Skip to main content
SpringerLink
Log in

Elastic field due to dislocation loops in isotropic multilayer system

  • Original Paper
  • Published:
Journal of Materials Science Aims and scope Submit manuscript

Abstract

A semi-analytical solution based on fast discrete Fourier transform is developed to analyze the elastic fields induced by dislocation loops within perfect bonding multilayers. Final elastic field is the linear superposition of bulk elastic field and correction elastic field, and the correction elastic field is induced by elastic modulus mismatch and lattice plane misorientation across interface planes. Final displacement and traction stress are continuous across the interface planes of multilayer system, and the case of perfect bonding two-layer system is elaborated in detail. Validity of the semi-analytical approach is tested by analyzing elastic fields due to dislocation loop within perfect bonding two-layer system. Interface elastic field profiles of perfect bonding two-layer Cu–Nb system are studied, demonstrating that final elastic field is the linear superposition of bulk stress and correction stress fields. Simulation results demonstrate that loop depth within thin film and modulus mismatch have a remarkable effect on the elastic field distribution.

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
Fig. 7
Fig. 8
Fig.. 9

Similar content being viewed by others

References

  1. Koehler JS (1970) Attempt to design a strong solid. Phys Rev B 2(2):547

    Article  Google Scholar 

  2. Anderson P, Lin IH, Thomson R (1992) Fracture in multilayers. Scr Metall Mater 27:687

    Article  Google Scholar 

  3. Lashmore DS, Thomson R (1992) Cracks and dislocations in face-centered cubic metallic multilayers. J Mater Res 7(9):2379

    Article  Google Scholar 

  4. Zhou Q, Xie JY, Wang F, Huang P, Xu KW, Lu TJ (2015) The mechanical behavior of nanoscale metallic multilayers: a survey. Acta Mech Sin 31(3):319

    Article  Google Scholar 

  5. Beyerlein IJ, Demkowicz MJ, Misra A, Uberuaga BP (2015) Defect-interface interactions. Prog Mater Sci 74:125

    Article  Google Scholar 

  6. Misra A, Hirth JP, Kung H (2002) Single-dislocation-based strengthening mechanisms in nanoscale metallic multilayers. Philos Mag A 82(16):2935

    Article  Google Scholar 

  7. Han X, Ghoniem NM (2005) Stress field and interaction forces of dislocations in anisotropic multilayer thin films. Philos Mag 85(11):1205

    Article  Google Scholar 

  8. Han W, Demkowicz MJ, Mara NA, Fu E, Sinha S, Rollett AD, Wang Y, Carpenter JS, Beyerlein IJ, Misra A (2013) Design of radiation tolerant materials via interface engineering. Adv Mater 25:6975

    Article  Google Scholar 

  9. Rao SI, Hazzledine PM (2000) Atomistic simulations of dislocation–interface interactions in the Cu–Ni multilayer system. Philos Mag A 80(9):2011

    Article  Google Scholar 

  10. Wang J, Misra A (2011) An overview of interface-dominated deformation mechanisms in metallic multilayers. Curr Opin Solid State Mater Sci 15:20

    Article  Google Scholar 

  11. Head AK (1953) Interaction of dislocations and boundaries. Philos Mag 44:92

    Article  Google Scholar 

  12. Head AK (1953) Edge dislocations in inhomogeneous media. Proc Phys Soc B 66(9):793

    Article  Google Scholar 

  13. Head AK (1960) The interaction of dislocations with boundaries and surface films. Aust J Phys 13(2):278

    Article  Google Scholar 

  14. Lee MS, Dundurs J (1973) Edge dislocation in a surface layer. Int J Eng Sci 11(1):87

    Article  Google Scholar 

  15. Kelly PA, O’Connor JJ, Hills DA (1995) The stress field due to dislocation in layered media. J Phys D Appl Phys 28:530

    Article  Google Scholar 

  16. Choi ST, Earmme YY (2002) Elastic study on singularities interacting with interfaces using alternating technique: part I. Anisotropic trimaterial. Int J Solids Struct 39:943

    Article  Google Scholar 

  17. Choi ST, Earmme YY (2002) Elastic study on singularities interacting with interfaces using alternating technique: part II. Isotropic trimaterial. Int J Solids Struct 39:1199

    Article  Google Scholar 

  18. Öveçoğlu ML, Doerner MF, Nix WD (1987) Elastic interactions of screw dislocations in thin films on substrates. Acta Metall 35(12):2947

    Article  Google Scholar 

  19. Kamat SV, Hirth JP, Carnahan B (1987) Image forces on screw dislocations in multilayer structures. Scripta Metall 21(11):1587

    Article  Google Scholar 

  20. Fivel MC, Gosling TJ, Canova GR (1996) Implementing image stresses in a 3D dislocation simulation. Modell Simul Mater Sci Eng 4:581

    Article  Google Scholar 

  21. Gosling TJ, Willis JR (1994) A line-integral representation for the stresses due to an arbitrary dislocation in an isotropic half-space. J Mech Phys Solids 42(8):1199

    Article  Google Scholar 

  22. Tan E, Sun L (2011) Dislocation-induced stress field in multilayered heterogeneous thin film system. J Nanomech Micromech 1(3):91

    Article  Google Scholar 

  23. Wen J, Wu MS (2007) Analysis of a line defect in a multilayered smart structure by the image method. Mech Mater 39(2):126

    Article  Google Scholar 

  24. Wang X, Pan E, Albrecht JD (2007) Anisotropic elasticity of multilayered crystals deformed by a biperiodic network of misfit dislocations. Phys Rev B 76(13):134112

    Article  Google Scholar 

  25. Kuo CH (2014) Elastic field due to an edge dislocation in a multilayered composite. Int J Solids Struct 51:1421

    Article  Google Scholar 

  26. Salamon NJ, Dundurs J (1971) Elastic fields of a dislocation loop in a two-phase material. J Elast 1(2):153

    Article  Google Scholar 

  27. Salamon NJ, Dundurs J (1977) A circular glide dislocation loop in a two-phase material. J Phys C 10:497

    Article  Google Scholar 

  28. Weinberger CR, Aubry S, Lee SW, Nix WD, Cai W (2009) Modelling dislocations in a free-standing thin film. Modell Simul Mater Sci Eng 17:075007

    Article  Google Scholar 

  29. Wu W, Schäublin R, Chen J (2012) General dislocation image stress of anisotropic cubic thin film. J Appl Phys 112:093522

    Article  Google Scholar 

  30. Ghoniem NM, Han X (2005) Dislocation motion in anisotropic multilayer materials. Philos Mag 85(24):2809

    Article  Google Scholar 

  31. Sobie C, McPhie MG, Capolungo L, Cherkaoui M (2014) The effect of interfaces on the mechanical behaviour of multilayered metallic laminates. Modell Simul Mater Sci Eng 22:045007

    Article  Google Scholar 

  32. Burgers JM (1939) Some considerations on the fields of stress connected with dislocations in a regular crystal lattice. Proc Kon Ned Akad Wetenschap 42:293

    Google Scholar 

  33. Peach MO, Koehler JS (1950) The forces exerted on dislocations and the stress fields produced by them. Phys Rev 80(3):436

    Article  Google Scholar 

  34. Volterra V (1907) Sur l’équilibre des corps élastiques multiplement connexes. Gauthier-Villars, Paris

    Google Scholar 

  35. Devincre B (1995) Three dimensional stress field expressions for straight dislocation segments. Solid State Commun 93(11):875

    Article  Google Scholar 

  36. Hirth JP, Lothe J (1982) Theory of dislocations, 2nd edn. Krieger Publishing Co., Malabar

    Google Scholar 

  37. Mura T (1987) Micromechanics of defects in solids, vol 2. Springer, Berlin

    Book  Google Scholar 

  38. Berbenni S, Berveiller M, Richeton T (2008) Intra-granular plastic slip heterogeneities: discrete vs. mean field approaches. Int J Solids Struct 45(14–15):4147

    Article  Google Scholar 

  39. Khraishi TA, Hirth JP, Zbib HM, Khaleel MA (2000) The displacement and strain-stress fields of a general circular volterra dislocation loop. Int J Eng Sci 38:251

    Article  Google Scholar 

  40. Akarapu S, Zbib HM (2009) Line-integral solution for the stress and displacement fields of an arbitrary dislocation segment in isotropic bi-materials in 3D space. Philos Mag 89(25):2149

    Article  Google Scholar 

  41. Zhernenkov M, Gill S, Stanic V, DiMasi E, Kisslinger K, Baldwin JK, Misra A, Demkowicz MJ, Ecker L (2014) Design of radiation resistant metallic multilayers for advanced nuclear systems. Appl Phys Lett 104:241906

    Article  Google Scholar 

  42. Beyerlein IJ, Mara NA, Carpenter JS, Nizolek T, Mook WM, Wynn TA, McCabe RJ, Mayeur JR, Kang K, Zheng S, Wang J, Pollock TM (2013) Interface-driven microstructure development and ultra high strength of bulk nanostructured Cu–Nb multilayers fabricated by severe plastic deformation. J Mater Res 28(13):1799

    Article  Google Scholar 

  43. Bower AF (2012) Applied mechanics of solids. CRC Press, London

    Google Scholar 

  44. Voigt W (1889) Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper. Ann Phys 38:573

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the China Postdoctoral Science Foundation No. 58; the Scientific Research and Development Fund of Tsinghua University under Grant No. 120002049; the National Natural Science Foundation of China under Grant No. 51305223; and State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics, MCMS-0414Y01).

Author information

Authors and Affiliations

  1. Key Laboratory of Hubei Province for Water Jet Theory and New Technology, Wuhan University, Wuhan, 430072, China

    Re Xia & Runni Wu

  2. State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing, 100084, People’s Republic of China

    Wenwang Wu

Authors
  1. Re Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenwang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Runni Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenwang Wu.

Appendix

Appendix

$$ N^{S} = \left[ {\begin{array}{*{20}l} {\left( {k_{x} z_{0} } \right) \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} & {\left( {k_{y} } \right) \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} & {\left( {ik_{x} } \right) \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ {\left( {k_{y} z_{0} } \right) \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} & {\left( { - k_{x} } \right) \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} & {\left( {ik_{y} } \right) \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ {\left\{ {\begin{array}{*{20}l} {\left( { - ik_{z} z_{0} } \right) \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ { + i\frac{\lambda + 3\mu }{\lambda + \mu } \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right\}} & 0 & {\left( {k_{z} } \right) \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right] $$
(34)
$$ N^{A} = \left[ {\begin{array}{*{20}l} {\left( {k_{x} z_{0} } \right) \cdot \cos h\left( {k_{z} z_{0} } \right)} & {\left( { - k_{y} } \right) \cdot \sin h\left( {k_{z} z_{0} } \right)} & {\left( {ik_{x} } \right) \cdot \sin h\left( {k_{z} z_{0} } \right)} \\ {\left( {k_{y} z_{0} } \right) \cdot \cos h\left( {k_{z} z_{0} } \right)} & {\left( {k_{x} } \right) \cdot \sin h\left( {k_{z} z_{0} } \right)} & {\left( {ik_{y} } \right) \cdot \sin h\left( {k_{z} z_{0} } \right)} \\ {\left\{ {\begin{array}{*{20}l} {\left( { - ik_{z} z_{0} } \right) \cdot \sin h\left( {k_{z} z_{0} } \right)} \\ { + i\frac{\lambda + 3\mu }{\lambda + \mu } \cdot \cos h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right\}} & 0 & {\left( {k_{z} } \right) \cdot \cos h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right] $$
(35)
$$ M^{S} = \left[ {\begin{array}{*{20}l} {2\mu k_{x} \left( {\begin{array}{*{20}l} {k_{z} z_{0} \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ { - \frac{\mu }{\lambda + \mu } \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right)} & {\mu k_{y} k_{z} \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} & {2i\mu k_{x} k_{z} \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ {2\mu k_{y} \left( {\begin{array}{*{20}l} {k_{z} z_{0} \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ { - \frac{\mu }{\lambda + \mu } \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right)} & { - \mu k_{x} k_{z} \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} & {2i\mu k_{y} k_{z} \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ {2i\mu \left( {\begin{array}{*{20}l} {\frac{\lambda + 2\mu }{\lambda + \mu }k_{z} \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ { - k_{z}^{2} z_{0} \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right)} & 0 & {2\mu k_{z}^{2} \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right] $$
(36)
$$ M^{A} = \left[ {\begin{array}{*{20}l} {2\mu k_{x} \left( {\begin{array}{*{20}l} {k_{z} z_{0} \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ { - \frac{\mu }{\lambda + \mu } \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right)} & { - \mu k_{y} k_{z} \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} & {2i\mu k_{x} k_{z} \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ {2\mu k_{y} \left( {\begin{array}{*{20}l} {k_{z} z_{0} \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ { - \frac{\mu }{\lambda + \mu } \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right)} & {\mu k_{x} k_{z} \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} & {2i\mu k_{y} k_{z} \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ {2i\mu \left( {\begin{array}{*{20}l} {\frac{\lambda + 2\mu }{\lambda + \mu }k_{z} \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ { - k_{z}^{2} z_{0} \cdot { \cos }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right)} & 0 & {2\mu k_{z}^{2} \cdot { \sin }\,h\left( {k_{z} z_{0} } \right)} \\ \end{array} } \right] $$
(37)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xia, R., Wu, W. & Wu, R. Elastic field due to dislocation loops in isotropic multilayer system. J Mater Sci 51, 2942–2957 (2016). https://doi.org/10.1007/s10853-015-9603-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10853-015-9603-y

Keywords

Search

Navigation