Research Papers

Analysis and Inversion of Contact Stress for the Finite Thickness Neo-Hookean Layer

[+] Author and Article Information
Heng Yang, Xue-Feng Yao, Shen Wang, Yu-Chao Ke, Sheng-Hao Huang, Ying-Hua Liu

Applied Mechanics Lab,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China

1Corresponding author.

Manuscript received April 5, 2018; final manuscript received June 19, 2018; published online July 5, 2018. Assoc. Editor: Yong Zhu.

J. Appl. Mech 85(10), 101008 (Jul 05, 2018) (9 pages) Paper No: JAM-18-1191; doi: 10.1115/1.4040598 History: Received April 05, 2018; Revised June 19, 2018

In this paper, the theoretical analysis and the inversion of the contact stress on the finite thickness rubber contact surface with the friction effect are investigated. First, an explicit expression of deformation and stress on the surface of rubber under a rigid spherical indenter is developed by means of theoretical model, dimensional analysis, and nonlinear finite element simulation. Second, the inverse approach for obtaining the contact stress on the finite thickness rubber contact surface is presented and verified theoretically. Also, the displacement, the stress field, and the friction coefficient are obtained by means of three-dimensional digital image correlation (3D DIC) method. Finally, the applicability to other hyperelastic models, general boundary conditions, and loading modes are discussed. The results will provide an important theoretical and experimental basis for evaluating the contact stress on the finite thickness rubber layer.

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Flitney, R. K. , 2011, Seals and Sealing Handbook, Butterworth-Heinemann, Oxford, UK. [PubMed] [PubMed]
Ke, Y. , Yao, X. , Yang, H. , and Liu, X. , 2014, “ Kinetic Friction Characterizations of the Tubular Rubber Seals,” Tribol. Int., 72, pp. 35–41.
Liu, Q. , Wang, Z. , Lou, Y. , and Suo, Z. , 2014, “ Elastic Leak of a Seal,” Extreme Mech. Lett., 1, pp. 54–61.
Ke, Y.-C. , Yao, X.-F. , Yang, H. , and Ma, Y.-J. , 2017, “ Gas Leakage Prediction of Contact Interface in Fabric Rubber Seal Based on a Rectangle Channel Model,” Tribol. Trans., 60(1), pp. 146–153.
K. L. Johnson, 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK. [PubMed] [PubMed]
Persson, B. , Albohr, O. , Tartaglino, U. , Volokitin, A. , and Tosatti, E. , 2005, “ On the Nature of Surface Roughness With Application to Contact Mechanics, Sealing, Rubber Friction and Adhesion,” J. Phys.: Condens. Matter, 17(1), p. R1. [PubMed]
Persson, B. N. J. , 2001, “ Theory of Rubber Friction and Contact Mechanics,” J. Chem. Phys., 115(8), p. 3840.
Lin, Y.-Y. , Chang, C.-F. , and Lee, W.-T. , 2008, “ Effects of Thickness on the Largely-Deformed JKR (Johnson–Kendall–Roberts) Test of Soft Elastic Layers,” Int. J. Solids. Struct, 45(7–8), pp. 2220–2232.
Zisis, T. , Zafiropoulou, V. , and Giannakopoulos, A. , 2011, “ The Adhesive Contact of a Flat Punch on a Hyperelastic Substrate Subject to a Pull-out Force or a Bending Moment,” Mech. Mater., 43(1), pp. 1–24.
Vasu, T. S. , and Bhandakkar, T. K. , 2016, “ A Study of the Contact of an Elastic Layer–Substrate System Indented by a Long Rigid Cylinder Incorporating Surface Effects,” ASME J. Appl. Mech., 83(6), p. 061009.
Ihara, T. , Shaw, M. , and Bhushan, B. , 1986, “ A Finite Element Analysis of Contact Stress and Strain in an Elastic Film on a Rigid Substrate—Part I: Zero Friction,” ASME J. Tribol., 108(4), pp. 527–533.
Scheibert, J. , Prevost, A. , Frelat, J. , Rey, P. , and Debrégeas, G. , 2008, “ Experimental Evidence of Non-Amontons Behaviour at a Multi-Contact Interface,” Europhys. Lett., 83(3), p. 34003.
Nguyen, D. T. , Paolino, P. , Audry, M. , Chateauminois, A. , Fretigny, C. , Le Chenadec, Y. , Portigliatti, M. , and Barthel, E. , 2011, “ Surface Pressure and Shear Stress Fields Within a Frictional Contact on Rubber,” J. Adhes., 87(3), pp. 235–250.
Prevost, A. , Scheibert, J. , and Debrégeas, G. , 2013, “ Probing the Micromechanics of a Multi-Contact Interface at the Onset of Frictional Sliding,” Eur. J. Phys. E, 36(17), pp. 1–12.
Eguchi, M. , Shibamiya, T. , and Yamamoto, T. , 2009, “ Measurement of Real Contact Area and Analysis of Stick/Slip Region,” Tribol. Int., 42(11–12), pp. 1781–1791.
Scheibert, J. , Prevost, A. , Debrégeas, G. , Katzav, E. , and Adda-Bedia, M. , 2009, “ Stress Field at a Sliding Frictional Contact: Experiments and Calculations,” J. Mech. Phys. Solids, 57(12), pp. 1921–1933.
Belforte, G. , Conte, M. , Bertetto, A. M. , Mazza, L. , and Visconte, C. , 2009, “ Experimental and Numerical Evaluation of Contact Pressure in Pneumatic Seals,” Tribol. Int., 42(1), pp. 169–175.
Treloar, L. , 1943, “ The Elasticity of a Network of Long-Chain Molecules—II,” Trans. Faraday Soc., 39, pp. 241–246.
Bertrand, J. , 1878, “ Sur L'homogénéité Dans Les Formules De Physique,” C. Rendus, 86(15), pp. 916–920.
Giannakopoulos, A. , and Panagiotopoulos, D. , 2009, “ Conical Indentation of Incompressible Rubber-like Materials,” Int. J. Solids. Struct., 46(6), pp. 1436–1447.
Corp., D. S, 2012, “ ABAQUS User's Manual, Version 6.12,” Dassault Systemes, Providence, RI.
Zhang, M.-G. , Chen, J. , Feng, X.-Q. , and Cao, Y. , 2014, “ On the Applicability of Sneddon's Solution for Interpreting the Indentation of Nonlinear Elastic Biopolymers,” ASME J. Appl. Mech., 81(9), p. 091011.
Zisis, T. , Zafiropoulou, V. , and Giannakopoulos, A. , 2015, “ Evaluation of Material Properties of Incompressible Hyperelastic Materials Based on Instrumented Indentation of an Equal-Biaxial Prestretched Substrate,” Int. J. Solids. Struct., 64–65, pp. 132–144.
Zhang, M.-G. , Cao, Y.-P. , Li, G.-Y. , and Feng, X.-Q. , 2014, “ Spherical Indentation Method for Determining the Constitutive Parameters of Hyperelastic Soft Materials,” Biomech. Model. Mech., 13(1), pp. 1–11.
Tiwari, V. , Sutton, M. A. , McNeill, S. R. , Xu, S. , Deng, X. , Fourney, W. L. , and Bretall, D. , 2009, “ Application of 3D Image Correlation for Full-Field Transient Plate Deformation Measurements During Blast Loading,” Int. J. Impact. Eng., 36(6), pp. 862–874.
Luo, P. , Chao, Y. , Sutton, M. , and Peters Iii, W. , 1993, “ Accurate Measurement of Three-Dimensional Deformations in Deformable and Rigid Bodies Using Computer Vision,” Exp. Mech., 33(2), pp. 123–132.
Helm, J. D. , McNeill, S. R. , and Sutton, M. A. , 1996, “ Improved Three‐Dimensional Image Correlation for Surface Displacement Measurement,” Opt. Eng., 35(7), pp. 1911–1920.
Guanchang, J. , Zhen, W. , Nikeng, B. , and Xuefeng, Y. , 2003, “ Digital Speckle Correlation Method With Compensation Technique for Strain Field Measurements,” Opt. Laser. Eng., 39(4), pp. 457–464.
Yao, X. , Meng, L. , Jin, J. , and Yeh, H. , 2005, “ Full-Field Deformation Measurement of Fiber Composite Pressure Vessel Using Digital Speckle Correlation Method,” Polym. Test, 24(2), pp. 245–251.
Hadamard, J. , 1923, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, New Haven, CT.
Cao, Y. P. , and Lu, J. , 2004, “ A New Method to Extract the Plastic Properties of Metal Materials From an Instrumented Spherical Indentation Loading Curve,” Acta Mater., 52(13), pp. 4023–4032.
Meng, L. , Jin, G. , Yao, X. , and Yeh, H. , 2006, “ 3D Full-Field Deformation Monitoring of Fiber Composite Pressure Vessel Using 3D Digital Speckle Correlation Method,” Polym. Test, 25(1), pp. 42–48.
Yang, H. , Yao, X.-F. , Ke, Y.-C. , Ma, Y-J. , and Liu, Y.-H. , 2016, “ Constitutive Behaviors and Mechanical Characterizations of Fabric Reinforced Rubber Composites,” Compos. Struct., 152, pp. 117–123.
Sutton, M. A. , Orteu, J. J. , and Schreier, H. , 2009, Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications, Springer Science & Business Media, New York.
Mooney, M. , 1940, “ A Theory of Large Elastic Deformation,” J. Appl. Phys., 11(9), pp. 582–592.
Rivlin, R. , 1948, “ Large Elastic Deformations of Isotropic Materials. IV. Further Developments of the General Theory,” Philos. Trans. R. Soc. A, 241(835), pp. 379–397.
Ogden, R. , 1972, “ Large Deformation Isotropic Elasticity-on the Correlation of Theory and Experiment for Incompressible Rubberlike Solids,” Proc. R. Soc. A, 326(1567), pp. 565–584.
Arruda, E. M. , and Boyce, M. C. , 1993, “ A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials,” J. Mech. Phys. Solids, 41(2), pp. 389–412.


Grahic Jump Location
Fig. 1

Schematic diagram for the finite thickness rubber contact problems

Grahic Jump Location
Fig. 2

Effects of friction on the contact stress p and the vertical displacement uz under the condition of F/(μ0R2) = 2.0, b/R = 0.5: (a) the vertical displacement uz and (b) the contact stress p

Grahic Jump Location
Fig. 3

The contact stress and vertical displacement on the frictionless contact surface of rubber for different thicknesses under F/(μ0R2)=3.0: (a) the vertical displacement and (b) the contact stress

Grahic Jump Location
Fig. 4

The inversion of the contact stress from displacement field for rubber contact problems: (a) spherical indenter rubber contact and (b) general rubber contact problem

Grahic Jump Location
Fig. 5

Condition number for the inversion of stress under various ratios f and r/a at (a) the thickness ratio b/R of 0.5 and (b) r/a = 0

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Fig. 9

The vertical displacement uz of the experiments and inversion results to obtain the friction coefficient in the lubrication interface under F = 50 N

Grahic Jump Location
Fig. 8

The vertical displacement uz and the contact stress p along the radius of the experiments and inversion: (a) the vertical displacement and (b) the contact stress

Grahic Jump Location
Fig. 7

The measurement results of the vertical displacement uz under the load of 33.0 N and 86.5 N: (a) the vertical displacement uz under the load of 33.0 N and (b) the vertical displacement uz under the load of 86.5 N

Grahic Jump Location
Fig. 6

The experimental setup for 3D DIC method and the speckle images of rubber surface: (a) test system on the rubber contact and (b) speckle field in measurement area



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