International Journal of Fracture

, Volume 216, Issue 2, pp 135–148 | Cite as

Intergranular fracture prediction and microstructure design

  • Shmuel Osovski
  • Alan Needleman
  • Ankit SrivastavaEmail author
Original Paper


A model based on discrete unit events coupled with a graph search algorithm is developed to predict intergranular fracture. The model is based on two hypotheses: (i) the key unit event associated with intergranular crack propagation is the interaction of a grain boundary crack with a grain boundary segment located at an angle with the initial crack plane; and (ii) for a given crack path, the overall crack growth resistance can be calculated using the crack growth resistance of a collection of unit events. Next, using a directed graph containing the connectivity of grain boundary junctions and the distances between them, and crack deflection versus crack growth resistance data, a directed graph in the J-resistance space is created. This graph contains information on the crack growth resistance for all possible crack paths in a given grain microstructure. Various crack growth resistance curves are then calculated including those corresponding to: (i) a local resistance minimum; (ii) a global minimum; and (iii) for verification, a path specified by microstructure-based finite element calculations. The results show that the proposed method based on discrete unit events and graph search can predict the crack path and the crack growth resistance for cracks that propagate from one grain boundary junction to another. The proposed computationally inexpensive model can be used to design material microstructures with improved intergranular fracture resistance, and/or to assess the overall crack growth resistance of materials with a known distribution of grain morphology.


Grain boundaries Crack path Fracture toughness Graph search Microstructure design 



The financial support provided by the Pazy foundation young researchers award Grant # 1176 (SO), U.S. National Science Foundation Grant CMMI - 1663130 (AS), and European Union’s Horizon2020 Programme (Excellent Science, Marie-Sklodowska - Curie Actions, H2020 - MSCA - RISE - 2017) under REA Grant agreement 777896 (Project QUANTIFY, SO and AS) are gratefully acknowledged. We are also grateful for the high performance research computing resources provided by Texas A&M University.


  1. Arafin MA, Szpunar JA (2009) A new understanding of intergranular stress corrosion cracking resistance of pipeline steel through grain boundary character and crystallographic texture studies. Corros Sci 51:119–128CrossRefGoogle Scholar
  2. Belytschko T, Chiapetta RL, Bartel HD (1976) Efficient large scale non-linear transient analysis by finite elements. Int J Numer Methods Eng 10:579–596CrossRefGoogle Scholar
  3. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224CrossRefGoogle Scholar
  4. Bendsøe MP, Sigmund O (2004) Topology optimization: theory, methods, and applications, 2nd edn. Springer, BerlinCrossRefGoogle Scholar
  5. Cantwell PR, Ma S, Bojarski SA, Rohrer GS, Harmer MP (2016) Expanding time temperature-transformation (TTT) diagrams to interfaces: a new approach for grain boundary engineering. Acta Mater 106:78–86CrossRefGoogle Scholar
  6. Chapman CD, Saitou K, Jakiela MJ (1994) Genetic algorithms as an approach to configuration and topology design. J Mech Des 116:1005–1012CrossRefGoogle Scholar
  7. Cheng G, Jiang Z (1992) Study on topology optimization with stress constraints. Eng Optim 20:129–148CrossRefGoogle Scholar
  8. Dijkstra EW (1959) A note on two problems in connexion with graphs. Numerische Mathematik 1:269–271CrossRefGoogle Scholar
  9. Frazier WE (2014) Metal additive manufacturing: a review. J Mater Eng Perform 23:1917–1928CrossRefGoogle Scholar
  10. Fullwood DT, Niezgoda SR, Adams BL, Kalidindi SR (2010) Microstructure sensitive design for performance optimization. Prog Mater Sci 55:477–562CrossRefGoogle Scholar
  11. Gurson AL (1975) Plastic flow and fracture behavior of ductile materials incorporating void nucleation, growth and interaction. Ph.D. thesis, Brown University, Providence, RIGoogle Scholar
  12. Hao S, Moran B, Liu WK, Olson GB (2003) A hierarchical multi-physics model for design of high toughness steels. J Comput Aided Mater Des 10:99–142CrossRefGoogle Scholar
  13. Hao S, Liu WK, Moran B, Vernerey F, Olson GB (2004) Multi-scale constitutive model and computational framework for the design of ultra-high strength, high toughness steels. Comput Methods Appl Mech Eng 193:1865–908CrossRefGoogle Scholar
  14. Herderick E (2011) Additive manufacturing of metals: a review. In: Proceedings of MS&T’11. Additive Manufacturing of Metals, Columbus, OHGoogle Scholar
  15. Hosokawa A, Wilkinson DS, Kang J, Maire E (2013) Onset of void coalescence in uniaxial tension studied by continuous X-ray tomography. Acta Mater 61:1021–1036CrossRefGoogle Scholar
  16. Hyun S, Torquato S (2001) Designing composite microstructures with targeted properties. J Mater Res 16:280–285CrossRefGoogle Scholar
  17. James KA, Waisman H (2014) Failure mitigation in optimal topology design using a coupled nonlinear continuum damage model. Comput Methods Appl Mech Eng 268:614–631CrossRefGoogle Scholar
  18. Kahziz M, Morgeneyer TF, Mazire M, Helfen L, Bouaziz O, Maire E (2016) In situ 3D synchrotron laminography assessment of edge fracture in dual-phase steels: quantitative and numerical analysis. Exp Mech 56:177–195CrossRefGoogle Scholar
  19. Kim T, Hong KT, Lee KS (2003) The relationship between the fracture toughness and grain boundary character distribution in polycrystalline NiAl. Intermetallics 11:33–39CrossRefGoogle Scholar
  20. Kim CS, Rollett AD, Rohrer GS (2006) Grain boundary planes: new dimensions in the grain boundary character distribution. Scr Mater 54:1005–1009CrossRefGoogle Scholar
  21. Kirsch U (1990) On singular topologies in optimum structural design. Struct Optim 2:133–142CrossRefGoogle Scholar
  22. Kobayashi S, Maruyama T, Tsurekawa S, Watanabe T (2012) Grain boundary engineering based on fractal analysis for control of segregation-induced intergranular brittle fracture in polycrystalline nickel. Acta Mater 60:6200–6212CrossRefGoogle Scholar
  23. Kobayashi S, Maruyama T, Saito S, Tsurekawa S, Watanabe T (2014) In situ observations of crack propagation and role of grain boundary microstructure in nickel embrittled by sulfur. J Mater Sci 49:4007–4017CrossRefGoogle Scholar
  24. Kulkarni AJ, Krishnamurthy K, Deshmukh S, Mishra R (2004) Microstructural optimization of alloys using a genetic algorithm. Mater Sci Eng A 372:213–20CrossRefGoogle Scholar
  25. Larsen UD, Signund O, Bouwsta S (1997) Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio. J Microelectromech Syst 6:99–106CrossRefGoogle Scholar
  26. Lee CY (1961) An algorithm for path connections and its applications. IRE Trans Electron Comput 3:346–365CrossRefGoogle Scholar
  27. Liu R, Kumar A, Chen Z, Agrawal A, Sundararaghavan V, Choudhary A (2015) A predictive machine learning approach for microstructure optimization and materials design. Sci Rep 5:11551CrossRefGoogle Scholar
  28. Lütjering G, Williams JC (2007) Titanium, 2nd edn. Springer, BerlinGoogle Scholar
  29. Mantri SA, Choudhuri D, Behera A, Cotton JD, Kumar N, Banerjee R (2015) Influence of fine-scale alpha precipitation on the mechanical properties of the beta titanium alloy beta-21S. Metall Mater Trans A 46:2803–2808CrossRefGoogle Scholar
  30. McDowell DL (2007) Simulation-assisted materials design for the concurrent design of materials and products. JOM 59:21–5CrossRefGoogle Scholar
  31. McDowell DL, Olson GB (2009) Concurrent design of hierarchical materials and structures. Scientific modeling and simulations. Springer, Dordrecht, pp 207–40Google Scholar
  32. Needleman A, Tvergaard V, Bouchaud E (2012) Prediction of ductile fracture surface roughness scaling. J Appl Mech 79:031015CrossRefGoogle Scholar
  33. Olson GB (1997) Computational design of hierarchically structured materials. Science 277:1237–42CrossRefGoogle Scholar
  34. Osovski S, Srivastava A, Williams JC, Needleman A (2015a) Grain boundary crack growth in metastable titanium \(\beta \) alloys. Acta Mater 82:167–178CrossRefGoogle Scholar
  35. Osovski S, Srivastava A, Ponson L, Bouchaud E, Tvergaard V, Ravi-Chandar K, Needleman A (2015b) The effect of loading rate on ductile fracture toughness and fracture surface roughness. J Mech Phys Solids 76:20–46CrossRefGoogle Scholar
  36. Pan J, Saje M, Needleman A (1983) Localization of deformation in rate sensitive porous plastic solids. Int J Fract 21:261–278CrossRefGoogle Scholar
  37. Paris PC, Tada H, Zahoor A, Ernst H (1979) The theory of instability of the tearing modes in elastic-plastic crack growth. In: Landes JD, Begley JA, Clarke GA (eds) Elastic-plastic fracture, ASTM STP 668. American Society for Testing and Materials, pp 5–36Google Scholar
  38. Peirce D, Shih CF, Needleman A (1984) A tangent modulus method for rate dependent solids. Comput Struct 18:875–887CrossRefGoogle Scholar
  39. Randle V, Owen G (2006) Mechanisms of grain boundary engineering. Acta Mater 54:1777–1783CrossRefGoogle Scholar
  40. Rice J (1968) A path-independant integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35:379–386CrossRefGoogle Scholar
  41. Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization method. In: Smart structures and materials’ 97. International Society for Optics and Photonics, pp 52–60Google Scholar
  42. Sigmund O, Torquato S, Aksay IA (1998) On the design of 1-3 piezocomposites using topology optimization. J Mater Res 13:1038–1048CrossRefGoogle Scholar
  43. Srivastava A, Ponson L, Osovski S, Bouchaud E, Tvergaard V, Needleman A (2014) Effect of inclusion density on ductile fracture toughness and roughness. J Mech Phys Solids 63:62–79CrossRefGoogle Scholar
  44. Srivastava A, Osovski S, Needleman A (2017) Engineering the crack path by controlling the microstructure. J Mech Phys Solids 100:1–20CrossRefGoogle Scholar
  45. Takahashi Y, Kondo H, Asano R, Arai S, Higuchi K, Yamamoto Y, Muto S, Tanaka N (2016) Direct evaluation of grain boundary hydrogen embrittlement: a micro-mechanical approach. Mater Sci Eng A 661:211–216CrossRefGoogle Scholar
  46. Torquato S (2005) Microstructure optimization. Handbook of materials modeling. Springer, Berlin, pp 2379–2396CrossRefGoogle Scholar
  47. Torquato S, Hyun S (2001) Effective-medium approximation for composite media: realizable single-scale dispersions. J Appl Phys 89:1725–1729CrossRefGoogle Scholar
  48. Tvergaard V (1981) Influence of voids on shear band instabilities under plane strain conditions. Int J Fract 17:389–407CrossRefGoogle Scholar
  49. Tvergaard V (1982a) On localization in ductile materials containing spherical voids. Int J Fract 18:237–252Google Scholar
  50. Tvergaard V (1982b) Influence of void nucleation on ductile shear fracture at a free surface. J Mech Phys Solids 30:399–425CrossRefGoogle Scholar
  51. Tvergaard V, Needleman A (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 32:157–169CrossRefGoogle Scholar
  52. Uchic MD, Holzer L, Inkson BJ, Principe EL, Munroe P (2007) Three-dimensional microstructural characterization using focused ion beam tomography. MRS Bull 32:408–416CrossRefGoogle Scholar
  53. Ueda T, Helfen L, Morgeneyer T (2014) In situ laminography study of three-dimensional individual void shape evolution at crack initiation and comparison with Gurson–Tvergaard–Needleman-type simulations. Acta Mater 78:254–270CrossRefGoogle Scholar
  54. van Dijk NP, Maute K, Langelaar M, Van Keulen F (2013) Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim 48:437–472CrossRefGoogle Scholar
  55. Williams JC, Froes FH, Chesnutt JC, Rhodes CG, Berryman RG (1978) Development of high fracture toughness titanium alloys. Toughness and fracture behavior of titanium, ASTM STP 651. American Society for Testing and Materials, pp 64–114Google Scholar
  56. Xia L, Breitkopf P (2015) Design of materials using topology optimization and energy-based homogenization approach in Matlab. Struct Multidiscip Optim 52:1229–1241CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Shmuel Osovski
    • 1
  • Alan Needleman
    • 2
  • Ankit Srivastava
    • 2
    Email author
  1. 1.Faculty of Mechanical EngineeringTechnion - Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of Materials Science and EngineeringTexas A&M UniversityCollege StationUSA

Personalised recommendations