae are derived for a punch whose planform has a shape of a polygon, a rectangle, a rhombus and a cross. An example of a finite rigid cylinder lying on its generator and pressed against an elastic half-space is considered in detail. The method allows to have singular stresses at the cylinder edges and zero stresses at the rest of the boundary of the contact domain. The last condition serves for defining the width of the domain of contact. All the formulae are checked against the solutions known in the literature, and a good accuracy is confirmed in a sufficiently wide range of the aspect ratio.With 3 Figures     

An asymmetric mixed boundary-value problem of the elastic layer: Translation of a rigid circular disc parallel to the contact plane

The problem considered is the deformation of an elastic layer produced by a uniform tangential displacement of a circular disc attached to its free surface. The other face of the layer is assumed to be fixed to a rigid foundation. The asymmetric mixed boundary-value problem is reduced to a pair of coupled Fredholm integral equations of the second kind: a numerical solution is given, as well as an expansion in powers of 1/λ (λ = layer thickness/disc radius). The surface traction underneath the disc and the force-displacement relation are evaluated as functions of λ.     

Elastic equilibrium of an interface crack between rigid stamp and orthotropic strip

Rupture of the interface between an absolutely rigid stamp and an orthotropic infinite strip is investigated. A plane elasticity problem for an interface crack formally leads to oscillatory singularities at the crack tip. In order to overcome this nonphysical solution, a model of an interface crack with frictionless contact zones near the crack tips and the corners of the stamp is developed. By using the method of integral Fourier transforms the problem is reduced to a system of three singular integral equations. The system is solved by the method of collocations with the points of collocation chosen at zeros of the Chebyshev polynomials. The stress intensity factors at the crack tips and the stamp corner points are evaluated.     

The frictionless rolling contact of a rigid circular cylinder on a semi-infinite granular material

The resulting flow and deformation of a semi-infinite granular material under a rolling, smooth rigid circular cylinder is investigated using a perturbation method. Based on the double-shearing theory of granular flow, complete stress and velocity fields, resistance to rolling and the permanent displacement of surface particles are determined to first order; when the internal friction angle is zero, the solutions reduce to those obtained in the corresponding analysis for Tresca or von-Mises materials. The solution scheme and the double-shearing model for granular flow both find their origins in the work of A.J.M. Spencer.     

Edge crack due to circular rigid punch in incomplete contact

A circular rigid punch with friction is assumed to be in contact with a half-plane and the punch has one end sliding on the half-plane and another end with a sharp corner. The contact length is determined by satisfying the finite stress condition at the sliding end of the punch. An oblique edge crack is assumed to be initiated near the end with the sharp corner where the infinite stresses exist. Coulomb's frictional force is assumed to act on the contact region. The cracked half-plane is mapped into a unit circle by using a rational mapping function and the problem is transformed into a standard Riemann–Hilbert problem, which is solved by introducing a Plemelj function. The contact length, the stress intensity factors of the crack and the resultant moment about the origin of the coordinates are calculated with different frictional coefficients and oblique angles of the crack. The stress distributions on the contact region are also shown.