Micro/Nanocontact Between a Rigid Ellipsoid and an Elastic Substrate With Surface Tension

2016 ◽  
Vol 84 (1) ◽  
Author(s):  
W. K. Yuan ◽  
J. M. Long ◽  
Y. Ding ◽  
G. F. Wang
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Singular Integral Equation ◽  
Contact Region ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Contact Radius ◽  
Elastic Substrate ◽  
Chebyshev Quadrature ◽  
Indent Depth

For micro/nanosized contact problems, the influence of surface tension becomes prominent. Based on the solution of a point force acting on an elastic half space with surface tension, we formulate the contact between a rigid ellipsoid and an elastic substrate. The corresponding singular integral equation is solved numerically by using the Gauss–Chebyshev quadrature formula. When the size of contact region is comparable with the elastocapillary length, surface tension significantly alters the distribution of contact pressure and decreases the contact area and indent depth, compared to the classical Hertzian prediction. We generalize the explicit expression of the equivalent contact radius, the indent depth, and the eccentricity of contact ellipse with respect to the external load, which provides the fundament for analyzing nanoindentation tests and contact of rough surfaces.

A Unified Treatment of Axisymmetric Adhesive Contact on a Power-Law Graded Elastic Half-Space

2013 ◽  
Vol 80 (6) ◽  
Author(s):  
Fan Jin ◽  
Xu Guo ◽  
Wei Zhang
Keyword(s):  
Half Space ◽  
Power Law ◽  
Contact Region ◽  
Adhesive Contact ◽  
Elastic Half Space ◽  
Hertzian Contact ◽  
Contact Radius ◽  
Abel Transformation ◽  
Numerical Solution Methods ◽  
Special Cases

In the present paper, axisymmetric frictionless adhesive contact between a rigid punch and a power-law graded elastic half-space is analytically investigated with use of Betti's reciprocity theorem and the generalized Abel transformation, a set of general closed-form solutions are derived to the Hertzian contact and Johnson–Kendall–Roberts (JKR)-type adhesive contact problems for an arbitrary punch profile within a circular contact region. These solutions provide analytical expressions of the surface stress, deformation fields, and equilibrium relations among the applied load, indentation depth, and contact radius. Based on these results, we then examine the combined effects of material inhomogeneities and punch surface morphologies on the adhesion behaviors of the considered contact system. The analytical results obtained in this paper include the corresponding solutions for homogeneous isotropic materials and the Gibson soil as special cases and, therefore, can also serve as the benchmarks for checking the validity of the numerical solution methods.

scholarly journals The ring delamination between bodies under the action of heat sinks distributed along a circle

2017 ◽  
pp. 55-62
Author(s):  
Maryana Mykytyn ◽  
Kristina Serednytska ◽  
Bohdan Monastyrskyy ◽  
Rostyslav Martynyak
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Half Space ◽  
Singular Integral ◽  
Elastic Half Space ◽  
Heat Sinks ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Normal Contact ◽  
Frictionless Contact

The frictionless contact an elastic half-space and a rigid thermo-insulated base with a local delamination between them on a ring domain under the action of heat sinks distributed uniformly along a circle and located in the half-space some distance away from its surface, is considered. The corresponding contact thermos-elasticity problem is reduced to a singular integral equation for a height of a ring gap. The solution of the singular integral equation and the internal and external radius of the ring are numerically determined using the method of collocation and the method of successive approximations. The dependence of the form of gap and normal contact stresses on the distance between the heat sinks and the surface of the half-space and the intensity of the heat sink are analyzed.

scholarly journals Indentation of a rigid sphere into an elastic substrate with surface tension and adhesion

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140727 ◽  
Author(s):  
Chung-Yuen Hui ◽  
Tianshu Liu ◽  
Thomas Salez ◽  
Elie Raphael ◽  
Anand Jagota
Keyword(s):  
Surface Tension ◽  
Small Strain ◽  
Work Of Adhesion ◽  
Contact Radius ◽  
Rigid Sphere ◽  
Elastic Substrate ◽  
Indentation Force ◽  
Resistance To Deformation ◽  
Sphere Radius ◽  
Force Displacement

The surface tension of compliant materials such as gels provides resistance to deformation in addition to and sometimes surpassing that owing to elasticity. This paper studies how surface tension changes the contact mechanics of a small hard sphere indenting a soft elastic substrate. Previous studies have examined the special case where the external load is zero, so contact is driven by adhesion alone. Here, we tackle the much more complicated problem where, in addition to adhesion, deformation is driven by an indentation force. We present an exact solution based on small strain theory. The relation between indentation force (displacement) and contact radius is found to depend on a single dimensionless parameter: ω = σ ( μR ) −2/3 ((9 π /4) W ad ) −1/3 , where σ and μ are the surface tension and shear modulus of the substrate, R is the sphere radius and W ad is the interfacial work of adhesion. Our theory reduces to the Johnson–Kendall–Roberts (JKR) theory and Young–Dupre equation in the limits of small and large ω , respectively, and compares well with existing experimental data. Our results show that, although surface tension can significantly affect the indentation force, the magnitude of the pull-off load in the partial wetting liquid-like limit is reduced only by one-third compared with the JKR limit and the pull-off behaviour is completely determined by ω .

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.

Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary

2021 ◽  
pp. 108128652110214
Author(s):  
Anna Y. Zemlyanova ◽  
Lauren M. White
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Surface Energy ◽  
Additional Condition ◽  
Hankel Transform ◽  
Integral Representations ◽  
Classic Problem ◽  
Rigid Stamp ◽  
Chebyshev Quadrature ◽  
Ill Posed

An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann–Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin–Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin–Murdoch model with a non-zero surface tension and for the general Steigmann–Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss–Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported.

General Relations of Indentations on Solids With Surface Tension

2017 ◽  
Vol 84 (5) ◽  
Author(s):  
Jianmin Long ◽  
Yue Ding ◽  
Weike Yuan ◽  
Wen Chen ◽  
Gangfeng Wang
Keyword(s):  
Surface Tension ◽  
Contact Mechanics ◽  
Integral Equations ◽  
Analytical Solutions ◽  
Integral Kernel ◽  
Singular Integral Equations ◽  
Elastic Half Space ◽  
Contact Radius ◽  
Rigid Sphere ◽  
General Relations

The conventional contact mechanics does not account for surface tension; however, it is important for micro- or nanosized contacts. In the present paper, the influences of surface tension on the indentations of an elastic half-space by a rigid sphere, cone, and flat-ended cylinder are investigated, and the corresponding singular integral equations are formulated. Due to the complicated structure of the integral kernel, it is difficult to obtain their analytical solutions. By using the Gauss–Chebyshev quadrature formula, the integral equations are solved numerically first. Then, for each indenter, the analytical solutions of two limit cases considering only the bulk elasticity or surface tension are presented. It is interesting to find that, through a simple combination of the solutions of two limit cases and fitting the direct numerical results, the dependence of load on contact radius or indent depth for general case can be given explicitly. The results incorporate the contribution of surface tension in contact mechanics and are helpful to understand contact phenomena at micro- and nanoscale.

On Some Unbonded Contact Problems in Plane Elasticity Theory

1976 ◽  
Vol 43 (2) ◽  
pp. 263-267 ◽  
Author(s):  
G. M. L. Gladwell
Keyword(s):  
Integral Equation ◽  
Elasticity Theory ◽  
Contact Pressure ◽  
Chebyshev Polynomials ◽  
Contact Region ◽  
Contact Problems ◽  
Elastic Cylinder ◽  
Plane Elasticity ◽  
Elastic Strip ◽  
Plane Elasticity Theory

Paper considers plane, frictionless, unbonded contact problems. It is shown that the integral equation relating the unknown contact pressure to the specified displacement in the contact region may be solved approximately by using an expansion in terms of Chebyshev polynomials. Three examples are chosen, a beam resting on a half plane, a rigid cylinder pressed into an elastic strip, and an elastic cylinder pressed between rigid planes. Graphs of results are presented.

scholarly journals Indentation load–depth relation for an elastic layer with surface tension

2018 ◽  
Vol 24 (4) ◽  
pp. 1147-1160 ◽  
Author(s):  
Shaoheng Li ◽  
Weike Yuan ◽  
Yue Ding ◽  
Gangfeng Wang
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Elastic Layer ◽  
Surface Deformation ◽  
Integral Transform ◽  
Indentation Load ◽  
Thin Layers ◽  
Contact Radius ◽  
Rigid Sphere ◽  
Practical Applications

The load–depth relation is a fundamental requisite in nanoindentation tests for thin layers; however, the effects of surface tension are seldom included. This paper concerns micro-/nano-sized indentation by a rigid sphere of a bonded elastic layer. The surface Green’s function with the incorporation of surface tension is first derived by applying the Hankel integral transform, and subsequently used to formulate the governing integral equation for the axisymmetric contact problem. By using a numerical method based on the Gauss–Chebyshev quadrature formula, the singular integral equation is solved efficiently. Several numerical results are presented to investigate the influences of surface tension and layer thickness on contact pressure, surface deformation, and bulk stress. It is found that when the size of contact is comparable to the ratio of surface tension to elastic modulus, the contribution of surface tension to the load–depth relation becomes quite prominent. With the help of a parametric study, explicit general expressions for the indentation load–depth relation as well as the load–contact radius relation are summarized, which provide the groundwork for practical applications.

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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