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.

Dynamic Response of a Rigid Footing Bonded to an Elastic Half Space

1972 ◽  
Vol 39 (2) ◽  
pp. 527-534 ◽  
Author(s):  
J. E. Luco ◽  
R. A. Westmann
Keyword(s):  
Dynamic Response ◽  
Integral Equations ◽  
Half Plane ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Elastic Half Space ◽  
Vertical Shear ◽  
Fredholm Integral Equations ◽  
Interface Conditions ◽  
Rigid Footing

The problem of determining the response of a rigid strip footing bonded to an elastic half plane is considered. The footing is subjected to vertical, shear, and moment forces with harmonic time-dependence; the bond to the half plane is complete. Using the theory of singular integral equations the problem is reduced to the numerical solution of two Fredholm integral equations. The results presented permit the evaluation of approximate footing models where assumptions are made about the interface conditions.

The Singularity at the Apex of a Bonded Wedge-Shaped Stamp

1979 ◽  
Vol 46 (3) ◽  
pp. 577-580 ◽  
Author(s):  
K. S. Parihar ◽  
L. M. Keer
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Solution ◽  
Integral Equations ◽  
Half Space ◽  
Singular Integral ◽  
Green's Functions ◽  
Green’S Functions ◽  
Singular Integral Equations ◽  
Elastic Half Space

The problem of determining the singularity at the apex of a rigid wedge bonded to an elastic half space is formulated by considerations of Green’s functions for the loaded half space. The eigenvalue problem is reduced to finding the solution of a coupled pair of singular integral equations. A numerical solution for small wedge angles is given.

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 .

Rapid Contact on a Prestressed Highly Elastic Half-Space: The Sliding Ellipsoid and Rolling Sphere

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
L. M. Brock
Keyword(s):  
Half Space ◽  
Contact Zone ◽  
Compressive Load ◽  
Reference Value ◽  
Singular Integral Equations ◽  
Elastic Half Space ◽  
Polar Coordinates ◽  
Rigid Sphere ◽  
Cauchy Stress ◽  
Specific Contact

A neo-Hookean half-space, in equilibrium under uniform Cauchy stress, undergoes contact by a sliding rigid ellipsoid or a rolling rigid sphere. Sliding is resisted by friction, and sliding or rolling speed is subcritical. It is assumed that a dynamic steady state is achieved and that deformation induced by contact is infinitesimal. Transform methods, modified by introduction of quasi-polar coordinates, are used to obtain classical singular integral equations for this deformation. Assumptions of specific contact zone shape are not required. Signorini conditions and the requirement that resultant compressive load is stationary with respect to contact zone stress give an equation for any contact zone span in terms of a reference value and an algebraic formula for the latter. Calculations show that prestress can significantly alter the ratio of spans parallel and normal to the direction of die travel, an effect enhanced by increasing die speed.

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.

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.

Contact Mechanics of Graded Materials: Analyses Using Singular Integral Equations

2008 ◽  
Author(s):  
Serkan Dag ◽  
Glaucio H. Paulino ◽  
Marek-Jerzy Pindera ◽  
Robert H. Dodds ◽  
Fernando A. Rochinha ◽  
...  
Keyword(s):  
Contact Mechanics ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Graded Materials
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