Strength of Materials

, Volume 50, Issue 6, pp 925–936 | Cite as

Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method

  • S. H. Mirmohammad
  • M. Safarabadi
  • M. Karimpour
  • M. R. M. AlihaEmail author
  • F. Berto

One of the most important challenges of storing fluids in thin walled pressure vessels under internal pressure is preventing crack propagation. At low temperatures, steel shows brittle crack propagation characteristic, which is highly dangerous. In this paper, a new numerical model is presented, in order to investigate the reinforcement of a cracked thin walled pressure vessel by composite patch. The extended finite element method (XFEM) technique is used to model brittle crack propagation through the thickness of a thin-walled pressure vessel utilizing the multi-scaling technique. Crack propagation in the thickness of a pressure vessel was studied utilizing the combination of XFEM approach in fracture mechanics and multi-scaling technique. Then, the critical energy, which is the maximum strain energy that the pressure vessel can absorb before the brittle crack starts to propagate, was calculated using the numerical techniques of XFEM. In order to increase the critical energy, cohesive elements and composite patches with different stacking sequence, which were extracted from previous experimental and analytical studies, were used, and the best stacking sequence was identified using the current XFEM code. Moreover, the optimization was carried out using the traditional optimization technique for reinforcing with composite patches, which was based on the optimum ratio of the increased critical energy to the thickness of the reinforcement. Results obtained show that, keeping constant the reinforcing thickness and changing the stacking angle, the maximum energy capacity is increased by 7–11%. Also, by increasing the thickness of the reinforcement, a significant growth in strain energy capacity (up to 40%) is observed. The Hashin damage criterion was used to ensure that none of the laminas’ damage during the crack propagation is critical.


thin-walled pressure vessel extended finite element method fiber composite reinforcement multi-scaling technique fracture mechanics Hashin damage criterion 


  1. 1.
    T. L. Anderson, Fracture Mechanics: Fundamentals and Applications, CRC Press (2005).Google Scholar
  2. 2.
    S. Mohammadi, Extended Finite Element Method: For Fracture Analysis of Structures, John Wiley & Sons (2008).Google Scholar
  3. 3.
    O. Rabinovitch and Y. Frostig, “Delamination failure of RC beams strengthened with FRP strips – a closed-form high-order and fracture mechanics approach,” J. Eng. Mech., 127, No. 8, 852–861 (2001).CrossRefGoogle Scholar
  4. 4.
    P. Colombi, “Reinforcement delamination of metallic beams strengthened by FRP strips: fracture mechanics based approach,” Eng. Fract. Mech., 73, No. 14, 1980– 1995 (2006).CrossRefGoogle Scholar
  5. 5.
    D. Bruno, R. Carpino, and F. Greco, “Modelling of mixed mode debonding in externally FRP reinforced beams,” Compos. Sci. Technol., 67, Nos. 7–8, 1459–1474 (2007).CrossRefGoogle Scholar
  6. 6.
    F. Greco, P. Lonetti, and P. N. Blasi, “An analytical investigation of debonding problems in beams strengthened using composite plates,” Eng. Fract. Mech., 74, No. 3, 346–372 (2007).CrossRefGoogle Scholar
  7. 7.
    A. J. Fawkes, D. R. J. Owen, and A. R. Luxmoore, “An assessment of crack tip singularity models for use with isoparametric elements,” Eng. Fract. Mech., 11, No. 1, 143–159 (1979).CrossRefGoogle Scholar
  8. 8.
    T. Belytschko and T. Black, “Elastic crack growth in finite elements with minimal remeshing,” Int. J. Num. Meth. Eng., 45, No. 5, 601–620 (1999).CrossRefGoogle Scholar
  9. 9.
    N. Moës, J. Dolbow, and T. Belytschko, “A finite element method for crack growth without remeshing,” Int. J. Num. Meth. Eng., 46, No. 1, 131–150 (1999).CrossRefGoogle Scholar
  10. 10.
    P. M. A. Areias and T. Belytschko, “Analysis of three-dimensional crack initiation and propagation using the extended finite element method,” Int. J. Num. Meth. Eng., 63, No. 5, 760–788 (2005).CrossRefGoogle Scholar
  11. 11.
    J. M. Melenk and I. Babuška, “The partition of unity finite element method: basic theory and applications,” Comput. Meth. Appl. M., 139, Nos. 1–4, 289–314 (1996).CrossRefGoogle Scholar
  12. 12.
    T. Belytschko, N. Moës, S. Usui, and C. Parimi, “Arbitrary discontinuities in finite elements,” Int. J. Num. Meth. Eng., 50, No. 4, 993–1013 (2001).CrossRefGoogle Scholar
  13. 13.
    O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method for Solid and Structural Mechanics, Butterworth-Heinemann (2005).Google Scholar
  14. 14.
    P. M. A. Areias and T. Belytschko, “Non-linear analysis of shells with arbitrary evolving cracks using XFEM,” Int. J. Num. Meth. Eng., 62, No. 3, 384–415 (2005).CrossRefGoogle Scholar
  15. 15.
    P. R. Budarapu, R. Gracie, S. P. A. Bordas, and T. Rabczuk “An adaptive multiscale method for quasi-static crack growth,” Comput. Mech., 53, No. 6, 1129–1148 (2014).CrossRefGoogle Scholar
  16. 16.
    K. Sharma, I. V. Singh, B. K. Mishra, and V. Bhasin, “Numerical modeling of part-through cracks in pipe and pipe bend using XFEM,” Proc. Mater. Sci., 6, 72–79 (2014).CrossRefGoogle Scholar
  17. 17.
    B. Zhang, C. Ye, B. Liang, et al., “Ductile failure analysis and crack behavior of X65 buried pipes using extended finite element method,” Eng. Fail. Anal., 45, 26–40 (2014).CrossRefGoogle Scholar
  18. 18.
    Z. Hashin, “Failure criteria for unidirectional fiber composites,” J. Appl. Mech., 47, No. 2, 329–334 (1980).CrossRefGoogle Scholar
  19. 19.
    M. F. S. Al-Khalil, P. D. Soden, R. Kitching, and M. J. Hinton, “The effects of radial stresses on the strength of thin-walled filament wound GRP composite pressure cylinders,” Int. J. Mech. Sci., 38, No. 1, 97–120 (1995).CrossRefGoogle Scholar
  20. 20.
    P. F. Liu, L. J. Xing, and J. Y. Zheng, “Failure analysis of carbon fiber/epoxy composite cylindrical laminates using explicit finite element method,” Compos. Part B-Eng., 56, 54–61 (2014).CrossRefGoogle Scholar
  21. 21.
    D. Roylance, Mechanical Properties of Materials, Massachusetts Institute of Technology (2008), pp. 51–78.Google Scholar
  22. 22.
    Abaqus User’s Guide, Abaqus 6.13 Documentation (2013).Google Scholar
  23. 23.
    H. T. Hahn and S. W. Tsai, Introduction to Composite Materials, CRC Press (1980).Google Scholar
  24. 24.
    M. M. Shahryarifard, M. Golzar, and M. Safarabadi, “Investigation of the geometrical parameters effect on laminated GFRP/steel circular tube joints,” in: Proc. of the Int. Conf. on Composites Pipes, Vessels & Tanks (Tehran, Iran, 2015).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • S. H. Mirmohammad
    • 1
  • M. Safarabadi
    • 1
  • M. Karimpour
    • 1
  • M. R. M. Aliha
    • 2
    Email author
  • F. Berto
    • 3
  1. 1.Mechanical Engineering DepartmentCollege of Engineering of the University of TehranTehranIran
  2. 2.Welding and Joining Research Center, School of Industrial EngineeringIran University of Science and Technology (IUST)TehranIran
  3. 3.Department of Engineering Design and MaterialsNorwegian University of Science and TechnologyTrondheimNorway

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