# Modeling the contact interaction of two transversally isotropic elastic half-spaces in the presence of a hard disk-like inclusion between them and pressure in the region of separation

## DOI:

https://doi.org/10.20535/SRIT.2308-8893.2020.1.10## Keywords:

mathematical modeling, transversely isotropic material, elastic half-space, hard disk-like inclusion, pressure in the region of delamination, contact interaction parameters## Abstract

A rigorous mathematical model is used for analyzing the contact interaction of two transversally isotropic elastic half-spaces with different properties under compression in the presence of a hard disk-like inclusion between them and pressure in the region of material separation. The solution to the problem is obtained by presenting a general solution of the system of equilibrium equations for a transversely isotropic body through harmonic functions, reducing the boundary-value problem to considering the integral equation, expanding the desired function in a small parameter. As a special case, the found contact results for the main contact parameters for two elastic isotropic half-spaces (in the presence of inclusion between them and pressure in the region of separation). Numerical results are obtained, the influence of the elastic properties of half-spaces, the geometric dimensions of the inclusion and loads on the parameters of contact interaction are studied.## References

Kirilyuk V.S. Modeling of contact interaction of piezoelectric half-space and elastic isotropic base with surface groove of circular section / V.S. Kirilyuk, O.I Levchuk // System research and information technologies. — 2016. — № 4. — P. 120–132.

Kirilyuk V.S. Mathematical modeling of contact interaction of two electroelastic half-spaces under compression with rigid disc-shaped inclusion between them / V.S. Kirilyuk, O.I. Levchuk, O.V. Gavrilenko // System research and information technologies. — 2018. — № 2. — P. 89–98. DOI: 10.20535/SRIT.2308-8893.2018.2.09

Podil’chuk Yu.N. Boundary value problems of statics of elastic bodies [in Russian] / Yu.N. Podil’chuk. — K.: Nauk. dumka, 1984. — 304 p.

Podil’chuk Yu.N. Exact analytic solutions of three-dimensional boundary-value problems of the statics of a transversely isotropic body of canonical form (Survey) / Yu.N. Podil’chuk // International Applied Mechanics. — 1997. — 33, № 10. — P. 763–787.

Borodich F.M. The JKR-type adhesive contact problems for transversely isotropic elastic solids / F.M. Borodich , B.A. Galanov , L.M. Keer , M.M. Suarez-Alvarez // Mechanics of Materials. — 2014. — 75. — P. 34–44.

Chai Y.S. Local tangential contact of elastically similar, transversely isotropic elastic bodies / Y.S. Chai, I.I. Argatov // Meccanica. — 2018. — 53, N 11–12. — P. 3137–3143.

Chen W.Q. 3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium / W.Q. Chen, C.W. Lim // Int. J. Fract. — 2005. — 131, N 3. — P. 231–246.

Chen W.Q. General solutions for elasticity of transversely isotropic materials with thermal and other effects: A review / W.Q. Chen, J. Zhu, X.Y. Li // Int. J. Mech. Sciencis. — 2019. — 151. — P. 471–497.

Dai L. Stress concentration at an elliptic hole in transversely isotropic piezoelectric solids / L. Dai, W. Guo, X. Wang // Int. J. Solids and Struct. — 2006. — 43, N 6. — P. 1818–1831.

Davtyan D.B. Action of an elliptic punch on a transversally isotropic half-space/ D.B. Davtyan , D.A. Pozharskii // Mechanics of Solids. — 2014. — 49, N 5. — P. 578–586.

Elliott H.A. Three-dimensional stress distributions in hexagonal aeolotropic crystals / H.A. Elliott, N.F. Mott // Mathematical Proceedings of the Cambridge Philosophical Society. — 1948. — 44, N 4. — P.522–533.

Fabrikant V.I. Contact problem for an arbitrarily oriented transversely isotropic half-space/ V.I. Fabrikant // Acta Mechanca. — 2017. — 228, N 4. — P. 1541–1560.

Freund L.B. Thin Film Materials / L.B. Freund, S. Suresh. — Cambridge: Cambridge University Press, 2003. — 802 p.

Gladwell G.M.L. On Inclusions at a Bi-Material Elastic Interface / G.M.L. Gladwell // Journal of Elasticity. — 1999. — 54, N 1. — P.27–41.

Hou P.F. Three-dimensional exact solutions of homogeneous transversely isotropic coated structures under spherical contact / P.F. Hou, W.H. Zhang, J.-Y. Chen // Int. J. Solids Structures. — 2019. — 161. — P. 136–173.

Kaloerov S.A. Problem of Electromagnet viscoelasticity for Multiply Connected Plates / S.A. Kaloerov, A.A. Samodurov // International Applied Mechanics. — 2015. — 51, N 6. — P. 623–639.

Kaloerov S.A. Determining the intensity factors for stresses, electric-flux density, and electric-field strength in multiply connected electroelastic anisotropic media / S.A. Kaloerov // Int. Appl. Mech. — 2007. — 43, N 6. — P. 631–637.

Kirilyuk V.S. On stressed state of transversely isotropic medium with an arbitraly orientated spheroidal void or penny-shaped crack under internal pressure / V.S. Kirilyuk, O.I. Levchuk // Strength of Materials. — 2005. — 37, N 5. — P. 480–488.

Kirilyuk V.S. Elastic state of a transversely isotropic piezoelectric body with an arbitrarily oriented elliptic crack / V.S. Kirilyuk // Int. Appl. Mech. — 2008. — 44, N 2. — P. 150–157.

Kirilyuk V.S. Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion / V.S. Kirilyuk // Int. Appl. Mech. — 2008. — 44, N 7. — P. 757–768.

Kotousov A. On a rigid inclusion pressed between two elastic half spaces / A. Kotousov, L.B. Neto, A. Khanna // Mechanics of Materials. — 2014. — 68, N 1. — P. 38–44.

Kumar R. Green's function for transversely isotropic thermoelastic diffusion bimaterials / R. Kumar , V. Gupta // Journal of Thermal Stresses. — 2014. — 37, N 10. — P. 1201–1229.

Marmo F. Analytical formulas and design charts for transversely isotropic half-spaces subject to linearly distributed pressures / F. Marmo, F. Toraldo, L. Rosati // Meccanica. — 2016. — 51, N 11. — P. 2909–2928.

Podil’chuk Yu.N. Representation of the general solution of statics equations of the electroelasticity of a transversally isotropic piezoceramic body in terms of harmonic functions / Yu.N. Podil’chuk // International Applied Mechanics. — 1998. — 34, N 7. — P. 623–628.

Selvadurai A.P.S. A unilateral contact problem for a rigid disc inclusion embedded between two dissimilar elastic half-spaces / A.P.S. Selvadurai // Q. J. Mech. Appl. Math. — 1994. — N 3. — P. 493–509.

Tokovyy Yu.V. Three-Dimensional Elastic Analysis of Transversely-Isotropic Composites / Yu.V. Tokovyy, C.C. Ma // Journal of Mechanics. — 2018. — 33, N 6. — P. 821–830.

Wang Y.J. The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material / Y.J. Wang, C.F. Gao, H.P. Song // Mechanics Research Communications. — 2015. — Vol. 65. — P. 17–23.

Wang Z.K. The general solution of three-dimension problems in piezoelectric media / Z.K. Wang, B.L. Zheng // Int. J. Solids Structures. — 1995. — 32, N 1. — P. 105–115.

Yu H.Y. A concise treatment of indentation problems in transversely isotropic half-spaces / H.Y. Yu // Int. J. Solids Struct. — 2001. — 38, N 7. — P. 2213–2232.

Zhao M. Three-dimensional steady-state general solution for transversely isotropic hygrothermoelastic media / M. Zhao, H. Dang, C. Fan, Z. Chen // Journal of Thermal Stresses. — 2018. — 41, N 8. — P. 951–972.

Zhao M.H. Singularity analysis of planar cracks in three-dimensional piezoelectric semiconductors via extended displacement discontinuity boundary integral equation method / M.H. Zhao, Y. Li, Y. Yan, C.Y. Fan // Engineering Analysis with Boundary Elements. — 2016. — Vol. 67. — P. 115–125.

Zhao M.H. Extended displacement discontinuity method for analysis of cracks in 2D piezoelectric semiconductors / M.H. Zhao, Y.B. Pan, C.Y. Fan, G.T. Xu // International Journal of Solids and Structures. — 2016. — Vol. 94–95. — P. 50–59.