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Inclusion problem for a generalized strain gradient elastic continuum

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Abstract

This paper treats an inclusion problem in the framework of a generalized strain gradient theory (GSGT) of elasticity. The GSGT used here consists of three material characteristic length scales for the evaluation of the elastic response. The Green’s function for the governing differential equations, involving stresses and higher-order stresses, is obtained considering the strain gradient model mentioned above. Then, Eshelby’s tensor is derived in the current framework for an inclusion of any arbitrary shape, under uniform eigenfields. This general form of the Eshelby’s tensor is used to derive closed-form expressions for a spherical inclusion and an infinitely long cylindrical inclusion, each embedded in an unbounded matrix to ignore the boundary effects. To determine the homogenized properties of the composites considering size-effects, the volume averages of the position-dependent Eshelby’s tensor are evaluated for both the inclusions. All the results from the current study match the corresponding results from classical elasticity, when the strain gradient effects are ignored. The homogenized properties of composites with either of the inclusions are strongly influenced by the gradient effects when the dimensions of the inclusions are comparable to the characteristic length scales. They approach the volume averages evaluated using classical elasticity, when the dimensions of the inclusion are very high in comparison with the characteristic lengths. The current study presents the importance of considering the size-effects using GSGT, without any assumption, for the evaluation of the effective properties of the micro- and nano-composites.

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References

  1. Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241(1226), 376–396 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  2. Eshelby, J.D.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Lond. A 252(1271), 561–569 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  3. Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21(5), 571–574 (1973)

    Article  Google Scholar 

  4. Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13(4), 213–222 (1965)

    Article  Google Scholar 

  5. Hill, R.: Theory of mechanical properties of fibre-strengthened materials—III. Self-consistent model. J. Mech. Phys. Solids 13(4), 189–198 (1965)

    Article  Google Scholar 

  6. Weng, G.J.: Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions. Int. J. Eng. Sci. 22(7), 845–856 (1984)

    Article  MATH  Google Scholar 

  7. Mura, T.: Micromechanics of Defects in Solids. Springer, Berlin (2013)

    Google Scholar 

  8. Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials, vol. 37. Elsevier, Amsterdam (2013)

    MATH  Google Scholar 

  9. Qu, J., Cherkaoui, M.: Fundamentals of Micromechanics of Solids. Wiley Online Library, New York (2006)

    Book  Google Scholar 

  10. Ramanathan, T., Abdala, A.A., Stankovich, S., Dikin, D.A., Herrera-Alonso, M., Piner, R.D., Adamson, D.H., Schniepp, H.C., Chen, X.R.R.S., Ruoff, R.S., et al.: Functionalized graphene sheets for polymer nanocomposites. Nat. Nanotechnol. 3(6), 327–331 (2008)

    Article  Google Scholar 

  11. Jordan, J., Jacob, K.I., Tannenbaum, R., Sharaf, M.A., Jasiuk, I.: Experimental trends in polymer nanocomposites—a review. Mater. Sci. Eng. A 393(1), 1–11 (2005)

    Article  Google Scholar 

  12. Yang, R., Chen, G.: Thermal conductivity modeling of periodic two-dimensional nanocomposites. Phys. Rev. B 69(19), 195316 (2004)

    Article  Google Scholar 

  13. McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15(5), 1060 (2005)

    Article  Google Scholar 

  14. Lloyd, D.J.: Particle reinforced aluminium and magnesium matrix composites. Int. Mater. Rev. 39(1), 1–23 (1994)

    Article  Google Scholar 

  15. Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 296–361 (1997)

    MATH  Google Scholar 

  16. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)

    Article  MATH  Google Scholar 

  17. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  18. Koiter, W.T.: General theorems for elastic–plastic solids, chapter 4. In: Sneddon, I.N., Hill, R. (eds.) Progress in Solid Mechanics, vol. 1, pp. 167–221. North-Holland, Amsterdam (1960)

    Google Scholar 

  19. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4(1), 109–124 (1968)

    Article  MATH  Google Scholar 

  22. 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(22), 7486–7499 (2007)

    Article  MATH  Google Scholar 

  23. Zhou, S., Li, A., Wang, B.: A reformulation of constitutive relations in the strain gradient elasticity theory for isotropic materials. Int. J. Solids Struct. 80, 28–37 (2016)

    Article  Google Scholar 

  24. Cheng, Z.-Q., He, L.-H.: Micropolar elastic fields due to a spherical inclusion. Int. J. Eng. Sci. 33(3), 389–397 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cheng, Z.-Q., He, L.-H.: Micropolar elastic fields due to a circular cylindrical inclusion. Int. J. Eng. Sci. 35(7), 659–668 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, X., Hu, G.: Inclusion problem of microstretch continuum. Int. J. Eng. Sci. 42(8), 849–860 (2004)

    Article  Google Scholar 

  27. Zhang, X., Sharma, P.: Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems. Int. J. Solids Struct. 42(13), 3833–3851 (2005)

    Article  MATH  Google Scholar 

  28. Sharma, P., Dasgupta, A.: Average elastic fields and scale-dependent overall properties of heterogeneous micropolar materials containing spherical and cylindrical inhomogeneities. Phys. Rev. B 66(22), 224110 (2002)

    Article  Google Scholar 

  29. Gao, X.-L., Ma, H.M.: Green’s function and Eshelby’s tensor based on a simplified strain gradient elasticity theory. Acta Mech. 207(3), 163–181 (2009)

    Article  MATH  Google Scholar 

  30. 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(1–2), 115–129 (2010)

    Article  MATH  Google Scholar 

  31. Gao, X.-L., Ma, H.M.: Strain gradient solution for Eshelby’s ellipsoidal inclusion problem. Proc. R. Soc. Lond. A 466(2120), 2425–2446 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Maranganti, R., Sharma, N.D., Sharma, P.: Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Greens function solutions and embedded inclusions. Phys. Rev. B 74(1), 014110 (2006)

    Article  Google Scholar 

  33. Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)

    Article  Google Scholar 

  34. Exadaktylos, G.E., Vardoulakis, I.: Surface instability in gradient elasticity with surface energy. Int. J. Solids Struct. 35(18), 2251–2281 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)

    Article  MATH  Google Scholar 

  36. Shaat, M.: Physical and mathematical representations of couple stress effects on micro/nanosolids. Int. J. Appl. Mech. 7(01), 1550012 (2015)

    Article  Google Scholar 

  37. Hadjesfandiari, A.R., Dargush, G.F.: Couple stress theory for solids. Int. J. Solids Struct. 48(18), 2496–2510 (2011)

    Article  Google Scholar 

  38. Münch, I., Neff, P., Madeo, A., Ghiba, I.-D.: The modified indeterminate couple stress model: why Yang et al.’s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless. ZAMM J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik 97(12), 1524–1554 (2017)

    Article  MathSciNet  Google Scholar 

  39. Sidhardh, S., Ray, M.C.: Exact solutions for elastic response in micro- and nano-beams considering strain gradient elasticity. Math. Mech. Solids (2018). https://doi.org/10.1177/1081286518761182

    Google Scholar 

  40. Sidhardh, S., Ray, M.C.: Element-free Galerkin model of nano-beams considering strain gradient elasticity. Acta Mech. (2018). https://doi.org/10.1007/s00707-018-2139-x

  41. Delfani, M.R., Latifi Shahandashti, M.: Elastic field of a spherical inclusion with non-uniform eigenfields in second strain gradient elasticity. Proc. R. Soc. Lond. A 473(2205), 20170254 (2017)

    Article  MathSciNet  Google Scholar 

  42. Luo, H.A., Weng, G.J.: On Eshelby’s S-tensor in a three-phase cylindrically concentric solid, and the elastic moduli of fiber-reinforced composites. Mech. Mater. 8(2–3), 77–88 (1989)

    Article  Google Scholar 

  43. Monchiet, V., Bonnet, G.: Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems. Proc. R. Soc. Lond. A 467(2126), 314–332 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Maranganti, R., Sharma, P.: Length scales at which classical elasticity breaks down for various materials. Phys. Rev. Lett. 98(19), 195504 (2007)

    Article  Google Scholar 

  45. Dontsov, E.V., Tokmashev, R.D., Guzina, B.B.: A physical perspective of the length scales in gradient elasticity through the prism of wave dispersion. Int. J. Solids Struct. 50(22), 3674–3684 (2013)

    Article  Google Scholar 

  46. Shodja, H.M., Zaheri, A., Tehranchi, A.: Ab initio calculations of characteristic lengths of crystalline materials in first strain gradient elasticity. Mech. Mater. 61, 73–78 (2013)

    Article  Google Scholar 

  47. Ma, H.M., Gao, X.-L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56(12), 3379–3391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, Kharagpur, 721302, India

    Sai Sidhardh & M. C. Ray

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  1. Sai Sidhardh
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Sidhardh, S., Ray, M.C. Inclusion problem for a generalized strain gradient elastic continuum. Acta Mech 229, 3813–3831 (2018). https://doi.org/10.1007/s00707-018-2199-y

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