We consider an axially symmetric contact problem of pressing of an absolutely rigid ball into an inhomogeneous half space formed by a homogeneous base and an inhomogeneous surface layer. The Poisson’s ratio of the layer is constant and its Young modulus is an exponential function of the distance from the surface of the half space. The solution of the problem of the theory of elasticity with continuous dependence of the Young modulus on the coordinate is compared with the solution of the problem in which the inhomogeneous layer is replaced with a package of homogeneous layers.
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References
B. G. Korenev, “A die lying on the elastic half space whose modulus of elasticity is a function of depth,” Dokl. Akad. Nauk SSSR, 112, No. 5, 823–826 (1957).
V. I. Mossakovskii, “Pressure of a circular die upon the elastic half space whose modulus of elasticity is a power function of depth,” Prikl. Mat. Mekh., 22, No. 1, 123–125 (1958).
B. I. Kogan and V. D. Zinchenko, “Stressed state of an inhomogeneous layer lying on the elastic half space,” Izv. Vyssh. Uchebn. Zaved., Ser. Stroit. Arkh., No. 3, 8–18 (1960).
A. K. Rakov and V. P. Rvachev, “Contact problem of the theory of elasticity for the half space whose modulus of elasticity is a power function of depth,” Dopov. Akad. Nauk Ukr. SSR, No. 3, 286–290 (1961).
L. N. Ter-Mkrtich’yan, “Some problems of the theory of elasticity for inhomogeneous elastic media,” Prikl. Mat. Mekh., 25, No. 6, 1120–1125 (1961).
N. A. Rostovtsev, “On the theory of elasticity for inhomogeneous media,” Prikl. Mat. Mekh., 28, No. 4, 601–611 (1964).
Yu. A. Shevlyakov, Yu. A. Naumov, and V. N. Chistyak, “On the numerical analysis of inhomogeneous bases,” Prikl. Mekh., 4, No. 9, 66–73 (1968).
G. Ya. Popov, “Axially symmetric contact problem for the elastic inhomogeneous half space in the presence of cohesion,” Prikl. Mat. Mekh., 37, No. 6, 1109–1116 (1973).
V. P. Plevako, “On the possibility of application of harmonic functions to the solution of the problems of elasticity theory for inhomogeneous media,” Prikl. Mat. Mekh., 36, No. 5, 886–894 (1972).
V. P. Plevako, “Inhomogeneous layer coupled with a half space under the action of internal and external forces,” Prikl. Mat. Mekh., 38, No. 5, 865–875 (1974).
M. K. Kassir and M. F. Chuaprasert, “A rigid punch in contact with a nonhomogeneous elastic solid,” Trans. ASME: J. Appl. Mech., 41, 1019–1024 (1974).
A. E. Giannakopoulos and S. Suresh, “Indentation of solids with gradients in elastic properties: Part II. Axisymmetric indenters,” Int. J. Solids Struct., 34, 2393–2428 (1997).
A. E. Giannakopoulos and P. Pallot, “Two-dimensional contact analysis of elastic graded materials,” J. Mech. Phys. Solids, 48, 1597–1631 (2000).
A. C. Fischer-Cripps, “Analysis of instrumented indentation test data for functionally graded materials,” Surf. Coat. Techn., 168, 136–141 (2003).
M. A. Guler and F. Erdogan, “Contact mechanics of graded coatings,” Int. J. Solids Struct., 41, 3865–3889 (2004).
M. A. Guler and F. Erdogan, “Contact mechanics of two deformable elastic solids with graded coatings,” Mech. Mater., 38, 633–647 (2006).
M. A. Guler and F. Erdogan, “The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded coatings,” Int. J. Mech. Sci., 49, 161–182 (2007).
S. M. Aizikovich, V. M. Alexandrov, J. J. Kalker, et al., “Analytic solution of the spherical indentation problem for a half space with gradients with the depth elastic properties,” Int. J. Solids Struct., 39, 2745–2772 (2002).
Liao-Liang Ke and Yue-Sheng Wang, “Two-dimensional contact mechanics of functionally graded materials with arbitrary spatial variations of material properties,” Int. J. Solids Struct., 43, 5779–5798 (2006).
Liao-Liang Ke and Yue-Sheng Wang, “Two-dimensional sliding frictional contact of functionally graded materials,” Eur. J. Mech. A. Solids, 26, 171–188 (2007).
Tie-Jun Liu, Yue-Sheng Wang, and Chuanzeng Zhang, “Axisymmetric frictionless contact of functionally graded materials,” Arch. Appl. Mech., 78, 267–282 (2008).
Tie-Jun Liu and Yue-Sheng Wang, “Axisymmetric frictionless contact problem of a functionally graded coating with exponentially varying modulus,” Acta Mech., 199, 151–165 (2008).
R. Kulchytsky-Zhyhailo and S. J. Matysiak, “On heat conduction problem in a semiinfinite periodically laminated layer,” Int. Comm. Heat Mass Transfer, 32, No. 1–2, 123–132 (2005).
R. Kulchytsky-Zhyhailo and S. J. Matysiak, “On some heat conduction problem in a periodically two-layered body. Comparative results,” Int. Comm. Heat Mass Transfer, 32, No. 3–4, 332–340 (2005).
R. Kulchytsky-Zhyhailo and W. Kolodziejczyk, “Stress field in a inhomogeneous half plane with periodic structure caused by the Hertz pressure,” Tren. Iznos, 26, No. 4, 358–366 (2005).
W. Kolodziejczyk and R. Kul’chyts’kyi-Zhyhailo, “Pressure of the lateral surface of a cylinder on a periodically layered half space,” Mater. Sci., 43, No. 3, 351–360 (2007).
R. Kulchytsky-Zhyhailo and W. Kolodziejczyk, “On axisymmetrical contact problem of pressure of a rigid sphere into a periodically twolayered semispace,” Int. J. Mech. Sci., 49, 704–711 (2007).
R. Kulchytsky-Zhyhailo, S. Matysiak, and D. Perkowski, “On displacements and stresses in a semiinfinite laminated layer: comparative results,” Meccanica, 42, 117–126 (2007).
K. L. Johnson, Contact Mechanics, Cambridge Univ. Press, Cambridge (1985).
S. Matysiak, A. A. Evtushenko, and R. D. Kul’chyts’kyi-Zhyhailo, “Contact problems of thermoelasticity for the half space made of a functionally graded material,” Mat. Met. Fiz.-Mekh. Polya, 41, No. 2, 45–56 (1998).
R. Kulczycki, Przestrzenne zagadnienia kontaktowe termosprężystości, Wydawnictwo PB, Białystok (2002).
M. Ozturk and F. Erdogan, “Axisymmetric crack problem in bonded materials with a graded interfacial region,” Int. J. Solids Struct., 33, 193–219 (1996).
P. K. Gupta and J. A. Walowit, “Contact stresses between an elastic cylinder and a layered elastic solid,” Trans ASME: J. Lubr. Technol., 96, 250–257 (1974).
W. T. Chen and P. A. Engel, “Impact and contact stress analysis in multilayer media,” Int. J. Solids Struct., 8, 1257–1281 (1972).
R. Kouitat Njiwa, R. Consiglio, and J. Stebut, “Boundary element modeling of coated materials in static and sliding ball-flat elastic contact,” Surf. Coat. Techn., 102, 148–153 (1998).
R. Kouitat Njiwa and J. Stebut, “Boundary element numerical modeling as a surface engineering tool: application to very thin coatings,” Surf. Coat. Techn., 116–119, 573–579 (1999).
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Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 45, No. 6, pp. 82–92, November–December, 2009.
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Kul’chyts’kyi-Zhyhailo, R., Rogowski, G. Axially symmetric contact problem of pressing of an absolutely rigid ball into an elastic half space with inhomogeneous coating. Mater Sci 45, 845–858 (2009). https://doi.org/10.1007/s11003-010-9251-y
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DOI: https://doi.org/10.1007/s11003-010-9251-y