Mathematical modeling of the contact interaction of two elastic transversely isotropic half-spaces, one of which contains a near-surface groove of an elliptical section
Keywords:mathematical modeling, transversely isotropic half-space, near-surface groove, elliptical section, parameters of contact interaction
On the basis of a mathematical model, the problem of compression of two elastic transversely isotropic half-spaces, one of which contains a shallow near-surface groove of an elliptical section, is considered. The solution to the problem is obtained using the Elliott representation for a transversely isotropic body in terms of harmonic functions, classical harmonic potentials and reducing the boundary value problem to considering an integro-differential equation with an unknown domain of integration. As a special case, the obtained analytical expressions yield the basic parameters of the contact of transversely isotropic half-spaces in the presence of an axisymmetric groove in one of them, as well as the parameters of the contact interaction of two elastic isotropic half-spaces, one of which contains an elliptical cross-section groove. Numerical results are obtained, the influence of elastic properties of half-spaces, geometrical parameters of groove and loading on contact interaction and closing of the gap between bodies is studied.
V.S. Kirilyuk, O.I. Levchuk, O.V. Gavrilenko, and M.B. Viter, “Simulation of contact interaction of a heated flat rigid elliptical stamp with a transversely isotropic half-space”, System research and information technologies, no. 3, pp.138–148, 2020. doi: 10.20535/SRIT.2308-8893.2020.3.10.
V.S. Kirilyuk and O.I. Levchuk, “Modeling of contact interaction of piezoelectric half-space and elastic isotropic base with surface groove of circle section”, System research and information technologies, no. 4, pp. 120–132, 2016.
V.S. Kirilyuk and O.I. Levchuk, “Simulation of the contact interaction of two transversely isotropic spring half-spaces for the presence of a hard disk-like inclusion between them and pressure on the stratification area”, System research and information technologies, no. 1, pp. 107–119, 2020. doi: 10.20535/SRIT.2308-8893.2020.1.10.
Yu.N. Podil’chuk, Boundary value problems of statics of elastic bodies. Kyiv: Nauk. dumka, 1984, 304 p.
Yu.N. Podil’chuk, “Exact analytical solutions of spatial boundary value problems of statics of a transversely isotropic body of canonical form (Review)”, Int. Appl. Mech., 33, no. 10, pp. 3–30, 1997.
F.M. Borodich, B.A. Galanov, L.M. Keer, and M.M. Suarez-Alvarez, “The JKR-type adhesive contact problems for transversely isotropic elastic solids”, Mechanics of Materials, 75, pp. 34–44, 2014.
Y.S. Chai and I.I. Argatov, “Local tangential contact of elastically similar, transversely isotropic elastic bodies”, Meccanica, 53, no. 11-12, pp. 3137–3143, 2018.
W.Q. Chen, J. Zhu, and X.Y. Li, “General solutions for elasticity of transversely isotropic materials with thermal and other effects: A review”, J. Thermal Stresses, 42, no. 1, pp. 90–106, 2019.
L. Dai, W. Guo, and X. Wang, “Stress concentration at an elliptic hole in transversely isotropic piezoelectric solids”, Int. J. Solids and Struct., 43, no. 6, pp. 1818–1831, 2006.
D.B. Davtyan and D.A. Pozharskii, “Action of an elliptic punch on a transversally isotropic half-space”, Mechanics of Solids, 49, no. 5, pp. 578–586, 2014.
H.A. Elliott and N.F. Mott, “Three-dimensional stress distributions in hexagonal aeolotropic crystals”, Mathematical Proceedings of the Cambridge Philosophical Society, 44, no. 4, pp. 522–533, 1948.
V.I. Fabrikant, “Contact problem for an arbitrarily oriented transversely isotropic half-space”, Acta Mechanca, 228, no. 4, pp. 1541–1560, 2017.
L.B. Freund and S. Suresh, Thin Film Materials. Cambridge: Cambridge University Press, 2003, 802 p.
G.M.L. Gladwell, “On Inclusions at a Bi-Material Elastic Interface”, Journal of Elasticity, 54, no. 1, pp. 27–41, 1999.
P.F. Hou, W.H. Zhang, and J.-Y. Chen, “Three-dimensional exact solutions of homogeneous transversely isotropic coated structures under spherical contact”, Int. J. Solids Structures, 161, pp. 136–173, 2019.
S.A. Kaloerov and A.A. Samodurov, “Problem of Electromagnetoviscoelasticity for Multiply Connected Plates”, International Applied Mechanics, 51, no. 6, pp. 623–639, 2015.
S.A. Kaloerov, “Determining the intensity factors for stresses, electric-flux density, and electric-field strength in multiply connected electroelastic anisotropic media”, Int. Appl. Mech., 43, no. 6, pp. 631–637, 2007.
V.S. Kirilyuk and O.I. Levchuk, “On stressed state of transversely isotropic medium with an arbitraly orientated spheroidal void or penny-shaped crack under internal pressure”, Strength of Materials, 37, no. 5, pp. 480–488, 2005.
V.S. Kirilyuk, “Elastic state of a transversely isotropic piezoelectric body with an arbitrarily oriented elliptic crack”, Int. Appl. Mech., 44, no. 2, pp. 150–157, 2008.
V.S. Kirilyuk, “Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion”, Int. Appl. Mech., 44, no. 7, pp. 757–768, 2008.
A. Kotousov, L.B. Neto, and A. Khanna, “On a rigid inclusion pressed between two elastic half spaces”, Mechanics of Materials, 68, no. 1, pp. 38–44, 2014.
R. Kumar and V. Gupta, “Green’s function for transversely isotropic thermoelastic diffusion bimaterials”, Journal of Thermal Stresses, 37, no. 10, pp. 1201–1229, 2014.
F. Marmo, F. Toraldo, and L. Rosati, “Analytical formulas and design charts for transversely isotropic half-spaces subject to linearly distributed pressures”, Meccanica, 51, no. 11, pp. 2909–2928, 2016.
Yu.N. Podil’chuk, “Representation of the general solution of statics equations of the electroelasticity of a transversally isotropic piezoceramic body in terms of harmonic functions”, International Applied Mechanics, 34, no. 7, pp. 623–628, 1998.
A.P.S. Selvadurai, “A unilateral contact problem for a rigid disc inclusion embedded between two dissimilar elastic half-spaces”, Q. J. Mech. Appl. Math., no. 3, pp. 493–509, 1994.
Yu.V. Tokovyy and C.C. Ma, “Three-Dimensional Elastic Analysis of Transversely-Isotropic Composites”, Journal of Mechanics, 33, no. 6, pp. 821–830, 2018.
Y.J. Wang, C.F. Gao, and H.P. Song, “The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material”, Mechanics Research Communications, 65, pp. 17–23, 2015.
Z.K. Wang and B.L. Zheng, “The general solution of three-dimension problems in piezoelectric media”, Int. J. Solids Structures, 32, no. 1, pp. 105–115, 1995.
M.H. Zhao, Y.B. Pan, C.Y. Fan, and G.T. Xu, “Extended displacement discontinuity method for analysis of cracks in 2D piezoelectricsemiconductors”, International Journal of Solids and Structures, vol. 94–95, pp. 50–59, 2016.
G.S. Keith and R.M. Martyniak, “Spatial contact problems for an elastic half-space and a rigid base with surface recesses”, Math. methods and physical and mechanical fields, 42, no. 6, pp. 7–11, 1999.
M.V. Hai, Two-dimensional integral equations of Newtonian potential and their applications. Kiev: Naukova dumka, 1993, 256 p.