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Solutions of a singular integro-differential equation related to the adhesive contact problems of elasticity theory

  • Nugzar Shavlakadze EMAIL logo and Otar Jokhadze
Published/Copyright: December 8, 2021
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Abstract

Exact and approximate solutions of a some type singular integro-differential equation related to problems of adhesive interaction between elastic thin half-infinite or finite homogeneous patch and elastic plate are investigated. For the patch loaded with vertical forces, there holds a standard model in which vertical elastic displacements are assumed to be constant. Using the theory of analytic functions, integral transforms and orthogonal polynomials, the singular integro-differential equation is reduced to a different boundary value problem of the theory of analytic functions or to an infinite system of linear algebraic equations. Exact or approximate solutions of such problems and asymptotic estimates of normal contact stresses are obtained.

MSC 2010: 74B05; 45E05; 30E25

References

[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau Stand. Appl. Math. Ser. 55, U.S. Government, Washington, 1964. 10.1115/1.3625776Search in Google Scholar

[2] V. M. Aleksandrov and S. M. Mkhitaryan, Contact Problems for Bodies with Thin Coverings and Layers (in Russian), “Nauka”, Moscow, 1983. Search in Google Scholar

[3] Y. A. Antipov, Effective solution of a Prandtl-type integro-differential equation on an interval and its application to contact problems for a strip (in Russian), Prikl. Mat. Mekh. 57 (1993), no. 3, 146–155; translation in J. Appl. Math. Mech. 57 (1993), no. 3, 547–556. Search in Google Scholar

[4] Y. A. Antipov and N. K. Arutyunyan, A contact problem for an elastic layer with cover plates in the presence of friction and adhesion (in Russian), Prikl. Mat. Mekh. 57 (1993), no. 1, 137–147; translation in J. Appl. Math. Mech. 57 (1993), no. 1, 159–170. Search in Google Scholar

[5] Y. A. Antipov and N. G. Moiseev, Exact solution of the two-dimensional problem for a composite plane with a cut that crosses the interface line (in Russian), Prikl. Mat. Mekh. 55 (1991), no. 4, 662–671; translation in J. Appl. Math. Mech. 55 (1991), no. 4, 531–539. Search in Google Scholar

[6] R. Bantsuri, A certain boundary value problem in analytic function theory (in Russian), Soobshch. Akad. Nauk Gruzin. SSR 73 (1974), 549–552. Search in Google Scholar

[7] R. Bantsuri, The contact problem for an anisotropic wedge with an elastic stringer (in Russian), Dokl. Akad. Nauk SSSR 222 (1975), no. 3, 568–571. Search in Google Scholar

[8] R. Bantsuri and N. N. Shavlakadze, A contact problem for an anisotropic wedge-shaped plate with elastic bracing of variable rigidity (in Russian), Prikl. Mat. Mekh. 66 (2002), no. 4, 663–669; translation in J. Appl. Math. Mech. 66 (2002), no. 4, 645–650. Search in Google Scholar

[9] F. D. Gahov and J. I. Čerskiĭ, Equations of Convolution Type (in Russian), “Nauka”, Moscow, 1978. Search in Google Scholar

[10] O. Jokhadze, S. Kharibegashvili and N. Shavlakadze, Approximate and exact solution of a singular integro-differential equation related to contact problem of elasticity theory, Prikl. Mat. i Mekh. 82 (2018), no. 1, 114–124; translation in J. Appl. Math. Mech. 82 (2018), no. 1, 114–124. Search in Google Scholar

[11] L. Kantorovich and G. Akilov, Functional Analysis (in Russian), Izdat. “Nauka”, Moscow, 1977. Search in Google Scholar

[12] L. Kantorovich and V. Krilov, Approximate Methods of Higher Analysis (in Russian), Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1962. Search in Google Scholar

[13] J. L. Lubkin and L. C. Lewis, Adhesive shear flow for an axially-loaded, finite stringer bonded to an infinite sheet, Quart. J. Mech. Appl. Math. 23 (1970), no. 4, 521–533. 10.1093/qjmam/23.4.521Search in Google Scholar

[14] N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Fundamental Equations, Plane Theory of Elasticity, Torsion and Bending, Noordhoff, Groningen, 1953. Search in Google Scholar

[15] N. I. Muskhelishvili, Singular Integral Equations. Boundary Problems of Function Theory and Their Application to Mathematical Physics, Wolters-Noordhoff, Groningen, 1972. Search in Google Scholar

[16] B. M. Nuller, On the deformation of an elastic wedge plate reinforced by a variable stiffness bar and a method of solving mixed problems (in Russian), Prikl. Mat. Mekh. 40 (1976), no. 2, 306–316; translation in J. Appl. Math. Mech. 40 (1976), no. 2, 280–291. 10.1016/0021-8928(76)90065-4Search in Google Scholar

[17] G. Y. Popov, Concentration of Elastic Stresses Near Punches, Cuts, Thin Inclusion and Supports (in Russian), “Nauka”, Moscow, 1983. Search in Google Scholar

[18] N. Shavlakadze, On singularities of contact stress upon tension and bending of plates with elastic inclusions, Proc. A. Razmadze Math. Inst. 120 (1999), 135–147. Search in Google Scholar

[19] N. Shavlakadze, The contact problem of bending of plate with thin fastener (in Russian), Izv. Ross. Akad. Nauk, Mekh. Tv. Tela. 2001 (2001), no. 3, 144–155; translation in Mech. Solids 36 (2001), no. 3, 122–127. Search in Google Scholar

[20] N. Shavlakadze, The contact problems of the mathematical theory of elasticity for plates with an elastic inclusion, Acta Appl. Math. 99 (2007), no. 1, 29–51. 10.1007/978-1-4020-8774-5_18Search in Google Scholar

[21] N. Shavlakadze, The solution of system of integral differential equations and its application in the theory of elasticity, ZAMM Z. Angew. Math. Mech. 91 (2011), no. 12, 979–992. 10.1002/zamm.201000220Search in Google Scholar

[22] N. Shavlakadze, Contact problem of electroelasticity for a piecewise homogeneous piezoelectric plate with an elastic coating (in Russian), Prikl. Mat. i Mekh. 81 (2017), no. 3, 337–347; translation in J. Appl. Math. Mech. 81 (2017), no. 3, 228–235. 10.1016/j.jappmathmech.2017.08.015Search in Google Scholar

[23] N. Shavlakadze, N. Odishelidze and F. Criado-Aldeanueva, The contact problem for a piecewise-homogeneous orthotropic plate with a finite inclusion of variable cross-section, Math. Mech. Solids 22 (2017), no. 6, 1326–1333. 10.1177/1081286516631160Search in Google Scholar

[24] T. C. T. Ting, Uniform stress inside an anisotropic elliptic inclusion with imperfect interface bonding, J. Elasticity 96 (2009), no. 1, 43–55. 10.1007/s10659-009-9197-1Search in Google Scholar

[25] T. C. T. Ting and P. Schiavone, Uniform antiplane shear stress inside an anisotropic elastic inclusion of arbitrary shape with perfect or imperfect interface bonding, Internat. J. Engrg. Sci. 48 (2010), no. 1, 67–77. 10.1016/j.ijengsci.2009.06.008Search in Google Scholar

[26] E. Tsuchida, T. Mura and J. Dundurs, The elastic field of an elliptic inclusion with a slipping interface, Trans. ASME J. Appl. Mech. 53 (1986), no. 1, 103–107. 10.1115/1.3171693Search in Google Scholar

Received: 2020-03-03
Revised: 2020-06-26
Accepted: 2020-10-15
Published Online: 2021-12-08
Published in Print: 2022-04-01

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