Skip to main content
Chinese Journal of Mechanical Engineering
Download PDF
  • Published:

Analytical solutions of cracks emanating from an elliptic hole in an infinite plate under tension

Chinese Journal of Mechanical Engineering volume 27, pages 1057–1063 (2014)Cite this article

An Erratum to this article was published on 12 December 2014

Abstract

It is a common phenomenon that the cracks originating from a hole can cause structural damage in engineering. However, the fracture mechanics studies of hole edge crack problems are not sufficient. The problem of an elliptical hole with two collinear edge cracks of unequal length in an infinite plate under uniform tension at infinity is investigated. Based on the complex variable method, the analytical solutions of stress functions and stress intensity factors are provided. The stress distribution along the axes and the edge of the elliptical hole is given graphically. The numerical results show that there is obvious stress concentration near the hole and cracks, and the stresses tend to applied loads at distances far from the defect, which conform to Saint-Venant’s principle. Hence, the stress functions are proved to be right. Under special conditions, the present configuration becomes the Griffith crack, two symmetrical cracks emanating from an elliptical hole, two cracks of unequal length emanating from a circular hole, a crack at the edge of a circular hole, or a crack emanating from an elliptical hole. Compared with available results, stress intensity factors for these special shapes of ellipses and cracks show good coincidence. The stress intensity factor for two cracks of unequal length at the edge of an elliptical hole increases with the crack length and the major-to-minor axis ratio of the elliptical hole. The stress distribution in an infinite plate containing an elliptic hole with unsymmetrical cracks is given for the first time.

References

  1. BOWIE O L. Analysis of an infinite plate containing radial cracks originating at the boundary of an internal circular hole[J]. Journal of Mathematics and Physics, 1956, 35: 60–71.

    Article  MathSciNet  MATH  Google Scholar 

  2. TWEED J, ROOKE D P. The distribution of stress near the tip of a radial crack at the edge of a circular hole[J]. International Journal of Engineering Science, 1973, 11(11): 1185–1195.

    Article  MATH  Google Scholar 

  3. TWEED J, ROOKE D P. The stress intensity factor for a crack at the edge of a loaded hole[J]. International Journal of Solids and Structures, 1979, 15(11): 899–906.

    Article  MATH  Google Scholar 

  4. SCHIJVE J, LAI J. The stress intensity factor of hole edge cracks in a finite width plate[J]. International Journal of Fracture, 1990, 46(3): 37–42.

    Article  Google Scholar 

  5. SCHIJVE J. Comparison between empirical and calculated stress intensity factors of hole edge cracks[J]. Engineering Fracture Mechanics, 1985, 22(1): 49–58.

    Article  Google Scholar 

  6. LIN X B, SMITH R A. Stress intensity factors for corner cracks emanating from fastener holes under tension[J]. Engineering Fracture Mechanics, 1999, 62(6): 535–553.

    Article  Google Scholar 

  7. ABDELMOULA R, SEMANI K, LI J. Analysis of cracks originating at the boundary of a circular hole in an infinite plate by using a new conformal mapping approach[J]. Applied Mathematics and Computation, 2007, 188(2): 1891–1896.

    Article  MathSciNet  MATH  Google Scholar 

  8. STEFANESCU D, EDWARDS L, FITZPATRICK M E. Stress intensity factor correction for asymmetric through-thickness fatigue cracks at holes[J]. International Journal of Fatigue, 2003, 25(7): 569–576.

    Article  Google Scholar 

  9. ZHAO J F, XIE L Y, LIU J Z, et al. A method for stress intensity factor calculation of infinite plate containing multiple hole-edge cracks[J]. International Journal of Fatigue, 2012, 35(1): 2–9.

    Article  Google Scholar 

  10. LAI J B, ZHANG X, SCHIJVE J. An investigation of a hole-edge crack problem by a combined complex variable and least square method[J]. Engineering Fracture Mechanics, 1991, 39(4): 713–737.

    Article  Google Scholar 

  11. ISIDA M, CHEN D H, NISITANI H. Plane problems of an arbitrary array of cracks emanating from the edge of an elliptical hole[J]. Engineering Fracture Mechanics, 1985, 21(5): 983–995.

    Article  Google Scholar 

  12. YAN X Q. A numerical analysis of cracks emanating from an elliptical hole in a 2-D elasticity plate[J]. European Journal of Mechanics. A: Solids, 2006, 25(1): 142–153.

    Article  MATH  Google Scholar 

  13. GUO H M, LIU G T, PI J D. Stress analysis of an ellipse hole with a straight edge-crack by complex variable method[J]. Acta Mechanica Solida Sinica, 2007, 28(3): 308–312. (in Chinese)

    Google Scholar 

  14. GUO J H, LIU G T. Stress analysis for an elliptical hole with two straight cracks[J]. Chinese Journal of Theoretical and Applied Mechanics, 2007, 39(5): 699–703. (in Chinese)

    Google Scholar 

  15. TWEED J, MELROSE G. Cracks of unequal length at the edge of an elliptic hole in out of plane shear[J]. Applied Mathematics Letters, 1989, 2(2): 171–174

    Article  MathSciNet  MATH  Google Scholar 

  16. TWEED J, MELROSE G. Cracks at the edge of an elliptic hole in out of plane shear[J]. Engineering Fracture Mechanics, 1989, 34(3): 743–747.

    Article  MATH  Google Scholar 

  17. GUO J H, LU Z X, HAN H T, et al. The behavior of two non-symmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid[J]. European Journal of Mechanics. A: Solids, 2010, 29(4): 654–663.

    Article  Google Scholar 

  18. GUO J H, LU Z X, HAN H T, et al. Exact solutions for anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material[J]. International Journal of Solids and Structures, 2009, 46(21): 3799–3809

    Article  MATH  Google Scholar 

  19. LIU S H, SHEN Y M, LIU J X. Exact solutions for piezoelectric materials with an elliptic hole or a crack under uniform internal pressure[J]. Chinese Journal of Mechanical Engineering, 2012, 25(4): 845–852.

    Article  MathSciNet  Google Scholar 

  20. LIU S H, LI Y Q, SHEN Y M. The electro-elastic fields of piezoelectric materials with an elliptic hole[J]. Engineering Mechanics, 2012, 29(12): 45–50. (in Chinese)

    Google Scholar 

  21. DU Y L, LIU S H, DUAN S J, et al. Electro-elastic fields of piezoelectric materials with an elliptic hole under uniform internal shearing forces[J]. Chinese Journal of Mechanical Engineering, 2013, 26(3): 467–474.

    Google Scholar 

  22. LIU S H, QI Y Q, FENG D D, et al. Numerical analysis of a crack emanating from an elliptical hole[J]. Journal of Mechanical Engineering, 2012, 48(20): 83–87. (in Chinese)

    Article  Google Scholar 

  23. MIAO C Q, WEI Y T, YAN X Q. Interactions of two collinear circular hole cracks subjected to internal pressure[J]. Applied Mathematics and Computation, 2013, 223: 216–224.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, China

    Shuhong Liu

  2. The Key Laboratory for Health Monitoring and Control of Large Structures of Hebei Province, Shijiazhuang, 050043, China

    Shuhong Liu

  3. Department of Adult Education, Shijiazhuang Tiedao University, Shijiazhuang, 050043, China

    Shijie Duan

Authors
  1. Shuhong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shijie Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuhong Liu.

Additional information

Supported by Hebei Provincial Natural Science Foundation of China(Grant No. A2011210033), and Foundation of Hebei Education Department of China(Grant No. ZH2011116)

LIU Shuhong, born in 1968, is currently a professor at Shijiazhuang Tiedao University, China. She received her PHD degree from Beijing Jiaotong University, China, in 2004. Her main research interests include solid mechanics, structural safety evaluation, and mechanical bearing capacity of intelligent materials.

DUAN Shijie, born in 1968, is currently an associate professor at Shijiazhuang Tiedao University, China. He received his bachelor degree from Shijiazhuang Tiedao University, China, in 1992. His research interest is computer application in engineering.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, S., Duan, S. Analytical solutions of cracks emanating from an elliptic hole in an infinite plate under tension. Chin. J. Mech. Eng. 27, 1057–1063 (2014). https://doi.org/10.3901/CJME.2014.0526.300

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3901/CJME.2014.0526.300

Keywords