Elastic-Plastic and Quasi-Brittle Fracture

  • Xiaozhi HuEmail author
  • Li Liang
Living reference work entry


This chapter presents a simple Elastic-Plastic and Quasi-Brittle Fracture Approach that can be used by graduate students and design engineers in their structural integrity analyses to determine safe design loads or to predict critical failure loads. The simple model can also be used to extrapolate the tensile strength and fracture toughness KIC of various metals and composites from elastic-plastic fracture results of small test samples, which otherwise would require impractically large specimens for the fracture toughness measurements and well-polished specimens for the strength measurements. Elastic-plastic fracture of metals and quasi-brittle fracture of coarse-structured brittle composites such as concrete and rock can also be fairly accurately predicted by the predictive model if both strength and toughness are known. The model is derived by a simple modification of the stress intensity factor commonly used in Linear Elastic Fracture Mechanics (LEFM). Experimental results of AISI-1040 like carbon steel and Aluminum Alloy 6061 and concrete are used to verify the new approach. The new design model can be applied to different fracture situations from strength-controlled failure to LEFM KIC-controlled fracture and the elastic-plastic and quasi-brittle fracture region (or non-LEFM fracture) between the two distinct strength and toughness territories. The model also explains the limitations of LEFM imposed by the stringent requirements for the fracture toughness KIC measurements and the reason why Elastic and Plastic Fracture Mechanics (EPFM) is necessary to most structural integrity analyses.


  1. 1.
    Atkins AG, Mai YW. Elastic and plastic fracture: metals, polymers, ceramics, composites, biological materials. Chichester: Ellis Horwood Limited; 1988.Google Scholar
  2. 2.
    Anderson TL. Fracture mechanics fundamentals and applications. Boca Raton: CRC Press; 1995.zbMATHGoogle Scholar
  3. 3.
    Hertzberg RW. Deformation and fracture mechanics of engineering materials. 4th ed. New York: Wiley; 1996.Google Scholar
  4. 4.
    Callister WD Jr, Rethwisch DG. Materials science and engineering an introduction. 8th ed. Hoboken: Wiley; 2010.Google Scholar
  5. 5.
    Dugdale DS. Yielding of steel sheets containing slits. J Mech Phys Solids. 1960;8:110–04.CrossRefGoogle Scholar
  6. 6.
    Well AA. Unstable crack propagation in metals: cleavage and fast fracture. In: Proceedings of the Crack Propagation Symposium, vol. 1, Paper 84, Cranfield; 1961.Google Scholar
  7. 7.
    Irwin GR. Plastic zone near a crack and fracture toughness. Sagamore Research Conference Proceedings, vol. 4; 1961.Google Scholar
  8. 8.
    Barenblatt GI. Mathematical theory of equilibrium cracks in brittle fracture. In: Advances in applied mechanics, vol. VII. New York: Academic; 1962.Google Scholar
  9. 9.
    Burdekin FM, Sone DEW. The crack opening displacement approach to fracture mechanics in yielding materials. J Strain Anal. 1966;1:145–53.CrossRefGoogle Scholar
  10. 10.
    Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech. 1968;35:379–86.CrossRefGoogle Scholar
  11. 11.
    Cotterell B, Reddel JK. The essential work of plane stress ductile fracture. Int J Fract. 1977;13:267–77.Google Scholar
  12. 12.
    Cotterell B. The past, present and future of fracture mechanics. Int J Fract. 2002;69:533–53.Google Scholar
  13. 13.
    Zhu XK, Joyce JA. Review of fracture toughness (G, K, J, CTOD, CTOA) testing and standardization. Eng Fract Mech. 2012;85:1–46.CrossRefGoogle Scholar
  14. 14.
    Wang YS, Hu XZ, Liang L, Zhu WC. Determination of tensile strength and fracture toughness of concrete using notched 3-p-b specimens. Eng Fract Mech. 2016;160:67–77.CrossRefGoogle Scholar
  15. 15.
  16. 16.
    ASTM E1820-15a. Standard test method for measurement of fracture toughness. West Conshohocken: ASTM International; 2015.Google Scholar
  17. 17.
    Hu XZ, Wittmann F. Size effect on toughness induced by crack close to free surface. Eng Fract Mech. 2000;65:209–21.CrossRefGoogle Scholar
  18. 18.
    Duan K, Hu XZ. Specimen boundary induced size effect on quasi-brittle fracture. Strength Fract Complex. 2004;2:47–68.Google Scholar
  19. 19.
    Duan K, Hu XZ, Wittmann F. Scaling of quasi-brittle fracture: boundary and size effect. Mech Mater. 2006;38:128–41.CrossRefGoogle Scholar
  20. 20.
    Hu XZ, Duan K. Size effect: influence of proximity of fracture process zone to specimen boundary. Eng Fract Mech. 2007;74:1093–100.CrossRefGoogle Scholar
  21. 21.
    Hu XZ, Duan K. Size effect and quasi-brittle fracture: the role of FPZ. Int J Fract. 2008;154:3–14.CrossRefGoogle Scholar
  22. 22.
    Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook. 3rd ed. New York: ASME Press; 2000.CrossRefGoogle Scholar
  23. 23.
    Hu XZ, Guan JF, Geng L. A simple elastic and plastic fracture model. Inter J Fract. 2017.Google Scholar
  24. 24.
    Cotterell B, Mai YW. Fracture mechanics of cementitious materials. London: Blackie Academic & Professional, An Imprint of Chapman & Hall; 1996.Google Scholar
  25. 25.
    Swartz SE, Yap ST. Evaluation of recently proposed recommendations for the determination of fracture parameters for concrete in bending. In: Proceedings of 8th International Conference on Experimental Stress Analysis; 1986. p. 233–44.Google Scholar
  26. 26.
    Swartz SE, Refai TME. Influence of size effects on opening mode fracture parameters for precracked concrete beams in bending. In: Shah SP, Swartz SE, editors. Fracture of concrete and rock; SEM-RILEM international conference, Houston. 17–19 June; 1987. p. 242–54.Google Scholar
  27. 27.
    Quality Certificate Y25-05-05. Anyang Iron and Steel Company Limited (
  28. 28.
    Amiri S, Lecis N, Manes A, Giglio M. A study of a micro-indentation technique for estimating the fracture toughness of Al6061-T6. Mech Res Commun. 2014;58:10–6.CrossRefGoogle Scholar
  29. 29.
    NASGRO 4.11. Reference Manual, NASA Johnson Space Center; 2004.Google Scholar
  30. 30.
    Hillerborg A, Modeer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res. 1976;6(6):773–82.CrossRefGoogle Scholar
  31. 31.
    Bažant ZP, Oh BH. Crack band theory for fracture concrete. Mater Struct. 1983;16(3):155–77.Google Scholar
  32. 32.
    Jenq YS, Shah SP. Two parameter fracture model for concrete. J Eng Mech. 1985;111(10): 1227–41.CrossRefGoogle Scholar
  33. 33.
    Karihaloo BL, Nallathambi P. An improved effective crack model for the determination of fracture toughness of concrete. Cem Concr Res. 1989;19(4):603–10.CrossRefGoogle Scholar
  34. 34.
    Hu XZ, Wittmann FH. Fracture energy and fracture process zone. Mater Struct. 1992;25:319–26.CrossRefGoogle Scholar
  35. 35.
    Carpinteri A. Fractal nature of material microstructure and size effects on apparent mechanical properties. Mech Mater. 1994;18(2):89–101.CrossRefGoogle Scholar
  36. 36.
    Karihaloo BL, Abdalla HM, Xiao QZ. Size effect in concrete beams. Eng Fract Mech. 2003;70:979–93.CrossRefGoogle Scholar
  37. 37.
    Abdalla HM, Karihaloo BL. Determination of size-independent specific fracture energy of concrete from three-point bend and wedge splitting tests. Mag Concr Res. 2003;55(2):133–42.CrossRefGoogle Scholar
  38. 38.
    Duan K, Hu XZ, Wittmann FH. Boundary effect on concrete fracture and non-constant fracture energy distribution. Eng Fract Mech. 2006;70(16):2257–68.CrossRefGoogle Scholar
  39. 39.
    Yang ST, Hu XZ, Wu ZM. Influence of local fracture energy distribution on maximum fracture load of three-point-bending notched concrete beams. Eng Fract Mech. 2011;78:3289–99.CrossRefGoogle Scholar
  40. 40.
    Karihaloo BL, Murthy AM, Iyer NR. Determination of size-independent specific fracture energy of concrete mixes by the tri-linear model. Cem Concr Res. 2013;49:82–8.CrossRefGoogle Scholar
  41. 41.
    Cifuentes H, Alcalde M, Medina F. Measuring the size-independent fracture energy of concrete. Strain. 2013;49:54–9.CrossRefGoogle Scholar
  42. 42.
    Hoover CG, Bazant ZP. Comparison of the Hu-Duan boundary effect model with the size-shape effect law for quasi-brittle fracture based on new comprehensive fracture tests. J Eng Mech ASCE. 2014;140:480–6.CrossRefGoogle Scholar
  43. 43.
    Caglar Y, Sener S. Size effect tests of different notch depth specimens with support rotation measurements. Eng Fract Mech. 2016;157:43–55.CrossRefGoogle Scholar
  44. 44.
    Guan JF, Hu XZ, Li QB. In-depth analysis of notched 3-p-b concrete fracture. Eng Fract Mech. 2016;165:57.CrossRefGoogle Scholar
  45. 45.
    Hu XZ. Size effect on tensile softening relation. Mater Struct. 2010;44:129–38.CrossRefGoogle Scholar

Copyright information

© Crown Copyright 2018

Authors and Affiliations

  1. 1.School of Mechanical and Chemical EngineeringUniversity of Western AustraliaPerthAustralia
  2. 2.School College of Resources and Civil EngineeringNortheastern UniversityShenyangPeople’s Republic of China

Personalised recommendations