Abstract
Based on the theory of complex variable functions, the stress field in an infinite matrix containing an arbitrary shape inclusion with a functionally graded coating is analyzed. The elastic properties in the functionally graded coating change continuously and arbitrarily along the normal direction of the inclusion. By using the method of piecewise homogeneous layers and the technique of conformal mapping, the complex potential functions in the matrix, coating and inclusion are derived in the form of Laurent series and Faber series, respectively. The influences of different varying Young’s modulus on the interfacial stresses are discussed by numerical examples for various shape inclusions, including ellipse, triangle, square and rectangle. It is shown that the magnitude and distribution of interfacial stresses for arbitrary shape inclusions can be successfully designed and controlled by adding a functionally graded coating with proper varying elastic properties along the normal direction. The results for some special cases are compared with previous literature and found in good agreement.
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Yang, Q., Gao, C.F.: Non-axisymmetric thermal stress of a functionally graded coated circular inclusion in an infinite matrix. Mech. Res. Commun. 50, 27–32 (2013)
Yang, Q., Gao, C.F.: Thermal stresses around a circular inclusion with functionally graded interphase in a finite matrix. Sci. China Phys. Mech. Astron. 57(10), 1927–1933 (2014)
Noda, N., Nakai, S., Tsuji, T.: Thermal stresses in functionally graded material of particle-reinforced composite. JSME Int. J. Ser. A 41(2), 178–184 (1998)
Ru, C.Q.: A new method for an inhomogeneity with stepwise graded interphase under thermomechanical loadings. J. Elast. 56(2), 107–127 (1999)
Ding, K., Weng, G.J.: The influence of moduli slope of a linearly graded matrix on the bulk moduli of some particle- and fiber-reinforced composites. J. Elast. 53(1), 1–22 (1999)
Li, J.Y.: Thermoelastic behavior of composites with functionally graded interphase: a multi-inclusion model. Int. J. Solids Struct. 37(39), 5579–5597 (2000)
You, L.H., You, X.Y.: A unified numerical approach for thermal analysis of transversely isotropic fiber-reinforced composites containing inhomogeneous interphase. Compos. Part A 36(6), 728–738 (2005)
Zhang, X.C., Xu, B.S., Wang, H.D., Jiang, Y., Wu, Y.X.: Prediction of three-dimensional residual stresses in the multilayer coating-based systems with cylindrical geometry. Compos. Sci. Technol. 66(13), 2249–2256 (2006)
Hatami-Marbini, H., Shodja, H.M.: Thermoelastic fields of a functionally graded coated inhomogeneity with sliding/perfect interfaces. ASME J. Appl. Mech. 74(3), 389–398 (2007)
Hatami-Marbini, H., Shodja, H.M.: On thermoelastic fields of a multi-phase inhomogeneity system with perfectly/imperfectly bonded interfaces. Int. J. Solids Struct. 45(22–23), 5831–5843 (2008)
Wang, X., Pan, E., Roy, A.K.: A functionally graded plane with a circular inclusion under uniform antiplane eigenstrain. ASME J. Appl. Mech. 75(1), 014501 (2008)
Abbasion, S., Rafsanjani, A., Irani, N., Farshidianfar, A.: Stress analysis for a coated fiber embedded in an infinite matrix subjected to body force. Eur. J. Mech. A Solid 28(4), 777–785 (2009)
Artioli, E., Bisegna, P.: Effective longitudinal shear moduli of periodic fibre-reinforced composites with functionally-graded fibre coatings. Int. J. Solids Struct. 50(7–8), 1154–1163 (2013)
Sabiston, T., Mohammadi, M., Cherkaoui, M., Lévesque, J., Inal, K.: Micromechanics for a long fibre reinforced composite model with a functionally graded interphase. Compos. Part B 84, 188–199 (2016)
Qiu, Y.P., Weng, G.J.: The influence of inclusion shape on the overall elastoplastic behavior of a two-phase isotropic composite. Int. J. Solids Struct. 27(12), 1537–1550 (1991)
Wang, Y.M., Weng, G.J.: The influence of inclusion shape on the overall viscoelastic behavior of composites. ASME J. Appl. Mech. 59(3), 510–518 (1992)
Luo, J.C., Gao, C.F.: Faber series method for plane problems of an arbitrarily shaped inclusion. Acta Mech. 208, 133–145 (2009)
Pan, E.: Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes. J. Mech. Phys. Solids 52, 567–589 (2004)
Sudak, L.J., Wang, X.: An irregular-shaped inclusion with imperfect interface in antiplane elasticity. Acta Mech. 224, 2009–2023 (2013)
Fang, X.Q., Liu, H.W., Liu, J.X., Nie, G.Q.: Interface energy effect on electromechanical response of piezoelectric composites with an arbitrary nano-inclusion under anti-plane shear. Acta Mech. 226, 2323–2333 (2015)
Lee, Y.G., Zou, W.N., Ren, H.H.: Eshelby’s problem of inclusion with arbitrary shape in an isotropic elastic half-plane. Int. J. Solids Struct. 81, 399–410 (2016)
Luo, J.C., Gao, C.F.: Stress field of a coated arbitrary shape inclusion. Meccanica 46, 1055–1071 (2011)
Shen, M.H., Chen, F.M., Hung, S.Y.: Piezoelectric study for a three-phase composite containing arbitrary inclusion. Int. J. Mech. Sci. 52, 561–571 (2010)
Shen, M.H., Hung, S.Y.: Piezoelectric screw dislocation in an arbitrarily shaped three-phase composite. Eur. J. Mech. A Solid 32, 13–20 (2012)
Wang, X., Schiavone, P.: Multi-coating an inclusion of arbitrary shape to achieve uniformity of interior stresses. Math. Mech. Solids 18(2), 218–227 (2012)
Wang, X., Chen, W.Q.: Three-phase inclusions of arbitrary shape with internal uniform hydrostatic thermal stresses. Z. Angew. Math. Phys. 64, 1399–1411 (2013)
Muskhelishvili, N.I.: Some Basic Problem of Mathematical Theory of Elasticity. Noordhoff, Groningen (1975)
Savin, G.N.: Stress Concentration Around Holes. Pergamon Press, London (1961)
England, A.H.: Complex Variable Methods in Elasticity. Wiley-Interscience, London (1971)
Sharma, D.S.: Stresses around hypotrochoidal hole in infinite isotropic plate. Int. J. Mech. Sci. 105, 32–40 (2016)
Goyat, V., Verma, S., Garg, R.K.: Reduction of stress concentration for a rounded rectangular hole by using a functionally graded material layer. Acta Mech. 228, 3695–3707 (2017)
Ashrafi, H., Asemi, K., Shariyat, M.: A three-dimensional boundary element stress and bending analysis of transversely/longitudinally graded plates with circular cutouts under biaxial loading. Eur. J. Mech. A Solid 42, 344–357 (2013)
Curtiss, J.H.: Faber polynomials and the Faber series. Am. Math. Mon. 78(6), 577–596 (1971)
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Yang, Q., Zhu, W., Li, Y. et al. Stress field of a functionally graded coated inclusion of arbitrary shape. Acta Mech 229, 1687–1701 (2018). https://doi.org/10.1007/s00707-017-2052-8
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DOI: https://doi.org/10.1007/s00707-017-2052-8