Abstract
Despite extensive studies of inclusions with simple shape, little effort has been devoted to inclusions of irregular shape. In this study, we consider an inclusion of irregular shape embedded within an infinite isotropic elastic matrix subject to antiplane shear deformations. The inclusion–matrix interface is assumed to be imperfect characterized by a single, non-negative, and constant interface parameter. Using complex variable techniques, the analytic function that is defined within the irregular-shaped inclusion is expanded into a Faber series, and in conjunction with the Fourier series, a set of linear algebraic equations for a finite number of unknown coefficients is determined. With this approach and without imposing any constraints on the stress distribution, a semi-analytical solution is derived for the elastic fields within the irregular-shaped inclusion and the surrounding matrix. The method is illustrated using three examples and verified, when possible, with existing solutions. The results from the calculations reveal that the stress distribution within the inclusion is highly non-uniform and depends on the inclusion shape and the weak mechanical contact at the inclusion/matrix boundary. In fact, the results illustrate that the imperfect interface parameter significantly influences the stress distribution.
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Antipov Y.A., Schiavone P.: On the uniformity of stresses inside an inhomogeneity of arbitrary shape. IMA J. Appl. Math. 68, 299–311 (2003)
Berezhnyts’kyi L.T., Kachur P.S., Mazurak L.P.: A method for the determination of the elastic equilibrium of isotropic bodies with curvilinear inclusions, part 3. Antiplane deformation. Mater. Sci. 35, 166–172 (1999)
Curtiss J.H.: Faber polynomials and the Faber series. Am. Math. Mon. 78, 577–596 (1971)
England A.H.: Complex Variable Methods in Elasticity. Wiley, London (1971)
Eshelby J.D.: The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A21, 376–396 (1957)
Eshelby J.D.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Lond. A 252, 561–569 (1959)
Gao J.: A circular inhomogeneity with imperfect interface: Eshelby’s tensor and related problems. J. Appl. Mech. 62, 860–866 (1995)
Gao C.F., Noda N.: Faber series method for two-dimensional problems of an arbitrarily shaped inclusion in piezoelectric materials. Acta Mech. 171, 1–13 (2004)
Gong S.X.: A unified treatment of the elastic elliptical inclusion under antiplane shear. Arch. Appl. Mech. 65, 55–64 (1995)
Gong S.X., Meguid S.A.: A general treatment of the elastic field of an elliptical inhomogeneity under antiplane shear. J. Appl. Mech. 59, S131–S135 (1992)
Hashin Z.: The spherical inhomogeneity with imperfect interface. J. Appl. Mech. 58, 444–449 (1991)
Luo J.C., Gao C.F.: Faber series method for plane problems of an arbitrarily shaped inclusion. Acta Mech. 208, 133–145 (2009)
Luo J.C., Gao C.F.: Stress field of a coated arbitrary shape inclusion. Meccanica 46, 1055–1071 (2011)
Mazurak L.P, Berezhnyts’kyi L.T., Kachur P.S.: Method for determination of elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 2. Plane problem. Fiz-Khim. Mekh. Mater. 34, 16–26 (1999)
Ru C.Q.: Analytic solution for Eshelby’s problem of an inclusion of arbitrary shape in a plane or half-plane. J. Appl. Mech. 66, 315–322 (1999)
Ru C.Q., Schiavone P.: On the elliptical inclusion in antiplane shear. Math. Mech. Solids 1, 327–333 (1996)
Shen H., Schiavone P., Ru C.Q., Mioduchowski A.: An elliptic inclusion with imperfect interface in antiplane shear. IJSS 37, 4557–4575 (2000)
Sudak L.J.: On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear. Mech. Res. Commun. 30, 53–59 (2003)
Sudak L.J., Ru C.Q., Schiavone P., Mioduchowski A.: A circular inclusion with inhomogeneously imperfect interface in plane elasticity. J. Elast. 55, 19–41 (1999)
Tsukrov L., Novak J.: Effective elastic properties of solids with two dimensional inclusions of irregular shape. Int. J. Solids Struct. 41, 6905–6924 (2004)
Vasudevan M., Schiavone P.: New results concerning the identification of neutral inhomogeneities in plane elasticity. Arch. Mech. 58, 45–58 (2006)
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Sudak, L.J. An irregular-shaped inclusion with imperfect interface in antiplane elasticity. Acta Mech 224, 2009–2023 (2013). https://doi.org/10.1007/s00707-013-0842-1
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DOI: https://doi.org/10.1007/s00707-013-0842-1