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Two circular inclusions with arbitrarily varied surface effects

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Abstract

We investigate the problem associated with two interacting circular inclusions in a composite subjected to anti-plane shear deformation. A version of the Gurtin–Murdoch model of surface/interface elasticity is incorporated into the description of the two inclusion–matrix interfaces. In contrast to previous investigations, the surface shear moduli are varied arbitrarily along the two circular interfaces. The composite is subjected to various loading mechanisms that include uniform stresses applied at infinity, uniform eigenstrains imposed on the two circular inclusions and a screw dislocation located either in the matrix or inside one of the two inclusions. Complex variable methods are used in conjunction with Fourier series expansions and matrix algebra to derive an analytical solution to the corresponding problem. The unknown coefficients in the ensuing three analytic functions are determined in an elegant manner. The size-dependent material forces acting on the two inclusions and the screw dislocation are also obtained.

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References

  1. Gurtin, M.E., Murdoch, A.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  2. Gurtin, M.E., Weissmuller, J., Larche, F.: A general theory of curved deformable interface in solids at equilibrium. Philos. Mag. A. 78, 1093–1109 (1998)

    Article  Google Scholar 

  3. Ru, C.Q.: Simple geometrical explanation of Gurtin–Murdoch model of surface elasticity with clarification of its related versions. Sci. China 53, 536–544 (2010)

    MathSciNet  Google Scholar 

  4. Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. Lond. A 453, 853–877 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, T., Dvorak, G.J., Yu, C.C.: Size-dependent elastic properties of unidirectional nano-composites with interface stresses. Acta Mech. 188, 39–54 (2007)

    Article  MATH  Google Scholar 

  6. Antipov, Y.A., Schiavone, P.: Integro-differential equation for a finite crack in a strip with surface effects. Q. J. Mech. Appl. Math. 64, 87–106 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Sigaeva, T., Schiavone, P.: Solvability of a theory of anti-plane shear with partially coated boundaries. Arch. Mech. 66, 113–125 (2014)

    MATH  MathSciNet  Google Scholar 

  8. Sigaeva, T., Schiavone, P.: Solvability of the Laplace equation in a solid with boundary reinforcement. Z. Angew. Math. Phys. 65, 809–815 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  9. Benveniste, Y.: Models of thin interphases with variable moduli in plane-strain elasticity. Math. Mech. Solids 18, 119–134 (2012)

    Article  MathSciNet  Google Scholar 

  10. Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535–537 (2003)

    Article  Google Scholar 

  11. Sharma, P., Ganti, S.: Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies. ASME J. Appl. Mech. 71, 663–671 (2004)

    Article  MATH  Google Scholar 

  12. Tian, L., Rajapakse, R.K.N.D.: Analytical solution for size-dependent elastic field of a nanoscale circular inhomogeneity. ASME J. Appl. Mech. 74, 568–574 (2007)

    Article  MATH  Google Scholar 

  13. Tian, L., Rajapakse, R.K.N.D.: Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity. Int. J. Solids Struct. 44, 7988–8005 (2007)

    Article  MATH  Google Scholar 

  14. Luo, J., Xiao, Z.M.: Analysis of a screw dislocation interacting with an elliptical nano inhomogeneity. Int. J. Eng. Sci. 47, 883–893 (2009)

    Article  MathSciNet  Google Scholar 

  15. Kim, C.I., Schiavone, P., Ru, C.Q.: The effects of surface elasticity on an elastic solid with mode-III crack: complete solution. ASME J. Appl. Mech. 77, 021011-1–021011-7 (2010)

    Google Scholar 

  16. Kim, C.I., Schiavone, P., Ru, C.Q.: The effects of surface elasticity on mode-III interface crack. Arch. Mech. 63, 267–286 (2011)

    MATH  MathSciNet  Google Scholar 

  17. Kim, C.I., Schiavone, P., Ru, C.Q.: Effect of surface elasticity on an interface crack in plane deformations. Proc. R. Soc. Lond. A 467, 3530–3549 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ru, C.Q., Schiavone, P.: A circular inclusion with circumferentially inhomogeneous interface in antiplane shear. Proc. R. Soc. Lond. A 453, 2551–2572 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Chen, T.: Thermal conduction of a circular inclusion with variable interface parameter. Int. J. Solids Struct. 38, 3081–3097 (2001)

    Article  MATH  Google Scholar 

  20. Wang, X., Shen, Y.P.: Two circular inclusions with circumferentially inhomogeneous interfaces interacting with a circular Eshelby inclusion in anti-plane shear. Acta Mech. 158(1–2), 67–84 (2002)

    Article  MATH  Google Scholar 

  21. Ting, T.C.T.: Anisotropic Elasticity—Theory and Applications. Oxford University Press, New York (1996)

    MATH  Google Scholar 

  22. Wang, X.: Two circular inclusions with rate-dependent interfacial sliding under longitudinal shear. Mech. Adv. Mater. Struct. 20, 603–623 (2013)

    Article  Google Scholar 

  23. Rice, J.R.: A path independent integral and the approximate analysis of strain concentration by notches and cracks. ASME J. Appl. Mech. 35, 379–386 (1968)

    Article  Google Scholar 

  24. Budiansky, B., Rice, J.R.: Conservation laws and energy-release rates. ASME J. Appl. Mech. 40, 201–203 (1973)

    Article  MATH  Google Scholar 

  25. Honein, E., Honein, T., Hermann, G.: Energies of two circular inclusions in anti-plane elastostatics. Int. J. Solids Struct. 37, 3667–3679 (2000)

    Article  MATH  Google Scholar 

  26. Knowles, J.K., Sternberg, E.: On a class of conservation laws in linearized and finite elastostatics. Arch. Rat. Mech. Anal. 44, 187–211 (1972)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China

    Xu Wang

  2. Department of Mechanical Engineering, University of Alberta, 4-9 Mechanical Engineering Building, Edmonton, AB, T6G 2G8, Canada

    Peter Schiavone

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  1. Xu Wang
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  2. Peter Schiavone
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Correspondence to Peter Schiavone.

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Wang, X., Schiavone, P. Two circular inclusions with arbitrarily varied surface effects. Acta Mech 226, 1471–1486 (2015). https://doi.org/10.1007/s00707-014-1264-4

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  • DOI: https://doi.org/10.1007/s00707-014-1264-4

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