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Research Papers: Contact Mechanics

Strain Hardening From Elastic–Perfectly Plastic to Perfectly Elastic Flattening Single Asperity Contact

[+] Author and Article Information
Hamid Ghaednia

Department of Mechanical Engineering,
William Marsh Rice University,
Houston, TX 77251
e-mail: hamid.ghaednia@rice.edu

Matthew R. W. Brake

Department of Mechanical Engineering,
William Marsh Rice University,
Houston, TX 77251
e-mail: brake@rice.edu

Michael Berryhill

Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: mtb0027@auburn.edu

Robert L. Jackson

Professor
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: jacksr7@auburn.edu

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received April 19, 2018; final manuscript received September 17, 2018; published online November 21, 2018. Assoc. Editor: Liming Chang.

J. Tribol 141(3), 031402 (Nov 21, 2018) (11 pages) Paper No: TRIB-18-1162; doi: 10.1115/1.4041537 History: Received April 19, 2018; Revised September 17, 2018

For elastic contact, an exact analytical solution for the stresses and strains within two contacting bodies has been known since the 1880s. Despite this, there is no similar solution for elastic–plastic contact due to the integral nature of plastic deformations, and the few models that do exist develop approximate solutions for the elastic–perfectly plastic material model. In this work, the full transition from elastic–perfectly plastic to elastic materials in contact is studied using a bilinear material model in a finite element environment for a frictionless dry flattening contact. Even though the contact is considered flattening, elastic deformations are allowed to happen on the flat. The real contact radius is found to converge to the elastic contact limit at a tangent modulus of elasticity around 20%. For the contact force, the results show a different trend in which there is a continual variation in forces across the entire range of material models studied. A new formulation has been developed based on the finite element results to predict the deformations, real contact area, and contact force. A second approach has been introduced to calculate the contact force based on the approximation of the Hertzian solution for the elastic deformations on the flat. The proposed formulation is verified for five different materials sets.

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References

Ghaednia, H. , and Marghitu, D. B. , 2016, “ Permanent Deformation During the Oblique Impact With Friction,” Arch. Appl. Mech., 86(1–2), pp. 121–134. [CrossRef]
Ghaednia, H. , Marghitu, D. B. , and Jackson, R. L. , 2015, “ Predicting the Permanent Deformation After the Impact of a Rod With a Flat Surface,” ASME J. Tribol., 137(1), p. 011403. [CrossRef]
Gheadnia, H. , Cermik, O. , and Marghitu, D. B. , 2015, “ Experimental and Theoretical Analysis of the Elasto-Plastic Oblique Impact of a Rod With a Flat,” Int. J. Impact Eng., 86, pp. 307–317. [CrossRef]
Brake, M. , 2016, Mechanics Jointed Structures, Springer, Berlin.
Brake, M. , 2014, “ The Role of Epistemic Uncertainty of Contact Models in the Design and Optimization of Mechanical Systems With Aleatoric Uncertainty,” Nonlinear Dyn., 77(3), pp. 899–922. [CrossRef]
Ghaednia, H. , Jackson, R. L. , and Gao, J. , 2014, “ A Third Body Contact Model for Particle Contaminated Electrical Contacts,” IEEE 60th Holm Conference on Electrical Contacts (Holm), New Orleans, LA, Oct. 12–15, pp. 1–5.
Jackson, R. L. , Ghaednia, H. , Elkady, Y. A. , Bhavnani, S. H. , and Knight, R. W. , 2012, “ A Closed-Form Multiscale Thermal Contact Resistance Model,” IEEE Trans. Compon., Packag. Manuf. Technol., 2(7), pp. 1158–1171. [CrossRef]
Golgoon, A. , Sadik, S. , and Yavari, A. , 2016, “ Circumferentially-Symmetric Finite Eigenstrains in Incompressible Isotropic Nonlinear Elastic Wedges,” Int. J. Non-Linear Mech., 84, pp. 116–129. [CrossRef]
Golgoon, A. , and Yavari, A. , 2018, “ Nonlinear Elastic Inclusions in Anisotropic Solids,” J. Elast., 130(2), pp. 239–269. [CrossRef]
Golgoon, A. , and Yavari, A. , 2017, “ On the Stress Field of a Nonlinear Elastic Solid Torus With a Toroidal Inclusion,” J. Elast., 128(1), pp. 115–145. [CrossRef]
Golgoon, A. , and Yavari, A. , 2018, “ Line and Point Defects in Nonlinear Anisotropic Solids,” Z. Für Angew. Math. Phys., 69(3), p. 81. [CrossRef]
Sadeghi, F. , Jalalahmadi, B. , Slack, T. S. , Raje, N. , and Arakere, N. K. , 2009, “ A Review of Rolling Contact Fatigue,” ASME J. Tribol., 131(4), p. 041403. [CrossRef]
Zhao, D. , Banks, S. A. , Mitchell, K. H. , D'Lima, D. D. , Colwell, C. W. , and Fregly, B. J. , 2007, “ Correlation Between the Knee Adduction Torque and Medial Contact Force for a Variety of Gait Patterns,” J. Orthop. Res., 25(6), pp. 789–797. [CrossRef] [PubMed]
Mollaeian, K. , Liu, Y. , Bi, S. , and Ren, J. , 2018, “ Atomic Force Microscopy Study Revealed Velocity-Dependence and Nonlinearity of Nanoscale Poroelasticity of Eukaryotic Cells,” J. Mech. Behav. Biomed. Mater., 78, pp. 65–73. [CrossRef] [PubMed]
Firrone, C. M. , and Zucca, S. , 2011, “ Modelling Friction Contacts in Structural Dynamics and Its Application to Turbine Bladed Disks,” Numerical Analysis—Theory and Application, J. Awrejcewicz, ed., InTech, Rijeka, Croatia.
Kardel, K. , Ghaednia, H. , Carrano, A. L. , and Marghitu, D. B. , 2017, “ Experimental and Theoretical Modeling of Behavior of 3D-Printed Polymers Under Collision With a Rigid Rod,” Addit. Manuf., 14, pp. 87–94. [CrossRef]
Pawlowski, A. E. , Cordero, Z. C. , French, M. R. , Muth, T. R. , Carver, J. K. , Dinwiddie, R. B. , Elliott, A. M. , Shyam, A. , and Splitter, D. A. , 2017, “ Damage-Tolerant Metallic Composites Via Melt Infiltration of Additively Manufactured Preforms,” Mater. Des., 127, pp. 346–351. [CrossRef]
Ghaednia, H. , Wang, X. , Saha, S. , Xu, Y. , Sharma, A. , and Jackson, R. L. , 2017, “ A Review of Elastic-Plastic Contact Mechanics,” ASME Appl. Mech. Rev., 69(6), p. 060804. [CrossRef]
Bhushan, B. , 1996, “ Contact Mechanics of Rough Surfaces in Tribology: Single Asperity Contact,” ASME Appl. Mech. Rev., 49(5), pp. 275–298. [CrossRef]
Bhushan, B. , 1998, “ Contact Mechanics of Rough Surfaces in Tribology: Multiple Asperity Contact,” Tribol. Lett., 4(1), pp. 1–35. [CrossRef]
Adams, G. , and Nosonovsky, M. , 2000, “ Contact Modeling-Forces,” Tribol. Int., 33(5–6), pp. 431–442. [CrossRef]
Johnson, K. L. , 1987, Contact Mechanics, Cambridge University Press, Cambridge, UK.
Jackson, R. L. , and Kogut, L. , 2006, “ A Comparison of Flattening and Indentation Approaches for Contact Mechanics Modeling of Single Asperity Contacts,” ASME J. Tribol., 128(1), pp. 209–212. [CrossRef]
Ghaednia, H. , Pope, S. A. , Jackson, R. L. , and Marghitu, D. B. , 2016, “ A Comprehensive Study of the Elasto-Plastic Contact of a Sphere and a Flat,” Tribol. Int., 93, pp. 78–90. [CrossRef]
Olsson, E. , and Larsson, P.-L. , 2016, “ A Unified Model for the Contact Behaviour Between Equal and Dissimilar Elastic–Plastic Spherical Bodies,” Int. J. Solids Struct., 81, pp. 23–32. [CrossRef]
Hertz, H. , 1882, “ Über Die Berührung Fester Elastischer Körper,” J. Für Die Reine Angew. Math., 92, pp. 156–171.
Jackson, R. L. , and Green, I. , 2005, “ A Finite Element Study of Elasto-Plastic Hemispherical Contact against a Rigid Flat,” ASME J. Tribol., 127(2), pp. 343–354. [CrossRef]
Chang, W.-R. , 1986, Contact, Adhesion, and Static Friction of Metallic Rough Surfaces, University of California, Berkeley, CA.
Hardy, C. , Baronet, C. , and Tordion, G. , 1971, “ The Elasto-Plastic Indentation of a Half-Space by a Rigid Sphere,” Int. J. Numer. Methods Eng., 3(4), pp. 451–462. [CrossRef]
Follansbee, P. , and Sinclair, G. , 1984, “ Quasi-Static Normal Indentation of an Elasto-Plastic Half-Space by a Rigid Sphere-I—Analysis,” Int. J. Solids Struct., 20(1), pp. 81–91. [CrossRef]
Sinclair, G. , Follansbee, P. , and Johnson, K. , 1985, “ Quasi-Static Normal Indentation of an Elasto-Plastic Half-Space by a Rigid Sphere-II—Results,” Int. J. Solids Struct., 21(8), pp. 865–888. [CrossRef]
Chang, W. , Etsion, I. , and Bogy, D. B. , 1987, “ An Elastic-Plastic Model for the Contact of Rough Surfaces,” ASME J. Tribol., 109(2), pp. 257–263. [CrossRef]
Kogut, L. , and Etsion, I. , 2002, “ Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat,” ASME J. Appl. Mech., 69(5), pp. 657–662. [CrossRef]
Zhao, B. , Zhang, S. , Wang, Q. , Zhang, Q. , and Wang, P. , 2015, “ Loading and Unloading of a Power-Law Hardening Spherical Contact Under Stick Contact Condition,” Int. J. Mech. Sci., 94, pp. 20–26. [CrossRef]
Li, L. , Wu, C. , and Thornton, C. , 2001, “ A Theoretical Model for the Contact of Elastoplastic Bodies,” Proc. Inst. Mech. Eng., Part C., 216(4), pp. 421–431. [CrossRef]
Shankar, S. , and Mayuram, M. , 2008, “ Effect of Strain Hardening in Elastic–Plastic Transition Behavior in a Hemisphere in Contact With a Rigid Flat,” Int. J. Solids Struct., 45(10), pp. 3009–3020. [CrossRef]
Shankar, S. , and Mayuram, M. , 2008, “ A Finite Element Based Study on the Elastic-Plastic Transition Behavior in a Hemisphere in Contact With a Rigid Flat,” ASME J. Tribol., 130(4), p. 044502. [CrossRef]
Lin, L. P. , and Lin, J. F. , 2006, “ A New Method for Elastic-Plastic Contact Analysis of a Deformable Sphere and a Rigid Flat,” ASME J. Tribol., 128(2), pp. 221–229. [CrossRef]
Brake, M. , 2012, “ An Analytical Elastic-Perfectly Plastic Contact Model,” Int. J. Solids Struct., 49(22), pp. 3129–3141. [CrossRef]
Brake, M. , 2015, “ An Analytical Elastic Plastic Contact Model With Strain Hardening and Frictional Effects for Normal and Oblique Impacts,” Int. J. Solids Struct., 62, pp. 104–123. [CrossRef]
Tabor, D. , 2000, The Hardness of Metals, Oxford University Press, Oxford, UK.
Ishlinsky, A. , 1944, “ The Problem of Plasticity With Axial Symmetry and Brinell's Test,” J. Appl. Math. Mech., 8, pp. 201–224.
Kogut, L. , and Komvopoulos, K. , 2004, “ Analysis of the Spherical Indentation Cycle for Elastic–Perfectly Plastic Solids,” J. Mater. Res., 19(12), pp. 3641–3653. [CrossRef]
Jackson, R. L. , Ghaednia, H. , and Pope, S. , 2015, “ A Solution of Rigid–Perfectly Plastic Deep Spherical Indentation Based on Slip-Line Theory,” Tribol. Lett., 58(3), p. 47. [CrossRef]
Rodriguez, S. A. , Alcala, J. , and Martins Souza, R. , 2011, “ Effects of Elastic Indenter Deformation on Spherical Instrumented Indentation Tests: The Reduced Elastic Modulus,” Philos. Mag., 91(7–9), pp. 1370–1386. [CrossRef]
Brake, M. , Reu, P. L. , and Aragon, D. S. , 2017, “ A Comprehensive Set of Impact Data for Common Aerospace Metals,” ASME J. Comput. Nonlinear Dyn., 12(6), p. 061011. [CrossRef]
Brinell, J. , 1900, “ Way of Determining the Hardness of Bodies and Some Applications of the Same,” Tek. Tidskr., 5, p. 69.
Meyer, E. , 1908, “ Investigations of Hardness Testing and Hardness,” Phys. Z, 9, p. 66.
Biwa, S. , and Storåkers, B. , 1995, “ An Analysis of Fully Plastic Brinell Indentation,” J. Mech. Phys. Solids, 43(8), pp. 1303–1333. [CrossRef]
Brizmer, V. , Zait, Y. , Kligerman, Y. , and Etsion, I. , 2006, “ The Effect of Contact Conditions and Material Properties on Elastic-Plastic Spherical Contact,” J. Mech. Mater. Struct., 1(5), pp. 865–879. [CrossRef]
Mesarovic, S. D. , and Fleck, N. A. , 2000, “ Frictionless Indentation of Dissimilar Elastic–Plastic Spheres,” Int. J. Solids Struct., 37(46–47), pp. 7071–7091. [CrossRef]
Sharma, A. , and Jackson, R. L. , 2017, “ A Finite Element Study of an Elasto-Plastic Disk or Cylindrical Contact against a Rigid Flat in Plane Stress With Bilinear Hardening,” Tribol. Lett., 65(3), p. 112. [CrossRef]
Jackson, R. L. , and Green, I. , 2003, “ A Finite Element Study of Elasto-Plastic Hemispherical Contact,” ASME Paper No. 2003-TRIB-0268.
Green, I. , 2005, “ Poisson Ratio Effects and Critical Valus in Spherical and Cylindrical Hertzian Contacts,” Appl. Mech. Eng., 10(3), p. 451. http://itzhak.green.gatech.edu/rotordynamics/Poisson_ratio_and_critical_values.pdf

Figures

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Fig. 1

Meshing of the FE model

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Fig. 2

von Mises stress distribution contour for a flattening contact, of a bilinear sphere and an elastic sphere

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Fig. 3

Deformation ratio of the sphere for, 0 ≤ Et/Es ≤ 1 and 0 < Δ/R ≤ 0.05 using FE analysis

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Fig. 4

Contact radius versus Δ/R and Et/Es, Hertz and JG model show upper and lower limits, respectively

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Fig. 5

Average contact pressure versus Δ/R and Et/Es. Hertz and JG models show upper and lower limits, respectively.

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Fig. 6

Contact force versus Δ/R and Et/Es. Hertz and JG model show upper and lower limits, respectively.

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Fig. 7

Comparison of deformation ratio's large deformation limit, δsp*, between FE results at Δ/R = 0.05 and Eq. (11)

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Fig. 8

Comparison of deformation ratio, δs*, contact radius, a, and contact force, F, between the FE modeling and proposed formulation for material set 1

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Fig. 9

Comparison of deformation ratio, δs*, contact radius, a, and contact force, F, between the FE modeling and proposed formulation for material set 2

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Fig. 10

Comparison of deformation ratio, δs*, contact radius, a, and contact force, F, between the FE modeling and proposed formulation for material set 3

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Fig. 11

Comparison of deformation ratio, δs*, contact radius, a, and contact force, F, between the FE modeling and proposed formulation for material set 4

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Fig. 12

Comparison of deformation ratio, δs*, contact radius, a, and contact force, F, between the FE modeling and proposed formulation for material set 5

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