Abstract
Starting from the classical theory of contact mechanics it is shown that the relationship between load, penetration and contact radius of any axisymmetric contact can be mapped exactly on a one-dimensional system, thus the reduction method of dimensionality is valid for conforming and non-conforming contacts. Furthermore the reduction method has been successfully extended to adhesive contact problems. The mapping of the classical theory of Johnson, Kendall and Roberts derived for spherical contacts as well as its application to axisymmetric contacts of arbitrary shape is possible; all results are reproduced precisely.
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References
H. Hertz, Über die Berührung fester elastischer Körper, J. für die reine und angewandte Mathematik, 92 (1882) 156.
K.L. Johnson, K. Kendall, and A.D. Roberts, Surface energy and the contact of elastic solids, Proc. Roy. Soc. Lond. A. Mat., 324 (1971) 301.
J. Boussinesq, Application des Potentiels a L’etude de L’equilibre et du Mouvement des Solides Elastiques, Gauthier-Villars, Paris, 1885.
I.N. Sneddon, The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile, Int. J. Eng. Sci., 3 (1965) 47.
W.C. Oliver and G.M. Pharr, Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology, J. Mater. Res., 19, No. 1 (2004) 3.
D. Maugis and M. Barquins, Fracture mechanics and the adherence of viscoelastic bodies, J. Phys. D, 11 (1978) 1989.
M. Barquins and D. Maugis, Adhesive contact of axisymmetric punches on an elastic half-space: The modified Hertz-Huber’s stress tensor for contacting spheres, J. Méc. Théor. Appl., 1 (1982) 331.
D. Maugis and M. Barquins, Adhesive contact of sectionally smooth-ended punches on elastic half-spaces: Theory and experiment, J. Phys. D, 16 (1983) 1843.
H. Yao and H. Gao, Optimal shapes for adhesive binding between two elastic bodies, J. Colloid Interf. Sci., 298, No. 2 (2006) 564.
T. Geike and V.L. Popov, Mapping of three-dimenional contact problems into one dimension, Phys. Rev. E, 76, No. 3 (2007) 036710.
V.L. Popov, Contact Mechanics and Friction. Physical Principles and Applications, Springer-Verlag, Berlin, 2010.
V.L. Popov and S.G. Psakhie, Numerical simulation methods in tribology, Tribol. Int., 40, No. 6 (2007) 916.
V.L. Popov and A. Dimaki, Using hierarchical memory to calculate friction force between fractal rough solid surface and elastomer with arbitrary linear rheological properties. Tech. Phys. Lett., 37, No. 1 (2011) 8.
M. Heß, Über die exakte Abbildung ausgewählter dreidimensionaler Kontakte auf Systeme mit niedrigerer räumlicher Dimension, Cuvillier-Verlag, Berlin, 2011.
E. Steuermann, To Hertz’s theory of local deformations in compressed elastic bodies, Dokl. AS URSS, 25 (1939) 359.
C.M. Segedin, The relation between load and penetration for a spherical punch, Mathematika, 4 (1957) 156.
G.M. Pharr, W.C. Oliver, and F.R. Brotzen, On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation, J. Mater. Res., 7, No. 3 (1992) 613.
R.E. Gibson, Some results concerning displacements and stresses in a non-homogeneous elastic half-space, Geotechnique, 17, No. 1 (1967) 58.
G.R. Irwin, Fracture, in Handbook of Physics, Springer-Verlag, Berlin, V. 6 (1958) 551.
A.A. Griffith, The phenomena of rapture and flow in solids, Philos. T. Roy. Soc. A, 221 (1921) 163.
D. Maugis, Contact, Adhesion and Rupture of Elastic Solids, Springer Verlag, Berlin, 2000.
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Original Text © M. Heß, 2012, published in Fiz. Mezomekh., 2012, Vol. 15, No. 4, pp. 19–24.
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Heß, M. On the reduction method of dimensionality: The exact mapping of axisymmetric contact problems with and without adhesion. Phys Mesomech 15, 264–269 (2012). https://doi.org/10.1134/S1029959912030034
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DOI: https://doi.org/10.1134/S1029959912030034