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Challenges Below the Grain Scale and Multiscale Models

  • Hussein M. Zbib
  • David F. Bahr
Chapter

Abstract

The main point in this chapter is the multiscale aspect of plastic deformation and strength in crystalline materials. Particular emphasis is placed on models and experiments below the grain level where the length scale is too small for classical continuum constitutive equations to be meaningful but yet is not small enough to be treated within an atomistic framework. More specifically, this chapter deals with experimental advances and theoretical models that have been developed recently to characterize dislocations at the subgrain level. Emphases is placed on recent developments in the nanoindentation technique to measure mechanical properties in small volumes, and on most recent experiments which attempt to measure mechanical properties using more traditional experiments such as tension and compression of microscale specimens. A discussion of a multiscale modeling framework is provided illuminating the important role discrete dislocation dynamics models play in thisarea.

Keywords

Burger Vector Slip Plane Representative Volume Element Screw Dislocation Crystal Plasticity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgment

The support from the Office of Basic Energy Science at the DOE under grant number DE-FG02–07ER46435 is gratefully acknowledged.

References

  1. Akarapu, A., H. M. Zbib and D. F. Bahr (2010). “Analysis of heterogeneous defromation and dislocation dynamics in single crystal micropillars under compression.” Int. J. Plast. 26: 239–257.CrossRefGoogle Scholar
  2. Akasheh, F., H. M. Zbib, J. P. Hirth, R. G. Hoagland and A. Misra (2007a). “Dislocation dynamics analysis of dislocation intersections in nanoscale multilayer metallic composites.” J. Appl. Phys. 101: 084314.CrossRefGoogle Scholar
  3. Akasheh, F., H. M. Zbib, J. P. Hirth, R. G. Hoagland and A. Misra (2007b). “Interactions between glide dislocations and parallel interfacial dislocations in nanoscale strained layers.” J. Appl. Phys. 102: 034324.CrossRefGoogle Scholar
  4. Al’shitz, V. I. (1992). “The phonon-dislocation interaction and its role in dislocation dragging and thermal resistivity.” Elastic Strain and Dislocation Mobility (ed. V. L. Indenbom and J. Lother), Chapter11 Elsevier Science Publishers B.V., Amsterdam.Google Scholar
  5. Ashmawi, W. M. and M. A. Zikry (2002). “Prediction of grain-boundary interfacial mechanisms in polycrystalline materials.” J. Eng. Mater. Tech. 124(1): 88–96.CrossRefGoogle Scholar
  6. Bahr, D. F. and W. W. Gerberich (1996). “Pile up and plastic zone size around indentations.” Met. Mater. Trans. 27A: 3793–3800.CrossRefGoogle Scholar
  7. Bahr, D. F., D. E. Kramer and W. W. Gerberich (1998). “Non-linear deformation mechanisms during nanoindentation.” Acta Mater. 46: 176–182.CrossRefGoogle Scholar
  8. Bahr, D. F., D. E. Wilson and D. A. Crowson (1999). “Energy considerations regarding yield points during indentation.” J. Mater. Res. 14: 2269–2275.CrossRefGoogle Scholar
  9. Benzerga, A. A. and N. F. Shaver (2006). “Size dependence of mechanical properties of single crystals under uniform deformation.” Scripta Mater. 54:1937.CrossRefGoogle Scholar
  10. Bower, A. F., N. A. Fleck, A. Needleman and N. Ogboma (1993). “Indentation of a power law creeping solid.” Proc. R. Soc. 441A:97–124.Google Scholar
  11. Bulatov, V., M. Tang and H. M. Zbib (2001). “Crystal plasticity from dislocation dynamics.” MRS Bull. 26 (191–195).Google Scholar
  12. Bull, S. J., T. F. Page and E. H. Yoffe (1989). “An explanation of the indentation size effect in ceramics.” Philos. Mag. Lett. 59: 281–288.CrossRefGoogle Scholar
  13. Canova, G. R., Y. Brechet and L. P. Kubin (1992). “3D dislocation simulation of plastic instabilities by work?softening in alloys.” Modelling of Plastic Deformation and Its Engineering Applications (ed. S.I. Anderson etal), Riso National Laboratory, Roskilde, Denmark.Google Scholar
  14. Chang, S. C. and H. C. Chen (1995). “The determination of F.C.C. crystal orientation by indentation.” Acta Metall. Mater. 43: 2501–2505.Google Scholar
  15. Choi, Y., K. J. V. Vliet, J. Li and S. Suresh (2003). “Size effects on the onset of plastic deformation during nanoindentation of thin films and patterned lines.” J. Appl. Phys. 94: 6050–6058.CrossRefGoogle Scholar
  16. Corcoran, S. G., R. J. Colton, E. T. Lilleodden and W. W. Gerberich (1997). “Anomalous plastic deformation at surfaces: Nanoindentation of gold single crystals” Phys. Rev. B 55: 16057–16060.CrossRefGoogle Scholar
  17. Cotrell, A. H. (1953). Dislocations and Plastic Flow in Crystals. Oxford, OxfordPress.Google Scholar
  18. Deshphande, V. S., A. Needleman and E. Van der Giessen (2005). “Plasticity size effects in tension and compression of single crystals.” J. Mech. Phys. Solids 53:2661.CrossRefGoogle Scholar
  19. Devincre, B. and L. P. Kubin (1997). Mater. Sci. Eng. A234–236: 8.CrossRefGoogle Scholar
  20. DeWit, R. (1960). “The continuum theory of stationary dislocations.” Solid State Phys 10: 249–292.CrossRefGoogle Scholar
  21. Diaz de la Rubia, T., H. M. Zbib, M. Victoria, A. Wright, T. Khraishi and M. Caturla (2000). “Flow localization in irradiated materials: a multiscale modeling approach.” Nature 406: 871–874.CrossRefGoogle Scholar
  22. Dimiduk, D. M., M. D. Uchic and T. A. Parthasarathy (2005). “Size-affected single slip behavior of pure nickel microcrystals.” Acta Mater. 53:4065.CrossRefGoogle Scholar
  23. Doerner, M. F. and W. D. Nix (1986). “A method for interpreting the data from depth-sensing indentation instruments.” J. Mater. Res. 1: 601–609.CrossRefGoogle Scholar
  24. Dugdale, D. S. (1954). “Cone indentation experiments.” J. Mech. Phys. Solids 2: 265–277.CrossRefGoogle Scholar
  25. Elmustafa, A. A., J. A. Eastman, M. N. Ritter, J. R. Weertman and D. S. Stone (2000). “Indentation size effect: large grained aluminum versus nanocrystalline aluminum-zirconium alloys.” Scripta Mater. 43: 951–955.CrossRefGoogle Scholar
  26. Field, J.S., Swain, M.V. (1993). “Simple predictive model for spherical indentation” J. Mater. Res. 8:297–306CrossRefGoogle Scholar
  27. Fischer-Cripps, A. C. (2004). Nanoindentation, Spriner, NewYork.Google Scholar
  28. Fischer-Cripps, A. C. (2006). “Critical review of analysis and interpretation of nanoindentation test data.” Surf. Coatings Tech 200: 4153–4165.CrossRefGoogle Scholar
  29. Fleck, N. A., G. M. Muller, M. F. Ashby and J. W. Hutchinson (1994). “Strain gradient plasticity: theory and experiment.” Acta Metall. Mater. 42: 475–487.CrossRefGoogle Scholar
  30. Frick, C. P., B. G. Clark, S. Orso, A. S. Schneider and E. Artz (2008). “Size effect on strength and strain hardening of small-scale [111] nickel compression pillars.” Mater. Sci. Eng.A.Google Scholar
  31. Gaillard, Y., C. Tromas and J. Woirgrad (2006). “Quantitative analysis of dislocation pile-ups nucleated during nanoindentation in MgO.” Acta Mater. 54: 1409–1417.CrossRefGoogle Scholar
  32. Gane, N. and F. P. Bowden (1968). “Microdeformation of solids.” J. Appl. Phys. 39: 1432–1435.CrossRefGoogle Scholar
  33. Gerberich, W. W., N. I. Tymiak, J. C. Grunlan, M. F. Horstemeyer and M. I. Baskes (2002). “Interpretations of indentation size effects.” J. Appl. Phys. 69: 433–442.MATHGoogle Scholar
  34. Greer, J. R. and W. D. Nix (2006). “Nanoscale gold pillars strengthened through dislocation starvation.” Phys. Rev. B 73: 245210.CrossRefGoogle Scholar
  35. Harvey, S., H. Huang, Vannkataraman, and W. W. Gerberich (1993). “Microscopy and microindentation mechanics of single crystal Fe-3 wt. %Si: Part I. Atomic force microscopy of a small indentation.” J. Mater. Res. 8: 1291–1299.Google Scholar
  36. Hiratani, M. and H. M. Zbib (2002). “Stochastic dislocation dynamics for dislocation-defects interaction.” J. Eng. Mater. Technol. 124: 335–341.CrossRefGoogle Scholar
  37. Hirth, J. P. (1992). Injection of Dislocations into Strained Multilayer Structures. Semiconductors and Semimetals, Academic, New York. 37: 267–292.Google Scholar
  38. Hirth, J. P. and J. Lothe (1982). Theory of Dislocations. New York,Wiley.Google Scholar
  39. Hirth, J. P., M. Rhee and H. M. Zbib (1996). “Modeling of deformation by a 3d simulation of multipole, curved dislocations.” J. Comput. Aided Mater Des. 3: 164–166.CrossRefGoogle Scholar
  40. Hirth, J. P., H. M. Zbib and J. Lothe (1998). “Forces on high velocity dislocations.” Model. Simul. Mater. Sci. Eng.. 6: 165–169.CrossRefGoogle Scholar
  41. Horstemeyer, M. F., M. I. Baskes and S. J. Plimpton (2001). Acta Mater. 49:4363.CrossRefGoogle Scholar
  42. Hoyt, S. L. (1924). “The ball indentation hardness test.” Trans. Am. Soc. Steel Treating 6:396.Google Scholar
  43. Huang, H., N. Ghoniem, T. Diaz de la Rubia, Rhee, Z. H.M. and J. P. Hirth (1999). “Development of physical rules for short range interactions in BCC crystals.” ASME-JEMT 121: 143–150.Google Scholar
  44. Johnson, K. L. (1970). “The correlation of indentation experiments.” J. Mech. Phys. Solids 18: 115–126.CrossRefGoogle Scholar
  45. Johnson, K. L. (1985). Contact Mechanics, Cambridge University Press, Cambridge.MATHGoogle Scholar
  46. Kelchner, C. L., S. J. Plimpton and J. C. Hamilton (1998). “Dislocation nucleation and defect structure during surface indentation” Phys. Rev. B 58: 11085–11088.CrossRefGoogle Scholar
  47. Khan, A., H. M. Zbib and D. A. Hughes (2001). Stress Patterns of Deformation Induced Planar Dislocation Boundaries. MRS, San Francisco.Google Scholar
  48. Khan, A., H. M. Zbib and D. A. Hughes (2004). “Modeling planar dislocation boundaries using a multi-scale approach.” Int. J. Plast. 20: 1059–1092.MATHCrossRefGoogle Scholar
  49. Khraishi, T. A. and H. M. Zbib (2002a). “Dislocation dynamics simulations of the interaction between a short rigid fiber and a glide dislocation pile-up.” Comput. Mater. Sci. 24: 310–322.CrossRefGoogle Scholar
  50. Khraishi, T. A. and H. M. Zbib (2002b). “Free-surface effects in 3d dislocation dynamics: formulation and modeling.” J. Eng. Mater. Tech.124: 342–351.CrossRefGoogle Scholar
  51. Khraishi, T., H. M. Zbib, T. Diaz de la Rubia and M. Victoria (2000). “Modeling of flow localization and hardening in irradiated materials using discrete dislocation dynamics (DD).” Acta. Metall.Google Scholar
  52. Khraishi, T., H. M. Zbib, T. Diaz de la Rubia and M. Victoria (2002). “Localized deformation and hardening in irradiated metals: three-dimensional discrete dislocation dynamics simulations.” Metall. Mater. Trans. 33B: 285–296.Google Scholar
  53. Khraishi, T. A. and Y.-L. Shen (2004). Int. J. Plast. 20: 1039.MATHCrossRefGoogle Scholar
  54. Kiely, J. D. and J. E. Houston (1998). “Nanomechanical properties of Au(111), (001), and (110) surfaces.” Phys. Rev. B57.Google Scholar
  55. Kiener, D., W. Grosinger, G. Dehm and R. Pippan (2008). “A further step towards an understanding of size-dependent crystal plasticity: In situ tension experiments of miniaturized single-crystal copper samples.” Acta Mater. 56:580.CrossRefGoogle Scholar
  56. King, R. B. (1987). “Elastic analysis of some punch problems for a layered medium.” Int. J. Solids Struct. 23: 1657–1664.MATHCrossRefGoogle Scholar
  57. Kramer, D., H. Huang, M. Kriese, J. Robach, J. Nelson, A. Wright, D. F. Bahr and W. W. Gerberich (1998). “Yield strength predictions from plastic zone around nanocontacts.” Acta Mater. 47: 333–343.CrossRefGoogle Scholar
  58. Lepinoux, J. and L. P. Kubin (1987). “The dynamic organization of dislocation structures: a simulation.” Scripta Metall. 21: 833–838.CrossRefGoogle Scholar
  59. Li, J., J. W. Morris, Jennerjohn, D. F. Bahr and Levin “in press.” J. Mater.Res.Google Scholar
  60. Li, X. and B. Bhushan (2002). “A review of nanoindentation continuous stiffness measurement technique and its applications.” Mater. Char. 48:11–36.CrossRefGoogle Scholar
  61. Ma, Q. and D. R. Clarke (1995). “Size dependent hardness of silver single crystals ” J. Mater. Res. 10: 853–863.CrossRefGoogle Scholar
  62. Mason, W. and D. MacDonald (1971). “Damping of dislocations in niobium by phonon viscosity.” J. Appl. Phys. 42:1836.CrossRefGoogle Scholar
  63. Meyer, E. (1908). Ziets d. Vereines Deustscher Ingenieure 52: 645.Google Scholar
  64. Michalske, T. A. and J. E. Houston (1998). “Dislocation nucleation at nano-scale mechanical contacts.” Acta Mater. 46: 391–396.CrossRefGoogle Scholar
  65. Minor, A. M., E. T. Lilleodden, E. A. Stach and J. W. Morris (2004). “Direct observations of incipient plasticity during nanoindentation of Al.” J. Mater. Res. 19: 176–182.CrossRefGoogle Scholar
  66. Minor, A. M., S. A. Syed-Asif, Z. Shan, E. A. Stach, E. Cyrankowski, T. J. Wyrobek and O.L.Warren(2006). “A new view of the onset of plasticity during the nanoindentation of aluminium.” Nat. Mater.5.Google Scholar
  67. Misra, A., Hirth, J.P., Hoagland, R.G. (2005). “Length scale dependent deformation mechanisms in incoherent metallic multilayered composites.” Acta Mater. 53:4817–4824.CrossRefGoogle Scholar
  68. Motz, C., D. Weygand, J. Senger and P. Gumbsch (2009). “Initial dislocation structures in 3-D discrete dislocation dynamics and their influence on microscale plasticity.” Acta Mater. 57: 1744–1754.CrossRefGoogle Scholar
  69. Ngan, A. H. W. and P. C. Wo (2006). “Delayed plasticity in nanoindentation of annealed crystals.” Phil. Mag. 86: 1287–1304.CrossRefGoogle Scholar
  70. Nibur, K. A., F. Akasheh and D. F. Bahr (2007). “Analysis of dislocation mechanisms around indentations through slip step observations.” J. Mater. Res. 42: 889–900.Google Scholar
  71. Nibur, K. A. and D. F. Bahr (2003). “Identifying slip systems around indentations in FCC metals.” Scripta Mater. 49: 1055–1060.CrossRefGoogle Scholar
  72. Nix, W. D. and H. Gao (1998). “Indentation size effects in crystalline materials: a law for strain gradient plasticity.” J. Mech. Phys. Solids 46: 411–425.MATHCrossRefGoogle Scholar
  73. Oliver, W. C. and G. M. Pharr (1992). “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments.” J. Mater. Res. 7: 1564–1583.CrossRefGoogle Scholar
  74. Oliver, W. C. and G. M. Pharr (2004). “Review: Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology.” J.Mater. Res. 19:3–20.CrossRefGoogle Scholar
  75. Page, T. F., W. C. Oliver and C. J. McHargue (1992). “Deformation behavior of ceramic crystals subjected to very low load (nano)indentations.” J. Mater. Res. 7: 450–473.CrossRefGoogle Scholar
  76. Pethica, J. B. and D. Tabor (1979). “Contact of characterized metal surfaces at very low loads: Deformation and adhesion.” Surf. Sci. 89: 182–190.CrossRefGoogle Scholar
  77. Raabe, D. (1998). “Introduction of a hybrid model for the discrete 3d simulation of dislocation dynamics.” Comput. Mater. Sci. 11:1–15.CrossRefGoogle Scholar
  78. Rhee, M., J. P. Hirth and H. M. Zbib (1994a). “On the bowed out tilt wall model of flow stress and size effects in metal matrix composites.” Scripta Metall. Mater. 31: 1321–1324.CrossRefGoogle Scholar
  79. Rhee, M., J. P. Hirth and H. M. Zbib (1994b). “A superdislocation model for the strengthening of metal matrix composites and the initiation and propagation of shear bands.” Acta Metall. Mater. 42: 2645–2655.CrossRefGoogle Scholar
  80. Rhee, M., H. M. Zbib, J. P. Hirth, H. Huang and T. D. d. L. Rubia (1998). “Models for long/short range interactions in 3D dislocatoin simulation.” Model. Simul. Mater. Sci. Eng. 6: 467–492.Google Scholar
  81. Samuels, L. E. and T. O. Mulheam (1957). “An experimental investigation of the deformed zone associated with indentation hardness impressions ” J. Mech. Phys. Solids 5: 125–134.CrossRefGoogle Scholar
  82. Sayed-Asif, S. A. and J. B. Pethica (1997). “Nanoindentation creep of single-crystal tungsten and gallium arsenide.” Philos. Mag. A 76(6): 1105–1118.CrossRefGoogle Scholar
  83. Schuh, C. A., J. K. Mason and A. C. Lund (2005). “Quantitative insight into dislocation nucleation from high-temperature nanoindentation experiments.” Nat. Mater. 4: 617–621.CrossRefGoogle Scholar
  84. Shaw, Z. N., R. K. Mishra, S. A. Syed-Asif, O. L. Warren and A. M. Minor (2008). “Mechanical annealing and source-limited deformation in submicrometre-diameter Ni crystals.” Nature 7:115.Google Scholar
  85. Shehadeh, M., E. M. Bringa, H. M. Zbib, J. M. McNaney and B. A. Remington (2006). “Simulation of shock-induced plasticity including homogeneous and Heterogeneous dislocation nucleation.” Appl. Phys. Lett. 89: 171918.CrossRefGoogle Scholar
  86. Shehadeh, M. A., H. M. Zbib and T. D. de la Rubia (2005a). “Modeling the dynamic deformation and patterning in FCC single crystals at high strain rates: dislocation dynamic plasticity analysis” Philos. Mag. A 85 1667–1684.Google Scholar
  87. Shehadeh, M. A., H. M. Zbib and T. D. de la Rubia (2005b). “Multiscale dislocation dynamics simulations of shock compressions in copper single crystal.” Int. J. Plast. 21: 2396–2390.CrossRefGoogle Scholar
  88. Sneddon (1965). “The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of Arbitrary Profile.” Int. J. Eng. Sci. 3:47–56.Google Scholar
  89. Stelmashenko, N.A., Walls, M.G., Brown, L.M., Milman, Y.V. (1993) “Microindentations on W and Mo oriented single crystals: An STM study.” Acta Metall. Mater. 41:2855–2865.CrossRefGoogle Scholar
  90. Stolken, J. and A. G. Evans (1998). “A microbend test method for measuring the plasticity length scale.” Acta Mater. 46: 5109–5115.CrossRefGoogle Scholar
  91. Strader, J. H., S. Shim, B. H., W. C. Oliver and G. M. Pharr (2006). “An experimental evaluation of the constant β relating the contact stiffness to the contact area in nanoindentation.” Philos. Mag. 86: 5285–5298.Google Scholar
  92. Tabor, D. (1951). The Hardnessof Metals, Oxford Press, Oxford.Google Scholar
  93. Tang, H., K. W. Schwartz and H. D. Espinosa (2007). “Dislocation escape related size effects in single-crystal micropillars under uniaxial compression.” Acta Mater. 55:1607.CrossRefGoogle Scholar
  94. Tang, H., K. W. Schwartz and H. D. Espinosa (2008). “Dislocation-source shutdown and the plastic behavior of single-crystal micropillars.” Phys. Rev. Lett. 100: 185503.CrossRefGoogle Scholar
  95. Tromas, C., Girard, J.C., Woigard, J. (2000). “Study by atomic force microscopy of elementary deformation mechanisms involved in low load indentations in MgO crystals.” Philos. Mag. A80:2325–2335.CrossRefGoogle Scholar
  96. Tromas, C., J. C. Girard, V. Audurier and J. Woirgrad (1999). “Study of the low stress plasticity in single-crystal MgO by nanoindentation and atomic force microscope.” J. Mater. Sci. 34: 5337–5342.CrossRefGoogle Scholar
  97. Uchic, M. D., D. M. Dimiduk, J. N. Florando and W. D. Vix (2004). “Sample dimensions influence strength and crystal plasticity.” Science 305: 986–989.CrossRefGoogle Scholar
  98. Urabe, N. and J. Weertman (1975). “Dislocation mobility in potassium and iron single crystals.” Mater. Sci. Eng. 18:41.CrossRefGoogle Scholar
  99. Van der Giessen, E. and A. Needleman (1995). “Discrete dislocation plasticity: a simple planar model.” Mater. Sci. Eng. 3: 689–735.Google Scholar
  100. VanLandingham, M. R. (2003). “Review of instrumented indentation.” J. Res. Natl. Inst. Stand. Technol. 108: 249–265.Google Scholar
  101. Volkert, C. A. and E. T. Lilleodden (2006). “Size effects in the deformation of sub-micron Au columns.” Philos. Mag. 86: 5567–5579.CrossRefGoogle Scholar
  102. von Blanckenhagen, B., P. Gumbsch and E. Artz (2003). “Dislocation sources and the flow stress of polycrystalline thin metal films.” Philos. Mag. Lett. 83:1–8.CrossRefGoogle Scholar
  103. Wasserbäch, W. (1986). “Plastic deformation and dislocation arrangemnet of Nb-34 at. % TA alloy crystals,.” Philos. Mag. A 53: 335–356.Google Scholar
  104. Weinberger, C. R., S. Aubry, S. W. Lee, W. D. Nix and W. Cai (2009). “Modelling dislocations in a free-standing thin film.” Model. Simul. Mater. Sci. Eng. 17:1–26.CrossRefGoogle Scholar
  105. Yasin, H., H. M. Zbib and M. A. Khaleel (2001). “Size and boundary effects in discrete dislocation dynamics: coupling with continuum finite element.” Mater. Sci. Eng. A309–310: 294–299.Google Scholar
  106. Yip, S., Ed. (2005). Handbook of Materials Modeling. Springer, NewYork.Google Scholar
  107. Zbib, H. M. and E. C. Aifantis (1988a). “A gradient-dependent model for the portevin-le chatelier effect.” Scripta Metall 22(8): 1331–1336.CrossRefGoogle Scholar
  108. Zbib, H. M. and E. C. Aifantis (1988b). “On the localization and post localization behavior of plastic deformation-i. on the initiation of shear bands.” Res. Mech., Int. J. Struct. Mech. Mater. Sci. 23: 261–277.Google Scholar
  109. Zbib, H. M. and E. C. Aifantis (1988c). “On the localization and post localization behavior of plastic deformation-ii. on the evolution and thickness of shear bands.” Res. Mech., Int. J. Struct. Mech. Mater. Sci. 23: 279–292.Google Scholar
  110. Zbib, H. M. and E. C. Aifantis (1988d). “On the structure and width of shear bands.” Scripta Metall 22(5): 703–708.CrossRefGoogle Scholar
  111. Zbib, H. M., S. Akarupa, F. Akasheh, C. Overman and D. F. Bahr (2009). “Deformation and size effects in small scale structures.” Plasticity 2009: Macro to nano Scale Inelastic behavior of Materials: Plasticity, Fatigue and Fracture St. Thomas.Google Scholar
  112. Zbib, H. M., T. D. de La Rubia, M. Rhee and J. P. Hirth (2000). “3D Dislocation dynamics: stress-strain behavior and hardening mechanisms in FCC and BCC metals.” J. Nucl. Mater. 276: 154–165.CrossRefGoogle Scholar
  113. Zbib, H. M. and T. Diaz de la Rubia (2002). “A multiscale model of plasticity.” Int. J. Plast. 18(9): 1133–1163.MATHCrossRefGoogle Scholar
  114. Zbib, H. M., T. Diaz de la Rubia and V. A. Bulatov (2002). “A multiscale model of plasticity based on discrete dislocation dynamics.” ASME J. Eng.. Mater. Tech.124:78–87.CrossRefGoogle Scholar
  115. Zbib, H. M. and T. A. Diaz de la Rubia (2001). “Multiscale Model of Plasticity: Patterning and Localization.” Material Science For the 21st Century, The Society of Materials Science, Japan Vol A: 341–347.Google Scholar
  116. Zbib, H. M., M. Hiratani and M. Shehadeh (2004). “Multiscale discrete dislocation dynamics plasticity. continuum scale simulation of engineering materials fundamentals - microstructures - process applications.” D. Raabe, D. Roters, F. Baralt and L.-Q. Chen, Wiley-VCH, Weinheim: 202–229.Google Scholar
  117. Zbib, H. M., M. Rhee and J. P. Hirth (1996). “3D simulation of curved dislocations: discretization and long range interactions.” Advances in Engineering Plasticity and its Applications (eds.T.Abe and T. Tsuta) Pergamon, NY:15–20.Google Scholar
  118. Zbib, H. M., M. Rhee and J. P. Hirth (1998). “On plastic deformation and the dynamcis of 3d dislocations.” Int. J. Mech. Sci. 40: 113–127.MATHCrossRefGoogle Scholar
  119. Zeng, X. H. and H. Hatmaier (2010). “Modeling size effects on fracture toughness by dislocation dynamics.” Acta Mater. 58: 301–310.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.School of Mechanical and Materials EngineeringWashington State UniversityPullmanUSA

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