Mechanics of adhesive contact on a power-law graded elastic half-space

We consider adhesive contact between a rigid sphere of radius R and a graded elastic half-space with Young's modulus varying with depth according to a power law E=E₀(z/c₀)^k (0<k<1) while Poisson's ratio ν remaining a constant. Closed-form analytical solutions are established for the critical force, the critical radius of contact area and the critical interfacial stress at pull-off. We highlight that the pull-off force has a simple solution of P_cr=−(k+3)πRΔγ/2 where Δγ is the work of adhesion and make further discussions with respect to three interesting limits: the classical JKR solution when k=0, the Gibson solid when k→1 and ν=0.5, and the strength limit in which the interfacial stress reaches the theoretical strength of adhesion at pull-off.     

Non-slipping adhesive contact of a rigid cylinder on an elastic power-law graded half-space

In this paper, adhesive contact of a rigid cylinder on an elastic power-law graded half-space is studied analytically with the theory of weakly singular integral equation and orthogonal polynomial method. Emphasis is placed on the coupling effect between tangential and normal directions which was often neglected in previous works. Our analysis shows that the coupling effect tends to reduce the contact area in the compressive regime. The effect of bending moment on the adhesion behavior is also examined. Like a pull-off force, there also exists a critical bending moment at which the cylinder can be bended apart from the substrate. However, unlike pull-off force, the critical bending moment is insensitive to the gradient exponent of the graded material.     

Orientation-dependent adhesion strength of a rigid cylinder in non-slipping contact with a transversely isotropic half-space

Recently, Chen and Gao [Chen, S., Gao, H., 2007. Bio-inspired mechanics of reversible adhesion: orientation-dependent adhesion strength for non-slipping adhesive contact with transversely isotropic elastic materials. J. Mech. Phys. solids 55, 1001–1015] studied the problem of a rigid cylinder in non-slipping adhesive contact with a transversely isotropic solid subjected to an inclined pulling force. An implicit assumption made in their study was that the contact region remains symmetric with respect to the center of the cylinder. This assumption is, however, not self-consistent because the resulting energy release rates at two contact edges, which are supposed to be identical, actually differ from each other. Here we revisit the original problem of Chen and Gao and derive the correct solution by removing this problematic assumption. The corrected solution provides a proper insight into the concept of orientation-dependent adhesion strength in anisotropic elastic solids.     

Bio-inspired mechanics of reversible adhesion: Orientation-dependent adhesion strength for non-slipping adhesive contact with transversely isotropic elastic materials

Geckos and many insects have evolved elastically anisotropic adhesive tissues with hierarchical structures that allow these animals not only to adhere robustly to rough surfaces but also to detach easily upon movement. In order to improve our understanding of the role of elastic anisotropy in reversible adhesion, here we extend the classical JKR model of adhesive contact mechanics to anisotropic materials. In particular, we consider the plane strain problem of a rigid cylinder in non-slipping adhesive contact with a transversely isotropic elastic half space with the axis of symmetry oriented at an angle inclined to the surface. The cylinder is then subjected to an arbitrarily oriented pulling force. The critical force and contact width at pull-off are calculated as a function of the pulling angle. The analysis shows that elastic anisotropy leads to an orientation-dependent adhesion strength which can vary strongly with the direction of pulling. This study may suggest possible mechanisms by which reversible adhesion devices can be designed for engineering applications.     

Unbonded contact of a Mindlin plate on an elastic half-space

Summary In this paper a penalty formulation of the frictionless unilateral contact problem between an elastic rectangular plate and an elastic half-space is presented. In order to take into account the effects of the shear stress, the Mindlin plate model is analyzed. Some numerical results, obtained via finite elements, are given.

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Sommario In questo lavoro viene presentata una formulazione penalty del problema di contatto unilaterale senza attrito tra una piastra rettangolare elastica ed un semispazio elastico. Per la piastra si utilizza il modello di Mindlin, che consente di tener conto dell'effetto delle tensioni da taglio. Si forniscono alcuni risultati numerici ottenuti mediante discretizzazione agli elementi finiti.

     

Response of an elastic half-space with power-law nonhomogeneity to static loads

In this paper a series of problems for an isotropic elastic half-space with power-law nonhomogeneity are considered. The action of surface vertical and horizontal forces applied to the half-space is studied. A part of the paper deals with the case of zero-valued surface shear modulus (for positive values of the power determining the nonhomogeneity). This condition leads to simple solutions for two-dimensional (2D) case when radial distribution of stresses exists for surface loads concentrated along an infinite line. Corresponding results for the three-dimensional (3D) case are constructed on the basis of the relationships between 2D and 3D solutions developed in the paper. A more complicated case, in which the shear modulus at the surface of the half-space differs from zero, is treated using fundamental solutions of the differential equations for Fourier–Bessel transformations of displacements. In the paper the fundamental solutions are built in the following two forms: (a) a combination of functions expressing displacements of the half-space under the action of vertical and horizontal forces in the case of zero surface shear modulus, and (b) a representation of the fundamental solutions using confluent hypergeometric functions. The results of numerical calculation given in the paper relate to Green functions for the surface vertical and horizontal point forces.     

The axisymmetric torsional contact problem of a functionally graded piezoelectric coated half-space

In this article, we study the axisymmetric tor-sional contact problem of a half-space coated with func-tionally graded piezoelectric material (FGPM) and subjected to a rigid circular punch. It is found that, along the thick-ness direction, the electromechanical properties of FGPMs change exponentially. We apply the Hankel integral trans-form technique and reduce the problem to a singular integral equation, and then numerically determine the unknown con-tact stress and electric displacement at the contact surface. The results show that the surface contact stress, surface azimuthal displacement, surface electric displacement, and inner electromechanical field are obviously dependent on the gradient index of the FGPM coating. It is found that we can adjust the gradient index of the FGPM coating to modify the distributions of the electric displacement and contact stress.     

Unbonded contact between a circular plate and an elastic half-space

The paper concerns the unbonded contact between a thin circular plate of finite radius, governed by Kirchhof or Reissner theory, pressed by means of rotationally symmetric distributed load and its own weight against the surface of an elastic half-space. The contact is assumed frictionless and unbonded. A Hankel transform solution is used for the half-space and the plate deflection is found by inverting the plate equation. The coefficients in a power expansion are obtained by equating plate and half-space deflections at a number of points in the contact region. The variation of contact radius with plate radius, the radius of the uniformly applied load, and the relative stiffness of plate and foundation, is displayed in a series of figures.     

Study of half-space elastic contact problems by caustics

The optical method of caustics has been successfully applied to several two dimensional problems of elasticity. Up to now, no complicated three dimensional problems of elasticity have ever been treated by this method. In this paper, the experimental technique of caustics is developed, the caustics are obtained by annealing the stress-frozen epoxy slices. In applying this technique to Boussinesq's problem of a normal force and Cerruti's problem of a tangential force on the plane surface of a half-space, the experimentally obtained caustics for these problems are seen to be in satisfactory agreement with the corresponding theoretical forms. The treatment of the rather complicated three dimensional elasticity problems, including crack problems, by the author's method is also possible.     

Loss of contact between an elastic layer and half-space

The plane problem of smooth contact between an elastic layer and a half-space is analyzed. The layer is pressed against the half-space by a uniform load applied over its entire surface except for one region of finite length. The solution is given as the sum of solutions to two plane problems, one of which is known a priori. The analysis of the other problem leads to an inhomogeneous Fredholm integral equation for an auxiliary function that is related to the contact pressure. The boundedness of the contact stress at the end of the contact is used to determine the region of separation. The integral equation is solved numerically and the relevant physical quantities are computed and presented in the form of curves.

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Résumé On analyse le problème du contact glissant entre une couche élastique et un demi-espace. La couche est pressée sur le demi-espace par une charge uniforme appliquée à sa surface entière, à l'exception d'une région de longueur finie. La solution est donnée comme une somme des solutions de deux problèmes plans, dont l'une est connue à priori. L'analyse de l'autre est amené à une équation intégrale non-homogène de Fredholm pour une fonction auxiliaire qui dépend de la pression de contact. Le fait que la pression du contact ne peu pas être infinie aux extrémités du contact est utilisé pour déterminer la région de séparation. L'équation intégrale est résolue de façon numérique et les quantités physiques importantes sont calculées et présentées en forme des courbes.

     

Axisymmetric contact problem for a layer on an elastic half-space with initial stresses

International Applied Mechanics -     

Adhesion in the contact of a spherical indenter with a layered elastic half-space

With the emergence of micro- and nano-technology, the contact mechanics of MEMS and NEMS devices and components is becoming more important. Thus it is important to gain a better understanding of the role of coatings and thin films on micro- and nano-scale contact phenomena, and to understand the interactions of measurement devices, such as an atomic force microscope (AFM), with layered media.More specifically, in this work the frictionless contact, with adhesion, between a spherical indenter and an elastic-layered medium is investigated. This configuration can be viewed as either a single contact model or as a building block of a multi-asperity rough surface contact model. As the scale decreases to the nano level, adhesion becomes an important issue. The presence of adhesion affects the relationships among the applied force, the penetration of the indenter, and the size of the contact area. This axisymmetric problem includes the effect of adhesion using a Maugis type of adhesion model. This model spans the range of the Tabor parameter between the JKR and DMT regions. The key parameters in this analysis are the elastic moduli ratio of the layer and the substrate, the dimensionless layer thickness, and the Maugis adhesion parameter. The results can be applied to a rigid or to an elastic indenter.     

Non-slipping adhesive contact between mismatched elastic cylinders

We have recently proposed a generalized JKR model for non-slipping adhesive contact between two elastic spheres subjected to a pair of pulling forces and a mismatch strain (Chen, S., Gao, H., 2006c. Non-slipping adhesive contact between mismatched elastic spheres: a model of adhesion mediated deformation sensor. J. Mech. Phys. Solids 54, 1548–1567). Here we extend this model to adhesion between two mismatched elastic cylinders. The attention is focused on how the mismatch strain affects the contact area and the pull-off force. It is found that there exists a critical mismatch strain at which the contact spontaneously dissociates. The analysis suggests possible mechanisms by which mechanical deformation can affect binding between cells and molecules in biology.     

Axisymmetric contact problem for an elastic half-space and a circular cover plate

The contact interaction problem for a thin circular rigid cover plate and an elastic half-space loaded at infinity by a tensile force directed in parallel to the boundary of the half-space is considered. It is assumed that the cover plate is not resistant to bending deformations. The problem can be reduced to an integral equation of the first kind whose kernel has a logarithmic singularity. The equation is solved approximately by the Multhopp-Kalandia method. The resulting approximate solution is compared with the previously obtained asymptotic solution.     

Exact solution of external tangential contact problem for a transversely isotropic elastic half-space

Summary  The following mixed boundary-value problem for a transversely isotropic elastic half-space is considered. Arbitrary tangential displacements are prescribed at the exterior of a circle, while the interior of the circle is free of tangential stress, and the normal stress vanishes all over the boundary. The governing integral equation is solved exactly, in closed form, and in terms of elementary functions. The method of continuation of solutions previously published by the author has been used here. Several examples are considered. No similar results has been reported before, even in the case of an isotropic body. Received 8 May 2000; accepted for publication 20 July 2000     

A generalized JKR-model for two-dimensional adhesive contact of transversely isotropic piezoelectric half-space

The aim of the present paper is to investigate the adhesive behavior between a transversely isotropic piezoelectric half-space and a cylinder punch subjected to combined mechanical and electric loads under plane-strain condition. The effect of adhesion is described by using a generalized JKR-model which can account for the non-slip condition in the contact regions. Analytical function theory is employed to find the solution of the resulting singular integral equations. Our analysis shows that the adhesive contact behavior for different types of piezoelectric materials may be quite different. The results obtained in this paper may be helpful to understand the contact mechanics of piezoelectric materials at micro-scale.