0
Research Papers

# Plane Contact and Adhesion of Two Elastic Solids With an Interface Groove

[+] Author and Article Information
Fan Jin

Institute of Systems Engineering,
Mianyang 621900, Sichuan, China
e-mail: jinfan2046@163.com

Xu Guo

State Key Laboratory of Structural
Analysis for Industrial Equipment,
Department of Engineering Mechanics,
Dalian University of Technology,
Dalian 116023, China

Qiang Wan

Institute of Systems Engineering,
Mianyang 621900, Sichuan, China

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 16, 2017; final manuscript received January 12, 2018; published online February 2, 2018. Assoc. Editor: Kyung-Suk Kim.

J. Appl. Mech 85(4), 041002 (Feb 02, 2018) (10 pages) Paper No: JAM-17-1516; doi: 10.1115/1.4039040 History: Received September 16, 2017; Revised January 12, 2018

## Abstract

A systematic study is performed on the plane contact and adhesion of two elastic solids with an interface groove. The nonadhesion and Johnson–Kendall–Roberts (JKR) adhesion solutions for a typical groove shape are obtained in closed form by solving singular integral equations and using energy release rate approaches. It is found that the JKR adhesion solution depends solely on a dimensionless parameter $α$ and the groove is predicted to be unstably flattened with no applied load when $α≥0.535$. Furthermore, the corresponding Maugis–Dugdale adhesion model has been revisited with three possible equilibrium states. By introducing the classical Tabor parameter $μ$, a complete transition between the nonadhesion and the JKR adhesion contact models is captured, which can be recovered as two limiting cases of the Maugis–Dugdale model. Depending on two nondimensional parameters $α$ and $μ$, where $α2$ represents the ratio of the surface energy in the groove to the elastic strain energy when the grooved surface is flattened, different transition processes among three equilibrium states are characterized by one or more jumps between partial and full contact. Larger values of $α$ and $μ$ tend to induce more energy loss due to adhesion hysteresis. Combination values of $α$ and $μ$ are also suggested to design self-healing interface grooves due to adhesion.

<>

## References

Etsion, I. , 2005, “ State of the Art in Laser Surface Texturing,” ASME J. Tribol., 127(1), pp. 248–253.
Nakano, M. , Korenaga, A. , Korenaga, A. , Miyake, R. , Murakami, T. , Ando, Y. , Usami, H. , and Sasaki, S. , 2007, “ Applying Micro-Texture to Cast Iron Surfaces to Reduce the Friction Coefficient Under Lubricated Conditions,” Tribol. Lett., 28(2), pp. 131–137.
Yong, X. , and Zhang, L. T. , 2009, “ Nanoscale Wetting on Grooved-Patterned Surfaces,” Langmuir, 25(9), pp. 5045–5053. [PubMed]
Sharp, K. G. , Blackman, G. S. , Glassmaker, N. J. , Jagota, A. , and Hui, C. Y. , 2004, “ Effect of Stamp Deformation on the Quality of Microcontact Printing: Theory and Experiment,” Langmuir, 20(15), pp. 6430–6438. [PubMed]
Hsia, K. J. , Huang, Y. , Menard, E. , Park, J. U. , Zhou, W. , Rogers, J. , and Fulton, J. M. , 2005, “ Collapse of Stamps for Soft Lithography Due to Interfacial Adhesion,” Appl. Phys. Lett., 86(15), p. 154106.
Li, T. , and Zhang, Z. , 2010, “ Substrate-Regulated Morphology of Grapheme,” J. Phys. D: Appl. Phys., 43(7), p. 075303.
Gao, W. , and Huang, Y. , 2011, “ Effect of Surface Roughness on Adhesion of Graphene Membranes,” J. Phys. D: Appl. Phys., 44(45), p. 452001.
Hu, D. , and Adams, G. G. , 2016, “ Adhesion of a Micro-/Nano- Beam/Plate to a Sinusoidal/Grooved Surface,” Int. J. Solids Struct., 99, pp. 40–47.
Gao, H. , and Yao, H. , 2004, “ Shape Insensitive Optimal Adhesion of Nanoscale Fibrillar Structures,” Proc. Natl. Acad. Sci. U.S.A., 101(21), pp. 7851–7856. [PubMed]
Waters, J. F. , Gao, H. , and Guduru, P. R. , 2011, “ On Adhesion Enhancement Due to Concave Surface Geometries,” J. Adhes., 87(3), pp. 194–213.
Jin, F. , Guo, X. , and Zhang, W. , 2013, “ A Unified Treatment of Axisymmetric Adhesive Contact on a Power-Law Graded Elastic Half-Space,” ASME J. Appl. Mech., 80(6), p. 061024.
Johnson, K. L. , Kendall, K. , and Roberts, A. D. , 1971, “ Surface Energy and the Contact of Elastic Solids,” Proc. R. Soc. London A, 324(1558), pp. 301–313.
Derjaguin, B. V. , Muller, V. M. , and Toporov, Y. P. , 1975, “ Effect of Contact Deformations on the Adhesion of Particles,” J. Colloid Interface Sci., 53(2), pp. 314–326.
Tabor, D. , 1977, “ Surface Forces and Surface Interactions,” J. Colloid Interface Sci., 58(1), pp. 2–13.
Maugis, D. , 1992, “ Adhesion of Spheres: The JKR-DMT Transition Using a Dugdale Model,” J. Colloid Interface Sci., 150 (1), pp. 243–269.
Baney, J. M. , and Hui, C. Y. , 1997, “ A Cohesive Zone Model for the Adhesion of Cylinders,” J. Adhes. Sci. Technol., 11(3), pp. 393–406.
Johnson, K. L. , and Greenwood, J. A. , 2008, “ A Maugis Analysis of Adhesive Line Contact,” J. Phys. D: Appl. Phys., 41(15), p. 155315.
Jin, F. , Zhang, W. , Guo, X. , and Zhang, S. , 2014, “ Adhesion Between Elastic Cylinders Based on the Double-Hertz Model,” Int. J. Solids Struct., 51(14), pp. 2706–2712.
Kesari, H. , and Lew, A. , 2011, “ Effective Macroscopic Adhesive Contact Behavior Induced by Small Surface Roughness,” J. Mech. Phys. Solids, 59(12), pp. 2488–2510.
Jin, C. , Khare, K. , Vajpayee, S. , Yang, S. , Jagota, A. , and Hui, C. Y. , 2011, “ Adhesive Contact Between a Rippled Elastic Surface and a Rigid Spherical Indenter: From Partial to Full Contact,” Soft Matter, 7(22), pp. 10728–10736.
Wu, J. J. , 2012, “ Numerical Simulation of the Adhesive Contact Between a Slightly Wavy Surface and a Half-Space,” J. Adhes. Sci. Technol., 26(1–3), pp. 331–351.
Carbone, G. , and Mangialardi, L. , 2004, “ Adhesion and Friction of an Elastic Half-Space in Contact With a Slightly Wavy Rigid Surface,” J. Mech. Phys. Solids, 52(6), pp. 1267–1287.
Waters, J. F. , Lee, S. , and Guduru, P. R. , 2009, “ Mechanics of Axisymmetric Wavy Surface Adhesion: JKR-DMT Transition Solution,” Int. J. Solids Struct., 46(5), pp. 1033–1042.
Jin, F. , Guo, X. , and Wan, Q. , 2016, “ Revisiting the Maugis-Dugdale Adhesion Model of Elastic Periodic Wavy Surfaces,” ASME J. Appl. Mech., 83(10), p. 101007.
McMeeking, R. M. , Ma, L. , and Arzt, E. , 2010, “ Bi-Stable Adhesion of a Surface With a Dimple,” Adv. Eng. Mater., 12(5), pp. 389–397.
Papangelo, A. , and Ciavarella, M. , 2017, “ A Maugis-Dugdale Cohesive Solution for Adhesion of a Surface With a Dimple,” J. R. Soc. Interface, 14(127), p. 20160996. [PubMed]
Martynyak, R. , 2001, “ The Contact of a Half-Space and an Uneven Base in the Presence of an Intercontact Gap Filled by an Ideal Gas,” J. Math. Sci., 107(1), pp. 3680–3685.
Chumak, K. , Chizhik, S. , and Martynyak, R. , 2014, “ Adhesion of Two Elastic Conforming Solids With a Single Interface Gap,” J. Adhes. Sci. Technol., 28(16), pp. 1568–1578.
Chumak, K. , 2016, “ Adhesive Contact Between Solids With Periodically Grooved Surfaces,” Int. J. Solids Struct, 78–79, pp. 70–76.
Malanchuk, N. , and Martynyak, R. , 2012, “ Contact Interaction of Two Solids With Surface Groove Under Proportional Loading,” Int. J. Solids Struct., 49(23–24), pp. 3422–3431.
Johnson, K. L. , 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK. [PubMed] [PubMed]
Johnson, K. L. , 1995, “ The Adhesion of Two Elastic Bodies With Slightly Wavy Surfaces,” Int. J. Solids Struct., 32(3–4), pp. 423–430.
Jin, F. , Wan, Q. , and Guo, X. , 2016, “ A Double-Westergaard Model for Adhesive Contact of a Wavy Surface,” Int. J. Solids Struct., 102–103, pp. 66–76.
Hills, D. A. , Nowell, D. , and Sackfield, A. , 1993, Mechanics of Elastic Contacts, Butterworth–Heinemann, Oxford, UK.

## Figures

Fig. 1

Schematics of adhesive contact between an elastic flat surface and an elastic surface with a single interface groove subjected to a remotely applied normal traction p¯ (positive when compressive, negative when tensile)

Fig. 2

Nonadhesive contact behaviors of the interface groove. (a) The groove half-width versus the applied compressive traction. (b) The groove shapes at some typical loading stages. (c) The contact pressure distributions at some typical loading stages.

Fig. 3

The normalized groove half-width versus the normalized applied normal traction in the JKR adhesion model for different values of α

Fig. 4

Schematics of the cohesive zone within the interfacial groove. Adhesive traction in the Maugis–Dugdale model is assumed to be constant σ0 acting over a cohesive zone with separation less than hc.

Fig. 5

Maugis–Dugdale adhesive contact behaviors of the interface groove for α=0.3 and μ=0.6. (a) The groove half-width versus the applied normal traction. The nonadhesion and JKR adhesion curves are also included for comparison. (b) The groove shapes at some typical loading stages. Curves A and E correspond to initial shape and full contact cases, respectively.

Fig. 6

The groove half-width versus the applied normal traction in the Maugis–Dugdale model with α=0.4 for (a) μ=0.01, (b) μ=0.7, (c) μ=1.2, and (d) μ=4. The corresponding nonadhesion and JKR adhesion curves are also included for comparison.

Fig. 7

The groove half-width versus the applied normal traction in the Maugis–Dugdale model with α=1 for (a) μ=0.01, (b) μ=0.4, (c) μ=1.2, and (d) μ=4. The corresponding nonadhesion and JKR adhesion curves are also included for comparison.

Fig. 8

Maugis–Dugdale and JKR adhesion regions of α and μ for self-healing interface grooves

Fig. 9

Critical compressive traction to achieve full contact as a function of the Tabor parameter for different values of α. The subplot (b) is an enlarged part of plot (a) where the behavior for α=0.4 can be seen in more detail.

Fig. 10

The critical load for gap initiation as a function of the Tabor parameter for different values of α

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections