Estimation of \(J{-}R\) Curves for Small-Sized COEC Specimens and Its Application Considering Crack-Tip Constraints

  • G. W. He
  • C. BaoEmail author
  • L. X. CaiEmail author
  • Y. J. Wu


The normalization method has been employed to estimate the J–resistance (\(J{-}R\)) curves for C-shaped outside edge-notched compression (COEC) specimens of SA-508 steel. The COEC specimens present the lowest \(J{-}R\) curves in comparison with other three types of specimens. Based on the \(J{-}Q{-}M\) method, the constraint parameter Q determined at the mid-plane in the through-thickness direction of the specimen is used to approximately quantify the three-dimensional crack-tip constraint for COEC specimens. Combining with the previous experiments on \(J{-}R\) curves, the constraint-corrected \(J{-}R\) curves for SA-508 steel specimens are given, which may be used to predict the fracture toughness of actual cracked components in structural integrity assessment.


Ductile fracture toughness COEC specimen Dimensionless load separation principle Normalization method Crack-tip constraint \(J{-}Q{-}M\) method 

List of Symbols


Crack length


Corrected crack length considering crack-tip blunting

\(a_{i}, a_{j}\)

Initial crack lengths of two different blunt cracked specimens


Remaining ligament


Specimen thickness


Net thickness


The specimen elastic load-line compliance


Crack opening displacement


A linearization factor


Young’s modulus


An integration constant

\(J_{\mathrm{0.2BL}}, J_{\mathrm{0.5BL}}\)

The intersections of the \(J{-}R\) curve and, respectively, the 0.2-mm and 0.5-mm offset blunt lines


Characteristic lengths (commonly set to be 1 mm)


Stress intensity factor


Resultant bending moment per unit thickness acting on the ligament


Strain hardening exponent


Applied load


Scale deviation in hydrostatic stress


Inner radius

(\(r,\vartheta \))

Polar coordinates centered at the crack-tip


Outer radius


Separation parameter


Plastic component of displacement


Specimen width

\(\Delta {\varvec{U}}_{{\varvec{p}}}\)

Incremental plastic work done

\(\varPi \)

Normalized load

\(\alpha \)

Strain hardening coefficient

\(\varepsilon _{0}\)

Yield strain

\(\sigma _{0}\)

Flow stress

\(\eta _{\mathrm{p}}\)

Plastic factor

\(\gamma \)

A factor that considers the effect of crack growth on the calculation of \(J_{\mathrm{p}}\)

\(\delta _{ij}\)

Kronecker delta

\(\tilde{\sigma }_{\mathrm{ij}} \left( {\theta , N} \right) \)

Dimensionless stress function



This work is financially supported by the National Natural Science Foundation of China (Grant No. 11872320). The authors appreciate the helpful comments and suggestions from the anonymous reviewers.


  1. 1.
    Yoshimoto K, Hirota T, Sakamoto H, et al. Applicability of miniature C (T) specimen to evaluation of fracture toughness for reactor pressure vessel steel. ASME paper no. PVP2013-97840.Google Scholar
  2. 2.
    Martínez-Pañeda E, García TE, Rodríguez C. Fracture toughness characterization through notched small punch test specimens. Mater Sci Eng A. 2016;657:422–30.CrossRefGoogle Scholar
  3. 3.
    ASTM E1820-15a. Standard test method for measurement of fracture toughness. Annual book of ASTM standards, West Conshohocken, PA: American Society for Testing and Materials; 2015.Google Scholar
  4. 4.
    BS7448. Part 1: method for determining of KIc, critical crack tip opening displacement (CTOD) and critical J values of fracture toughness for metallic materials under displacement controlled monotonic loading at quasistatic rates. London: British Standards Institution; 1991.Google Scholar
  5. 5.
    Bao C, Cai LX, Dan C. Estimation of fatigue crack growth behavior for small-sized C-shaped inside edge-notched tension (CIET) specimen using compliance technique. Int J Fatigue. 2015;81:202–12.CrossRefGoogle Scholar
  6. 6.
    Zhao XH, Cai LX, Bao C. Testing method study on fatigue crack propagation behavior of CRO specimen. Eng Mech. 2016;33:20–8 (In Chinese).CrossRefGoogle Scholar
  7. 7.
    Herrera R, Landes JD. Direct JR curve analysis: a guide to the methodology. Fracture mechanics: twenty-first symposium. West Conshohocken: ASTM International; 1990.Google Scholar
  8. 8.
    Ernst HA, Paris PC, Landes JD. Estimations on J-integral and tearing modulus T from a single specimen test record. Fracture mechanics. West Conshohocken: ASTM International; 1981.Google Scholar
  9. 9.
    Betegon C, Hancock JW. Two-parameter characterization of elastic–plastic crack-tip fields. J Appl Mech. 1991;58(1):104–10.CrossRefGoogle Scholar
  10. 10.
    O’dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter—I. Structure of fields. J Mech Phys Solids. 1991;39(8):989–1015.CrossRefGoogle Scholar
  11. 11.
    O’dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter—II. Fracture applications. J Mech Phys Solids. 1992;40(5):939–63.CrossRefGoogle Scholar
  12. 12.
    O’dowd NP, Shih CF. Two-parameter fracture mechanics: theory and applications. Fracture mechanics: twenty-fourth volume. West Conshohocken: ASTM International; 1994.Google Scholar
  13. 13.
    Hutchinson JW. Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids. 1968;16(1):13–31.CrossRefzbMATHGoogle Scholar
  14. 14.
    Rice JR, Rosengren GF. Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids. 1968;16(1):1–12.CrossRefzbMATHGoogle Scholar
  15. 15.
    Yang S, Chao YJ, Sutton MA. Complete theoretical analysis for higher order asymptotic terms and the HRR zone at a crack tip for mode I and mode II loading of a hardening material. Acta Mech. 1993;98(1–4):79–98.CrossRefzbMATHGoogle Scholar
  16. 16.
    Yang S, Chao YJ, Sutton MA. Higher order asymptotic crack tip fields in a power-law hardening material. Eng Fract Mech. 1993;45(1):1–20.CrossRefGoogle Scholar
  17. 17.
    Nikishkov GP. An algorithm and a computer program for the three-term asymptotic expansion of elastic–plastic crack tip stress and displacement fields. Eng Fract Mech. 1995;50(1):65–83.CrossRefGoogle Scholar
  18. 18.
    Nikishkov GP, Brückner-Foit A, Munz D. Calculation of the second fracture parameter for finite cracked bodies using a three-term elastic–plastic asymptotic expansion. Eng Fract Mech. 1995;52(4):685–701.CrossRefGoogle Scholar
  19. 19.
    Pluvinage G, Capelle J, Hadj Méliani M. A review of fracture toughness transferability with constraint and stress gradient. Fatigue Fract Eng Mater Struct. 2014;37(11):1165–85.CrossRefGoogle Scholar
  20. 20.
    Nikishkov GP, Matvienko YG. Elastic-plastic constraint parameter A for test specimens with thickness variation. Fatigue Fract Eng Mater Struct. 2016;39(8):939–49.CrossRefGoogle Scholar
  21. 21.
    Kotousov A, Wang CH. Effect of plate thickness on crack-tip plasticity. Int J Fract. 2001;111(3):53–8.CrossRefGoogle Scholar
  22. 22.
    Guo W. Elastoplastic three dimensional crack border field—I. Singular structure of the field. Eng Fract Mech. 1993;46(1):93–104.CrossRefGoogle Scholar
  23. 23.
    Guo W. Elastoplastic three dimensional crack border field—II. Asymptotic solution for the field. Eng Fract Mech. 1993;46(1):105–13.CrossRefGoogle Scholar
  24. 24.
    Guo W. Elasto-plastic three-dimensional crack border field—III. Fracture parameters. Eng Fract Mech. 1995;51(1):51–71.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mostafavi M, Smith DJ, Pavier MJ. Reduction of measured toughness due to outofplane constraint in ductile fracture of aluminium alloy specimens. Fatigue Fract Eng Mater Struct. 2010;33(11):724–39.Google Scholar
  26. 26.
    Mostafavi M, Smith DJ, Pavier MJ. A micromechanical fracture criterion accounting for in-plane and out-of-plane constraint. Comput Mater Sci. 2011;50(10):2759–70.CrossRefGoogle Scholar
  27. 27.
    Yang J, Wang GZ, Xuan FZ, et al. Unified characterisation of inplane and outofplane constraint based on cracktip equivalent plastic strain. Fatigue Fract Eng Mater Struct. 2013;36(6):504–14.CrossRefGoogle Scholar
  28. 28.
    Yang J, Wang GZ, Xuan FZ, et al. Unified correlation of inplane and outofplane constraints with fracture toughness. Fatigue Fract Eng Mater Struct. 2014;37(2):132–45.CrossRefGoogle Scholar
  29. 29.
    Parks DM. Three-dimensional aspects of HRR-dominance. EGF9; 1991.Google Scholar
  30. 30.
    Shih CF, O’Dowd NP, Kirk MT. A framework for quantifying crack tip constraint. Constraint effects in fracture. West Conshohocken: ASTM International; 1993.Google Scholar
  31. 31.
    O’dowd NP. Applications of two parameter approaches in elastic–plastic fracture mechanics. Eng Fract Mech. 1995;52(3):445–65.CrossRefGoogle Scholar
  32. 32.
    Zhu XK, Jang SK. J–R curves corrected by load-independent constraint parameter in ductile crack growth. Eng Fract Mech. 2001;68(3):285–301.CrossRefGoogle Scholar
  33. 33.
    Bao C, Cai LX, He GW, et al. Normalization method for evaluating J–resistance curves of small-sized CIET specimen and crack front constraints. Int J Solids Struct. 2016;94:60–75.CrossRefGoogle Scholar
  34. 34.
    Huang Y, Zhou W, Wang E. Constraintcorrected J–R curve based on threedimensional finite element analyses. Fatigue Fract Eng Mater Struct. 2014;37(10):1101–15.CrossRefGoogle Scholar
  35. 35.
    Chao YJ, Zhu XK. Constraint-modified J–R curves and its application to ductile crack growth. Int J Fract. 2000;106(2):135–60.CrossRefGoogle Scholar
  36. 36.
    Lam PS, Chao YJ, Zhu XK, et al. Determination of constraint-modified JR curves for carbon steel storage tanks. In: ASME 2002 pressure vessels and piping conference. American Society of Mechanical Engineers. 2002; p. 133–42.Google Scholar
  37. 37.
    Wang ZX, Chao YJ, Lam PS. Quantification of ductile crack growth in 18G2A steel at different constraint levels. Int J Press Vessels Pip. 2009;86(2–3):221–7.CrossRefGoogle Scholar
  38. 38.
    Sharobeam MH, Landes JD. The load separation criterion and methodology in ductile fracture mechanics. Int J Fract. 1991;47(2):81–104.CrossRefGoogle Scholar
  39. 39.
    Bao C, Cai L, Shi K, et al. Improved normalization method for ductile fracture toughness determination based on dimensionless load separation principle. Acta Mech Solida Sin. 2015;28(2):168–81.CrossRefGoogle Scholar
  40. 40.
    Chao YJ, Zhu XK, Kim Y, et al. Characterization of crack-tip field and constraint for bending specimens under large-scale yielding. Int J Fract. 2004;127(3):283–302.CrossRefzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics 2019

Authors and Affiliations

  1. 1.Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and EngineeringSouthwest Jiaotong UniversityChengduChina

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