ISRM-Suggested Method for Determining the Mode I Static Fracture Toughness Using Semi-Circular Bend Specimen

Abstract

The International Society for Rock Mechanics has so far developed two standard methods for the determination of static fracture toughness of rock. They used three different core-based specimens and tests were to be performed on a typical laboratory compression or tension load frame. Another method to determine the mode I fracture toughness of rock using semi-circular bend specimen is herein presented. The specimen is semi-circular in shape and made from typical cores taken from the rock with any relative material directions noted. The specimens are tested in three-point bending using a laboratory compression test instrument. The failure load along with its dimensions is used to determine the fracture toughness. Most sedimentary rocks which are layered in structure may exhibit fracture properties that depend on the orientation and therefore measurements in more than one material direction may be necessary. The fracture toughness measurements are expected to yield a size-independent material property if certain minimum specimen size requirements are satisfied.

Appendix: Details of Numerical Analysis Used for Deriving Eq. (2)

The SCB specimens of different crack lengths were simulated and analyzed using eight-node plane-strain elements in the finite element code Abaqus Unified FEA (2012). The loading, the boundary conditions and a typical finite element mesh used for the simulations are shown in Fig. 6. Singular elements with nodes at quarter-point positions were used for the first ring of elements around the crack tip. In the circular partitions surrounding the crack tip where the contour integrals are calculated, the mesh was biased toward the crack tip. The stress intensity factors K _I were extracted directly from ABAQUS which makes use of the J-integral method to compute the stress intensity factors. The numerical results showed that there was negligible variation in the J-integral values calculated for successive contours surrounding the crack tip.

Fig. 6

A sample mesh pattern used for simulating the SCB specimen

Using a fixed arbitrary load P, the stress intensity factor K _I was determined for each set of β and $\frac{s}{2R}$, and the non-dimensional stress intensity factor $Y^{'}$ was calculated from

$$ Y^{'} (\beta ,\frac{s}{2R}) = \frac{{2RBK_{\text{I}} }}{{P\sqrt {\pi a} }} $$

(A1)

then Eq. (2) was derived by fitting a second order polynomial to the numerical results obtained for $Y^{'}$. Tutluoglu and Keles (2011) recently reported limited numerical results for $Y^{'}$ in the SCB specimen. As shown in Table 3, very good agreement exists between the present results and those reported by Tutluoglu and Keles (2011). Table 3 can also be considered as validation for the finite element results obtained in this study, particularly for the ranges 0.4 ≤ β ≤ 0.6 and 0.5 ≤ s/2R ≤ 0.8, as suggested in Sect. 6.

Table 3 Numerical values of $Y^{'}$, present results compared with those of Tutluoglu and Keles (2011)

It is noteworthy that a number of investigators have presented mode I stress intensity factors of the SCB specimen (Chong et al. 1987; Lim et al. 1994; Basham 1989). For instance, Lim et al. (1994) extracted the stress intensity factors of the SCB specimen from finite element analysis and suggested a fifth order polynomial for $Y^{'}$ as.

$$ Y^{'} = \frac{s}{2R}\left( {2.91 + 54.39\beta - 391.4\beta^{2} + 1210.6\beta^{3} - 1650\beta^{4} + 875.9\beta^{5} } \right) $$

(A2)

Figure 7 shows a comparison between the curves plotted based on Eqs. (2) and (A2) for different values of β and $\frac{s}{2R}$. Significant discrepancies can be seen between these two sets of results.

Fig. 7

The curves plotted based on Eqs. (2) and (A2)

Having checked our finite element results by different mesh designs and element numbers, we concluded that the observed discrepancy can be due to less accurate method used by Lim et al. (1994) for determining the stress intensity factors of the SCB specimen. The displacement/stress extrapolation method employed by Lim et al. was a common technique in the 1990s for deriving stress intensity factors from finite element results. But, later more accurate methods were proposed like the contour integral techniques (e.g. J-integral method). It is now well established that the numerical errors in the region of high stress gradient around the crack tip affects the J-integral method much less than the displacement/stress extrapolation technique.