In this study, we solve the continuous contact problem of two layers with different material properties, loaded with two rigid flat blocks and resting on a rigid plane, by analytical and finite element methods. All surfaces are assumed to be frictionless. The upper layer is the functionally graded layer (FG), the height h1, the underlying layer is homogeneous, and the height is h2. It is assumed that the FG layer is isotropic and that the shear modulus and density change exponentially. The FG layer is loaded by means of two rigid flat blocks whose external loads are Q and P (per unit thickness in \( z \) direction). In the analytical solution of the problem, stress and displacement expressions are substituted in the boundary conditions and the problem is demoted to singular integral equations where the contact stresses are the unknown functions. These singular integral equations are solved numerically using the Gauss–Chebyshev integration formulas. Two-dimensional finite element analysis of the problem is performed using the ANSYS package program. The analyses are performed for different shear modulus ratios (μ2/μ-h1), distances between the blocks ((c − b)/h1) and inhomogeneity parameters (βh1). The contact stresses under the blocks, the normal stresses (σy) between interfaces of the FG layer and the homogeneous layer and between the homogeneous layer and rigid plane are also measured. In addition, the initial separation load and the initial separation distances (χcr, λcr) of the FG layer and homogeneous layer are determined. The results are presented comparatively as graphs and tables.
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Galin, L.A.: Contact problems in the theory of elasticity, İngilizce Çeviri: Moss, H., North Carolina State College Translation Series, Raleigh, North Carolina (1961)
G.G. Adams, D.B. ve Bogy, The plane symmetric contact problem for dissimilar elastic semi-ınfinite strips of different widths. ASME J Appl Mech 44(4), 604–610 (1977)
Nowell, D. ve Hills, D.A.: Contact problems incorporating elastic layers, International Journal of Solids and Structures, 24, 1, 105-111 (1988)
O. Aksogan, S. Akavcı, A.A. ve Becker, A comperative study of the contact problem of an elastic layer supported by two elastic quarter planes. J Fac Eng Arhit Çukurova Univ 11(1), 25–31 (1996)
Aksogan, O., Akavcı S, ve Becker A.A.: The solution of the nonsymmetrical contact problem of an elastic layer supported by two elastic quarter planes using three different methods, Journal of Faculty of Engineering and Arhitecture of Çukurova Universty, 12 1-2 1-14 (1997)
Ozsahin, T. S. and Taskıner, O.: Contact problem for an elastic layer on an elastic half plane loaded by means of three rigid flat punches, Mathematical Problems in Engineering, Volume 2013, Article ID 137427, 14 pages (2013)
Polat, A., Kaya, Y., and Özşahin, T. Ş. Analytical solution to continuous contact problem for a functionally graded layer loaded through two dissimilar rigid punches. Meccanica, 1-13 (2018)
S. El-Borgi, R. Abdelmoula, L. Keer, A receding contact problem between a functionally graded layer and a homogeneous substrate. Int. J. Solids Struct. 43, 658–674 (2006)
S. El-Borgi, S. Usman, M.A. Guler, A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate. Int. J. Solids Struct. 51, 4462–4476 (2014)
I. Çömez, Contact problem of a functionally graded layer resting on a Winkler foundation. Acta Mech. 224, 2833–2843 (2013)
Adiyaman G., Öner E., Birinci A.: Continuous and discontinuous contact problem of a functionally graded layer resting on a rigid foundation, Acta Mechanica, vol.0, pp.1-10 (2017)
I. Çömez, M.A. Guler, The contact problem of a rigid punch sliding over a functionally graded bilayer. Acta Mech. 228, 2237–2249 (2017)
S.K. Chan, I.S. Tuba, A finite element method for contact problems of solid bodies–Part I. Theory and validation. Int. J Mech. Sci. 13, 519–530 (1971)
X.C. Zhang, B.S. Xu, H.D. Wang, Y.X. Wu, Y. Jiang, Hertzian contact response of single-layer, functionally graded and sandwich coatings. Mater. Des. 28, 47–54 (2007)
Yaylaci M., Oner E., Birinci A.: Comparison between analytical and ANSYS calculations for a receding contact problem”, Journal of Engineering Mechanics-ASCE, vol.140 (2014)
M.N. Abhilash, H. Murthy, Finite element analysis of 2-d elastic contacts involving FGMs. Int J Comput Methods Eng Sci Mech 15(3), 253–257 (2014)
A. Birinci, G. Adiyaman, M. Yaylaci, E. Öner, Analysis of continuous and discontinuous cases of a contact problem using analytical method and FEM. Latin Am J Solids Struct 12, 1771–1789 (2015)
M.Y. Korkmaz, D. Coker, Finite element analysis of fretting contact for dissimilar and nonhomogeneous materials. Procedia Struct Integr 5, 452–459 (2017)
M.A. Güler, A. Kucuksucu, K.B. Yilmaz, B. Yildirim, On the analytical and finite element solution of plane contact problem of a rigid cylindrical punch sliding over a functionally graded orthotropic medium. Int. J. Mech. Sci. 120, 12–29 (2017)
A. Polat, Y. Kaya, T.S. Ozsahin, Analysis of frictionless contact problem for a layer on an elastic half plane using FEM. Duzce Univ J Sci Technol 6(2), 357–368 (2018)
Y. Kaya, A. Polat, T.Ş. Özşahin, Comparıson of FEM solution with analytical solution of continuous and discontinuous contact problem. Sigma J Eng Nat Sci 36(4), 977–992 (2018)
F. Erdogan, G. Gupta, On the numerical solutions of singular integral equations. Quart. J. Appl. Math. 29, 525–534 (1972)
ANSYS, Swanson Analysis Systems Inc., Houston PA, USA (2015)
A. Polat, Y. Kaya, B. Kouider, T.Ş. Özşahin, Frictionless contact problem for a functionally graded layer loaded through two rigid punches using finite element method. J. Mech. (2019). https://doi.org/10.1017/jmech.2018.55
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Kaya, Y., Polat, A. & Özşahin, T.Ş. Analytical and finite element solutions of continuous contact problem in functionally graded layer. Eur. Phys. J. Plus 135, 89 (2020). https://doi.org/10.1140/epjp/s13360-020-00138-9