Introduction of a Scaling Factor for Fracture Toughness Measurement of Rocks Using the Semi-circular Bend Test

Three distinct rock types are employed for this study, all of which were acquired from different mining sites in Iran. The first type, chosen from sedimentary rocks, is a limestone excavated from Fars province. The second type, from the metamorphic category, is a white marble extracted from West Azerbaijan Province. Finally, the third type is an igneous granitic rock also mined from West Azerbaijan Province. All these rock materials are initially extracted in the form of large blocks at the mines, followed by cutting into slabs with predefined thicknesses in a stone factory. We prepared Brazilian disk and SCB samples from slabs of 2 cm thickness. Only the largest SCBs, i.e. with \(R=300\) mm, were extracted from slabs with 3 cm thickness. To reduce the potential influence of the heterogeneity and anisotropy, all the samples from each rock type were prepared from adjacent slabs in a single block and were tested in a predefined direction.

To measure the tensile strength of the rock types, Brazilian disks (BDs) with the radius of \(R=50\) mm and the thickness of \(t=20\) mm were prepared using the waterjet cutting method. For each rock type, four repetitions were considered, meaning that twelve BD samples were tested overall. Figure 1a pictures a BD specimen of limestone during the test. Circular jaws were used for the exertion of diametrical compressive force, and polymethylmethacrylate (PMMA) cushions with the same curvature to that of jaws were utilised for smoother and better-distributed load transfer to the specimen. In fact, this configuration facilitates a proper execution of the BD test, in which fracturing initiates from a central zone, minimising the potential for failure of disks due to crack initiation near the loading points. Figure 1b–d show sample fractured BD specimens of the three rock materials. As is seen in these photographs, the limestone and granite samples show minimal/no damage at loading points, with minor eccentricity of fracture path observed in the granitic disk. For the marble specimen, however, the surfaces in contact with loading cushions are more pronouncedly damaged as a result of rock fragmentation close to the critical load. This issue was only spotted to exist in the tested marble owing to its crystalline texture, for minimisation of which we employed PMMA flexible jaws. To further diminish undesirable fractures at the loading area, one may use longer flexible jaws with more circumferential coverage, or alternatively, use less stiff wooden jaws.

After conducting the BD test, the peak load \(P_c\) corresponding to the fracturing of the disk was recorded by the universal testing machine whereby the indirect tensile strength is calculated from \(\sigma _t=0.636 P_c/(2Rt)\) (Bieniawski and Hawkes 1978). The final measurements of the tensile strength are as follows: \(5.42 \pm 0.26\) MPa for limestone, \(11.41 \pm 0.42\) MPa for marble and \(9.50 \pm 0.68\) MPa for granite. Figure 2a visualises typical load-displacement curves obtained for different tested rock materials, emphasising the superiority of marble’s indirect tensile strength. Note that the non-linear behaviour in these curves is mainly attributed to the large non-linear deformation of the PMMA cushions.

It should be noticed that the BD test is an ISRM-suggested method for the determination of indirect tensile strength. According to ASTM-International (2016), tests other than straight pull are categorised as indirect test methods, which are less difficult and more affordable to be performed compared to direct experiments. Despite the advantages of indirect tensile tests, they are referred to as methods which overestimate the tensile strength of rock materials. As a frequently used specimen in rock mechanics applications, BD has been reported to yield overestimations for tensile strength of about \(10\%\) for metamorphic, \(20\%\) for igneous and \(30\%\) for sedimentary rocks, for practical applications (Perras and Diederichs 2014). However, these empirical figures are not definitive and may vary from one rock material to another, therefore they should be used cautiously.

We also note that the tensile strength used in FPZ characterization models may also in essence differ from the true tensile strength of rock measured via a direct method. This is because the rock material in the process zone is under a biaxial state of stress and not a uniaxial one. In fact, the stress component parallel to the crack, called T-stress, is negative for the SCB specimen configuration used in this study. Hence, the nature of the failures in the SCB and BD are in fact similar in the sense that a compressive stress is applied parallel to the failure plane. In this case, strength obtained from the BD test may be more representative of the failure in the FPZ rather than the true tensile strength obtained from direct tests.

Semi-circular bend test is a standard testing method, suggested by the International Society for Rock Mechanics (ISRM), for the measurement of the mode \(\mathrm {I}\) static fracture toughness of rock materials (Kuruppu et al. 2014). This test is conducted under the three-point bend loading configuration, and features the following noticeable virtues: (a) usage of a small amount of core material; (b) requiring relatively simple machining process; and (c) suitability for both isotropic and anisotropic rocks. Hence, the SCB specimen is regarded as a prospective candidate for the measurements of the mode \(\mathrm {I}\) fracture toughness. Figure 3a illustrates a 3-D schematic view of the test configuration.

In this research, a total number of 93 SCB specimens were manufactured, with a large range of radii of \(R=25\), 50, 75, 100, 150, 200 and 300 mm, and taking account of at least four repetitions per size. The manufacturing process involved water jet cutting to extract semi-circular disks from stone slabs as well as generating notches of length \(a/R=0.5\) in the SCB samples. In order to maintain the perpendicularity of the cutting plane with respect to the front (test) plane, and to enhance surface-finishing quality especially at loading and support points, specimen thicknesses were kept at \(t=2\) cm for all the specimens with \(R=25\) to 200 mm. The only exception was for the largest SCBs of radius \(R=300\) mm, which had the thickness \(t=3\) cm, to ensure the stability of specimens under three-point bend compression. Note that fracture toughness of rock materials are reported to be independent of the specimen thickness in numerous experimental studies in the literature (Schmidt and Lutz 1979; Laqueche et al. 1986; Kobayashi et al. 1986; Nath Singh and Sun 1990; Haberfield and Johnston 1990; Lim et al. 1994; Khan and Al-Shayea 2000; Chang et al. 2002). Given that our tested rocks have isotropic elasticity response, the roller supports were placed symmetrically to apply pure mode \(\mathrm {I}\) loading, with the span length ratio adjusted to \(S/R=0.6\) (Bahrami et al. 2019; Sedighi et al. 2020). It is noteworthy that asymmetrical support spans are required to impose mode \(\mathrm {I}\) loading for SCB specimens made of anisotropic rocks (Nejati et al. 2019, 2020b).

After the preparation of the test samples, fracture tests were carried out using Santam STM-150 universal testing machine in the displacement-control mode with the crosshead speed of 0.1 mm/min (see Fig. 3b). The fracturing process of the tested rock samples was instantaneous and the load–displacement behaviour was linear up the peak load. Hence, the LEFM theory can be adopted securely to theoretically evaluate the size effects in this study. Figure 4a–c portray samples of fractured specimens for all sizes of the three rock types. As evidenced, the notch had almost always propagated in a self-similar manner as an inherent response of mode \(\mathrm {I}\) fracture tests of isotropic materials.

In line with the ISRM-suggested method for the SCB test (Kuruppu et al. 2014), once the peak load \(P_c\) is measured (from the load-displacement curves for SCB fracture tests as in Fig. 2b), the fracture toughness is calculated from \(K_\mathrm {Ic}=K^*_\mathrm {I} P_c \sqrt{\pi a}/(2Rt)\), where R and t represent the radius and thickness of the SCB specimen, and a is the length of the notch. Moreover, the geometry factor \(K^*_\mathrm {I}\) is a numerically calculated dimensionless parameter, that depends on the specimen geometry and loading configuration, and is adopted from the results by Nejati et al. (2019, 2020b) for the isotropic case. Figure 5a presents the measured values of \(K_\mathrm {Ic}\) against the SCB radius, R. It is observed that, within the entire size range, marble is the strongest material with the largest fracture toughness values, while limestone acquires the least values overall and seems to be rather weaker. Granite, on the other hand, appears to be moderately resistant to fracture growth with \(K_\mathrm {Ic}\) values falling in between marble’s and limestone’s. Figure 2b illustrates these trends.

The general trend for all three types of rocks is that the fracture toughness values grow as the specimen radius increases (see Fig. 5a). This phenomenon, referred to as the size effect, occurs due to the formation of a relatively large fracture process zone in front of the crack tip. The FPZ region embodies a zone of highly localised deformation that occurs due to extensive micro-cracking processes. Since \(K_\mathrm {Ic}\) is a measure valid only inside the K-dominant zone, the specimen dimensions (and subsequently K-zone) should be enlarged enough so that the FPZ fits inside the K-dominant region (Nejati et al. 2020b). A large FPZ may partially stay out of the K-dominant zone in small samples, whereby inducing size-dependent fracture toughness values (Wei et al. 2016b, 2017a, 2018a, b; Dutler et al. 2018). A clear illustration of this process is given in Nejati et al. (2020b). As our experiments suggest, even by enlarging the specimen radius to \(R=200-300\) mm, the fracture toughness values may not still experience a plateau, and a further increase is yet anticipated.

To have experimental reference values for the true scale-independent fracture toughness of the rock materials, \(K_\mathrm {Ic}^\infty\), we perform a least squares analysis over the test data utilising an arbitrary exponential model: \(K_\mathrm {Ic}=K_\mathrm {Ic}^\infty \left[ {1 - {\lambda _1}\exp \left( { - {\lambda _2}R} \right) } \right]\). Here, \(K_\mathrm {Ic}^\infty\), \(\lambda _1\) and \(\lambda _2\) are unknown parameters that are computed by means of the least squares method with their values given in Fig. 5a. It should be noted that this exponential relation is an arbitrary choice among many possible alternatives which can model a gradually stabilising behaviour, whereby projecting a representative value for the true fracture toughness. We also point out that this estimate for the scale-independent fracture toughness, \(K_\mathrm {Ic}^\infty\), will not be used in our theoretical model presented and employed in Sects. 3 and 4. Alternative models also yield relatively similar values to the ones reported for \(K_\mathrm {Ic}^\infty\).

A comparison of the results in Fig. 5a reveals that the fracture toughness of the largest specimens of limestone, marble and granite are, respectively, 2.01, 1.41 and 1.49 times greater than the ones of the smallest samples. A comparison between the smallest specimens and \(K_\mathrm {Ic}^\infty\) also reveals even greater respective figures of 2.30, 1.45 and 1.58. This significant growth in the fracture toughness values implies a serious effect of size on the fracture toughness that must be taken into account in rock mechanics and engineering applications.

The true (scale-independent) fracture toughness value \(K_\mathrm {Ic}^\infty\) for each rock type is then employed to normalise the fracture toughness experimental data, i.e. \(K_\mathrm {Ic}/K_\mathrm {Ic}^\infty\), which is plotted versus the normalised disk radius in Fig. 5b. Presenting the data in a normalised manner shows that all the data points irrespective of the rock type follow a universal trend. This behaviour indicates that a proper size effect rule must in fact be material independent. It is noteworthy that if the experimental data are normalised by the fracture toughness of the largest specimen rather than by \(K_\mathrm {Ic}^\infty\), again a universal, material-independent pattern similar to that of Fig. 5b will be obtained. Also explicitly shown in Fig. 5b is that \(K_\mathrm {Ic}\) values reach a plateau only at larger sizes than \(R/(K_\mathrm {Ic}^\infty /\sigma _t)^2 \approx 10\). This finding seriously questions the validity of the minimum size requirement recommended by the ISRM standard, which reads \(R \ge (K_\mathrm {Ic}^\infty /\sigma _t)^2\) (Kuruppu et al. 2014). This insufficiency in the minimum size requirement for the SCB specimen has also been pointed out by previous research studies (Nath Singh and Sun 1990; Akbardoost et al. 2014; Wei et al. 2016a, 2017b; Nejati et al. 2020b), all of which urging either the usage of larger specimens or the introduction of correction factors.

We employed the DIC technique to measure the length of FPZ in the tested samples. To prepare the samples for DIC measurements, the surfaces of the samples were painted using a white spray, followed by creating a speckled pattern using a black spray. To prevent the light reflection, matte black and white sprays were employed. The DIC set-up, shown in Fig. 3b, includes a Canon CMOS 600D camera with a mounted macro-lens EF 100 mm f/2.8 to capture images, and two 11-Watt LED lamps to provide an appropriate and stable illumination for the tests. The images corresponding to the undeformed state of the specimens were captured at almost zero load, and images of the deformed samples were taken automatically with intervals of 5 seconds. The deformation states associated with 90% of peak load before fracturing were used to measure the FPZ length.

The Match ID software was used to perform correlation analyses between the reference image (at zero load) and the ones associated to the deformed states at different load levels. The correlation analyses were conducted on a rectangular region of interest in front of the crack tip with the subset size of \(55 \times 55\) pixels and the step size of 1 pixel. Also, the normalised sum of squared differences (NSSD) was used as the correlation criterion. More details about the DIC parameters can be found in Bahrami et al. (2020a). Figure 6a, b show samples of displacement and strain fields in front of the crack tip, that represent an area of high displacement gradients and strain concentrations. The formation of a semi-elliptical-shaped FPZ shown in Fig. 6b agrees well with the FPZ shape reported in Dutler et al. (2018); Moazzami et al. (2020); Zhang et al. (2020). It is noteworthy that the strain field is a secondary output of the DIC measurements, that is derived from the gradients of the directly measured displacements by means of numerical smoothing techniques (Pan et al. 2009). We therefore use the displacement field to evaluate the fracture process zone due to its higher accuracy compared to the strain field. Figure 6c demonstrates the variations of the horizontal displacement component u extracted along several parallel horizontal lines in front of the crack tip. As seen, a displacement jump exists along those lines situated adjacent to the crack tip. These jumps indicate highly localised deformation that are indications of the formation of the FPZ in that region. The jump in u decreases and finally vanishes at lines further away from the crack tip. The closest line from the crack tip within which no perceptible displacement jump is observed, indicates the end of FPZ, and the vertical distance between the crack tip and such line is measured as the length of the fracture process zone, i.e. \(L_\mathrm {FPZ}\).

Figure 7a displays the variations of the FPZ length versus the specimen radius, for all three types of the rock materials. It should be noted that we did not carry out as many DIC tests as the fracture toughness tests, and thus smaller number of experimental data are available in this plot. The scatter of the test data may be attributed to the uncertainty of FPZ length measurement via the DIC technique. Despite fluctuations of \(L_\mathrm {FPZ}\) data for limestone that is the weakest rock among the three, the general trend of the data suggests a substantial growth in the FPZ length with specimen size. In the entire size range, granite almost always attains the biggest fracture process zones with values as high as of almost 18 mm for the biggest samples. Similar large fracture process zones in coarse-grained granite  have also been identified by previous experimental studies (Wei et al. 2018c; Dutler et al. 2018). A comparison between the FPZ lengths of the largest and smallest specimens shows ratios of 5.01, 4.15 and 6.27 for limestone, marble and granite, respectively. These figures are indicative of the profound influence of sample size on the development of the FPZ. Small samples significantly constrain the full development of the fracture process zone, consider granite as an example with an average \(L_\mathrm {FPZ}=2.8\) mm for the SCB specimen of radius \(R=25\) mm. Ayatollahi and Aliha (2008) have confirmed FPZ length of 3.2 mm for specimens with radius \(R=37.5\) mm, while Wei et al. (2017b) have obtained \(L_\mathrm {FPZ}=2.8~\)mm for same-sized granite samples.

Similar to the normalisation procedure taken in Fig. 5b, we also present here the normalised values of the FPZ length \(L_\mathrm {FPZ}/a\) against the normalised radius \(R/(K_\mathrm {Ic}/\sigma _t)^2\) in Fig. 7b. As seen, the normalisation fades away the highly scattered data especially for limestone, yielding a relatively smooth descending trend of the normalised FPZ length with sample size. The large range chosen for the vertical axis of this figure will be later helpful to demonstrate the curves for theoretical predictions.

Let us consider the FPZ as a cohesive zone ahead of the crack tip as shown in Fig. 8. Employing the superposition principle, one may write the force equilibrium along a segment of the ligament, starting from the crack tip and extending to \(m \times (R-a-L_{FPZ})\) along the notch bisector of the SCB specimen (see Fig. 8a):where t is the specimen thickness, \(\sigma _p\) is the stress induced by the load P when no closure stress exists, i.e. case (b) in Fig. 8, and \(\sigma _{cl}\) is the stress induced by the closure stress applied at the FPZ, i.e. case (c) in Fig. 8. We consider the force equilibrium in Eq. (1) being satisfied over half of the ligament, i.e. \(m=0.5\), which is a reasonable choice for preserving adequate distance away from the load-point stress concentration area. Inserting the first five terms of the stress expression for \(\sigma _y\) from Eq. (16) into Eq. (1) and evaluating at \(\theta =0^{\circ }\) yields Eq. (2a), that readily simplifies to Eq. (2b) as

Note that the even terms of the stress field (\(n=2,4, ...\) in Eq. (16)) vanish along the crack bisector. Eq. (2b) satisfies the force equilibrium along half of the ligament at any load level P. As the load increases, a larger FPZ is developed to satisfy the force equilibrium. This is because the closure stress is limited to the tensile strength of the material, and a compensation for the additional forces applied by P has to be associated with an extension of the FPZ. At the onset of the fracture propagation, the FPZ can no longer extend (no more potential for the energy dissipation within the FPZ), and therefore, the violation of the static equilibrium leads to the fracture growth. It should be noted that the crack length equals \(a_\mathrm {eff}=a+L_\mathrm {FPZ}\) in Eq. (2). We refer to the crack parameters as \({\tilde{A}}_n\) in the case associated to Fig 8b and \({\tilde{A}}_{n,cl}\) in the case associated to Fig. 8c. The respective normalised forms are also referred to as \({\tilde{A}}^*_n\) and \({\tilde{A}}^*_{n,cl}\), that are determined numerically as elaborated in Appendix A, and are related to \({\tilde{A}}_n\) and \({\tilde{A}}_{n,cl}\) as

Here, \(\sigma _t\) is the tensile strength (we assume that stress is limited to \(\sigma _t\) in the FPZ) and \(\sigma =P/(2Rt)\). We employ two different models to characterise the distribution of cohesive stresses over the fracture process zone: uniform and linear traction models. The former is first introduced by Barenblatt (1959) and Dugdale (1960), and the latter is originally put forward by Labuz et al. (1985), both of which founded upon the singular crack tip stress field. Substituting Eqs. (3a) and (3b) into Eq. (2b), while ignoring the higher order terms of \(\sigma _{cl}\), i.e. \({\tilde{A}}_{3,cl}={\tilde{A}}_{5,cl}=0\) (refer to Appendix A.3 for details), delivers the strip-yield model at the incipience of the crack growthwhere dividing the entire expression into \({A^*_1} {\sigma _c} \sqrt{{a_\mathrm {eff}}}\) and using the effective crack length \(a_\mathrm {eff}=a+L_\mathrm {FPZ}\) yieldsRecalling that \({A^*_1}{\sigma _c}\sqrt{a}=A_\mathrm {1c}=K_\mathrm {Ic}/\sqrt{2\pi }\) and defining the following coefficients Eq. (5) finally readsEquation (7) depicts the strip-yield model developed for the SCB specimen by considering higher-order parameters of the crack tip stress field, and is in fact an extension of Dugdale’s strip-yield model for plastic zone (Dugdale 1960). By specifying a valid range for the parameter \({\mathcal {R}}^*_\infty\) (as in Fig. 5b), there will remain only two unknowns, namely \({\mathcal {L}}^*\) and \({\mathcal {K}}^*\). In the next section, we develop a second equation based on the energy release rate concept which in hand with Eq. (7) can determine the two unknowns.

Let us consider the energy release rate (ERR) due to crack extension, defined as the amount of potential energy, \(\Pi\), released per unit area of crack extension (Anderson 2017):Here, A is the crack face area and U is the elastic strain energy. Note that a decrease in the potential energy \(\Pi\) is associated to an equivalent increase in the strain energy U in a system that is subjected to a fixed load during crack propagation (Anderson 2017). This increase in the strain energy is due to the additional work put into the system by the fixed load. In the present problem, the incremental crack growth can be considered equivalent to the length of the fracture process zone, i.e. \(\Delta a=L_\mathrm {FPZ}\), and \(\Delta U\) can be formulated as: where the opening displacement v (Fig. 9a) and the normal stress \(\sigma _y\) (Fig. 9b) should be acquired through Eqs. (16) and (17). It must be noted that the strain energy U characterised by Eq. (9) is in fact associated to the formation of the FPZ essentially prior to fracture growth, that includes only a portion of fracturing energy dissipation. This is followed by a complete de-cohesioning of the material and the creation of new fracture surfaces that complete the energy dissipation due to fracturing. Therefore, the energy release rate calculated in Eq. (9) may underestimate the actual fracture  energy of the material. It is also noteworthy that the crack length is equal to \(a_\mathrm {eff}\) in the analysis of displacements in Fig. 9a, which means that crack parameters must be named as \({\tilde{A}}_n\), with even terms vanishing along the crack flanks. The ERR in pure mode \(\mathrm {I}\) can, therefore, be formed aswhere \({\hat{E}}=E\) and \({\hat{E}}=E/(1-\nu ^2)\) are for plane-stress and plane-strain conditions, respectively. Implementing \(A_n={A^*_n} \, \sigma \, a^{1-n/2}\) in line with Eqs. (3a) and (6) and collecting \({A^*_1} \,{\sigma }\sqrt{a}=K_\mathrm {I}/\sqrt{2\pi }\) gives the critical ERR as the following analytically integrable expression: with I, as a function of \({\mathcal {L}}^*\), representing the integral expression. For a fairly large specimen, i.e. mathematically of infinite size, the critical ERR is obtained by the limit of Eq. (11), as given in Eq. (12a), and the ratio \({{\mathcal {G}}_\mathrm {Ic}}/{\mathcal {G}}_\mathrm {Ic}^\infty\) is thereafter achieved as in Eq. (12b):

On the other hand, according to Bažant and Kazemi (1991), the brittleness number introduced in Eq. (13a) can be employed to define the ratio \({{\mathcal {G}}_\mathrm {Ic}}/{\mathcal {G}}_\mathrm {Ic}^\infty\) as in Eq. (13b): where \({{\dot{M}}_{\mathcal {G}}} = {{\mathcal {G}}^*}'/{\mathcal {G}}^*\), \({{\ddot{M}}_{\mathcal {G}}} = {{\mathcal {G}}^*}''/{{\mathcal {G}}^*}'\), and \({\mathcal {G}}^*(\alpha )\) is the normalised form of the energy release rate, i.e. \({\mathcal {G}} ={\mathcal {G}}^*P^2/(4{\hat{E}}Rt^2)\), which is computable via the finite element method (FEM) as explained in Appendix A. Eventually, by equating Eqs. (12b) and (13b), one may obtain the ratio of the measured fracture toughness to the true scale-independent fracture toughness, \({\mathcal {K}}^*\), as a function of the normalised FPZ length \({\mathcal {L}}^*\):Using Eqs. (7) and (14) together, it is now feasible to solve for \({\mathcal {L}}^*\) and \({\mathcal {K}}^*\) simultaneously, over a desired range of \({\mathcal {R}}^*_\infty\). Using Eq. (6b), we also define \({\mathcal {R}}^*={\mathcal {R}}^*_\infty /{{\mathcal {K}}^*}^2=R/(K_\mathrm {Ic}/\sigma _t)^2\) as a dimensionless size measure that is next used to plot figures.