×

Mathematical models for characterizing non-Hertzian contacts. (English) Zbl 1481.74573

Summary: In soft and conformal contacts, the assumptions made in the Hertz theory are violated to some extent, leading to inaccurate outcomes. An alternative contact approach is the Kelvin-Voigt model that suffers from a discontinuity existing in its constitutive law. The finite element method is also expensive computationally to be used for contact simulation. The present study introduces a concept to simulate either soft or conformal contacts and develops mathematically closed-form contact models, which are nonlinear, promising, and easy-to-implement while resolving the discontinuity issue with the Kelvin-Voigt model. Two demonstrative applications, i.e. a ball-on-plate contact and a spherical joint, are considered. The developed approaches are integrated into forward dynamics algorithms to be assessed and compared against available contact approaches in the literature. Moreover, a finite element analysis is constructed for comparison purposes. It can be concluded that the proposed contact models are robust and easy-to-implement for non-Hertzian soft and conformal contacts.

MSC:

74M15 Contact in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Johnson, K. L., Contact Mechanics (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0599.73108
[2] Popov, V. L., Contact Mechanics and Friction. Physical Principles and Applications (2010), Springer-Verlag · Zbl 1193.74001
[3] Hibbeler, R. C., Engineering Mechanics Dynamics (2010), Pearson Prentice Hall: Pearson Prentice Hall New Jersey · Zbl 0313.70002
[4] Gilardi, G.; Sharf, I., Literature survey of contact dynamics modelling, Mech. Mach. Theory, 37, 10, 1213-1239 (2002) · Zbl 1062.70553
[5] Haneef, M. D.; Randall, R. B.; Peng, Z., Wear profile prediction of IC engine bearings by dynamic simulation, Wear, 364-365, 84-102 (2016)
[6] Mattei, L.; Di Puccio, F.; Ciulli, E., A comparative study of wear laws for soft-on-hard hip implants using a mathematical wear model, Tribol. Int., 63, 66-77 (2013)
[7] Flores, P., Modeling and simulation of wear in revolute clearance joints in multibody systems, Mech. Mach. Theory, 44, 1211-1222 (2009) · Zbl 1178.70022
[8] Antunes, P.; Magalhães, H.; Ambrósio, J.; Pombo, J.; Costa, J., A co-simulation approach to the wheel-rail contact with flexible railway track, Multibody Syst. Dyn., 45, 245-272 (2019)
[9] Hachani, M.; Fourment, M., A smoothing procedure based on quasi-C1 interpolation for 3D contact mechanics with applications to metal forming, Comput. Struct., 128, 1-13 (2013)
[10] Hertz, H., Ueber die Berührung fester elastischer Körper (On the contact of elastic solids), Journal für die reine und angewandte Mathematik, 92, 156-171 (1882) · JFM 14.0807.01
[11] Goldsmith, W., Impact, The Theory and Physical Behaviour of Colliding Solids (1960), Edward Arnold: Edward Arnold Sevenoaks · Zbl 0122.42501
[12] Lankarani, H. M.; Nikravesh, P. E., A contact force model with hysteresis damping for impact analysis of multibody systems, J. Mech. Des., 12, 369-376 (1990)
[13] Flores, P.; Machado, M.; Silva, M. T.; Martins, J. M., On the continuous contact force models for soft materials in multibody dynamics, Multibody Syst. Dyn., 25, 3, 357-375 (2011) · Zbl 1263.70007
[14] Blanco-Lorenzo, J.; Santamaria, J.; Vadillo, E. G.; Correa, N., On the influence of conformity on wheel-rail rolling contact mechanics, Tribol. Int., 103, 647-667 (2016)
[15] Goodman, L. E.; Keer, L. M., The contact stress problem for an elastic sphere indenting an elastic cavity, Int. J. Solids Struct. I, 407 (1965)
[16] Wilson, H. B.; Hill, J. L., Non-Hertzian contact stresses in a smoothly cradled heavy cylinder, J. Elast., 2, 2, 143-151 (1972)
[17] Askari, E.; Andersen, M. S., A closed-form formulation for the conformal articulation of metal-on-polyethylene hip prostheses: contact mechanics and sliding distance, Proc. Inst. Mech. Eng. Part H: J. Eng. Med., 232, 12, 1-13 (2018)
[18] Johnson, K.; Kendall, K.; Roberts, A. D., Surface energy and the contact of elastic solids, Proc. R. Soc. A, 324, 301 (1971)
[19] Spence, D. A., Self similar solutions to adhesive contact problems with incremental loading, Proc. R. Soc. A., 305, 55-80 (1968) · Zbl 0172.49903
[20] Hunt, K. H.; Crossley, F. R.E., Coefficient of restitution interpreted as damping in vibro-impact, J. Appl. Mech., 7, 440-445 (1975)
[21] Anderson, R. W.G.; Long, A. D.; Serre, T., Phenomenological continuous contact-impact modelling for multibody simulations of pedestrian-vehicle contact interactions based on experimental data, Nonlinear Dyn., 58, 1-2, 199-208 (2009) · Zbl 1183.74181
[22] Zhang, Y.; Sharf, I., Validation of nonlinear viscoelastic contact force models for low speed impact, J. Appl. Mech., 76, 5, Article 051002 pp. (2009)
[23] Silva, P. C.; Silva, M. T.; Martins, J. M., Evaluation of the contact forces developed in the lower limb/orthosis interface for comfort design, Multibody Syst. Dyn., 24, 3, 367-388 (2010) · Zbl 1376.70018
[24] Marhefka, D. W.; Orin, D. E., A compliant contact model with nonlinear damping for simulation of robotic systems, IEEE Trans. Syst. Man Cybern. Part A: Syst. Hum., 29, 566-572 (1999)
[25] Herbert, R. G.; McWhannell, D. C., Shape and frequency composition of pulses from an impact pair, Am. Soc. Mech. Eng., 99, 513-518 (1977)
[26] Lee, T. W.; Wang, A. C., On the dynamics of intermittent-motion mechanisms, Part 1: dynamic model and response, J. Mech. Transm. Autom. Des., 105, 534-540 (1983)
[27] Gonthier, Y.; McPhee, J.; Lange, C.; Piedboeuf, J. C., A regularized contact model with asymmetric damping and dwell-time dependent friction, Multibody Syst. Dyn., 11, 209-233 (2004) · Zbl 1143.74344
[28] Papetti, S.; Avanzini, F.; Rocchesso, D., Numerical methods for a nonlinear impact model: a comparative study with closed-form corrections, IEEE Trans. Audio, Speech Lang., 19, 2146-2158 (2011)
[29] Di Puccio, F.; Mattei, L., Biotribology of artificial hip joints, World J. Orthop., 6, 1, 77-94 (2015)
[30] Machado, M.; Flores, P.; Claro, P.; Ambrósio, J.; Silva, M.; Completo, A.; Lankarani, H. M., Development of a planar multibody model of the human knee joint, Nonlinear Dyn., 60, 459-478 (2010) · Zbl 1189.92008
[31] Zdero, R.; Bagheri, Z. S.; Rezaey, M.; Schemitsch, E. H.; Bougherara, H., The biomechanical effect of loading speed on metal-on-UHMWPE contact mechanics, The Open Biomed. Eng. J., 8, 28-34 (2014)
[32] Kruggel-Emden, H.; Wirtz, S.; Simsek, E.; Scherer, V., Modeling of granular flow and combined heat transfer in hoppers by the discrete element method (DEM), J. Pressure Vess. Technol., 128, 3, 439 (2006)
[33] Limtrakul, S.; Boonsrirat, A.; Vatanatham, T., DEM modeling and simulation of a catalytic gas-solid fluidized bed reactor: a spouted bed as a case study, Chem. Eng. Sci., 59, 22-23, 5225-5231 (2004)
[34] Latzel, M.; Luding, S.; Herrmann, H. J.; Howell, D. W.; Behringer, R. P., Comparing simulation and experiment of a 2D granular Couette shear device, Eur. Phys. J. E, 11, 4, 325-333 (2003)
[35] E. Winkler, Die Lehre von der Elasticitaet und Festigkei. Prag, Dominicus; 1867.
[36] Flodin, A.; Andersson, S., Simulation of mild wear in spur gears, Wear, 207, 1-2, 16-23 (1997)
[37] Põdra, P.; Andersson, S., Wear simulation with the Winkler surface model, Wear, 207, 79-85 (1997)
[38] Kerr, A. D., Elastic and viscoelastic foundation models, J. Appl. Mech., 31, 3, 491-498 (1964) · Zbl 0134.44303
[39] Younesian, D.; Hosseinkhani, A.; Askari, H.; Esmailzadeh, E., Elastic and viscoelastic foundations: a review on linear and nonlinear vibration modeling and applications, Nonlinear Dyn., 97, 853-895 (2019) · Zbl 1430.74004
[40] Findley, W. N.; Lai, J. S.; Onaran, K., Creep and Relaxation of Nonlinear Viscoelastic Materials (1976), Dover Publications, INC.: Dover Publications, INC. New York · Zbl 0345.73034
[41] Bartel, D. L.; Burstein, A. H.; Toda, M. D.; Edwards, D. L., The effect of conformity and plastic thickness on contact stress in metal-backed plastic implants, Trans. ASME, J. Biomech. Eng., 107, 193-199 (1985)
[42] Jin, Z. M.; Dowson, D.; Fisher, J., A parametric analysis of the contact stress in ultra-high molecular weight poly ethylene acetabular cups, Med. Eng. Phys., 16, 5, 398-405 (1994)
[43] Khulief, Y. A.; Shabana, A. A., A continuous force model for the impact analysis of flexible multibody systems, Mech. Mach. Theory, 22, 213-224 (1987)
[44] Bei, Y.; Fregly, B. J., Multibody dynamic simulation of knee contact mechanics, Med. Eng. Phys., 26, 777-789 (2004)
[45] Askari, E.; Andersen, M. S., A dynamic model of polyethylene damage in dry total hip arthroplasties: wear and creep, Multibody Syst. Dyn., 45, 4, 403-429 (2019) · Zbl 1461.74073
[46] Askari, E.; Andersen, M. S., A modification on velocity terms of Reynolds equation in a spherical coordinate system, Tribol. Int., 131, 15-23 (2019)
[47] Askari, E.; Flores, P., Coupling multi-body dynamics and fluid dynamics to model lubricated spherical joints, Arch. Appl. Mech., 90, 2091-2111 (2020)
[48] Askari, E.; Flores, P.; Dabirrahmani, D.; Appleyard, R., Dynamic modeling and analysis of wear in spatial hard-on-hard couple hip replacements using multibody systems methodologies, Nonlinear Dyn., 76, 1365-1377 (2014)
[49] Ye, K.; Li, L.; Zhu, H., A note on the Hertz contact model with nonlinear damping for pounding simulation, Earthquake Eng. Struct. Dyn., 38, 1135-1142 (2009)
[50] Skrinjar, L.; Slavič, J.; Boltežar, M., A review of continuous contact-force models in multibody dynamics, Int. J. Mech. Sci., 145, 171-187 (2018)
[51] Hu, S.; Guo, X., A dissipative contact force model for impact analysis in multibody dynamics, Multibody Syst. Dyn., 35, 2, 131-151 (2015) · Zbl 1361.74031
[52] Gharib, M.; Hurmuzlu, Y., A new contact force model for low Coefficient of restitution impact, J. Appl. Mech., 79, 6, Article 064506 pp. (2012)
[53] Ramamurti, B.; Bragdon, C. R.; O’Connor, D. O.; Lowenstein, J. D.; Jasty, M.; Estok, D. M.; Harris, W. H., Loci of movement of selected points on the femoral head during normal gait: three-dimensional computer simulation, J. Arthoplasty, 11, 7, 845-852 (1996)
[54] Askari, E.; Flores, P.; Dabirrahmani, D.; Appleyard, R., Nonlinear vibration and dynamics of ceramic on ceramic artificial hip joints: a spatial multibody modelling, Nonlinear Dyn., 76, 2, 1365-1377 (2014)
[55] Mattei, L.; Di Puccio, F.; Ciulli, E., A comparative study of wear laws for soft-on-hard hip implants using a mathematical wear model, Tribol. Int., 63, 66-77 (2013)
[56] Jourdan, F.; Samida, A., An implicit numerical method for wear modelling applied to a hip joint prosthesis problem, Comput. Methods Appl. Mech. Eng., 198, 2209-2217 (2009) · Zbl 1227.74035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.