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 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.

Self similar solutions to adhesive contact problems with incremental loading

1968 ◽  
Vol 305 (1480) ◽  
pp. 55-80 ◽  
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Half Space ◽  
Boundary Value ◽  
Contact Problems ◽  
Adhesive Contact ◽  
Elastic Half Space ◽  
The Body ◽  
Similar Solutions ◽  
Dimensional Argument

The boundary-value problem for axisymmetric distortion of an elastic half space by a rigid indentor is formulated. A dimensional argument is used to infer the form of the distribution of radial displacement within the contact circle in terms of the shape of the body, assuming the load to be applied progressively, with interfacial friction sufficient to prevent any slip taking place between the indentor and the half space. This obviates the need for solving a preliminary integral equation for the boundary conditions, as proposed by Goodman (1962) and Mossakovski (1963). The resulting boundary-value problem is cast in the form of an integral equation of Wiener-Hopf type, which has been solved in a separate paper (Spence 1968, referred to as II). The solution is used to calculate stresses, displacements and contact radii for adhesive indentation by (i) a flat faced cylinder, (ii) an almost flat conical indentor and (iii) a sphere. The results are compared with those for frictionless indentation, for a range of values of Poisson’s ratio (iv). Adhesive indentation of a half space by a sphere of radius R rolling with angular velocity ω and linear velocity V (excluding dynamical effects) is also treated, and a value found for the creep 1 ( V / R ω in the absence of torsional or tractive forces.

scholarly journals The role of adhesion in contact mechanics

2019 ◽  
Vol 16 (151) ◽  
pp. 20180738 ◽  
Author(s):  
M. Ciavarella ◽  
J. Joe ◽  
A. Papangelo ◽  
J. R. Barber
Keyword(s):  
Contact Mechanics ◽  
Contact Problems ◽  
Soft Materials ◽  
Adhesive Contact ◽  
Industrial Systems ◽  
Linear Elastic ◽  
Elastic Fracture ◽  
To Come ◽  
Interacting Surfaces

Adhesive (e.g. van der Waals) forces were not generally taken into account in contact mechanics until 1971, when Johnson, Kendall and Roberts (JKR) generalized Hertz’ solution for an elastic sphere using an energetic argument which we now recognize to be analogous to that used in linear elastic fracture mechanics. A significant result is that the load–displacement relation exhibits instabilities in which approaching bodies ‘jump in’ to contact, whereas separated bodies ‘jump out’ at a tensile ‘pull-off force’. The JKR approach has since been widely used in other geometries, but at small length scales or for stiffer materials it is found to be less accurate. In conformal contact problems, other instabilities can occur, characterized by the development of regular patterns of regions of large and small traction. All these instabilities result in differences between loading and unloading curves and consequent hysteretic energy losses. Adhesive contact mechanics has become increasingly important in recent years with the focus on soft materials (which generally permit larger areas of the interacting surfaces to come within the range of adhesive forces), nano-devices and the analysis of bio-systems. Applications are found in nature, such as insect attachment forces, in nano-manufacturing, and more generally in industrial systems involving rubber or polymer contacts. In this paper, we review the strengths and limitations of various methods for analysing contact problems involving adhesive tractions, with particular reference to the effect of the inevitable roughness of the contacting surfaces.

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.

The Analysis of Contact Stresses in Rolling Element Bearings

1979 ◽  
Vol 101 (1) ◽  
pp. 105-109 ◽  
Author(s):  
M. J. Hartnett
Keyword(s):  
Three Dimensional ◽  
Contact Problems ◽  
Rolling Element Bearing ◽  
Rolling Element Bearings ◽  
Algebraic Equations ◽  
Linear Elastic ◽  
Linear Algebraic Equations ◽  
Rolling Element ◽  
Contact Symmetry ◽  
Force Displacement

A numerical solution is presented which can be used to analyze the complete range of frictionless contact problems found in rolling element bearings. A three dimensional, linear elastic solution to the problem is developed by combining the Boussinesq force-displacement relationships for a half-space with a modified flexibility method. In this manner a stable system of linear algebraic equations in terms of the unknown surface pressures is formed, with no restrictions placed upon either contact symmetry or material connectivity. Several numerical examples of common but hitherto unsolved contact problems prevalent in rolling element bearing applications are also presented.

scholarly journals FROM WINKLER’S FOUNDATION TO POPOV’S FOUNDATION

2019 ◽  
Vol 17 (2) ◽  
pp. 181 ◽  
Author(s):  
Ivan Argatov
Keyword(s):  
Contact Problems ◽  
Adhesive Contact ◽  
Elastic Half Space ◽  
Elastic Contact ◽  
Contact Interface ◽  
Modeling Framework ◽  
One Dimensional ◽  
Method Of Dimensionality Reduction ◽  
Elastic Bodies ◽  
Contact Configuration

In recent years, the method of dimensionality reduction (MDR) has started to figure as a very convenient tool for dealing with a wide class of elastic contact problems. The MDR modeling framework introduces an equivalent punch profile and a one-dimensional Winkler-type elastic foundation, called henceforth Popov’s foundation. While the former mainly accounts for the geometry of contact configuration, the Popov foundation inherits the main characteristics of both the contact interface (like friction and adhesion) and the contacting elastic bodies (e.g., anisotropy, viscoelasticity or inhomogeneity). The discussion is illustrated with an example of the Kendall-type adhesive contact for an isotropic elastic half-space.

Semi-inverse method for predicting stress–strain relationship from cone indentations

2003 ◽  
Vol 18 (9) ◽  
pp. 2068-2078 ◽  
Author(s):  
A. DiCarlo ◽  
H. T. Y. Yang ◽  
S. Chandrasekar
Keyword(s):  
A Priori ◽  
Model Material ◽  
Material Behavior ◽  
The Finite Element Method ◽  
Stress Strain ◽  
Indentation Tests ◽  
Strain Relationship ◽  
Initial Force ◽  
Relationship Of ◽  
Force Displacement

A method for determining the stress–strain relationship of a material from hardness values H obtained from cone indentation tests with various apical angles is presented. The materials studied were assumed to exhibit power-law hardening. As a result, the properties of importance are the Young's modulus E, yield strength Y, and the work-hardening exponent n. Previous work [W.C. Oliver and G.M. Pharr, J. Mater. Res. 7, 1564 (1992)] showed that E can be determined from initial force–displacement data collected while unloading the indenter from the material. Consequently, the properties that need to be determined are Y and n. Dimensional analysis was used to generalize H/E so that it was a function of Y/E and n [Y-T. Cheng and C-M. Cheng, J. Appl. Phys. 84, 1284 (1999); Philos. Mag. Lett. 77, 39 (1998)]. A parametric study of Y/E and n was conducted using the finite element method to model material behavior. Regression analysis was used to correlate the H/E findings from the simulations to Y/E and n. With the a priori knowledge of E, this correlation was used to estimate Y and n.

scholarly journals Extracting real-crack properties from non-linear elastic behaviour of rocks: abundance of cracks with dominating normal compliance and rocks with negative Poisson ratios

2017 ◽  
Vol 24 (3) ◽  
pp. 543-551 ◽  
Author(s):  
Vladimir Y. Zaitsev ◽  
Andrey V. Radostin ◽  
Elena Pasternak ◽  
Arcady Dyskin
Keyword(s):  
Experimental Data ◽  
Poisson Ratio ◽  
A Priori ◽  
Elastic Behaviour ◽  
Crack Model ◽  
Normal Compliance ◽  
S Wave ◽  
Linear Elastic ◽  
Non Linear ◽  
Linear Elastic Behaviour

Abstract. Results of examination of experimental data on non-linear elasticity of rocks using experimentally determined pressure dependences of P- and S-wave velocities from various literature sources are presented. Overall, over 90 rock samples are considered. Interpretation of the data is performed using an effective-medium description in which cracks are considered as compliant defects with explicitly introduced shear and normal compliances without specifying a particular crack model with an a priori given ratio of the compliances. Comparison with the experimental data indicated abundance (∼ 80 %) of cracks with the normal-to-shear compliance ratios that significantly exceed the values typical of conventionally used crack models (such as penny-shaped cuts or thin ellipsoidal cracks). Correspondingly, rocks with such cracks demonstrate a strongly decreased Poisson ratio including a significant (∼ 45 %) portion of rocks exhibiting negative Poisson ratios at lower pressures, for which the concentration of not yet closed cracks is maximal. The obtained results indicate the necessity for further development of crack models to account for the revealed numerous examples of cracks with strong domination of normal compliance. Discovering such a significant number of naturally auxetic rocks is in contrast to the conventional viewpoint that occurrence of a negative Poisson ratio is an exotic fact that is mostly discussed for artificial structures.

Sign in / Sign up

Export Citation Format

Share Document