International Journal of Fracture

, Volume 190, Issue 1–2, pp 53–74 | Cite as

A discrete damage zone model for mixed-mode delamination of composites under high-cycle fatigue

  • Stephen Jimenez
  • Xia Liu
  • Ravindra Duddu
  • Haim Waisman
Original Paper


A discrete damage zone model is developed to describe the mode-mix ratio and temperature dependent delamination of laminated composite materials under high cycle fatigue loading within the framework of the finite element method. In this approach, discrete nonlinear spring elements are placed at the finite element nodes of the laminate interface, and a combination of static and fatigue damage growth laws is used to define its constitutive behavior. The model is implemented in the commercial software Abaqus using the user element subroutine. The static damage model parameters are estimated from fracture mechanics principles, whereas the fatigue damage model parameters are calibrated by fitting the numerical results to published experimental data. A quadratic relation is proposed to describe the non-monotonic variation of fatigue damage model parameters with mode-mix ratio. Next, an Arrhenius relation is proposed for the temperature dependence of fatigue damage, in addition to the incorporation of the temperature dependence of critical fracture energy. The model is convergent upon mesh refinement; however, for accurate prediction the mesh size used for model calibration should be sufficiently small. The model predicted fatigue crack growth rates are in agreement with those obtained from a quadratic relation for the Paris law parameters for variable mode mix conditions, thus verifying the approach. While the model captures the temperature effects on delamination for mode I and 50 % mode II, our prediction deviates from experiments for pure mode II, since the corresponding damage mechanism entirely changes with temperature.


High-cycle fatigue Discrete cohesive zone model Damage mechanics Mixed mode delamination Composite materials 



We gratefully acknowledge the funding from our sponsors: SKJ and RD were supported by the Vanderbilt University Discovery Grant Program; XL and HW were supported by the National Science Foundation under grant CMMI-0856161.


  1. Alfano G, Crisfield MA (2001) Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues. Int J Numer Methods Eng 50:1701–1736CrossRefGoogle Scholar
  2. Alfano M, Furgiuele F, Leonardi A, Maletta C, Paulino GH (2009) Mode I fracture of adhesive joints using tailored cohesive zone models. Int J Fract 157:193–204CrossRefGoogle Scholar
  3. Asp LE (1998) The effects of moisture and temperature on the interlaminar delamination toughness of a carbon/epoxy composite. J Compos Technol Res 58(6):967–977CrossRefGoogle Scholar
  4. Asp LE, Sjögren A, Greenhalgh ES (2001) Delamination growth and thresholds in a carbon/epoxy composites under fatigue loading. J Compos Technol Res 23(2):55–68CrossRefGoogle Scholar
  5. Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129CrossRefGoogle Scholar
  6. Bazant ZP (1999) Size effect on structural strength: a review. Arch Appl Mech 69(9–10):703–725Google Scholar
  7. Blackman B, Dear JP, Kinloch AJ, Osiyemi S (1991) The calculation of adhesive fracture energies from double-cantilever beam test specimens. J Mater Sci Lett 10:253–256CrossRefGoogle Scholar
  8. Blackman BRK, Hadavinia H, Kinloch AJ, Paraschi M, Williams JG (2003a) The calculation of adhesive fracture energies in mode I: revisiting the tapered double cantilever beam (TDCB) test. Eng Fract Mech 70:233–248Google Scholar
  9. Blackman BRK, Hadavinia H, Kinloch AJ, Williams JG (2003b) The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints. Int J Fract 119:25–46Google Scholar
  10. Blackman BRK, Kinloch AJ, Paraschi M, Teo WS (2003c) Measuring the mode I adhesive fracture energy, \(G_{\rm IC}\), of structural adhesive joints: the results of an international round-robin. Int J Adhes Adhes 23:293–305Google Scholar
  11. Blanco N, Gamstedt EK, Asp LE, Costa J (2004) Mixed-mode delamination growth in carbon-fibre composite laminates under cyclic loading. Int J Solids Struct 41(15):4219–4235CrossRefGoogle Scholar
  12. Camanho PP, Dávila CG, De Moura MF (2003) Numerical simulation of mixed-mode progressive delamination in composite materials. J Compos Mater 37(16):1415–1424CrossRefGoogle Scholar
  13. Chandra N, Li H, Shet C, Ghonem H (2002) Some issues in the application of cohesive zone models for metal-ceramic interfaces. Int J Solids Struct 39:2827–2855CrossRefGoogle Scholar
  14. Cui WC, Wisnom MR (1993) A combined stress-based and fracture-mechanics-based model for predicting delamination in composites. Composites 24(6):467–474CrossRefGoogle Scholar
  15. de Andres A, Perez JL, Ortiz M (1999) Elastoplastic finite element analysis of three-dimensional fatigue crack growth in aluminium shafts subjected to axial loading. Int J Solids Struct 36(15):2231–2258CrossRefGoogle Scholar
  16. Do BC, Liu W, Yang QD, Su XY (2013) Improved cohesive stress integration schemes for cohesive zone elements. Eng Fract Mech, 107:14–28, ISSN 0013–7944Google Scholar
  17. Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104CrossRefGoogle Scholar
  18. Goyal VK, Johnson ER, Davila CG (2004) Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities. Compos Struct 65(3–4):434–446Google Scholar
  19. Goyal VK, Johnson ER, Goyal VK (2008) Predictive strength-fracture model for composite bonded joints. Compos Struct 82(3):434–446CrossRefGoogle Scholar
  20. Harper PW, Hallet SR (2008) Cohesive zone length in numerical simulations of composite delamination. Eng Fract Mech 75(16):4774–4792CrossRefGoogle Scholar
  21. Harper PW, Hallet SR (2010) A fatigue degradation law for cohesive interface elements development and application to composite materials. Int J Fatigue 32(11):1774–1787CrossRefGoogle Scholar
  22. Hillerborg A, Modéer M, Petersson PE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 6:773–782CrossRefGoogle Scholar
  23. Jiang W-G, Hallet SR, Green BG, Wisnom MR (2007) A concise interface constitutive law for analysis of delamination and splitting in composite materials and its application to scaled notched tensile specimens. Int J Numer Methods Eng 69(9):1982–1995CrossRefGoogle Scholar
  24. Juntti M, Asp LE, Olsson R (1999) Assessment of evaluation methods for the mixed-mode bending test. J Compos Technol Res 21(1):37–48CrossRefGoogle Scholar
  25. Kawashita Luiz F, Hallett Stephen R (2012) A crack tip tracking algorithm for cohesive interface element analysis of fatigue delamination propagation in composite materials. Int J Solids Struct 49(21):2898–2913, ISSN 0020–7683Google Scholar
  26. Khan R, Khan Z, Al-Sulaiman F, Merah N (2002) Fatigue life estimates in woven carbon fabric/epoxy composites at non-ambient temperatures. J Compos Mater 36(22):2517–2535CrossRefGoogle Scholar
  27. Landry B, LaPlante G (2012) Modeling delamination growth in composites under fatigue loadings of varying amplitudes. Compos Part B Eng 43(2):533–541CrossRefGoogle Scholar
  28. Li S, Ghosh S (2006) Multiple cohesive crack growth in brittle materials by the extended Voronoi cell finite element model. Int J Fract 141:373–393CrossRefGoogle Scholar
  29. Li S, Thouless MD, Waas AM, Schroeder JA, Zavattieri PD (2006) Mixed-mode cohesive-zone models for fracture of an adhesively bonded polymer–matrix composite. Eng Fract Mech 73(1):64–78CrossRefGoogle Scholar
  30. Liljedahl CDM, Crocombe AD, Wahab MA, Ashcroft IA (2006) Damage modelling of adhesively bonded joints. Int J Fract 141:147–161CrossRefGoogle Scholar
  31. Liu X, Duddu R, Waisman H (2012) Discrete damage zone model for fracture initiation and propagation. Eng Fract Mech 92:1–18CrossRefGoogle Scholar
  32. Mazars J (1986) A description of micro- and macroscale damage of concrete structures. Eng Fract Mech 25:729–737CrossRefGoogle Scholar
  33. Mi Y, Crisfield MA, Davies GAO, Hellweg HB (1998) Progressive delamination using interface elements. J Compos Mater 32:1246–1272CrossRefGoogle Scholar
  34. Munoz JJ, Galvanetto U, Robinson P (2006) On the numerical simulation of fatigue-driven delamination using interface elements. Int J Fatigue 28(10):1136–1146CrossRefGoogle Scholar
  35. Neu RW, Sehitoglu H (1989) Thermomechanical fatigue, oxidation, and creep: II. Life prediction. Metall Trans A Phys Metall Mater Sci 20(9):1769–1783CrossRefGoogle Scholar
  36. Nguyen O, Repetto EA, Ortiz M, Radovitzky RA (2001) A cohesive model of fatigue crack growth. Int J Fract 110(4):351–369CrossRefGoogle Scholar
  37. Nilsson K-F, Asp LE, Alpman JE, Nystedt L (2001) Delamination buckling and growth for delaminations at different depths in a slender composite panel. Int J Solids Struct 38(17):3039–3071CrossRefGoogle Scholar
  38. Paas MHJW, Schreurs PJG, Brekelmans WAM (1993) A continuum approach to brittle and fatigue damage: theory and numerical procedures. Int J Solids Struct 30(4):579–599CrossRefGoogle Scholar
  39. Paris P, Erdogan F (1963) Critical analysis of propagation laws. J Basic Eng 85:528–534CrossRefGoogle Scholar
  40. Paris P, Gomez M, Anderson W (1961) A rational analytical theory of fatigue. Trend Eng 13:9–14Google Scholar
  41. Park K, Paulino GH, Roesler JR (2009) A unified potential-based cohesive model of mixed-mode fracture. J Mech Phys Solids 57:891–908CrossRefGoogle Scholar
  42. Payan J, Hochard C (2002) Damage modelling of laminated carbon/epoxy composites under static and fatigue loadings. Int J Fatigue 24(2–4):299–306CrossRefGoogle Scholar
  43. Peerlings RHJ, Brekelmans WAM, de Borst R, Geers MGD (2000) Gradient-enhanced damage modelling of high-cyclic fatigue. Int J Numer Methods Eng 49(12):1547–1569CrossRefGoogle Scholar
  44. Pettersson KB, Neumeister JM, Kristofer Gamstedt E, Oberg H (2006) Stiffness reduction, creep, and irreversible strains in fiber composites tested in repeated interlaminar shear. Compos Struct 76(12):151–161, 2006.Google Scholar
  45. Qiu Y, Crisfield MA, Alfano G (2001) An interface element formulation for the simulation of delamination with buckling. Eng Fract Mech 68:1755–1776CrossRefGoogle Scholar
  46. Reeder JR, Crews JR (1990) Mixed-mode bending method for delamination testing. AIAA J 28(7):1270–1276CrossRefGoogle Scholar
  47. Robinson P, Besant T, Hitchings D (2000) Delamination growth prediction using a finite element approach. In: 2nd ESIS TC4 conference on fracture of polymers, composites and adhesives, Switzerland, 27:135–147Google Scholar
  48. Robinson P, Galvanetto U, Tumino D, Bellucci G, Violeau D (2005) Numerical simulation of fatigue–driven delamination using interface elements. Int J Numer Methods Eng 63(13):1824–1848CrossRefGoogle Scholar
  49. Roe KL, Siegmund T (2003) An irreversible cohesive zone model for interface fatigue crack growth simulation. Eng Fract Mech 70(2):209–232CrossRefGoogle Scholar
  50. Schellenkens JCJ, de Borst R (1993) A non-linear finite element approach for the analysis of mode-i free edge delamination in composites. Int J Solids Struct 30(9):1239–1253CrossRefGoogle Scholar
  51. Siegmund T (2004) A numerical study of transient fatigue crack growth by use of an irreversible cohesive zone model. Int J Fatigue 26(9):929–939CrossRefGoogle Scholar
  52. Sjogren A, Asp LE (2002) Effects of temperature on delamination growth in a carbon/epoxy composite under fatigue loading. Int J Fatigue 24(2–4):179–184CrossRefGoogle Scholar
  53. Skallerud B, Zhang ZL (1997) A 3D numerical study of ductile tearing and fatigue crack growth under nominal cyclic plasticity. Int J Solids Struct 34(24):3141–3161CrossRefGoogle Scholar
  54. Turon A, Costa J, Camanho PP, Davila CG (2007) Simulation of delamination in composites under high-cycle fatigue. Compos: Part A 38(11):2270–2282CrossRefGoogle Scholar
  55. van den Bosch MJ, Schreurs PJG, Geers MGD (2006) An improved description of the exponential xu and needleman cohesive zone law for mixed-mode decohesion. Int J Fract 73:1220–1234Google Scholar
  56. Walker EK (1970) Analysis of stresses and strains near the end of a crack traversing a plate the effect of stress ratio during crack propagation and fatigue for 2024–T3 and 7076–T6 aluminum. In: Effect of environment and complex load history on fatigue life, ASTM STP 462. Philadelphia: American Society for Testing and Materials, 24:1–14Google Scholar
  57. Wang Y, Waisman H (2014) Progressive delamination analysis of composite materials using xfem and a discrete damage zone model. Comput Mech. doi: 10.1007/s00466-014-1079-0
  58. Williams JG (1988) On the calculation of energy release rates for cracked laminates. Int J Fract 36(2):101–119CrossRefGoogle Scholar
  59. Xie D, Waas AM (2006) Discrete cohesive zone model for mixed-mode fracture using finite element analysis. Eng Fract Mech 73:1783–1796Google Scholar
  60. Xie D, Salvi AG, Sun C, Waas AM (2006) Discrete cohesive zone model to simulate static fracture in 2D triaxially braided carbon fiber composites. J Compos Mater 40(22):2025–2046Google Scholar
  61. Xu X-P, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42(9):1397–1434CrossRefGoogle Scholar
  62. Yang B, Ravi-Chandar KA (1998) Antiplane shear crack growth under quasistatic loading in a damaging material. Int J Solids Struct 35(28–29):3695–3715CrossRefGoogle Scholar
  63. Yang B, Mall S, Ravi-Chandar KA (2001) A cohesive zone model for fatigue crack growth in quasibrittle materials. Int J Solids Struct 38(22–23):3927–4394CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Stephen Jimenez
    • 1
  • Xia Liu
    • 2
  • Ravindra Duddu
    • 1
  • Haim Waisman
    • 2
  1. 1.Department of Civil and Environmental EngineeringVanderbilt UniversityNashvilleUSA
  2. 2.Department of Civil Engineering and Engineering MechanicsColumbia UniversityNew YorkUSA

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