Skip to main content
SpringerLink
Log in

Size-dependent elastic field of nano-inhomogeneity: from interface effect to interphase effect

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

Due to the large surface/interface-to-volume ratio, surface/interface effect in nanomaterials begins to play an important role in changing the constitutive law seen in classical elasticity theory. This work investigates the size-dependent elastic field of nano-inhomogeneity due to interface and interphase effect, respectively. Closed-form solution to the elastic field of nano-inhomogeneity with interface and interphase effect is found. For comparison, the numerical result of InAs/GaAs system is provided as an example, which demonstrates that interface and interphase effect could have significant influence on the elastic field of nano-inhomogeneity. The continuity/discontinuity of the elastic field of nano-inhomogeneity with interface and interphase effect is discussed in detail. It is verified both numerically and analytically that the elastic field from interphase effect tends to converge to that from interface effect when the interphase thickness shrinks to zero. Furthermore, all results converge to the classical solution at positions far away from the interfacial region, so it is unnecessary to distinguish interface or interphase effect at positions far away from the interfacial region in practical situations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Duan, H., Karihaloo, B.: Thermo-elastic properties of heterogeneous materials with imperfect interfaces: generalized Levin’s formula and Hill’s connections. J. Mech. Phys. Solids 55(5), 1036–1052 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L.: Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J. Mech. Phys. Solids 53(7), 1574–1596 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Chu, H.J., Pan, E., Ramsey, J.J., Wang, J., Xue, C.X.: A general perturbation method for inhomogeneities in anisotropic and piezoelectric solids with applications to quantum-dot nanostructures. Int. J. Solids Struct. 48(5), 673–679 (2011)

    MATH  Google Scholar 

  6. Huang, M.J., Fang, X.Q., Liu, J.X., Feng, W.J., Zhao, Y.M.: Size-dependent effective properties of anisotropic piezoelectric composites with piezoelectric nano-particles. Smart Mater. Struct. 24(1), 15005–15013(9) (2015)

    Google Scholar 

  7. Feng, X., Fang, D., Soh, A.K., Hwang, K.C.: Predicting effective magnetostriction and moduli of magnetostrictive composites by using the double-inclusion method. Mech. Mater. 35(7), 623–631 (2003)

    Google Scholar 

  8. Feng, X., Fang, D.N., Hwang, K.C.: Closed-form solutions for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix. Eur. J. Mech. A Solids 23(6), 1007–1019 (2004)

    MATH  Google Scholar 

  9. Ahmadzadeh-Bakhshayesh, H., Gutkin, M.Y., Shodja, H.M.: Surface/interface effects on elastic behavior of a screw dislocation in an eccentric core–shell nanowire. Int. J. Solids Struct. 49(13), 1665–1675 (2012)

    Google Scholar 

  10. Fang, Q.H., Liu, Y.W.: Size-dependent elastic interaction of a screw dislocation with a circular nano-inhomogeneity incorporating interface stress. Scripta Mater. 55(1), 99–102 (2006)

    Google Scholar 

  11. Fang, Q.H., Liu, Y.W., Jin, B., Wen, P.H.: Interaction between a dislocation and a core–shell nanowire with interface effects. Int. J. Solids Struct. 46(6), 1539–1546 (2009)

    MATH  Google Scholar 

  12. Liu, Y., Ye, W.: Elastic and piezoelectric field around a quantum dot with interface effect. Phys. E 89, 5–9 (2017)

    Google Scholar 

  13. Müller, P.: Elastic effects on surface physics. Surf. Sci. Rep. 54(5–8), 157–258 (2004)

    Google Scholar 

  14. Shodja, H.M., Gutkin, M.Y., Moeini-Ardakani, S.S.: Effect of surface stresses on elastic behavior of a screw dislocation inside the wall of a nanotube. Phys. Status Solidi B 248(6), 1437–1441 (2011)

    Google Scholar 

  15. Ghahremani, F.: Effect of grain boundary sliding on steady creep of polycrystals. Int. J. Solids Struct. 16(9), 847–862 (1980)

    MATH  Google Scholar 

  16. Hashin, Z.: The spherical inclusion with imperfect interface. J. Appl. Mech. 58(2), 444–449 (1991)

    Google Scholar 

  17. Yu, H.Y.: A new dislocation-like model for imperfect interfaces and their effect on load transfer. Compos. A Appl. Sci. Manuf. 29(9), 1057–1062 (1998)

    Google Scholar 

  18. Gurtin, M.E., Weissmuller, J., Larche, F.: A general theory of curved deformable interfaces in solids at equilibrium. Philos. Mag. A 78(5), 1093–1109 (1998)

    Google Scholar 

  19. Firooz, S., Javili, A.: Understanding the role of general interfaces in the overall behavior of composites and size effects. Comput. Mater. Sci. 162, 245–254 (2019)

    Google Scholar 

  20. Firooz, S., Chatzigeorgiou, G., Meraghni, F., Javili, A.: Bounds on size effects in composites via homogenization accounting for general interfaces. Continuum Mech. Thermodyn. 32(1), 173–206 (2020)

    MathSciNet  Google Scholar 

  21. Firooz, S., Chatzigeorgiou, G., Meraghni, F., Javili, A.: Homogenization accounting for size effects in particulate composites due to general interfaces. Mech. Mater. 139, 103204 (2019)

    Google Scholar 

  22. Javili, A., Steinmann, P., Mosler, J.: Micro-to-macro transition accounting for general imperfect interfaces. Comput. Methods Appl. Mech. Eng. 317, 274–317 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Chatzigeorgiou, G., Meraghni, F., Javili, A.: Generalized interfacial energy and size effects in composites. J. Mech. Phys. Solids 106, 257–282 (2017)

    MathSciNet  Google Scholar 

  24. Shuttleworth, R.: The surface tension of solids. Proc. Phys. Soc. Sect. A 63, 444 (1950)

    Google Scholar 

  25. Gurtin, M.E., Murdoch, A.I.: Continuum theory of elastic-material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)

    MathSciNet  MATH  Google Scholar 

  26. Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139 (2000)

    Google Scholar 

  27. Dingreville, R., Qu, J.: Interfacial excess energy, excess stress and excess strain in elastic solids: planar interfaces. J. Mech. Phys. Solids 56(5), 1944–1954 (2008)

    MathSciNet  MATH  Google Scholar 

  28. Dingreville, R., Qu, J.M., Cherkaoui, M.: Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J. Mech. Phys. Solids 53(8), 1827–1854 (2005)

    MathSciNet  MATH  Google Scholar 

  29. Ye, W., Liu, Y.: Elastic and piezoelectric fields around a quantum wire of zincblende heterostructures with interface elasticity effect. Appl. Phys. A 124(4), 285 (2018)

    Google Scholar 

  30. Javili, A., dell’Isola, F., Steinmann, P.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids 61(12), 2381–2401 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Rahali, Y., Eremeyev, V., Ganghoffer, J.: Surface effects of network materials based on strain gradient homogenized media. Math. Mech. Solids 25(2), 389–406 (2020)

    MathSciNet  Google Scholar 

  32. Berbenni, S., Cherkaoui, M.: Homogenization of multicoated inclusion-reinforced linear elastic composites with eigenstrains: application to thermoelastic behavior. Philos. Mag. 90(22), 3003–3026 (2010)

    Google Scholar 

  33. Koutsawa, Y.: Multi-coating inhomogeneities approach for composite materials with temperature-dependent constituents under small strain and finite thermal perturbation assumptions. Compos. B Eng. 112, 137–147 (2017)

    Google Scholar 

  34. Lipinski, P., Barhdadi, E.H., Cherkaoui, M.: Micromechanical modelling of an arbitrary ellipsoidal multi-coated inclusion. Philos. Mag. 86(10), 1305–1326 (2006)

    Google Scholar 

  35. Yu, J., Lacy, T.E., Toghiani, H., Pittman, C.U., Hwang, Y.: Classical micromechanics modeling of nanocomposites with carbon nanofibers and interphase. J. Compos. Mater. 45(23), 2401–2413 (2011)

    Google Scholar 

  36. Dingreville, R., Qu, J.: A semi-analytical method to compute surface elastic properties. Acta Mater. 55(1), 141–147 (2007)

    Google Scholar 

  37. Dingreville, R., Qu, J.: A semi-analytical method to estimate interface elastic properties. Comput. Mater. Sci. 46(1), 83–91 (2009)

    Google Scholar 

  38. Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B 71(9), 094104–11 (2005)

    Google Scholar 

  39. Tang, Z., Chen, Y., Ye, W.: Calculation of surface properties of cubic and hexagonal crystals through molecular statics simulations. Crystals 10(4), 329 (2020)

    Google Scholar 

  40. Ye, W., Chen, B.: Investigation of the surface elasticity of GaN by atomistic simulations and its application to the elastic relaxation of GaN nanoisland. Mater. Lett. 141, 245–247 (2015)

    Google Scholar 

  41. Paliwal, B., Cherkaoui, M.: Atomistic-continuum interphase model for effective properties of composite materials containing nano-inhomogeneities. Philos. Mag. 91(30), 3905–3930 (2011)

    Google Scholar 

  42. Paliwal, B., Cherkaoui, M.: Estimation of anisotropic elastic properties of nanocomposites using atomistic-continuum interphase model. Int. J. Solids Struct. 49(18), 2424–2438 (2012)

    Google Scholar 

  43. Paliwal, B., Cherkaoui, M., Fassi-Fehri, O.: Effective elastic properties of nanocomposites using a novel atomistic-continuum interphase model. C.R. Mec. 340(4–5), 296–306 (2012)

    Google Scholar 

  44. Hashin, Z.: Thin interphase/imperfect interface in elasticity with application to coated fiber composites. J. Mech. Phys. Solids 50(12), 2509–2537 (2002)

    MathSciNet  MATH  Google Scholar 

  45. Wang, J., Duan, H.L., Zhang, Z., Huang, Z.P.: An anti-interpenetration model and connections between interphase and interface models in particle-reinforced composites. Int. J. Mech. Sci. 47(4), 701–718 (2005)

    MATH  Google Scholar 

  46. Davies, J.H.: Elastic and piezoelectric fields around a buried quantum dot: a simple picture. J. Appl. Phys. 84(3), 1358–1365 (1998)

    Google Scholar 

  47. Pan, E.: Elastic and piezoelectric fields around a quantum dot: fully coupled or semicoupled model? J. Appl. Phys. 91(6), 3785–3796 (2002)

    Google Scholar 

  48. Sharma, P., Ganti, S.: Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies. J. Appl. Mech. 71(5), 663–671 (2004)

    MATH  Google Scholar 

  49. Sharma, P., Ganti, S.: Interfacial elasticity corrections to size-dependent strain-state of embedded quantum dots. Phys. Status Solidi B Basic Res. 234(3), R10–R12 (2002)

    Google Scholar 

  50. Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82(4), 535–537 (2003)

    Google Scholar 

  51. Ellaway, S.W., Faux, D.A.: Effective elastic stiffnesses of InAs under uniform strain. J. Appl. Phys. 92(6), 3027–3033 (2002)

    Google Scholar 

  52. Vurgaftman, I., Meyer, J.R., Ram-Mohan, L.R.: Band parameters for III–V compound semiconductors and their alloys. J. Appl. Phys. 89(11), 5815–5875 (2001)

    Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 11702041) and Fundamental Research Funds for the Central Universities (No. 2018CDXYHK0016).

Author information

Authors and Affiliations

  1. College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China

    Mian Wang & Wei Ye

  2. Chongqing Key Laboratory of Heterogeneous Material Mechanics, Chongqing University, Chongqing, 400044, China

    Wei Ye

Authors
  1. Mian Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Ye.

Ethics declarations

Conflict of interest

The authors declare that they have no Conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, M., Ye, W. Size-dependent elastic field of nano-inhomogeneity: from interface effect to interphase effect. Arch Appl Mech 90, 2319–2333 (2020). https://doi.org/10.1007/s00419-020-01722-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-020-01722-2

Keywords

Search

Navigation