Abstract
In the following analysis, we present a rigorous solution for the problem of a circular elastic inclusion surrounded by an infinite elastic matrix in finite plane elastostatics. The inclusion and matrix are separated by a circumferentially inhomogeneous imperfect interface characterized by the linear spring-type imperfect interface model where the interface is such that the same degree of imperfection is realized in both the normal and tangential directions. Through the use of analytic continuation, a set of first-order coupled ordinary differential equations with variable coefficients are developed for two analytic potential functions. The unknown coefficients of the potential functions are determined from their analyticity requirements and some additional problem-specific constraints. An example is then presented for a specific class of interface where the inclusion mean stress is contrasted between the homogeneous interface and inhomogeneous interface models. It is shown that, for circumstances where a homogeneously imperfect interface may not be warranted, the inhomogeneous model has a pronounced effect on the mean stress within the inclusion.
Similar content being viewed by others
References
Chen F.C., Young K.: Inclusions of arbitrary shape in an elastic medium. J. Math. Phys. 18, 1412–1416 (1977)
Constanda C.: A Mathematical Analysis of Bending of Plates with Transverse Shear Deformation. Longman Scientific and Technical, Harlow (1990)
Eshelby J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241, 376–396 (1957)
Gao J.: A circular inclusion with imperfect interface: Eshelby’s tensor and related problems. J. Appl. Mech. 62, 860–866 (1995)
Hashin Z.: The spherical inclusion with imperfect interface. J. Appl. Mech. 58, 444–449 (1991)
John F.: Plane strain problems for a perfectly elastic material of harmonic type. Commun. Pure Appl. Math. XIII, 239–290 (1960)
Kim K., Sudak L.J.: Interaction between a radial matrix crack and a three-phase circular inclusion with imperfect interface in plane elasticity. Int. J. Fract. 131, 155–172 (2005)
Knowles J.K., Sternberg E.: On the singularity induced by certain mixed boundary conditions in linearized and nonlinear elastostatics. Int. J. Solids Struct. 11, 1173–1201 (1975)
Li X., Steigmann D.J.: Finite plane twist of an annular membrane. Q. J. Mech. Appl. Math. 46, 601–625 (1993)
Muskhelishvili N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1963)
Ogden R.W., Isherwood D.A.: Solution of some finite plane-strain problems for compressible elastic solids. Q. J. Mech. Appl. Math. XXXI, 219–249 (1978)
Ru C.Q., Schiavone P.: A circular inclusion with circumferentially inhomogeneous interface in antiplane shear. Proc. R. Soc. Lond. A 453, 2551–2572 (1997)
Ru C.Q.: A circular inclusion with circumferentially inhomogeneous sliding interface in plane elastostatics. ASME J. Appl. Mech. 65, 30–38 (1998)
Ru C.Q.: On complex-variable formulation for finite plane elastostatics of harmonic materials. Acta Mech. 156, 219–234 (2002)
Ru C.Q., Schiavone P., Sudak L.J., Mioduchowski A.: Uniformity of stresses inside an elliptical inclusion in finite elastostatics. Int. J. Nonlinear Mech. 40, 281–287 (2005)
Shen H., Schiavone P., Ru C.Q., Mioduchowski A.: Stress analysis of an elliptic inclusion with imperfect interface in plane elasticity. J. Elast. 62, 25–46 (2001)
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)
Varley E., Cumberbatch E.: Finite deformation of elastic materials surrounding cylindrical holes. J. Elast. 10, 341–405 (1980)
Wang X.: Three-phase elliptical inclusions with internal uniform hydrostatic stresses in finite plane elastostatics. Acta Mech. 219, 77–90 (2011)
Wang X.: A circular inclusion with imperfect interface in finite plane elastostatics. Acta Mech. 223, 481–491 (2012)
Wang X., Pan E.: On partially debonded circular inclusions in finite plane elastostatics of harmonic materials. ASME J. Appl. Mech. 76, 011012-1–011012-5 (2009)
Wang X., Schiavone P.: Three-phase inclusions of arbitrary shape with internal uniform hydrostatic stresses in finite elasticity. J. Appl. Mech. 79, 041012–041018 (2012)
Wang, X., Schiavone, P.: Harmonic three-phase circular inclusions in finite elasticity. Cont. Mech. Therm. (2014). doi:10.1007/s00161-014-0349-6
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.
Rights and permissions
About this article
Cite this article
McArthur, D.R., Sudak, L.J. A circular inclusion with circumferentially inhomogeneous imperfect interface in harmonic materials. Continuum Mech. Thermodyn. 28, 317–329 (2016). https://doi.org/10.1007/s00161-015-0430-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-015-0430-9