Abstract
This paper is concerned with the method of memory diagrams developed for solving a problem of frictional elastic contact. Our goal is to establish a link between contact force and displacement for a general plane contact between two arbitrarily-shaped bodies; rolling and torsion are not considered. Description of mechanical interactions between two solids in contact in the presence of friction is a nontrivial task since the desired force-displacement relationships have hysteretic (memory-dependent) character. Arbitrarily changing applied force (or displacement) creates a cumbersome shear stress distribution in the contact zone that has to be adequately parameterized and accounted for. In that regard, it is suggested to consider, instead of complex shear stress distributions, a simpler functional form called memory diagram that contains the same memory information. We have established two integral relationships that link the force and displacement vectors with that internal functional dependency. The integral relationships are supplemented with two other evolution rules for memory diagrams that eventually follow from the Coulomb friction law. The memory diagram is updated with the help of these rules following a given force history. Then the calculated memory diagram is used to update the history of displacement i.e. to produce the desired force-displacement relationship.
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References
Landau, L.D. and Lifshitz, E.M., Theory of Elasticity, Oxford: Pergamon Press, 1993.
Cattaneo, C., Sul Contatto di Due Corpi Elastici: Distribuzione Locale Degli Sforzi, Rend. Accad. Naz. Lincei, 1938, vol. 27, pp. 342–348. 434-436, 474-478.
Mindlin, R.D., Compliance of Elastic Bodies in Contact, J. Appl. Mech., 1949, vol. 16, pp. 259–268.
Mindlin, R.D. and Deresiewicz, H., Elastic Spheres in Contact under Varying Oblique Forces, J. Appl. Mech., 1953, vol. 20, pp. 327–344.
Jäger, J., Axisymmetric Bodies of Equal Material in Contact under Torsion or Shift, Arch. Appl. Mech., 1995, vol. 65, pp. 478–487.
Jäger, J., Half-Planes without Coupling under Contact Loading, Arch. Appl. Mech., 1997, vol. 67, pp. 247–259.
Jäger, J., Properties of Equal Bodies in Contact with Friction, Int. J. Solids Struct., 2003, vol. 40, pp. 5051–5061.
Ciavarella, M., The Generalized Cattaneo Partial Slip Plane Contact Problem. I. Theory, II.Examples, Int. J. Solids Struct., 1998, vol. 35, pp. 2349–2362.
Ciavarella, M., Tangential Loading of General 3D Contacts, ASME J. Appl. Mech., 1998, vol. 65, pp. 998–1003.
Aleshin, V. and van Den Abeele, K., Preisach Analysis of the Hertz-Mindlin System, J. Mech. Phys. Solids, 2009, vol. 57, pp. 657–672.
Aleshin, V. and van Den Abeele, K., Hertz-Mindlin Problem for Arbitrary Oblique 2D Loading: General Solution by Memory Diagrams,J. Mech. Phys. Solids, 2012, vol. 60, pp. 14–36.
Aleshin, V. and van Den Abeele, K., General Solution to the Hertz-Mindlin Problem via Preisach Formalism, Int. J. Non-Linear Mech., 2013, vol. 49, pp. 15–30.
Aleshin, V.V., Bou Matar, O., and van Den Abeele, K., Method of Memory Diagrams for Mechanical Frictional Contacts Subject to Arbitrary 2D Loading, Int. J. Sol. Struct., 2015, vol. 60-61, pp. 84–95.
Boltachev, G.Sh. and Aleshin, V.L., Shift and Torsion Contact Problems for Arbitrary Axisymmetric Normal Stress Distributions, Int. J. Solids Struct., 2013, vol. 50, pp. 2894–2900.
Popov, V.L. and Heß, M., Method of Dimensionality Reduction. Contact Mechanics and Friction, Berlin: Springer, 2014.
Johnson, K.L., Contact Mechanics, Cambridge: Cambridge University Press, 1985.
Hyun, S. and Robbins, M.O., Elastic Contact between Rough Surfaces: Effect of Roughness at Large and Small Wavelengths, Tribol. Int., 2007, vol. 40, pp. 1413–1422.
Carbone, G. and Bottiglione, F., Asperity Contact Theories: Do They Predict Linearity between Contact Area and Load? J. Mech. Phys. Solids, 2008, vol. 56, pp. 2555–2572.
Paggi, M., Pohrt, R., and Popov, VL., Partial-Slip Frictional Response of Rough Surfaces, Sci. Rep., 2014, vol. 4, p. 5178.
Paggi, M. and Ciavarella, M., The Coefficient of Proportionality k between Real Contact Area and Load, with New Asperity Models, Wear, 2010, vol. 268, pp. 10201029.
Pohrt, R. and Popov, V.L., Contact Mechanics of Rough Spheres: Crossover from Fractal to Hertzian Behavior, Adv. Tribol., 2013, p. 974178.
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Original Text © V.V. Aleshin, O. Bou Matar, 2015, published in Fizicheskaya Mezomekhanika, 2015, Vol. 18, No. 4, pp. 18-23.
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Aleshin, V.V., Bou Matar, O. Solution to the frictional contact problem via the method of memory diagrams for general 3D loading histories. Phys Mesomech 19, 130–135 (2016). https://doi.org/10.1134/S102995991602003X
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DOI: https://doi.org/10.1134/S102995991602003X