Peierls Framework for Dislocation Nucleation from a Crack Tip

  • J. R. Rice
  • G. E. Beltz
  • Y. Sun

Abstract

Dislocation nucleation from a stressed crack tip is analyzed based on the Peierls concept, in which a periodic relation between shear stress and atomic shear displacement is assumed to hold along a slip plane emanating from a crack tip. This approach allows some small slip displacement to occur near the tip in response to small applied loading and, with increase in loading, the incipient dislocation configuration becomes unstable and leads to a fully formed dislocation which is driven away from the crack. An exact solution for the loading at that nucleation instability was developed using the J-integral for the case when the crack and slip planes coincide (Rice, 1992). Solutions are discussed here for cases when they do not. The results were initially derived for isotropic materials and some generalizations to take into account anisotropic elasticity are noted here. Solutions are also given for emission of dissociated dislocations, especially partial dislocation pairs in fee crystals. The level of applied stress intensity factors required for dislocation nucleation is shown to be proportional to \(\sqrt {{\gamma _{us}}}\) where γus, the unstable stacking energy, is a new solid state parameter identified by the analysis. It is the maximum energy encountered in the block-like sliding along a slip plane, in the Burgers vector direction, of one half of a crystal relative to the other. Approximate estimates of γus are summarized, and the results are used to evaluate brittle versus ductile response in fee and bee metals in terms of the competition between dislocation nucleation and Griffith cleavage at a crack tip. The analysis also reveals features of the near-tip slip distribution corresponding to the saddle point energy configuration for cracks that are loaded below the nucleation threshold, and some implications for thermal activation are summarized. Additionally, the analysis of dislocation nucleation is discussed in connection with the emission from cracks along bimaterial interfaces, in order to understand recent experiments on copper bicrystals and copper/sapphire interfaces, and we discuss the coupled effects of tension and shear stresses along slip planes at a crack tip, leading to shear softening and eased nucleation.

Keywords

Fatigue Carbide Lithium Argon Brittle 

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REFERENCES

  1. Anderson, P. M. (1986). Ductile and brittle crack tip response, Ph.D. Thesis, Div. of Applied Sciences, Harvard University, Cambridge, MA, USA.Google Scholar
  2. Argon, A. S. (1987). Brittle to ductile transition in cleavage fracture, Acta Met., 35:185–196.CrossRefGoogle Scholar
  3. Armstrong, R. W. (1966). Cleavage crack propagation within crystals by the Griffith mechanism versus a dislocation mechanism, Mater. Sci. Eng., 1:251–256.CrossRefGoogle Scholar
  4. Ashcroft, N. W. and Mermin, N. D. (1976). Solid State Physics, Holt, Rinehart and Winston, New York.Google Scholar
  5. Barnett, D. and Asaro, R. J. (1972). The fracture mechanics of slit-like cracks in anisotropic elastic media, J. Mech. Phys. Solids, 20: 353–366.MATHCrossRefGoogle Scholar
  6. Beltz, G. E. (1991). Unpublished research, on the emission of dislocations on inclined slip planes in the Fe/TiC system.Google Scholar
  7. Beltz, G. E. and Rice, J. R. (1991). Dislocation nucleation versus cleavage decohesion at crack tips, In Lowe, T. C., Rollett, A. D., Follansbee, P. S. and Daehn, G. S., editors, Modeling the Deformation of Crystalline Solids, TMS, pages 457–480. Google Scholar
  8. Beltz, G. E., and Rice, J. R. (1992a). Dislocation nucleation at metal/ceramic interfaces, Acta Met., in press.Google Scholar
  9. Beltz, G.E., and Rice, J. R. (1992b). Research in progress, on the 2D and 3D calculations of the activation energy for dislocation nucleation.Google Scholar
  10. Beltz, G. E., and Wang, J.-S. (1992). Crack direction effects along cop-per/sapphire interfaces, Acta Met., in press.Google Scholar
  11. Brandes, E. A. (1983). Smithells Metals Reference Book, 6th ed., Butterworths, London.Google Scholar
  12. Burns, S. J. (1986). Crack tip dislocation nucleation observations in bulk specimens, Scripta Met., 20:1489–1494.CrossRefGoogle Scholar
  13. Cheung, K. (1990). Atomistic study of dislocation nucleation at a crack tip, Ph.D. Thesis, Dept. of Nuclear Engineering, MIT, Cambridge, MA, USA.Google Scholar
  14. Cheung, K., Yip, S. and Argon, A. S. (1991). Activation analysis of dislocation nucleation from a crack tip in α-Fe, J. Appl. Phys., 69:2088–2096.CrossRefGoogle Scholar
  15. Chiao, Y.-H., and Clarke, D. R. (1989). Direct observation of dislocation emission from crack tips in silicon at high temperatures, Acta Met., 37:203–219.CrossRefGoogle Scholar
  16. Daw, M. S., and Baskes, M. I. (1984). Embedded-atom method: Deriva¬tion and application to impurities and other defects in metals, Phys. Rev. B, 29:6443–6453.CrossRefGoogle Scholar
  17. Dragone, T. L., and Nix, W. D. (1988). Crack tip stress fields and dislo-cation nucleation in anisotropic materials, Scripta Met., 22:431–435.CrossRefGoogle Scholar
  18. Duesbery, M. S., Michel, D. J., Kaxiras, E. and Joos, B. (1991). Molecular dynamics studies of defects in Si, In Bristowe, P. D., Epperson, J. E., Griffith, J. E. and Liliental-Weber, Z., editors, Defects in Materials, Materials Research Society, 209:125–130.Google Scholar
  19. Eshelby, J. D. (1970). Energy relations and the energy-momentum tensor in continuum mechanics, In Kanninen, M. F., Adler, W. F., Rosenfield, A. R. and Jaffee, R. I., editors, Inelastic Behavior of Solids, McGraw- Hill, New York, pages 77–115. Google Scholar
  20. Foiles, S. M., Baskes, M. I. and Daw, M. S. (1986). Embedded-atom- method functions for the fee metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys, Phys. Rev. B., 33:7983–7991.CrossRefGoogle Scholar
  21. Foiles, S. M., and Daw, M. S. (1987). Application of the embedded atom method to Ni3Al, J. Mater. Res., 2:5–15.CrossRefGoogle Scholar
  22. Harrison, R. J., Spaepen, F., Voter, A. F. and Chen, A. F. (1990). Structure of grain boundaries in iron, In Olson, G. B., Azrin, M. and Wright, E. S., editors, Innovations in Ultrahigh-Strength Steel Technology, Plenum Press, pages 651–675. Google Scholar
  23. Hirth, J. P., and Lothe, J. (1982). Theory of Dislocations, 2nd Edition, McGraw Hill, New York.Google Scholar
  24. Hoagland, R. G., Daw, M. S., Foiles, S. M. and Baskes, M. I. (1990). An atomic model of crack tip deformation in aluminum using an embedded atom potential, J. Mater. Res., 5:313–324.CrossRefGoogle Scholar
  25. Kelly, A., Tyson, W. R. and Cotfcrell, A. H. (1967). Ductile and brittle crystals, Phil. Mag. 15, pages 567–586. CrossRefGoogle Scholar
  26. Lin, I.-H., and Thomson, R. (1986). Cleavage, dislocation emission, and shielding for cracks under general loading, Acta Met., 34:187–206.CrossRefGoogle Scholar
  27. Michot, G., and George, A. (1986). Dislocation emission from cracks — observations by x-ray topography in silicon, Scripta Met., 20:1495–1500.CrossRefGoogle Scholar
  28. Mohan, R, Ortiz, M and Shih, C. F. (1991). Crack-tip fields in ductile single crystals and bicrystals, In Lowe, T. C., Rollett, A. D., Follansbee, P. S. and Daehn, G. S., editors, Modeling the Deformation of Crystalline Solids, TMS, pages 481–498. Google Scholar
  29. Nabarro, F. R. N. (1947). Dislocations in a simple cubic lattice, Proc. Phys. Soc., 59:256–272.CrossRefGoogle Scholar
  30. Ohr, S. M. (1985). An electron microscope study of crack tip deformation and its impact on the dislocation theory of fracture, Mat. Sci. and Engr., 72:1–35.CrossRefGoogle Scholar
  31. Ohr, S. M. (1986). Electron microscope studies of dislocation emission from cracks, Scripta MetalL, 20:1501–1506.CrossRefGoogle Scholar
  32. Paxton, A. T., Gumbsch, P. and Methfessel, M. (1991). A quantum mechanical calculation of the theoretical strength of metals, Phil. Mag. Lett., 63:267–274.CrossRefGoogle Scholar
  33. Peierls, R. E. (1940). The size of a dislocation, Proc. Phys. Soc., 52:34–37.CrossRefGoogle Scholar
  34. Rice, J. R. (1968a). A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., 35:379–386.Google Scholar
  35. Rice, J. R. (1968b). Mathematical analysis in the mechanics of fracture, Ch. 3 of Liebowitz, H., editor, Fracture: An Advanced Treatise (vol. 2, Mathematical Fundamentals), Academic Press, NY, pages 191–311. Google Scholar
  36. Rice, J. R. (1985). Conserved integrals and energetic forces, In Bilby, B. A., Miller, K. J. and Willis, J. R., Fundamentals of Deformation and Fracture (Eshelby Memorial Symposium), Cambridge University Press, pages 33–56. Google Scholar
  37. Rice, J. R. (1987). Mechanics of brittle cracking of crystal lattices and interfaces, In Latanision, R. M. and Jones, R. H., editors, Chemistry and Physics of Fracture, Martinus Nijhoff Publishers, Dordrecht, pages 23–43. Google Scholar
  38. Rice, J. R. (1988). Elastic fracture mechanics concepts for interfacial cracks, In J. Appl. Mech., 55:98–103.CrossRefGoogle Scholar
  39. Rice, J. R. (1992). Dislocation nucleation from a crack tip: an analysis based on the Peierls concept, to be published in J. Mech. Phys. Solids, 40:239–271.CrossRefGoogle Scholar
  40. Rice J. R., Suo, Z. and Wang, J.-S. (1990). Mechanics and thermodynamics of brittle interfacial failure in bimaterial systems, In Rühle, M., Evans, A. G., Ashby, M. F. and Hirth, J. P., editors, Metal-Ceramic Interfaces, Pergamon Press, Oxford, pages 269–294.Google Scholar
  41. Rice, J. R., and Thomson, R. M. (1974). Ductile vs. brittle behavior of crystals, Phil. Mag., 29:73–97. CrossRefGoogle Scholar
  42. Rice, J. R., and Wang, J.-S. (1989). Embrittlement of interfaces by solute segregation, Mat. Sci. and Engr., A107:23–40.CrossRefGoogle Scholar
  43. Saeedvafa, M. (1991). Orientation dependence of fracture in copper bicrystals with symmetric tilt boundaries, submitted to Mech. Mat.Google Scholar
  44. Schoeck, G. (1991). Dislocation emission from crack tips, Phil. Mag., 63:111–120.CrossRefGoogle Scholar
  45. Stroh, A. H. (1958). Dislocations and cracks in anisotropic elasticity, Phil. Mag. 3:625–646.MathSciNetMATHCrossRefGoogle Scholar
  46. Suo, Z. (1989). Mechanics of interface fracture, Ph.D. Thesis, Div. of Applied Sciences, Harvard University, Cambridge, MA, USA.Google Scholar
  47. Sun, Y. (1991). Unpublished work, on EAM fits for a-Fe, Al, and Ni.Google Scholar
  48. Sun, Y., Beltz, G. E. and Rice, J. R. (1992). Research in progress, on embedded atom models as a basis for estimating normal stress effects in dislocation nucleation.Google Scholar
  49. Sun, Y., and Rice, J. R. (1992). Research in progress, on the anisotropic elastic formulation of dislocation nucleation.Google Scholar
  50. Sun, Y., Rice, J. R. and Truskinovsky, L. (1991). Dislocation nucleation versus cleavage in Ni3Al and Ni, In Johnson, L. A., Pope, D. T. and Stiegler, J. O., editors, High-Temperature Ordered Intermetallic Alloys, Materials Research Society, 213:243–248.Google Scholar
  51. Tyson, W. R. (1975). Surface energies of solid metals, Canadian Metal-lurgical Quarterly, 14:307–314.Google Scholar
  52. Vitek, V. (1968). Intrinsic stacking faults in body-centered cubic crystals, Phil. Mag., 18:773–786.CrossRefGoogle Scholar
  53. Vitek, V., Lejcek, L. and Bowen, D. K. (1972). On the factors controlling the structure of dislocation cores in bcc crystals, In Gehlen, P. C., Beeler, J. R. and Jaffee, R. I., editors, Interatomic Potentials and Simulation of Lattice Defects, Plenum Press, New York, pages 493–508.Google Scholar
  54. Wang, J.-S., and Anderson, P. M. (1991). fVacture behavior of embrittled fee metal bicrystals and its misorientation dependence, Acta Met., 39:779–789.CrossRefGoogle Scholar
  55. Weertman, J. (1981). Crack tip blunting by dislocation pair creation and separation, Phil. Mag., 43:1103–1123.CrossRefGoogle Scholar
  56. Willis, J. R. (1967). A comparison of the fracture criteria of Griffith and Barenblatt, J. Mech. Phys. of Solids, 15:151–162.CrossRefGoogle Scholar
  57. Yamaguchi, M., Vitek, V. and Pope, D. (1981). Planar faults in the lattice, stability and structure, Phil. Mag., 43, 1027–1044.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • J. R. Rice
  • G. E. Beltz
  • Y. Sun

There are no affiliations available

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