Rock Mechanics and Rock Engineering

, Volume 46, Issue 5, pp 1023–1034 | Cite as

Application of Cracked Triangular Specimen Subjected to Three-Point Bending for Investigating Fracture Behavior of Rock Materials

  • M. R. M. Aliha
  • Gh. R. Hosseinpour
  • M. R. Ayatollahi
Original Paper

Abstract

Numerical and experimental studies were performed on a new fracture test configuration called the edge cracked triangular (ECT) specimen. Using several finite-element analyses, the fracture parameters (i.e., K I, K II, and T-stress) were obtained for different combinations of modes I and II. The finite-element results show that the ECT specimen is able to provide pure mode I, pure mode II, and any mixed-mode loading conditions in between. Also, a series of mixed-mode fracture experiments were conducted on Neiriz marble rock using the proposed specimen. Furthermore, the generalized maximum tangential stress (GMTS) criterion was used to predict the experimental results. The GMTS criterion makes use of a three-parameter model (based on K I, K II, and T) for describing the crack tip stresses. Due to the significant positive T-stresses that exist in the ECT specimen, typical minimum fracture toughness values were expected to be obtained when the ECT specimen is used. The direction of fracture initiation and the path of fracture growth were also obtained theoretically using the GMTS criterion, and good agreement was observed between the experimental fracture path and theoretical simulations. The fracture study of this specimen reveals that the ECT specimen can be also used in mixed-mode fracture studies of rock materials in addition to the conventional circular or rectangular beam test samples.

Keywords

Neiriz marble Triangular specimen Finite-element analyses Mixed-mode loading Fracture experiments 

Abbreviations

MTS

Maximum tangential stress

ECT

Edge cracked triangular

FE

Finite element

GMTS

Generalized maximum tangential stress

CZM

Cohesive zone model

SED

Strain energy density

List of Symbols

a

Crack length for ECT specimen

a/W

Crack length ratio in the ECT specimen

r, θ

Crack tip coordinates

t

Specimen thickness

E

Elastic modulus

T

T-Stress

T*

Normalized T-stress

2W

Edge length of square plate

Greek Letters

θ0

Crack initiation direction

σθθ

Tangential stress component

ν

Poisson’s ratio

KI

Mode I stress intensity factor

KII

Mode II stress intensity factor

KIc

Mode I fracture toughness

P

Applied load

Pcr

Critical fracture load

rc

Critical distance from the crack tip

YI

Mode I geometry factor

YII

Mode II geometry factor

α

Crack inclination angle

σθθc

Critical tangential stress

σt

Tensile strength

References

  1. Aliha MRM, Ayatollahi MR (2009) Brittle fracture evaluation of a fine grain cement mortar in combined tensile-shear deformation. Fatigue Fract Eng Mater Struct 32:987–994CrossRefGoogle Scholar
  2. Aliha MRM, Ayatollahi MR (2010) Geometry effects on fracture behavior of polymethyl methaacrelate. Mater Sci Eng A 527(3):526–530CrossRefGoogle Scholar
  3. Aliha MRM, Ayatollahi MR, Ashtari R (2006) Mode I and mode II fracture toughness testing for a coarse grain marble. Appl Mech Mater 5–6:181–188CrossRefGoogle Scholar
  4. Aliha MRM, Ayatollahi MR, Pakzad R (2008) Brittle fracture analysis using a ring shape specimen containing two angled cracks. Int J Fract 153:63–68CrossRefGoogle Scholar
  5. Aliha MRM, Ayatollahi MR, Kharrazi B (2009) Numerical and experimental investigation of mixed mode fracture in granite using four point bend specimen. Damage Fract Mech 46(1):275–283CrossRefGoogle Scholar
  6. Aliha MRM, Ayatollahi MR, Smith DJ, Pavier MJ (2010) Geometry and size effects on fracture trajectory in a limestone rock under mixed mode loading. Eng Fract Mech 77:2200–2212CrossRefGoogle Scholar
  7. Aliha MRM, Ayatollahi MR, Akbardoost J (2012) Typical upper bound-lower bound mixed mode fracture resistance envelopes for rock material. Rock Mech Rock Eng 45:65–74CrossRefGoogle Scholar
  8. Arcan M, Hashin Z, Volosnin A (1978) A method to produce uniform plane-stress states with application to fibre-reinforced materials. Exp Mech 18:141–146CrossRefGoogle Scholar
  9. Atkinson C, Smelser RE, Sanchez J (1982) Combined mode fracture via the cracked Brazilian disc test. Int J Fract 18:279–291Google Scholar
  10. Awaji H, Sato S (1978) Combined mode fracture toughness measurement by the disc test. J Eng Mater Tech 100:175–182CrossRefGoogle Scholar
  11. Ayatollahi MR, Aliha MRM (2007) Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading. Comp Mater Sci 38:660–670CrossRefGoogle Scholar
  12. Ayatollahi MR, Aliha MRM (2008a) On the use of Brazilian disc specimen for calculating mixed mode I-II fracture toughness of rock materials. Eng Fract Mech 75:4631–4641CrossRefGoogle Scholar
  13. Ayatollahi MR, Aliha MRM (2008b) Mixed mode fracture analysis of polycrystalline graphite—a modified MTS criterion. Carbon 46:1302–1308CrossRefGoogle Scholar
  14. Ayatollahi MR, Aliha MRM (2009) Analysis of a new specimen for mixed mode fracture tests on brittle materials. Eng Fract Mech 76:1563–1573CrossRefGoogle Scholar
  15. Ayatollahi MR, Aliha MRM (2011) On the use of anti-symmetric four-point bend specimen for mode II fracture experiments. Fatigue Fract Eng Mater Struct 34:898–907CrossRefGoogle Scholar
  16. Ayatollahi MR, Aliha MRM, Hassani MM (2006) Mixed mode brittle fracture in PMMA—an experimental study using SCB specimens. Mater Sci Eng A 417(1–2):348–356Google Scholar
  17. Ayatollahi MR, Aliha MRM, Saghafi H (2011) An improved semi-circular bend specimen for investigating mixed mode brittle fracture. Eng Fract Mech 78(1):110–123CrossRefGoogle Scholar
  18. Banks-Sills L, Bortman Y (1986) A mixed mode fracture specimen; analysis and testing. Int J Fract 30:181–201CrossRefGoogle Scholar
  19. Buchholz FG, Pirro PJM, Richard HA, Dreyer KH (1987) Numerical and experimental mixed-mode analysis of a compact tension-shear specimen. Proceedings of the fourth International Conference on Numerical methods in fracture mechanics, Pineridge Press, Swansea, p 641–56Google Scholar
  20. Chang SH, Lee CI, Jeon S (2002) Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimen. Eng Geol 66:79–97CrossRefGoogle Scholar
  21. Choi SR, Zhu D, Miller RA (2005) Fracture behavior under mixed-mode loading of ceramic plasma-sprayed thermal barrier coatings at ambient and elevated temperatures. Eng Fract Mech 72(13):2144–2158CrossRefGoogle Scholar
  22. Chong KP, Kuruppu MD (1984) New specimen for fracture toughness determination for rock and other materials. Int J Fract 26:R59–R62CrossRefGoogle Scholar
  23. Chong KP, Kuruppu MD, Kuszmual JS (1987) Fracture toughness determination of layered materials. Eng Fract Mech 28(1):43–54CrossRefGoogle Scholar
  24. Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng Trans ASME 85:519–525CrossRefGoogle Scholar
  25. Fett T (2001) Stress intensity factors and T-stress for internally cracked circular disks under various boundary conditions. Eng Fract Mech 68:1119–1136CrossRefGoogle Scholar
  26. Fett T, Gerteisen G, Hahnenberger S, Martin G, Munz D (1995) Fracture tests for ceramics under mode-I, mode-II and mixed-mode loading. J Eur Ceram Soc 15:307–312CrossRefGoogle Scholar
  27. Gómez FJ, Elices M, Berto F, Lazzarin P (2009) Fracture of U-notched specimens under mixed mode: experimental results and numerical predictions. Eng Fract Mech 76(2):236–249CrossRefGoogle Scholar
  28. Hasanpour R, Choupani N (2009) Rock fracture characterization using the modified Arcan test specimen. Int J Rock Mech Min Sci 46(2):346–354CrossRefGoogle Scholar
  29. Hussain MA, Pu SL, Underwood J (1974) Strain energy release rate for a crack under combined mode I and mode II. Fracture analysis ASTM STP 560. American Society for Testing and Materials, Philadelphia, pp 22–28Google Scholar
  30. ISRM (2007) The complete ISRM suggested methods for rock characterization, testing and monitoring: 1974–2006. Suggested methods prepared by the commission on testing methods. In: Ulusay R, Hudson JA (eds) International Society for rock mechanics commission on testing methods. Compilation arranged by the ISRM Turkish National Group, AnkaraGoogle Scholar
  31. ISRM, Fowell RJ (1995) Suggested methods for determining mode I fracture toughness using cracked chevron notched Brazilian disk (CCNBD) specimens. Int J Rock Mech Min Sci Geomech Abstr 32:57–64CrossRefGoogle Scholar
  32. Khan K, Al-Shayea NA (2000) Effect of specimen geometry and testing method on mixed mode I-II fracture toughness of a limestone rock from Saudi Arabia. Rock Mech Rock Eng 33:179–206CrossRefGoogle Scholar
  33. Krishnan GR, Xl Zhao, Zaman M, Roegiers JC (1998) Fracture toughness of a soft sandstone. Int J Rock Mech Min Sci 35(6):695–710CrossRefGoogle Scholar
  34. Li FZ, Shih CF, Needleman A (1985) A comparison of methods for calculating energy release rate. Eng Fract Mech 21:405–421CrossRefGoogle Scholar
  35. Lim IL, Johnston IW, Choi SK, Boland JN (1994) Fracture testing of a soft rock with semi-circular specimens under three-point bending, part 2—mixed mode. Int J Rock Mech Min Sci Geomech Abstr 31(3):199–212CrossRefGoogle Scholar
  36. Maccagno TM, Knott JF (1989) The fracture behavior of PMMA in mixed modes I and II. Eng Fract Mech 34(1):65–86CrossRefGoogle Scholar
  37. Mahajan RV, Ravi-Chandar K (1989) An experimental investigation of mixed-mode fracture. Int J Fract 41:235–252CrossRefGoogle Scholar
  38. Maiti SKM, Prasad KSR (1980) A study on the theories of unstable crack extension for the prediction of crack trajectories. Int J Solids Struct 16:563–574CrossRefGoogle Scholar
  39. Richard HA, Benitz K (1983) A loading device for the creation of mixed mode in fracture mechanics. Int J Fract 22:R55–R58CrossRefGoogle Scholar
  40. Saghafi H, Ayatollahi MR, Sistaninia M (2010) A modified MTS criterion (MMTS) for mixed-mode fracture toughness assessment of brittle materials. Mater Sci Eng A 527(21–22):5624–5630Google Scholar
  41. Schmidt RA (1980) A microcrack model and its significance to hydraulic fracturing and fracture toughness testing. Proceedings of 21st US symposium on rock mechanics, p 581–590Google Scholar
  42. Seed GM, Nowell D (1994) Use of the distributed dislocations method to determine the T-stress. Fatigue Fract Eng Mater Struct 17(5):605–618CrossRefGoogle Scholar
  43. Shetty DK, Rosenfield AR, Duckworth WH (1987) Mixed-mode fracture in biaxial stress state: application of the diametral-compression (Brazilian disk) test. Eng Fract Mech 26(6):825–840CrossRefGoogle Scholar
  44. Shiryaev AM, Kotkis AM (1982) Methods for determining fracture toughness of brittle porous materials. Indus Lab 48(9):917–918Google Scholar
  45. Sih GC (1974) Strain-energy-density factor applied to mixed mode crack problems. Int J Fract 10:305–321CrossRefGoogle Scholar
  46. Singh RN, Sun GX (1990) A numerical and experimental investigation for determining fracture toughness of Welsh limestone. Min Sci Tech 10:61–70CrossRefGoogle Scholar
  47. Sistaninia M (2009) Mode II fracture study of rock materials using cracked Brazilian disc specimen (MSc theses). Iran University of Science and Technology, IranGoogle Scholar
  48. Smith DJ, Ayatollahi MR, Pavier MJ (2001) The role of T-stress in brittle fracture for linear elastic materials under mixed mode loading. Fatigue Fract Eng Mater Struct 24:137–150CrossRefGoogle Scholar
  49. Sumi Y (1985) Computational crack path prediction. Theor Appl Fract Mech 4:149–156CrossRefGoogle Scholar
  50. Suresh S, Shih CF, Morrone A, O-Dowd NP (1990) Mixed-mode fracture toughness of ceramic materials. J Am Ceram Soc 73(5):1257–1267CrossRefGoogle Scholar
  51. Tikare V, Choi SR (1993) Combined mode I and mode II fracture of monolithic ceramics. J Am Ceram Soc 76(9):2265–2272CrossRefGoogle Scholar
  52. Whittaker BN, Singh RN, Sun GX (1992) Rock fracture mechanics-principles, design and applications. Elsevier, AmsterdamGoogle Scholar
  53. Williams ML (1957) On the stress distribution at the base of a stationary crack. J Appl Mech 24:109–114Google Scholar
  54. Williams JG, Ewing PD (1972) Fracture under complex stress—the angled crack problem. Int J Fract 8:441–446Google Scholar
  55. Xeidakis GS, Samaras IS, Zacharopoulos DA, Papakalitakis GE (1996) Crack growth in a mixed-mode loading on marble beams under three point bending. Int J Fract 79:197–208CrossRefGoogle Scholar
  56. Zipf RK, Bieniawski ZT (1986) Mixed mode testing for fracture testing of coal based on critical energy density. Proceedings of 27th US symposium on rock mechanics, p 16–23Google Scholar

Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  • M. R. M. Aliha
    • 1
    • 2
  • Gh. R. Hosseinpour
    • 2
  • M. R. Ayatollahi
    • 2
  1. 1.Welding and Joining Research CenterIran University of Science and TechnologyTehranIran
  2. 2.Fatigue and Fracture LabIran University of Science and TechnologyTehranIran

Personalised recommendations