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The axisymmetric contact interaction of an infinite elastic plate with an absolutely rigid inclusion. (English) Zbl 1317.74072

Summary: In the proposed paper, the analytical solution of the problem on an axisymmetric stress-strength state of an infinite elastic layer (a plate) with an absolutely rigid inclusion, coupled with this plate, is solved. The upper plate plane side is under the axisymmetric compressive load. The bottom side of the plate could be in different conditions with the absolutely rigid base: it can be the conditions of a smooth contact or the conditions of a full adhesion. The integral Weber-type transformation is applied to the axisymmetric Lamé equations for the displacements and stress field construction. It leads to a one-dimensional vector inhomogeneous boundary problem. With the help of this problem solution, after satisfying a boundary condition, the initial problem is reduced by solving an integral singular equation on the finite interval. The equation is solved approximately by the orthogonal polynomial method with the previous extraction of the solution’s singularities on the interval ends.

MSC:

74M15 Contact in solid mechanics
74K20 Plates
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
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[1] Kotousov A., Harding S., Codrington J.: Deformation and fracture of plates of finite thickness. J. Phys. Conf. Ser. 181, 012059 (2009) · doi:10.1088/1742-6596/181/1/012059
[2] Youngdahl C.K., Stenberg E.: Three-dimensional stress concentration around a cylindrical hole in a semi-infinite elastic body. Trans. ASME J. Appl. Mech. 33, 855-865 (1966) · Zbl 0151.36504 · doi:10.1115/1.3625193
[3] Malitc, P., Privarnikov, A.: The applying of Weber type transformations to the solving of the elasticity problems for the layered mediums with an cylindrical hole. (in Russian) Voprosy prochnosti i plastichnosti, Dnepropetrovsk (1971) · Zbl 1196.74064
[4] Arutynyan N., Abramyan B.: Some axisymmetrical problems for the halfspace and elastic layer with a vertical cylindrical hole. Izvestiya Acad. Nauk Arm. SSR. 22, 3-13 (1969) (in Russian)
[5] Grinchenko V., Ulitko A.: The exact solution of problem on the stresses’ distribution near the circular hole in the elastic layer. Prikl. Mech. 4, 38-45 (1968)
[6] Koyalovich B.: The investigation of the infinite systems of the linear algebraic equations. Izv. fiz.-mat. in-ta Steklova. 3, 41-67 (1930)
[7] Folias E., Wang J.: On the three-dimensional stress field around a circular hole in a plate of arbitrary thickness. Comput. Mech. 6, 379-391 (1990) · doi:10.1007/BF00350419
[8] Babeshko V., Babeshko O., Evdokimova O.: Certain general properties of block elements. Doklady Phys. 57, 14-17 (2012) · Zbl 1238.82047 · doi:10.1134/S1028335812010016
[9] Vorovich, I., Alexandrov, V., Babeshko, V.: The non classic mixed problems of elasticity. Nauka (1974) (in Russian) · Zbl 1233.65025
[10] Mykhas’kiv, V., Khay, O., Zhang, Ch., Bostrom, A.: Effective dynamic properties of 3D composite materials containing rigid penny-shaped inclusions. Waves Random Complex Media 20, 491-510 (2010) · Zbl 1267.74063
[11] Ostrik V., Fil’shtinsky L.: The dynamical problems of magnetooelasticity for a layer and semilayer with the tunnel holes and defects of longitudinal share. Izvestiya Acad. Nauk Arm. SSR. 44, 34-45 (1991) (in Russian)
[12] Williams M.: Stress singularities resulting from various boundary conditions in angular corners of strips in extension. J. Appl. Mech. 74, 526-528 (1952)
[13] Kotousov A.: On stress singularities at angular corners of plates of arbitrary thickness under tension. Int. J. Fract. 132, L29-L36 (2005) · Zbl 1196.74064 · doi:10.1007/s10704-005-4481-y
[14] Uflyand, Ya.: The integral transformations in elasticity problems. Nauka (1967) (in Russian)
[15] Yahnioglu, N., Babuscu Yesil, U.: Forced vibration of an initial stressed rectangular composite thick plate with a cylindrical hole. In: ASME 2009 international Mechanical engineering congress and exposition IMECE09, November 13-19, 2009, Lake Buena Vista, Florida, USA (2009) · Zbl 1164.74354
[16] Jain N., Mittal N.: Finite element analysis for stress concentration and deflection in isotropic, orthtropic and laminated composite plates with central circular hole under transverse static loading. Mater. Sci. Eng. A 498, 115-124 (2008) · doi:10.1016/j.msea.2008.04.078
[17] Zheng Y., Chang-Boo K., Chongdu C., Hyeon Gyu B.: The concentration of stress and strain in finite thickness elastic plate containing a circular hole. Int. J. Solids Struct. 45, 713-731 (2008) · Zbl 1167.74435 · doi:10.1016/j.ijsolstr.2008.02.020
[18] Popov G.: On new transformations of the elasticity resolving equations and new integral transformations and their application to the boundary problems of mechanics. Prikl. Mech. 12, 46-73 (2003) · Zbl 1130.74324
[19] Vaysfel’d N., Popov G., Reut V.: The axisymmetrical mixed problem of elasticity theory for a cone clamped along its side surface with attached spherical segment. J. Appl. Math. Mech. 77, 70-78 (2013) · Zbl 1282.74012 · doi:10.1016/j.jappmathmech.2013.04.009
[20] Popov, G.: The concentration of elastic stresses near stamps, cuts, thin inclusion and reinforcements. Nauka (1982) (in Russian)
[21] Shuhuang X.: Numerical Quadrature for Bessel transformations with high oscillations. Numer. Anal. Appl. Lect. Notes Comput. Sci. 5435, 588-595 (2009) · Zbl 1233.65025
[22] Filon, L.: On a quadrature formula for trigonometric integrals. Proc. R. Soc. Edin. XLIX, 38-47 (1928-1929) · JFM 55.0946.02
[23] Popov, G., Abdymanapov, S., Efimov, V.: The Green’s function and matrices. Ruan (1999) (in Russian)
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