Skip to main content
SpringerLink
Log in

Size and Geometry Effects on Rock Fracture Toughness: Mode I Fracture

  • Original Paper
  • Published:
Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

Abstract

In this paper, the effects of specimen size and geometry on the apparent mode I fracture toughness (K c) of an Iranian white marble (Neyriz) are studied. A number of fracture tests were conducted on center-cracked circular disk (CCCD) specimens with different radii to investigate the size effects on K c. The experimental results demonstrate that the apparent fracture toughness increases in bigger specimens. In order to explain the experimental results, the modified maximum tangential stress (MMTS) criterion is used, where higher order terms of the Williams’ series expansion are included in the maximum tangential stress criterion. It is shown that the MMTS criterion provides good estimates for the apparent fracture toughness of Neyriz marble, obtained from fracture tests of edge-cracked triangular specimens. It is, therefore, concluded that the proposed criterion is able to account for the size and geometry effects on the fracture resistance of rocks simultaneously.

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

Similar content being viewed by others

Abbreviations

a :

Crack length/semi-crack length

\( A_{1}^{*} , A_{3}^{*} \) :

Non-dimensional forms of A 1 and A 3

A 3c :

Critical value of the 3rd term in the Williams’ expansion

A n :

Coefficients of the crack tip asymptotic field

C, R*:

Coefficients of Bazant’s size effect law

CCCD:

Center-cracked circular disk

ECT:

Edge-cracked triangular

FEOD:

Finite element over-deterministic method

FPZ:

Fracture process zone

f t :

Tensile strength of material

K c :

Apparent fracture toughness

K Ic :

Inherent fracture toughness

L :

Characteristic dimension

n :

Order of term in the Williams’ series expansion

P :

Applied load

R :

Radius of CCCD specimens

r, θ :

Crack tip coordinates

r c :

FPZ length

r c∞ :

FPZ length for an infinitely large specimen

t :

Specimen thickness

σ N :

Nominal stress

σ θθ :

Tangential stress component

σ θθc :

Critical value of the tangential stress component

References

  • 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(12):987–994. doi:10.1111/j.1460-2695.2009.01402.x

    Article  Google Scholar 

  • Aliha MRM, Ashtari R, Ayatollahi MR (2006) Mode I and mode II fracture toughness testing for a coarse grain marble. Appl Mech Mater 5–6:181–188

    Article  Google Scholar 

  • 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(11):2200–2212. doi:10.1016/j.engfracmech.2010.03.009

    Article  Google Scholar 

  • Al-Shayea NA (2005) Crack propagation trajectories for rocks under mixed mode I–II fracture. Eng Geol 81(1):84–97. doi:10.1016/j.enggeo.2005.07.013

    Article  Google Scholar 

  • Arslan A, Ince R (1996) The neural network approximation to the size effect in fracture of cementitious materials. Eng Fract Mech 54(2):249–261

    Article  Google Scholar 

  • Awaji H, Sato S (1978) Combined mode fracture toughness measurement by the disc test. J Eng Mater Technol 100:175–182

    Article  Google Scholar 

  • Ayatollahi MR, Aliha MRM (2008) On the use of Brazilian disc specimen for calculating mixed mode I–II fracture toughness of rock materials. Eng Fract Mech 75(16):4631–4641. doi:10.1016/j.engfracmech.2008.06.018

    Article  Google Scholar 

  • Ayatollahi MR, Aliha MRM (2011) Fracture analysis of some ceramics under mixed mode loading. J Am Ceram Soc 94(2):561–569. doi:10.1111/j.1551-2916.2010.04076.x

    Article  Google Scholar 

  • Ayatollahi MR, Nejati M (2011) An over-deterministic method for calculation of coefficients of crack tip asymptotic field from finite element analysis. Fatigue Fract Eng Mater Struct 34(3):159–176. doi:10.1111/j.1460-2695.2010.01504.x

    Article  Google Scholar 

  • Ayatollahi MR, Hosseinpour GR, Aliha MRM (2010) Using a new tensile–shear cracked specimen for investigating fracture behavior of rock materials. Paper presented in the proceedings of the international conference on experimental mechanics 2010 (ICEM 2010), Kuala Lumpur, Malaysia, November/December 2010

  • Bazant ZP (1982) Crack band model for fracture of geomaterials. In: Eisenstein Z (ed) Proceedings of the 4th international conference on numerical methods in geomechanics, Edmonton, Alberta, Canada, May/June 1982, pp 1137–1152

  • Bazant ZP (1984) Size effect in blunt fracture: concrete, rock, metal. J Eng Mech 110(4):518–535

    Article  Google Scholar 

  • Bazant ZP (2000) Size effect. Int J Solids Struct 37(1–2):69–80. doi:10.1016/s0020-7683(99)00077-3

    Article  Google Scholar 

  • Bazant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca Raton

    Google Scholar 

  • Bazant ZP, Kim J-K, Pfeiffer PA (1986) Nonlinear fracture properties from size effect tests. J Struct Eng 112(2):289–307

    Article  Google Scholar 

  • Bazant ZP, Gettu R, Kazemi MT (1991) Identification of nonlinear fracture properties from size effect tests and structural analysis based on geometry-dependent R-curves. Int J Rock Mech Min Sci 28(1):43–51

    Article  Google Scholar 

  • Carpinteri A (1989) Decrease of apparent tensile and bending strength with specimen size: two different explanations based on fracture mechanics. Int J Solids Struct 25(4):407–429

    Article  Google Scholar 

  • Chang S-H, Lee C-I, Jeon S (2002a) Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Eng Geol 66(1–2):79–97. doi:10.1016/s0013-7952(02)00033-9

    Article  Google Scholar 

  • Chang SH, Lee CI, Jeon S (2002b) Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Eng Geol 66:79–97

    Article  Google Scholar 

  • Chao YJ, Liu S, Broviak BJ (2001) Brittle fracture: variation of fracture toughness with constraint and crack curving under mode I conditions. Exp Mech 41(3):232–241

    Article  Google Scholar 

  • Cornetti P, Pugno N, Carpinteri A, Taylor D (2006) Finite fracture mechanics: a coupled stress and energy failure criterion. Eng Fract Mech 73(14):2021–2033

    Article  Google Scholar 

  • De Lorenzis L, Zavarise G (2009) Cohesive zone modeling of interfacial stresses in plated beams. Int J Solids Struct 46(24):4181–4191

    Article  Google Scholar 

  • Duan K, Hu X, Wittmann FH (2007) Size effect on specific fracture energy of concrete. Eng Fract Mech 74(1–2):87–96

    Article  Google Scholar 

  • Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng 85:519–525

    Article  Google Scholar 

  • Fowell RJ (1995) Suggested method for determining mode I fracture toughness using cracked chevron notched Brazilian disc (CCNBD) specimens. Int J Rock Mech Min Sci Geomech Abstr 32(1):57–64. doi:10.1016/0148-9062(94)00015-U

    Article  Google Scholar 

  • Griffith AA (1921) The phenomena of rupture and flow in solids. Phil Trans R Soc Lond A 221:163–198

    Article  Google Scholar 

  • Gutiérrez MA, De Borst R (1999) Deterministic and stochastic analysis of size effects and damage evolution in quasi-brittle materials. Arch Appl Mech 69(9–10):655–676

    Google Scholar 

  • Hillerborg A, Modéer M, Petersson PE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 6(6):773–781

    Article  Google Scholar 

  • Hu X, Duan K (2005) Size effect on fracture of MEMS materials. J Mater Sci Technol 21(Suppl 1):47–50

    Google Scholar 

  • Jenq YS, Shah SP (1985) Two parameter fracture model for concrete. J Eng Mech 111:1227–1241

    Article  Google Scholar 

  • Karihaloo BL (1995) Tension softening diagrams and longitudinally reinforced beams. In: Baker G, Karihaloo BL (eds) Fracture of brittle disordered materials: concrete, rock and ceramic. E & FN Spon, London, pp 35–50

    Google Scholar 

  • Karihaloo BL (1999) Size effect in shallow and deep notched quasi-brittle structures. Int J Fract 95(1–4):379–390

    Article  Google Scholar 

  • Karihaloo BL, Xiao QZ (2001a) Higher order terms of the crack tip asymptotic field for a wedge-splitting specimen. Int J Fract 112(2):129–137

    Article  Google Scholar 

  • Karihaloo BL, Xiao QZ (2001b) Higher order terms of the crack tip asymptotic field for a notched three-point bend beam. Int J Fract 112(2):111–128

    Article  Google Scholar 

  • Karihaloo BL, Abdalla HM, Xiao QZ (2003) Size effect in concrete beams. Eng Fract Mech 70(7–8):979–993

    Article  Google Scholar 

  • 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(3):179–206

    Article  Google Scholar 

  • Kim JK, Yi ST, Park CK, Eo SH (1999) Size effect on compressive strength of plain and spirally reinforced concrete cylinders. ACI Struct J 96(1):88–94

    Google Scholar 

  • Krishnan GR, Zhao XL, Zaman M, Roegiers J-C (1998) Fracture toughness of a soft sandstone. Int J Rock Mech Min Sci 35(6):695–710

    Article  Google Scholar 

  • Leicester RH (1969) The size effect of notches. In: Proceedings of the second Australasian conference on mechanics of materials and structures, Melbourne, Australia, pp 1–20

  • Li J, Zhang XB (2005) A criterion study for non-singular stress concentrations with size effect. Strength Fract Complex 3(2–4):205–215

    Google Scholar 

  • Morel S (2008) Size effect in quasibrittle fracture: derivation of the energetic size effect law from equivalent LEFM and asymptotic analysis. Int J Fract 154(1–2):15–26. doi:10.1007/s10704-008-9291-6

    Article  Google Scholar 

  • Ouchterlony F (1988) ISRM suggested methods for determining fracture toughness of rock. Int J Rock Mech Min Sci Geomech Abstr 25:71–96

    Google Scholar 

  • Planas J, Guinea GV, Elices M (1997) Generalized size effect equation for quasibrittle materials. Fatigue Fract Eng Mater Struct 20(5):671–687. doi:10.1111/j.1460-2695.1997.tb00300.x

    Article  Google Scholar 

  • Rocco C, Guinea GV, Planas J, Elices M (1999) Size effect and boundary conditions in the Brazilian test: experimental verification. Mater Struct 32(3):210–217. doi:10.1007/bf02481517

    Article  Google Scholar 

  • Saouma VE, Natekar D, Hansen E (2003) Cohesive stresses and size effects in elasto-plastic and quasi-brittle materials. Int J Fract 119(3):287–298. doi:10.1023/a:1023968010028

    Article  Google Scholar 

  • Schmidt RA (1980) A microcrack model and its significance to hydraulic fracturing and fracture toughness testing. In: Proceedings of the 21st US symposium on rock mechanics, Rolla, Missouri, May 1980, pp 581–590

  • van Vliet MRA, van Mier JGM (2000) Experimental investigation of size effect in concrete and sandstone under uniaxial tension. Eng Fract Mech 65(2–3):165–188

    Article  Google Scholar 

  • Weibull W (1939) The phenomenon of rupture in solids. Ing Vetenskaps Akad Handl 153:1–55

    Google Scholar 

  • Williams ML (1957) On the stress distribution at the base of a stationary crack. J Appl Mech 24:109–114

    Google Scholar 

  • Wittmann FH, Hu X (1991) Fracture process zone in cementitious materials. Int J Fract 51(1):3–18

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fatigue and Fracture Lab., Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846, Tehran, Iran

    M. R. Ayatollahi & J. Akbardoost

Authors
  1. M. R. Ayatollahi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Akbardoost
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. R. Ayatollahi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ayatollahi, M.R., Akbardoost, J. Size and Geometry Effects on Rock Fracture Toughness: Mode I Fracture. Rock Mech Rock Eng 47, 677–687 (2014). https://doi.org/10.1007/s00603-013-0430-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00603-013-0430-7

Keywords

Search

Navigation