Skip to main content
SpringerLink
Log in

T-stress for the central cracked Brazilian disk under non-uniformly distributed pressure

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

As the non-singular term in the Williams series expansion of the stress field at the crack tip, T-stress has a great influence on the fracture mode and crack propagation behavior of brittle materials. Based on the contact theory, the real load distribution on the disk–jaw interface in the Brazilian test is considered to be non-uniform. Using the weight function and integration methods, the analytical solutions for T-stress in the central cracked Brazilian disk subjected to the non-uniform pressures are achieved. The precise distribution of contact pressure is simulated, and the accuracy of the analytic solution is verified by the finite element method. The effects of loading angle, relative crack length, load distribution angle, and distribution type on T-stress are analyzed. The analyses indicate that the effects of load distribution on T-stress should be considered in the case of large contact angles and long cracks. The formulae of T-stress for uniform, non-uniform, and concentrated loads are compared, and the applicability ranges for each formula are given according to the comparison results. When the relative crack length α = 0.4–0.6, the solutions under non-uniformly distributed loads should be adopted if γ ≥ 8.65°.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Markides, C.F., Kourkoulis, S.K.: An alternative analytic approach to the Brazilian disc test with friction at the disc–jaw interface. Arch. Appl. Mech. 83, 743–763 (2013)

    Article  MATH  Google Scholar 

  2. Yu, J.H., Shang, X.C., Wu, P.F.: Influence of pressure distribution and friction on determining mechanical properties in the Brazilian test: theory and experiment. Int. J. Solids Struct. 161, 11–22 (2019)

    Article  Google Scholar 

  3. Gutiérrez-Moizant, R., Ramírez-Berasategui, M., Sánchez-Sanz, S., Santos-Cuadros, S.: Experimental verification of the boundary conditions in the success of the Brazilian test with loading arcs. An uncertainty approach using concrete disks. Int. J. Rock. Mech. Min. Sci 132, 104380 (2020)

    Article  Google Scholar 

  4. Hua, W., Dong, S., Li, Y., Wang, Q.: Effect of cyclic wetting and drying on the pure mode II fracture toughness of sandstone. Eng. Fract. Mech. 153, 143–150 (2016)

    Article  Google Scholar 

  5. Sahlabadi, M., Soltani, N.: Experimental and numerical investigations of mixed-mode ductile fracture in high-density polyethylene. Arch. Appl. Mech. 88, 933–942 (2018)

    Article  Google Scholar 

  6. Bahrami, B., Nejati, M., Ayatollahi, M.R., Driesner, T.: Theory and experiment on true mode II fracturing of rocks. Eng. Fract. Mech. 240, 107314 (2020)

    Article  MATH  Google Scholar 

  7. Miarka, P., Seitl, S., Horňáková, M., Lehner, P., Konečný, P., Sucharda, O., et al.: Influence of chlorides on the fracture toughness and fracture resistance under the mixed mode I/II of high-performance concrete. Theoret. Appl. Fract. Mech. 110, 102812 (2020)

    Article  Google Scholar 

  8. ASTM: D3967-95a Standard Test Method for Splitting Tensile Strength of Intact Rock Core Specimens. ASTM, West Conshohocken (1995)

  9. Bieniawski, Z.T., Hawkes, I.: Suggested methods for determining tensile strength of rock materials. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 15, 99–103 (1978)

    Article  Google Scholar 

  10. Dong, S., Wang, Y., Xia, Y.: Stress intensity factors for central cracked circular disk subjected to compression. Eng. Fract. Mech. 71, 1135–1148 (2004)

    Article  Google Scholar 

  11. Hua, W., Li, Y.F., Dong, S.M., Li, N.B., Wang, Q.Y.: T-stress for a centrally cracked Brazilian disk under confining pressure. Eng. Fract. Mech. 149, 37–44 (2015)

    Article  Google Scholar 

  12. Markides, C.F., Kourkoulis, S.K.: The stress field in a standardized Brazilian disc: the influence of the loading type acting on the actual contact length. Rock Mech. Rock Eng. 45, 145–158 (2012)

    Article  Google Scholar 

  13. Kourkoulis, S.K., Markides, C.F., Chatzistergos, P.E.: The Brazilian disc under parabolically varying load: theoretical and experimental study of the displacement field. Int. J. Solids Struct. 49, 959–972 (2012)

    Article  Google Scholar 

  14. Japaridze, L.: Stress-deformed state of cylindrical specimens during indirect tensile strength testing. J. Rock Mech. Geotech. Eng. 7, 509–518 (2015)

    Article  Google Scholar 

  15. Guerrero-Miguel, D.J., Alvarez-Fernandez, M.I., Garcia-Fernandez, C.C., Gonzalez-Nicieza, C., Menendez-Fernandez, C.: Analytical and numerical stress field solutions in the Brazilian Test subjected to radial load distributions and their stress effects at the centre of the disk. J. Eng. Math. 116, 29–48 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Markides, C.F., Pazis, D.N., Kourkoulis, S.K.: The Brazilian disc under non-uniform distribution of radial pressure and friction. Int. J. Rock Mech. Min. Sci. 50, 47–55 (2012)

    Article  Google Scholar 

  17. Yu, J.H., Shang, X.C.: Analysis of the influence of boundary pressure and friction on determining fracture toughness of shale using cracked Brazilian disc test. Eng. Fract. Mech. 212, 57–69 (2019)

    Article  Google Scholar 

  18. Tang, H.Z., Huang, J.Z., He, J.Y., Hua, W., Dong, S.M.: Stress intensity factors for a centrally cracked Brazilian disk under non-uniformly distributed pressure. Theoret. Appl. Fract. Mech. 114, 11 (2021)

    Article  Google Scholar 

  19. Markides, C.F., Kourkoulis, S.K.: The finite circular disc with a central elliptic hole under parabolic pressure. Acta Mech. 226, 1929–1955 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Markides, C.F., Kourkoulis, S.K.: ‘Mathematical’ cracks versus artificial slits: implications in the determination of fracture toughness. Rock Mech. Rock Eng. 49, 707–729 (2016)

    Article  Google Scholar 

  21. Ayatollahi, M.R., Pavier, M.J., Smith, D.J.: Mode I cracks subjected to large T-stresses. Int. J. Fract. 117, 159–174 (2002)

    Article  Google Scholar 

  22. Li, X.F., Lee, K.Y., Tang, G.J.: Kink angle and fracture load for an angled crack subjected to far-field compressive loading. Eng. Fract. Mech. 82, 172–184 (2012)

    Article  Google Scholar 

  23. Huang, J.Z., Pan, X., Li, J.X., Dong, S.M., Hua, W.: Numerical investigation on crack propagation for a central cracked Brazilian disk concerning friction. Appl. Sci. Basel 11, 15 (2021)

    Google Scholar 

  24. Tang, H., Qin, C., Huang, L., Xu, Y., Hua, W., Dong, S.: T-stress for the double-edge cracked Brazilian disc under compression. Theoret. Appl. Fract. Mech. 119, 103379 (2022)

    Article  Google Scholar 

  25. Smith, D.J., Ayatollahi, M.R., Pavier, M.J.: The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract. Eng. Mater. Struct. 24, 137–150 (2001)

    Article  Google Scholar 

  26. Matvienko, Y.G.: Maximum average tangential stress criterion for prediction of the crack path. Int J Fract 176, 113–118 (2012)

    Article  Google Scholar 

  27. Ayatollahi, M.R., Moghaddam, M.R., Berto, F.: A generalized strain energy density criterion for mixed mode fracture analysis in brittle and quasi-brittle materials. Theoret. Appl. Fract. Mech. 79, 70–76 (2015)

    Article  Google Scholar 

  28. Hua, W., Dong, S., Pan, X., Wang, Q.: Mixed mode fracture analysis of CCBD specimens based on the extended maximum tangential strain criterion. Fatigue Fract. Eng. Mater. Struct. 40, 2118–2127 (2017)

    Article  Google Scholar 

  29. Hua, W., Huang, J.Z., Pan, X., Li, J.X., Dong, S.M.: An extended maximum tangential strain energy density criterion considering T-stress for combined mode I–III brittle fracture. Fatigue Fract. Eng. Mater. Struct. 44, 169–181 (2021)

    Article  Google Scholar 

  30. Bueckner, H.F.: A novel principle for the computation of stress intensity factors. Z. Angew. Math. Mech. 50, 529–546 (1970)

    MathSciNet  MATH  Google Scholar 

  31. Fett, T.: T-stresses in rectangular plates and circular disks. Eng. Fract. Mech. 60, 631–652 (1998)

    Article  Google Scholar 

  32. Fett, T.: Stress intensity factors and T-stress for internally cracked circular disks under various boundary conditions. Eng. Fract. Mech. 68, 1119–1136 (2001)

    Article  Google Scholar 

  33. Kourkoulis, S.K., Markides, C.F., Chatzistergos, P.E.: The standardized Brazilian disc test as a contact problem. Int. J. Rock Mech. Min. Sci. 57, 132–141 (2013)

    Article  Google Scholar 

  34. Yu, J.H., Shang, X.C., Wang, G.: Theoretical analysis and experimental identification of contact pressure in Brazilian disc. Rock Mech. Rock Eng. 55, 799–811 (2022)

    Article  Google Scholar 

  35. Yu, J.H.: Theoretical Analysis and Numerical Computation of Fluid–Solid Coupling Nonlinear Seepage in Shale Gas Horizontal. University of Science and Technology Beijing, Beijing (2019)

    Google Scholar 

  36. Ayatollahi, M.R., Torabi, A.R., Firoozabadi, M.: Theoretical and experimental investigation of brittle fracture in V-notched PMMA specimens under compressive loading. Eng. Fract. Mech. 135, 187–205 (2015)

    Article  Google Scholar 

  37. Hua, W., Xu, J.G., Dong, S.M., Song, J.Z., Wang, Q.Y.: Effect of confining pressure on stress intensity factors for cracked Brazilian disk. Int. J. .Appl. Mech. 7, 9 (2015)

    Article  Google Scholar 

  38. Tang, Z., Yao, W., Zhang, J., Xu, Q., Xia, K.: Experimental evaluation of PMMA simulated tunnel stability under dynamic disturbance using digital image correlation. Tunn. Undergr. Space Technol. 92, 103039 (2019)

    Article  Google Scholar 

  39. Kishi, K., Yanagimoto, F., Fukui, T., Matsumoto, T., Shibanuma, K.: Analysis of rapid crack arrestability enhancement by structural factors in cross-joint components using a transparent elastic solid. Int. J. Mech. Sci. 174, 105502 (2020)

    Article  Google Scholar 

  40. Pan, X., Huang, J.Z., Gan, Z.Q., Dong, S.M., Hua, W.: Analysis of mixed-mode I/II/III fracture toughness based on a three-point bending sandstone specimen with an inclined crack. Appl. Sci. Basel 11, 23 (2021)

    Google Scholar 

  41. Aminzadeh, A., Fahimifar, A., Nejati, M.: On Brazilian disk test for mixed-mode I/II fracture toughness experiments of anisotropic rocks. Theoret. Appl. Fract. Mech. 102, 222–238 (2019)

    Article  Google Scholar 

  42. Huang, J.Z., Li, J.X., Pan, X., Xie, T.Z., Hua, W., Dong, S.M.: Numerical investigation on mixed mode (I–II) fracture propagation of CCBD specimens under confining pressure. Int. J. Appl. Mech. 12, 20 (2020)

    Google Scholar 

  43. Wang, L.F., Zhou, X.P.: Phase field model for simulating the fracture behaviors of some disc-type specimens. Eng. Fract. Mech. 226, 23 (2020)

    Google Scholar 

  44. Nakamura, T., Parks, D.M.: Determination of elastic T-stress along 3-dimensional crack fronts using an interaction integral. Int. J. Solids Struct. 29, 1597–1611 (1992)

    Article  MATH  Google Scholar 

  45. Zhao, L.G., Tong, J., Byrne, J.: Stress intensity factor K and the elastic T-stress for corner cracks. Int. J. Fract. 109, 209–225 (2001)

    Article  Google Scholar 

  46. Dong, S.M.: Theoretical analysis of the effects of relative crack length and loading angle on the experimental results for cracked Brazilian disk testing. Eng. Fract. Mech. 75, 2575–2581 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11872042 and 12132019) and the China Postdoctoral Science Foundation (2019M653395).

Author information

Author notes

  1. Huaizi Tang and Lin Huang contributed equally to this work.

Authors and Affiliations

  1. Key Laboratory of Deep Underground Science and Engineering, Ministry of Education, College of Architecture and Environment, Sichuan University, Chengdu, 610065, China

    Huaizi Tang, Lin Huang, Xin Pan, Jiuzhou Huang, Wen Hua & Shiming Dong

  2. Failure Mechanics and Engineering Disaster Prevention Key Laboratory of Sichuan Province, Sichuan University, Chengdu, 610065, China

    Wen Hua & Shiming Dong

Authors
  1. Huaizi Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lin Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xin Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiuzhou Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Wen Hua
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Shiming Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

HT performed writing—original draft. LH performed writing—original draft and investigation. XP carried out formal analysis. JH contributed to data curation. WH performed writing—review and editing. SD contributed to project administration and writing—review and editing.

Corresponding author

Correspondence to Shiming Dong.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, H., Huang, L., Pan, X. et al. T-stress for the central cracked Brazilian disk under non-uniformly distributed pressure. Arch Appl Mech 92, 2859–2880 (2022). https://doi.org/10.1007/s00419-022-02200-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-022-02200-7

Keywords

Search

Navigation