scholarly journals Adhesive contact problems for a thin elastic layer: Asymptotic analysis and the JKR theory

2018 ◽  
Vol 24 (5) ◽  
pp. 1405-1424 ◽  
Author(s):  
Feodor M. Borodich ◽  
Boris A. Galanov ◽  
Nikolay V. Perepelkin ◽  
Danila A. Prikazchikov
Keyword(s):  
Elastic Foundation ◽  
Elastic Layer ◽  
Order Approximation ◽  
Contact Problems ◽  
Adhesive Contact ◽  
Contact Radius ◽  
Direct Derivation ◽  
Compressible Elastic Layer ◽  
Jkr Theory ◽  
Thin Elastic Layer

Contact problems for a thin compressible elastic layer attached to a rigid support are studied. Assuming that the thickness of the layer is much less than the characteristic dimension of the contact area, a direct derivation of asymptotic relations for displacements and stress is presented. The proposed approach is compared with other published approaches. The cases are established when the leading-order approximation to the non-adhesive contact problems is equivalent to contact problem for a Winkler–Fuss elastic foundation. For this elastic foundation, the axisymmetric adhesive contact is studied in the framework of the Johnson–Kendall–Roberts (JKR) theory. The JKR approach has been generalized to the case of the punch shape being described by an arbitrary blunt axisymmetric indenter. Connections of the results obtained to problems of nanoindentation in the case that the indenter shape near the tip has some deviation from its nominal shape are discussed. For indenters whose shape is described by power-law functions, the explicit expressions are derived for the values of the pull-off force and for the corresponding critical contact radius.

Adhesive Contact of Bodies Having an Elliptical Contact Area

2005 ◽  
Author(s):  
K. L. Johnson ◽  
J. A. Greenwood
Keyword(s):  
Closed Form ◽  
Contact Area ◽  
Adhesive Contact ◽  
Contact Radius ◽  
General Shape ◽  
Elliptical Contact ◽  
Jkr Theory ◽  
Two Bodies

The so-called JKR theory of adhesion between elastic spheres in contact (Johnson, Kendall & Roberts 1971, Sperling 1964) has been widely used in micro-tribology. In this paper the theory is extended to solids of general shape and curvature. It is assumed that the area of contact is elliptical which turns out to be approximately true, though the eccentricity is different from that for non-adhesive contact. Closed form expressions are found for the variation with load of contact radius and displacement, as a function of the ratio of principal relative curvatures of the two bodies in contact. The pull-off force is found to decrease with increasing eccentricity from its value of 3πΔγR/2 in the case of contact of spheres of radius R.

scholarly journals Contact probing of prestressed adhesive membranes of living cells

2021 ◽  
Vol 379 (2203) ◽  
pp. 20200289 ◽  
Author(s):  
Feodor M. Borodich ◽  
Boris A. Galanov ◽  
Leon M. Keer ◽  
Maria M. Suarez-Alvarez
Keyword(s):  
Mechanical Properties ◽  
Cell Membranes ◽  
Contact Problems ◽  
Adhesive Contact ◽  
Living Cells ◽  
Work Of Adhesion ◽  
Biological Cells ◽  
Crucial Importance ◽  
Adhesive Properties ◽  
Jkr Theory

Atomic force microscopy (AFM) studies of living biological cells is one of main experimental tools that enable quantitative measurements of deformation of the cells and extraction of information about their structural and mechanical properties. However, proper modelling of AFM probing and related adhesive contact problems are of crucial importance for interpretation of experimental data. The Johnson–Kendall–Roberts (JKR) theory of adhesive contact has often been used as a basis for modelling of various phenomena including cell-cell interactions. However, strictly speaking the original JKR theory is valid only for contact of isotropic linearly elastic spheres, while the cell membranes are often prestressed. For the first time, effects caused by molecular adhesion for living cells are analytically studied taking into account the mechanical properties of cell membranes whose stiffness depends on the level of the tensile prestress. Another important question is how one can extract the work of adhesion between the probe and the cell. An extended version of the Borodich-Galanov method for non-direct extraction of elastic and adhesive properties of contacted materials is proposed to apply to experiments of cell probing. Evidently, the proposed models of adhesive contact for cells with prestressed membranes do not cover all types of biological cells because the structure and properties of the cells may vary considerably. However, the obtained results can be applied to many types of smooth cells and can be used to describe initial stages of contact and various other processes when effects of adhesion are of crucial importance. This article is part of a discussion meeting issue ‘A cracking approach to inventing new tough materials: fracture stranger than friction’.

scholarly journals Contact probing of stretched membranes and adhesive interactions: graphene and other two-dimensional materials

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160550 ◽  
Author(s):  
Feodor M. Borodich ◽  
Boris A. Galanov
Keyword(s):  
2D Materials ◽  
Contact Problems ◽  
Adhesive Contact ◽  
Contact Radius ◽  
Elastic Membrane ◽  
Displacement Curve ◽  
Circular Membrane ◽  
Two Dimensional ◽  
Adhesive Interactions ◽  
Force Displacement

Contact probing is the preferable method for studying mechanical properties of thin two-dimensional (2D) materials. These studies are based on analysis of experimental force–displacement curves obtained by loading of a stretched membrane by a probe of an atomic force microscope or a nanoindenter. Both non-adhesive and adhesive contact interactions between such a probe and a 2D membrane are studied. As an example of the 2D materials, we consider a graphene crystal monolayer whose discrete structure is modelled as a 2D isotropic elastic membrane. Initially, for contact between a punch and the stretched circular membrane, we formulate and solve problems that are analogies to the Hertz-type and Boussinesq frictionless contact problems. A general statement for the slope of the force–displacement curve is formulated and proved. Then analogies to the JKR (Johnson, Kendall and Roberts) and the Boussinesq–Kendall contact problems in the presence of adhesive interactions are formulated. General nonlinear relations among the actual force, displacements and contact radius between a sticky membrane and an arbitrary axisymmetric indenter are derived. The dimensionless form of the equations for power-law shaped indenters has been analysed, and the explicit expressions are derived for the values of the pull-off force and corresponding critical contact radius.

scholarly journals Explicit transformation between non-adhesive and adhesive contact problems by means of the classical Johnson–Kendall–Roberts formalism

2021 ◽  
Vol 379 (2203) ◽  
pp. 20200374
Author(s):  
Nikolay V. Perepelkin ◽  
Feodor M. Borodich
Keyword(s):  
Rotational Symmetry ◽  
A Priori ◽  
Contact Problems ◽  
Adhesive Contact ◽  
Elastic Half Space ◽  
Principle Of Superposition ◽  
Linear Elastic ◽  
Thin Elastic Layer ◽  
Force Displacement ◽  
Superposition Of Solutions

The classic Johnson–Kendall–Roberts (JKR) contact theory was developed for frictionless adhesive contact between two isotropic elastic spheres. The advantage of the classical JKR formalism is the use of the principle of superposition of solutions to non-adhesive axisymmetric contact problems. In the recent years, the JKR formalism has been extended to other cases, including problems of contact between an arbitrary-shaped blunt axisymmetric indenter and a linear elastic half-space obeying rotational symmetry of its elastic properties. Here the most general form of the JKR formalism using the minimal number of a priori conditions is studied. The corresponding condition of energy balance is developed. For the axisymmetric case and a convex indenter, the condition is reduced to a set of expressions allowing explicit transformation of force–displacement curves from non-adhesive to corresponding adhesive cases. The implementation of the developed theory is demonstrated by presentation of a two-term asymptotic adhesive solution of the contact between a thin elastic layer and a rigid punch of arbitrary axisymmetric shape. Some aspects of numerical implementation of the theory by means of Finite-Element Method are also discussed. This article is part of a discussion meeting issue ‘A cracking approach to inventing new tough materials: fracture stranger than friction’.

scholarly journals SOLUTION OF ADHESIVE CONTACT PROBLEM ON THE BASIS OF THE KNOWN SOLUTION FOR NON-ADHESIVE ONE

2018 ◽  
Vol 16 (1) ◽  
pp. 93 ◽  
Author(s):  
Valentin L. Popov
Keyword(s):  
Contact Problem ◽  
Material Properties ◽  
Present Method ◽  
Additional Assumption ◽  
Short Communication ◽  
Contact Problems ◽  
Adhesive Contact ◽  
Contact Shape

The well-known procedure of reducing an adhesive contact problem to the corresponding non-adhesive one is generalized in this short communication to contacts with an arbitrary contact shape and arbitrary material properties (e.g. non homogeneous or gradient media). The only additional assumption is that the sequence of contact configurations in an adhesive contact should be exactly the same as that of contact configurations in a non-adhesive one. This assumption restricts the applicability of the present method. Nonetheless, the method can be applied to many classes of contact problems exactly and also be used for approximate analyses.

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