Abstract
The solution for the Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The Green’s function based on the SSGET for an infinite three-dimensional elastic body undergoing anti-plane strain deformations is first obtained by employing Fourier transforms. The Eshelby tensor is then analytically derived in a general form for an anti-plane strain inclusion of arbitrary cross-sectional shape using the Green’s function method. By applying this general form, the Eshelby tensor for a circular cylindrical inclusion is obtained explicitly, which is separated into a classical part and a gradient part. The former does not contain any classical elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle size effect. The components of the new Eshelby tensor vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. For homogenization applications, the average of this Eshelby tensor over the circular cross-sectional area of the inclusion is obtained in a closed form. Numerical results reveal that when the inclusion radius is small, the contribution of the gradient part is significantly large and should not be ignored. Also, it is found that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. These components approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.
Similar content being viewed by others
References
Arfken G.B., Weber H.-J.: Mathematical Methods for Physicists, 6th edn. Elsevier, San Diego (2005)
Cheng Z.-Q., He L.-H.: Micropolar elastic fields due to a spherical inclusion. Int. J. Eng. Sci. 33, 389–397 (1995)
Cheng Z.-Q., He L.-H.: Micropolar elastic fields due to a circular cylindrical inclusion. Int. J. Eng. Sci. 35, 659–668 (1997)
Cho J., Joshi M.S., Sun C.T.: Effect of inclusion size on mechanical properties of polymeric composites with micro and nano particles. Compos. Sci. Tech. 66, 1941–1952 (2006)
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)
Eshelby J.D.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Lond. A 252, 561–569 (1959)
Gao X.-L.: A mathematical analysis of the elasto-plastic anti-plane shear problem of a power-law material and one class of closed-form solutions. Int. J. Solids Struct. 33, 2213–2223 (1996)
Gao X.-L., Li K.: A shear-lag model for carbon nanotube-reinforced polymer composites. Int. J. Solids Struct. 42, 1649–1667 (2005)
Gao X.-L., Ma H.M.: Green’s function and Eshelby’s tensor based on a simplified strain gradient elasticity theory. Acta Mech. 207, 163–181 (2009)
Gao X.-L., Ma H.M.: Solution of Eshelby’s inclusion problem with a bounded domain and Eshelby’s tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory. J. Mech. Phys. Solids 58, 779–797 (2010)
Gao X.-L., Ma H.M.: Strain gradient solution for Eshelby’s ellipsoidal inclusion problem. Proc. R. Soc. A 466, 2425–2446 (2010)
Gao X.-L., Park S.K.: Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int. J. Solids Struct. 44, 7486–7499 (2007)
Gao X.-L., Rowlands R.E.: Hybrid method for stress analysis of finite three-dimensional elastic components. Int. J. Solids Struct. 37, 2727–2751 (2000)
Haftbaradaran H., Shodja H.M.: Elliptic inhomogeneities and inclusions in anti-plane couple stress elasticity with application to nano-composites. Int. J. Solids Struct. 46, 2978–2987 (2009)
Kiris A., Inan E.: Eshelby tensors for a spherical inclusion in microstretch elastic fields. Int. J. Solids Struct. 43, 4720–4738 (2006)
Le Quang H., He Q.-C., Zheng Q.-S.: Some general properties of Eshelby’s tensor fields in transport phenomena and anti-plane elasticity. Int. J. Solids Struct. 45, 3845–3857 (2008)
Liu X.N., Hu G.K.: Inclusion problem of microstretch continuum. Int. J. Eng. Sci. 42, 849–860 (2004)
Lubarda V.A.: Circular inclusions in anti-plane strain couple stress elasticity. Int. J. Solids Struct. 40, 3827–3851 (2003)
Ma H.M., Gao X.-L.: Eshelby’s tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory. Acta Mech. 211, 115–129 (2010)
Ma H.M., Gao X.-L.: Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix. Int. J. Solids Struct. 48, 44–55 (2011)
Ma H.S., Hu G.K.: Eshelby tensors for an ellipsoidal inclusion in a micropolar material. Int. J. Eng. Sci. 44, 595–605 (2006)
Ma H.S., Hu G.K.: Eshelby tensors for an ellipsoidal inclusion in a microstretch material. Int. J. Solids Struct. 44, 3049–3061 (2007)
Mindlin R.D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–78 (1964)
Mindlin R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)
Mindlin R.D., Eshel N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)
Pak Y.E.: Circular inclusion problem in antiplane piezoelectricity. Int. J. Solids Struct. 29, 2403–2419 (1992)
Timoshenko S.P., Goodier J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)
Vollenberg P.H.T., Heikens D.: Particle size dependence of the Young’s modulus of filled polymers: 1 preliminary experiments. Polymer 30, 1656–1662 (1989)
Xu B.X., Wang M.Z.: The arithmetic mean theorem for the N-fold rotational symmetrical inclusion in anti-plane elasticity. Acta Mech. 194, 233–242 (2007)
Zheng Q.-S., Zhao Z.-H.: Green’s function and Eshelby’s fields in couple-stress elasticity. Int. J. Multiscale Comput. Eng. 2, 15–27 (2004)
Zou W.-N., Zheng Q.-S., He Q.-C.: Solutions to Eshelby’s problems of non-elliptical thermal inclusions and cylindrical elastic inclusions of non-elliptical cross section. Proc. R. Soc. A 467, 607–626 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gao, X.L., Ma, H.M. Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem. Acta Mech 223, 1067–1080 (2012). https://doi.org/10.1007/s00707-012-0614-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-012-0614-3