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Effectiveness of the stress solutions of 3-D V-notched/cracked structures by using extended boundary element method


In this paper, the extended boundary element method (XBEM) is proposed to calculate the stress fields of the three-dimensional (3-D) V-notched/cracked structure based on the linear elasticity theory. The 3-D V-notched/cracked structure contains two parts, a small sector column around the 3-D V-notch/crack tip and the outer region without the tip region. The stress fields in a small sector column are firstly expressed by asymptotic series expansions. Then the expressions of the small sector column are embedded into the boundary integral equation for the outer region. Finally, the whole displacement and stress fields of the 3-D V-notched/cracked structure are obtained by the XBEM. Two numerical examples are given to demonstrate the effectiveness and accuracy of the proposed 3-D XBEM.

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  1. Lu S, Huang QQ (2002) Dual BEM for sif analysis of 3D structures with complex loads. Acta Mech Sin 34(5):715–725

    Google Scholar 

  2. Noda N (2004) Stress intensity formulas for three-dimensional crack sin-homogeneous and bonded dissimilar materials. Eng Fract Mech 71(2004):1–15

    Article  Google Scholar 

  3. Raju IS, Newman JC (1979) Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Eng Fract Mech 11(2):817–829

    Article  Google Scholar 

  4. Hedia HSM (2006) The effect of the crack/bondline separation distance on the stress intensity factor for opening mode: a three-dimensional finite element analysis and experimental work. Materialprufung 48(11–12):551–557

    Google Scholar 

  5. Fehl BD, Truman KZ (1999) An evaluation of fracture mechanics quarter point displacement techniques used for computing stress intensity factors. Eng Struct 21(5):406–415

    Article  Google Scholar 

  6. Xie D, Qian Q, Li CA (2009) Numerical calculation method and engineering in fracture mechanics. Science Press, Beijing

    Google Scholar 

  7. Raju IS, Newman JC (1977) Three-dimensional finite-element analysis of finite-thickness fracture specimens. NASA TN D-8414

  8. Sternberg E, Sadowsky MA, Chicago ILL (1949) Three-dimensional solution for the stress concentration around a circular hole in a plate of arbitrary thickness. J Appl Mech 16(1):27–36

    MathSciNet  Article  Google Scholar 

  9. Krishnaswamy S, Jin ZH, Batra RC (1998) Stress concentration in an elastic crossest plate undergoing extensional deformations. J Appl Mech 65(1):66–70

    Article  Google Scholar 

  10. Yang Z, Huo CY, Zhao XW, Guo WL (2004) Three-dimensional elastic stress fields in finite thickness plate with a hole. J Xi’an Jiaotong Univ 38(9):971–974 (in Chinese)

    Google Scholar 

  11. Toribio J, Kharin V (2009) Finite-deformation analysis of the crack-tip fields under cyclic loading. Int J Solids Struct 46(9):1937–1952

    Article  Google Scholar 

  12. Khosravani MR, Zolfagharian A (2020) Fracture and load-carrying capacity of 3D-printed cracked components. Extreme Mech Lett 37:100692

    Article  Google Scholar 

  13. Aliha M, Bahmani A, Akhondi S (2015) Fracture and fatigue analysis for a cracked carabiner using 3D finite element simulations. Strength Mater 47:890–902

    Article  Google Scholar 

  14. Pierres E, Ba Ietto MC, Gravouil A (2010) A two-scale extended finite element method for modelling 3D crack growth with interfacial contact. Comput Methods Appl Mech Eng 199(17–20):1165–1177

    MathSciNet  Article  Google Scholar 

  15. Sukumar N, Chopp DL, Béchet E, Moës N (2010) Three-dimensional non-planar crack growth by a coupled extended finite element and fast marching method. Int J Numer Methods Eng 76(5):727–748

    MathSciNet  Article  Google Scholar 

  16. Li C, Niu ZR, Hu ZJ, Hu B, Cheng CZ (2019) Effectiveness of the stress solutions in notch/crack tip regions by using extended boundary element method. Eng Anal Bound Elem 108(1):1–13

    MathSciNet  MATH  Google Scholar 

  17. Niu ZR, Cheng CZ, Ye JQ, Naman R (2009) A new boundary element approach of modeling singular stress fields of plane V-notch problems. Int J Solids Struct 46:2999–3008

    MathSciNet  Article  Google Scholar 

  18. Yosibash Z, Szabó BA (1996) A note on numerically computed eigen-functions and generalized stress intensity factors associated with singular points. Eng Fract Mech 54:593–595

    Article  Google Scholar 

  19. Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plates in tension. J Appl Mech 19:526–528

    Article  Google Scholar 

  20. Niu ZR, Ge DL, Cheng CZ, Ye JQ, Naman R (2009) Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials. Appl Math Model 33(1):1776–1792

    MathSciNet  Article  Google Scholar 

  21. Cheng CZ, Ge RY, Niu ZR, Zhou HL (2012) Evaluation of the stress singularity order for three-dimensional V-notch. Chin J Solid Mech 33(6):623–629 (in Chinese)

    Google Scholar 

  22. Qian J, Long YQ (1994) The expression of stress and strain at the tip of three dimensional notch. Appl Math Mech 15(3):199–208 (in Chinese)

    MathSciNet  MATH  Google Scholar 

  23. Niu ZR, Wang XX, Zhou HL (2014) A regularization algorithm for the nearly singular integrals in 3-D BEM. Chin J Theor Appl Mech 46(3):417–427 (in Chinese)

    Google Scholar 

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This work was supported by the National Natural Science Foundation of China (No. 11272111) and the Doctoral Initiative Fund (No. 2020QDZ08).

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Correspondence to Cong Li.

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Li, C., Hu, B. & Niu, Z. Effectiveness of the stress solutions of 3-D V-notched/cracked structures by using extended boundary element method. J Eng Math 135, 5 (2022).

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  • Asymptotic series expansion
  • Extended boundary element method
  • Stress field
  • Three-dimensional
  • V-notch/crack