International Journal of Fracture

, Volume 193, Issue 1, pp 59–75 | Cite as

Analytical homogenization modeling and computational simulation of intergranular fracture in polycrystals

  • L. Benabou
  • Z. Sun
Original Paper


Failure at grain boundaries has a critical effect on the overall fracture behaviour of polycrystalline aggregates, as is the case in many metals. In the case of dynamic embrittlement, segregation of impurities occurs at grain boundaries, lowering their cohesive strength. The material response is then dominantly determined by grain boundary properties. The self-consistent scheme is extended to account for grain boundary decohesion by using a cohesive law to represent crack initiation and propagation. After introducing the imperfect interface conditions into the Eshelby’s equivalent inclusion solution, the effective tensile response and brittle intergranular fracture of a Cu–Ni–Si alloy is predicted. The proposed analytical model allows for the identification of parameters for both crystal plasticity and cohesive constitutive laws, from a single macroscopic tensile curve. Afterwards, multiscale computations of artificial microstructures are done using the analytically calibrated values of material parameters. Comparison of the results with experimental data shows a satisfactory agreement.


Intergranular fracture Mean-field modeling Imperfect interfaces Full-field simulation Finite element method 



The authors wish to express their gratitude to Albert Cerrone, of the Cornell Fracture Group in Cornell University, for his valuable advice on the use of DREAM3D and his help for the insertion of cohesive elements in the 3D synthetic microstructures.


  1. Achenbach JD, Zhu H (1989) Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites. J Mech Phys Solids 37:381–393CrossRefGoogle Scholar
  2. Benveniste Y (1985) The effective mechanical behaviour of composite materials with imperfect contact between the constituents. Mech Mater 4:197–208CrossRefGoogle Scholar
  3. Berveiller M, Zaoui A (1979) An extension of the self-consistent scheme to plastically flowing polycrystal. J Mech Phys Solids 26:325–344CrossRefGoogle Scholar
  4. Cerrone AR, Heber G, Wawrzynek PA, Paulino GH, Ingraffea AR (2011) Modeling microstructurally small fatigue cracking processes in an aluminum alloy with the PPR cohesive zone model. In: Proceedings of EMI 2011, BostonGoogle Scholar
  5. Cavalcante-Neto JB, Wawrzynek PA, Carvalho TM, Martha LF, Ingraffea AR (2001) An algorithm for three-dimensional mesh generation for arbitrary regions with cracks. Eng Comput 17:75–91CrossRefGoogle Scholar
  6. Cavalcante-Neto JB, Martha LF, Wawrzynek PA, Ingraffea AR (2005) A back-tracking procedure for optimization of simplex meshes. Commun Numer Meth En 21(12):711–722Google Scholar
  7. Cuitino AM, Ortiz M (1993) Computational modelling of single crystals. Model Simul Mater Sci 1:225–263CrossRefGoogle Scholar
  8. Dugdale S (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8:100–104CrossRefGoogle Scholar
  9. Duvaut G, Lions JL (1972) Les inéquations en mécanique et en physique. Dunod, ParisGoogle Scholar
  10. Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc Lond A Math 241:376–396CrossRefGoogle Scholar
  11. Field DA (1988) Laplacian smoothing and Delaunay triangulations. Commun Appl Numer Methods 4(6):709–712CrossRefGoogle Scholar
  12. Groeber M, Ghosh S, Uchic MD, Dimiduk DM (2008a) A framework for automated analysis and simulation of 3D polycrystalline microstructures. Part 1: statistical characterization. Acta Mater 56(6):1257–1273CrossRefGoogle Scholar
  13. Groeber M, Ghosh S, Uchic MD, Dimiduk DM (2008b) A framework for automated analysis and simulation of 3D polycrystalline microstructures. Part 2: synthetic structure generation. Acta Mater 56(6):1274–1287CrossRefGoogle Scholar
  14. Hashin Z (1991) The spherical inclusion with imperfect interface. J Appl Mech T ASME 58:444–449CrossRefGoogle Scholar
  15. Hill R (1965a) A self-consistent mechanics of composite materials. J Mech Phys Solids 13:213–222CrossRefGoogle Scholar
  16. Hill R (1965b) Continuum micro-mechanisms of elastoplastic polycrystals. J Mech Phys Solids 13:89–101CrossRefGoogle Scholar
  17. Kröner E (1958) Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einskristalls. Zeitschrift fur Physik 151:504–518CrossRefGoogle Scholar
  18. Kröner E (1961) Zur plastischen verformung des Vielkristalls. Acta Metall 9:155–161CrossRefGoogle Scholar
  19. Lee HK, Pyo SH (2007) Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces. Int J Solids Struct 44:8390–8406CrossRefGoogle Scholar
  20. Méric L, Cailletaud G (1991) Single crystal modeling for structural calculations: part 2-finite element implementation. J Eng Mater-T ASME 113:171–182CrossRefGoogle Scholar
  21. Misra RDK, Prasad VS, Rao PR (1996) Dynamic embrittlement in an age-hardenable copper-chromium. Scr Mater 35:129–133CrossRefGoogle Scholar
  22. Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21:571–574CrossRefGoogle Scholar
  23. Mura T (1987) Micromechanics of defects in solids. Kluwer Academic Publishers, DordrechtGoogle Scholar
  24. Nathani H, Misra RDK (2004) Characteristics of intermediate temperature dynamic embrittlement of age hardenable copper-chromium alloys. Mater Sci Technol Ser 20:546–549CrossRefGoogle Scholar
  25. Othmani Y, Delannay L, Doghri I (2011) Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law. Int J Solids Struct 48:3326–3335CrossRefGoogle Scholar
  26. Priester L (2011) Joints de grains et plasticité cristalline. Hermes, ParisCrossRefGoogle Scholar
  27. Qu J (1993a) Eshelby tensor for an elastic inclusion with slightly weakened interface. J Appl Mech-T ASME 60:1048–1050CrossRefGoogle Scholar
  28. Qu J (1993b) The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mech Mater 14:269–281CrossRefGoogle Scholar
  29. Rose JH, Ferrante J, Smith JR (1981) Universal binding energy curves for metals and bimetallic interfaces. Phys Rev Lett 47:675–678CrossRefGoogle Scholar
  30. Simo JC, Kennedy JG, Govindjee S (1988) Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms. Int J Numer Methods Eng 26:2161–2185CrossRefGoogle Scholar
  31. Sun Z (2008) Contribution à l’étude des propriétés mécaniques des alliages de cuivre à durcissement structural Cu–Ni–Si et Cu–Cr–Zr : influence de la microstructure et des conditions d’utilisation. PhD Thesis, LyonGoogle Scholar
  32. Sun Z, Laitem C, Vincent A (2008) Dynamic embrittlement at intermediate temperature in a Cu–Ni–Si alloy. Mater Sci Eng A 477:145–152CrossRefGoogle Scholar
  33. Tan H, Huang Y, Liu C, Geubelle PH (2005) The Mori-Tanaka method for composite materials with nonlinear interface debonding. Int J Plast 21:1890–1918CrossRefGoogle Scholar
  34. Tan H, Huang Y, Liu C, Geubellle PH (2006) Effect of nonlinear interface debonding on the constitutive model of composite materials. Int J Multiscale Comput Eng 4:147–167CrossRefGoogle Scholar
  35. Tan H, Huang Y, Liu C, Ravichandran G, Paulino GH (2007) Constitutive behaviors of composites with interface debonding: the extended Mori-Tanaka method for uniaxial tension. Int J Fract 146:139–148CrossRefGoogle Scholar
  36. Tvergaard V, Hutchinson JW (1992) The relation between crack growth resistance and fracture process parameters in elastic–plastic solids. J Mech Phys Solids 40:1377–1397CrossRefGoogle Scholar
  37. Van der Ven A, Ceder G (2003) Impurity-induced van der Waals transition during decohesion. Phys Rev B 67:060101Google Scholar
  38. Wu Z, Sullivan JM (2003) Multiple material marching cubes algorithm. Int J Numer Methods Eng 58(2):189–207CrossRefGoogle Scholar
  39. Zhong Z, Yu XB, Meguid SA (2004) 3D micromechanical modeling of particulate composite materials with imperfect interface. Int J Multiscale Comp Eng 2:172–187CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.LISVUniversité de Versailles Saint Quentin-en-YvelinesVersaillesFrance
  2. 2.LASMISUniversité de Technologie de TroyesTroyesFrance

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