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General solution for inhomogeneous line inclusion with non-uniform eigenstrain

  • Lifeng MaEmail author
  • Yike Qiu
  • Yumei Zhang
  • Guang Li
Original
  • 27 Downloads

Abstract

The inhomogeneous line inclusion problem has various backgrounds in practical application such as graphene sheet-reinforced composites, and hydrogen embrittlement, grain boundary segregation in metallic materials. Due to the long-standing mathematical difficulty, there is no explicit analytical solution obtained except for the thin ellipsoidal inhomogeneity and rigid line inhomogeneity. In this paper, to find the deformation state due to the presence of such kind of elastic inhomogeneities, the inhomogeneous line inclusion problem is tackled in the framework of plane deformation. Firstly, the fundamental solution for a point-wise residual strain is presented and its deformation strain field is derived. By using Green’s function method, the homogeneous line inclusion problem with non-uniform eigenstrain is formulated and an Eshelby tensor-like line inclusion tensor is derived. From the line inclusion concept, the classical edge dislocation is revisited. Also, by virtue of this model, some elementary line homogenous inclusion problems are explored. Secondly, based on the homogeneous line inclusion solution, the inhomogeneous line inclusion problem is formulated using the equivalent eigenstrain principle, and its general solution is derived. Then, an inhomogeneous edge dislocation model is proposed and its analytical solution is presented. Furthermore, to demonstrate the application of the proposed inhomogeneous line inclusion model, a typical thin inclusion under remote load is studied. This study provides a general solution for inhomogeneous thin inclusion problems. The models and their solutions introduced here will also find application in the mechanics of composites analysis, heterogeneous material modeling, etc.

Keywords

Inhomogeneous line inclusion Line inclusion tensor The equivalent eigenstrain principle Inhomogeneous edge dislocation Non-uniform eigenstrain distribution 

Notes

Acknowledgements

This work is partially supported by National Natural Science Foundation of China (Grant No.: 41630634).

References

  1. 1.
    Aguilar, S., Tabares, R., Serna, C.: Microstructural transformations of dissimilar austenite–ferrite stainless steels welded joints. J. Mater. Phys. Chem. 1, 65–68 (2013)Google Scholar
  2. 2.
    Atkinson, C.: Some ribbon-like inclusion problems. Int. J. Eng. Sci. 11, 243–266 (1973)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bakshi, S.R., Lahiri, D., Agarwal, A.: Carbon nanotube reinforced metal matrix composites—a review. Int. Mater. Rev. 55, 41–64 (2010)CrossRefGoogle Scholar
  4. 4.
    Ballarini, R.: An integral-equation approach for rigid line inhomogeneity problems. Int. J. Fract. 33, R23–R26 (1987)Google Scholar
  5. 5.
    Ballarini, R.: A rigid line inclusion at a bimaterial interface. Eng. Fract. Mech. 37, 1–5 (1990)CrossRefGoogle Scholar
  6. 6.
    Brussat, T.R., Westmann, R.A.: A Westergaard-type stress function for line inclusion problems. Int. J. Solids Struct. 11, 665–677 (1975)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cantwell, P.R., Tang, M., Dillon, S.J., Luo, J., Rohrer, G.S., Harmer, M.P.: Grain boundary complexions. Acta Mater. 62, 1–48 (2014)CrossRefGoogle Scholar
  8. 8.
    Christensen, R.M.: Mechanics of Composite Materials. Wiley, New York (1979)Google Scholar
  9. 9.
    Claussen, N., Ruehle, M., Heuer, A. H. (eds.): Advances in ceramics, Vol. 12, p. 352. Science and technology of zirconia II. The American Ceramic Society, Columbus, OH (1984)Google Scholar
  10. 10.
    Deng, W., Meguid, S.A.: Analysis of conducting rigid inclusion at the interface of two dissimilar piezoelectric materials. J. Appl. Mech. 65, 76–84 (1998)CrossRefGoogle Scholar
  11. 11.
    Du, Y.A., Ismer, L., Rogal, J., Hickel, T., Neugebauer, J., Drautz, R.: First-principles study on the interaction of H interstitials with grain boundaries in \(\alpha \)- and \(\beta \)-Fe. Phys. Rev. B 84, 667–673 (2011)CrossRefGoogle Scholar
  12. 12.
    Dundurs, J., Markenscoff, X.: A Green’s function formulation of anticracks and their interaction with load-induced singularities. J. Appl. Mech. 56, 550–555 (1989)CrossRefzbMATHGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Eshelby, J.D.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Lond. A 252, 561–569 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Frolov, T., Olmsted, D.L., Asta, M., Mishin, Y.: Structural phase transformations in metallic grain boundaries. Nat. Commun. 4, 1899 (2012)CrossRefGoogle Scholar
  16. 16.
    Gorbatikh, L., Lomov, S.V., Verpoest, I.: Original mechanism of failure initiation revealed through modelling of naturally occurring microstructures. J. Mech. Phys. Solids 58, 735–750 (2010)CrossRefGoogle Scholar
  17. 17.
    Hatano, M., Fujinami, M., Arai, K., Fujii, H., Nagumo, M.: Hydrogen embrittlement of austenitic stainless steels revealed by deformation microstructures and strain-induced creation of vacancies. Acta Mater. 67, 342–353 (2014)CrossRefGoogle Scholar
  18. 18.
    Herbig, M., Kuzmina, M., Haase, C., Marceau, R.K.W., Gutierrez-Urrutia, I., Haley, D., Molodov, D.A., Choi, P., Raabe, D.: Grain boundary segregation in Fe–Mn–C twinning-induced plasticity steels studied by correlative electron backscatter diffraction and atom probe tomography. Acta Mater. 83, 37–47 (2015)CrossRefGoogle Scholar
  19. 19.
    Hickel, T., Nazarov, R., McEniry, E.J., Leyson, G., Grabowski, B., Neugebauer, J.: Ab Initio based understanding of the segregation and diffusion mechanisms of hydrogen in steels. J. Met. 66, 1399–1405 (2014)Google Scholar
  20. 20.
    Hu, J., Shi, Y.N., Sauvage, X., Sha, G., Lu, K.: Grain boundary stability governs hardening and softening in extremely fine nanograined metals. Science 355, 1292–1296 (2017)CrossRefGoogle Scholar
  21. 21.
    Hull, D., Bacon, D.J.: Introduction to Dislocations, 5th edn. Elsevier, Kidlington (2011)Google Scholar
  22. 22.
    Hurtado, J.A., Dundurs, J., Mura, T.: Lamellar inhomogeneites in a uniform stress field. J. Mech. Phys. Solids 44, 1–21 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Inglis, C.E.: Stresses in a plate due to the presence of cracks and sharp corners. Inst. Naval Archit. Lond. 55, 219–230 (1913)Google Scholar
  24. 24.
    Ioakimidis, N.I., Theocaris, P.S.: The second fundamental crack problem and the rigid line inclusion problem in plane elasticity. Acta Mech. 34, 51–61 (1979)CrossRefzbMATHGoogle Scholar
  25. 25.
    Kaczynski, A., Kozłowski, W.: Thermal stresses in an elastic space with a perfectly rigid flat inclusion under perpendicular heat flow. Int. J. Solids Struct. 46, 1772–1777 (2009)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kitahara, H., Ueji, R., Tsuji, N., Minamino, Y.: Crystallographic features of lath martensite in low-carbon steel. Acta Mater. 54, 1279–1288 (2006)CrossRefGoogle Scholar
  27. 27.
    Koyama, M., Akiyama, E., Tsuzaki, K.: Hydrogen embrittlement in a Fe–Mn–C ternary twinning-induced plasticity steel. Corros. Sci. 54, 1–4 (2012)CrossRefGoogle Scholar
  28. 28.
    Li, H., Xia, S., Zhou, B., Liu, W.: C–Cr segregation at grain boundary before the carbide nucleation in Alloy 690. Mater. Charact. 66, 68–74 (2012)CrossRefGoogle Scholar
  29. 29.
    Li, Q., Ting, T.C.T.: Line inclusions in anisotropic elastic solids. J. Appl. Mech. 56, 556–563 (1989)CrossRefzbMATHGoogle Scholar
  30. 30.
    Liddicoat, P.V., Liao, X.Z., Zhao, Y., Zhu, Y., Murashkin, M.Y., Lavernia, E.J., Valiev, R.Z., Ringer, S.P.: Nanostructural hierarchy increases the strength of aluminium alloys. Nat. Commun. 1, 63 (2010)CrossRefGoogle Scholar
  31. 31.
    Ma, L.F.: Fundamental formulation for transformation toughening. Int. J. Solids Struct. 47, 3214–3220 (2010)CrossRefzbMATHGoogle Scholar
  32. 32.
    Ma, L.F., Korsunsky, A.M.: The principle of equivalent eigenstrain for inhomogeneous inclusion problems. Int. J. Solids Struct. 51, 4477–4484 (2014)CrossRefGoogle Scholar
  33. 33.
    Ma, L.F., Wang, B., Korsunsky, A.M.: Plane deformation of circular inhomogeneous inclusion problems with non-uniform symmetrical dilatational eigenstrain. Mater. Des. 86, 809–817 (2015)CrossRefGoogle Scholar
  34. 34.
    Ma, L.F., Wang, B., Korsunsky, A.M.: Complex variable formulation for a rigid line inclusion interacting with a generalized singularity. Arch. Appl. Mech. 88, 613–627 (2018)CrossRefGoogle Scholar
  35. 35.
    Miracle, D.B.: Metal matrix composites—from science to technological significance. Compos. Sci. Technol. 65, 2526–2540 (2005)CrossRefGoogle Scholar
  36. 36.
    Mura, T.: Micromechanics of Defects in Solids, 2nd edn. Martinus Nijhoff, Dordrecht (1987)CrossRefzbMATHGoogle Scholar
  37. 37.
    Mura, T.: Inclusion problems. Appl. Mech. Rev. 41, 15–20 (1988)CrossRefGoogle Scholar
  38. 38.
    Mura, T., Shodja, H.M., Hirose, Y.: Inclusion problems. Appl. Mech. Rev. 49, S118–S127 (1996)CrossRefGoogle Scholar
  39. 39.
    Muskhelishvili, N.I.: Some Problems of Mathematical Theory of Elasticity (English translation from the third Russian edition). Noordhoff Ltd., Groningen (1953)Google Scholar
  40. 40.
    Nemat-Nasser, S., Hori, M.: Micromechanics: Overall properties of heterogeneous materials, 2nd edn. Elsevier, Amsterdam (1999)zbMATHGoogle Scholar
  41. 41.
    Sakaguchi, N., Watanabe, S., Takahashi, H.: Heterogeneous dislocation formation and solute redistribution near grain boundaries in austenitic stainless steel under electron irradiation. Acta Mater. 49, 1129–1137 (2001)CrossRefGoogle Scholar
  42. 42.
    Shodja, H.M., Ojaghnezhad, F.: A general unified treatment of lamellar inhomogeneities. Eng. Fract. Mech. 74, 1499–1510 (2007)CrossRefGoogle Scholar
  43. 43.
    Song, J., Curtin, W.A.: Atomic mechanism and prediction of hydrogen embrittlement in iron. Nat. Mater. 12, 145–151 (2012)CrossRefGoogle Scholar
  44. 44.
    Stankovich, S., Dikin, D.A., Dommett, G.H.B., Kohlhaas, K.M., Zimney, E.J., Stach, E.A., Piner, R.D., Nguyen, S.T., Ruoff, R.S.: Graphene-based composite materials. Nature 442, 282–286 (2006)CrossRefGoogle Scholar
  45. 45.
    Stoller, R.E., Maziasz, P.J., Rowcliffe, A.F., Tanaka, M.P.: Swelling behavior of austenitic stainless steels in a spectrally tailored reactor experiment: implications for near-term fusion machines. J. Nucl. Mater. 155, 1328–1334 (1988)CrossRefGoogle Scholar
  46. 46.
    Tang, M., Carter, W.C., Cannon, R.M.: Diffuse interface model for structural transitions of grain boundaries. Phys. Rev. B 73, 024102 (2006)CrossRefGoogle Scholar
  47. 47.
    Wang, J., Li, Z., Fan, G., Pan, H., Chen, Z., Zhang, D.: Reinforcement with graphene nanosheets in aluminum matrix composites. Scr. Mater. 66, 594–597 (2012)CrossRefGoogle Scholar
  48. 48.
    Wang, Z.Y., Zhang, H.T., Chou, Y.T.: Characteristics of the elastic field of a rigid line inhomogeneity. J. Appl. Mech. 52, 818–822 (1985)CrossRefGoogle Scholar
  49. 49.
    Wu, K.C.: Line inclusions at anisotropic bimaterial interface. Mech. Mater. 10, 173–182 (1990)CrossRefGoogle Scholar
  50. 50.
    Zhou, K., Hoh, H.J., Wang, X., Keer, L.M., Pang, J.H.L., Song, B., Wang, Q.J.: A review of recent works on inclusions. Mech. Mater. 60, 144–158 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.S&V Lab, Department of Engineering MechanicsXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Materials Science and Engineering, State Key Laboratory of Advanced Processing and Recycling of Non-ferrous MetalsLanzhou University of TechnologyLanzhouChina

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