Abstract
A transversely isotropic rock, slate, was utilized to investigate the size effect and anisotropy on its deformation, tensile strength, and failure mechanism. A series of Brazilian tests were conducted on slate samples of six different sizes from 25 to 100 mm in diameter at seven different loadingfoliation angles from 0° to 90°. The results indicate that the Young’s modulus in the plane of transverse isotropy increases, while the Young’s modulus and shear modulus perpendicular to the plane of transverse isotropy decrease with specimen size. The tensile strength of the slate increases with increasing loadingfoliation angle, the variation of which is well captured by the Nova–Zaninetti criterion. Furthermore, the tensile strength of the slate increases with specimen size at loadingfoliation angles from 0° to 45°, while it increases first and then decreases with specimen size at loadingfoliation angles from 60° to 90°. A unified sizeeffect relation including two equations is proposed and verified against the experimental data on slate. The sizeeffect relation reveals the relationship among the tensile strength, specimen size, and loadingfoliation angle for the transversely isotropic rock. Finally, the slate samples exhibit an increased brittle failure with specimen size, which is consistent with the observations in various isotropic rocks. It is also found that the specimen size, loadingfoliation angle, and loading configuration together control the failure mechanism of transversely isotropic rocks in the Brazilian test.
Introduction
In recent years, the behavior of transversely isotropic rocks, e.g., gneiss, schist, slate, phyllite, shale, mudstone, and layered sandstone, has attracted increased attention (Fu et al. 2018; Kundu et al. 2018; Sesetty and Ghassemi 2018; Setiawan and Zimmerman 2018; Xu et al. 2018). In general, anisotropic characteristics originate from the stratification in sedimentary rocks, mineral foliation in metamorphic rocks, and discontinuities in rock masses (Cho et al. 2012). The anisotropy is one of the most distinct features that must be considered in this kind of rock, and is widely encountered in civil, mining, petroleum, geothermal, and geoenvironmental engineering (Ma et al. 2018).
The tensile strength plays an important role and is often the most vital role in rock engineering, because rocks are usually weaker in tension than in compression or shear (Dan et al. 2013). Moreover, tensile failure greatly influences many rock engineering activities, such as drilling, cutting and blasting of rocks, hydraulic fracturing of a wellbore or a tunnel, exploitation of rock slopes, and excavation of underground structures (Chen and Hsu 2001; Goodman 1989). Hence, for engineering practice, the determination of the tensile strength of rocks is indispensable. Compared to the high requirement for experimentation with direct tensile tests (Liao et al. 1997; Shang et al. 2016) or the high requirement for sample preparation with ring tests (Barla and Innaurato 1973; Chen and Hsu 2001), the Brazilian test is a more common and easy method for measuring the tensile strength of rock. However, the use of the formula for the Brazilian tensile strength requires the material to be isotropic. To improve the Brazilian test, so that it can be useful for determining the tensile strength of transversely isotropic rocks, much work has been done by theoretical, experimental, and numerical methods (Aliabadian et al. 2017; Cai and Kaiser 2004; Chen et al. 1998; Claesson and Bohloli 2002; Exadaktylos and Kaklis 2001). Among these, a reasonably accurate equation for the principal tension at the rock disk center based on elastic constants and anisotropic angle was proposed (Claesson and Bohloli 2002), which approximates well to the tensile strength of transversely isotropic rocks.
The pioneering work on the anisotropy of tensile strength for transversely isotropic rocks was performed by Hobbs (1964), who conducted Brazilian tests on laminated siltstone, sandstone, and mudstone to measure their tensile strengths. Since then, many studies have been carried out on this issue (Ma et al. 2018). The published results showed that the tensile strength of transversely isotropic rocks greatly depended on the angle (β) between the loading direction and the transversely isotropic plane. Based on Brazilian test results of various transversely isotropic rocks, Vervoort et al. (2014) classified four trends for the variation in the tensile strength with β: (1) trend I for which the tensile strength remains constant; (2) trend II for which the tensile strength increases first and then remains constant; (3) trend III for which the tensile strength increases systematically; and (4) trend IV for which the tensile strength remains constant first and then increases linearly. Recently, a Ushaped distribution exhibited by a slate was added into the classification as trend V (Xu et al. 2018). In addition, three typical failure modes have been observed (Dan et al. 2013; Tavallali and Vervoort 2010, 2013): (1) fractures along the transversely isotropic planes in the low β range; (2) fractures across the isotropic planes in the high β range; and (3) mixed fractures along and across the isotropic planes in the intermediate β range. Furthermore, Hu et al. (2017) revealed the main cause for the three failure modes by means of scanning electron microscopy (SEM). The results indicated that with increasing β, the fracture morphology is changed from an intergranular fracture pattern along the bedding plane to that across the rock matrix, and coupled the intergranular fracture with the transgranular fracture when β = 45°.
When upscaling the strength and elasticity properties obtained from the laboratory to practical engineering design, the size effect must be taken into consideration (Li et al. 2018). As a key input parameter in engineering applications, the size effect on the tensile strength of transversely isotropic rocks must be understood. Nevertheless, the existing sizeeffect relations are almost all derived from isotropic rock or rocklike materials (Bažant 1984, 1997; Carpinteri et al. 1995; Hoek and Brown 1980; Masoumi et al. 2015), neglecting the influence of rock anisotropy. Consequently, determination of a suitable relation capturing both the size effect and the anisotropy of the tensile strength of transversely isotropic rocks is urgently needed. For this purpose, Brazilian tests were conducted on slate specimens of six different sizes (25–100 mm) at seven various β (0°–90°) in this study. During the test, the load and strain data and fracture conditions were recorded. Furthermore, the size effects on the elastic properties, tensile strength, and failure mechanism were investigated in detail.
Theoretical Background
Constitutive Model of Transversely Isotropic Rocks
As presented in Fig. 1, the disk of a transversely isotropic material under diametral loading (Brazilian test) has a diameter D and a thickness t. The angle between the global (x, y, z) and local (x′, y′, z′) coordinate systems is \({\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern0pt} 2}  \beta\), as shown in Fig. 1. The local system is attached to the transversely isotropic plane, with the x′axis and y′axis parallel to and perpendicular to the isotropic plane, respectively, and the z′axis coinciding with the zaxis. If the disk is loaded with force P, the stresses \(\sigma_{x}\), \(\sigma_{y}\), and \(\tau_{xy}\) within the disk can be expressed in the form of stress concentration factors (SCFs) \(q_{xx}\), \(q_{yy}\), and \(q_{xy}\) (Amadei 1996):
A generalized plane stress formulation was used, and the constitutive law is expressed as follows:
As the medium in the direction parallel to the transversely isotropic plane is postulated to be linearly elastic, homogeneous, and continuous, Amadei (2012) proposed the expression of \(a_{ij}\) as follows:
where \(E\) and \(E^{\prime}\) represent the elastic moduli parallel to and perpendicular to the plane of transverse isotropy, respectively; \(\nu^{\prime}\) and \(G^{\prime}\) are the Poisson’s ratio and shear modulus in the direction normal to the transversely isotropic plane, respectively. Note that the parameters \(a_{ij}\) in Eq. (3) only depend on \(E\), \(E^{\prime}\), \(\nu^{\prime}\), and \(G^{\prime}\), and are independent of \(\nu\), which is Poisson’s ratio in the plane of transverse isotropy.
Determination of Elastic Constants for Transversely Isotropic Rocks
LoureiroPinto (1979) first determined the elastic constants of anisotropic rocks by means of Brazilian tests. Later, Pinto’s procedure was revised by Amadei (1996), and a more accurate solution was given by Chen et al. (1998), who combined the Brazilian test and the generalized reduced gradient (GRG) method. However, the theorem and mathematical computation procedure in the approach proposed by Chen et al. (1998) are rather complicated. To overcome this problem, Chou and Chen (2008) developed a more convenient method. This approach combines the Brazilian test and a commercial numerical program (e.g., finitedifference program, finiteelement program, or boundary element program), and the iterative solution procedure is shown in Fig. 2 and described briefly as follows:

1.
Brazilian tests are performed on two types of specimens: Type N (the central axis normal to the transversely isotropic plane) and Type P (the central axis parallel to the transversely isotropic plane). The strains at the disk center are obtained by Eq. (4), and \(\varepsilon_{H}\), \(\varepsilon_{45}\), and \(\varepsilon_{V}\) are measured by 45° strain rosettes in the test. The parameters \(E\) and \(\nu\) are calculated by Eq. (5).

2.
The temporary \(E^{\prime}\), \(\nu^{\prime}\), and \(G^{\prime}\) are computed by substituting \(\varepsilon_{x} {{\pi Dt} \mathord{\left/ {\vphantom {{\pi Dt} P}} \right. \kern0pt} P}\), \(\varepsilon_{y} {{\pi Dt} \mathord{\left/ {\vphantom {{\pi Dt} P}} \right. \kern0pt} P}\), \(\gamma_{xy} {{\pi Dt} \mathord{\left/ {\vphantom {{\pi Dt} P}} \right. \kern0pt} P}\), and SCFs into Eq. (2).

3.
The five elastic constants obtained by the previous steps are applied in the numerical simulation of the Brazilian tests, and the software of Fast Lagrangian Analysis of Continua (FLAC3D) is adopted in this study. The new SCFs are computed by Eq. (1).

4.
Steps (2) and (3) are repeated until \(E^{\prime}\), \(\nu^{\prime}\) and \(G^{\prime}\) have converged, and the difference between two successive cycles below 0.1% is adopted as the termination criterion:
Meanwhile, to determine the elastic constants for a transversely isotropic medium, \(E\), \(\nu\), \(E^{\prime}\), \(\nu^{\prime}\), and \(G^{\prime}\) must satisfy the following thermodynamic constraints (Amadei 1996; Chen et al. 1998):
Typical Tensile Failure Criteria for Transversely Isotropic Rocks
Regarding the variation in the tensile strength of transversely isotropic rocks with loading direction, Hobbs (1967) proposed the first failure criterion based on the Griffith crack theory. Afterwards, an anisotropic tensile strength criterion was proposed by Barron (1971) using the modified Griffith’s theory. In essence, the two criteria are consistent and combined together as the Hobbs–Barron (H–B) criterion (Ma et al. 2017). Subsequently, Nova and Zaninetti (1990) presented a continuous tensile strength criterion, called the Nova–Zaninetti (N–Z) criterion, which reproduced the variation in the direct tensile strength of the Luserna gneiss with reasonably good accuracy. Recently, another tensile failure criterion was proposed by Lee and Pietruszczak (2015) employing the single plane of weakness (SPW) theory, so the criterion is called the SPW criterion. Meanwhile, based on the tensile strength tensor, Lee and Pietruszczak (2015) also proposed another tensile failure criterion, named the Lee–Pietruszczak (L–P) criterion, which tends to oversimplify the real tensile strength features. These criteria are expressed in detail as follows:
with \(\cos \beta^{ * } (1 + \cos \beta^{ * } ) = \frac{{2T_{\text{b}} }}{{T_{\text{m}} }}\).

2.
N–Z criterion (Nova and Zaninetti 1990):

3.
SPW criterion (Lee and Pietruszczak 2015):
with \(\beta^{ * } = \cos^{  1} \sqrt {\frac{{T_{\text{b}} }}{{T_{\text{m}} }}} .\).

4.
L–P criterion (Lee and Pietruszczak 2015):
where \(T(\beta )\) denotes the tensile strength of a specimen when the angle between the loading direction and the transversely isotropic plane is \(\beta\); \(T_{\text{m}}\) and \(T_{\text{b}}\) represent the tensile strengths of the rock matrix and the weak planes (e.g., bedding plane, foliation plane, and discontinuity plane), respectively.
SizeEffect Models
When the sample shape, e.g., lengthtodiameter ratio, is prescribed, the sample diameter can be selected as the sizedependent representative parameter, since the variation in the diameter is directly related to the volumetric change in the sample (Masoumi et al. 2018). To our knowledge, although the existing sizeeffect models reviewed extensively by Masoumi et al. (2015) are all derived from isotropic materials, they are very important references to deduce the sizeeffect model for transversely isotropic rocks.
Based on the statistical theory, viz., the weakestlink theory, a statistical model was first proposed to describe the relationship between material strength and sample size (Weibull 1951). Furthermore, from the viewpoint of fracture energy, a sizeeffect law (SEL) was put forward by Bažant (1984), who explained that the size dependence in brittle and quasibrittle materials, e.g., concrete and rock, was caused by the blunting of microcracking prior to the fracture. Subsequently, sizeeffect studies were carried out by Carpinteri et al. (1995) on Brazilian tensile strength data of concrete specimens, and a multifractal scaling law (MFSL) was proposed. In addition, the concept of fractals was introduced into fracture energy by Bažant (1997), who presented a fractal fracture sizeeffect law (FFSEL). The former three sizeeffect models are the descending type, but the last one is the only ascending type. All four sizeeffect models are given in the following:

1.
Statistical model (Weibull 1951):

2.
SEL (Bažant 1984):

3.
MFSL (Carpinteri et al. 1995):

(4)
FFSEL (Bažant 1997):
where \(\sigma_{\text{t}}\) is the tensile strength; \(d\) is the sample diameter; \(k\) is a positive constant; \(\sigma_{{{\text{t}}50}}\) denotes the tensile strength obtained from a specimen of 50 mm diameter; \(B\) and \(\lambda\) are dimensionless material constants; \(f_{\text{t}}\) represents the strength of a specimen with an infinitesimal size; \(d_{0}\) represents the maximum aggregate size; \(f_{\text{c}}\) denotes the strength of a specimen with an infinite size; \(l\) is a material constant with unit of length; \(\sigma_{0}\) has the same meaning as \(f_{\text{t}}\); and \(d_{\text{f}}\) denotes the fractal dimension.
Materials and Methods
Sample Preparation
To minimize the influence of discreteness, five blocks of slate were taken from the same location in a slate quarry in Jiujiang, Jiangxi Province, China. The slate is a metamorphic Precambrian rock from sedimentary rocks. The slate, which exhibits dark graytolight gray colors, possesses a welldeveloped slaty structure with very straight layers (Fig. 3a), compared to the rippled layers of shale. Moreover, a thin section of the slate normal to the foliation planes is observed by optical microscopy and shown in Fig. 3b. The sample has a layered texture composed of granular calcite (5%), flaky sericite (25–30%), angular feldspar and quartz (65–70%) with a very fine grain size in the range of 0.01–0.05 mm. Figure 3c further shows using SEM that the slate is tightly packed with strongly oriented rock fragments. The natural density of the slate is 2759 ± 5 kg/m^{3}.
The core specimens were drilled parallel to the foliations (Type P) and perpendicular to the foliations (Type N) with diameters of 25, 38, 50, 63, 75, and 100 mm, and were then cut into disk specimens. The thickness (t) of each disk specimen was fixed at 0.5 times the diameter (D) of the specimen. The end surfaces of the disk specimens were polished to satisfy the standard for the tests (ASTM 2016). The Pwave velocities of the slate samples were tested with inclination angles (β) of 0°, 15°, 30°, 45°, 60°, 75°, and 90° using a Tektronix DPO 2012B oscilloscope, OLYMPUS 5077PR wave pulser/receiver and two OLYMPUS V195 ultrasound probes. The variation in the Pwave velocities (\(v_{\text{p}}\)) with β is depicted in Fig. 4. The results indicate that the \(v_{\text{p}}\) decreases from 6002 to 4760 m/s with increasing β because of the increased influence of foliations on the ultrasound transmission within the slate sample. Based on the anisotropy classification proposed by Tsidzi (1997), using a velocity anisotropy index (VA) expressed as Eq. (15), the slate (VA = 22.9) is classified as a highly anisotropic rock:
in which \(v_{\hbox{max} }\), \(v_{\hbox{min} }\), and \(v_{\text{mean}}\) are the maximum, minimum, and average ultrasonic wave velocities, respectively.
Test Procedure
To ensure accuracy, a VJ tech machine with a low loading capacity of 100 kN was employed to conduct the Brazilian tests. The loading rate was controlled at 0.3 mm/min. Among the typical loading configurations suggested by ASTM (2016) and ISRM (1978), the flat loading platens were chosen for this sizeeffect study. The reason behind this is that the ratio of the steel rod diameter to the specimen diameter (for flat loading platens with two smalldiameter steel rods) or the contact angle (for curved loading jaws) is a function of the specimen diameter, which will greatly affect the measured tensile strength (Komurlu and Kesimal 2015; Markides and Kourkoulis 2012; Rocco et al. 1999). In addition, strains at the disk center of the specimen were obtained by a 45° strain rosette with a length of 3 mm, which should not exceed 10% of the diameter of the disk (Chen et al. 1998). During the test, the load and strains were recorded simultaneously using a Kyowa datalogger. The fracture conditions of the specimens in the test were recorded through a Photron FASTCAM SAZ highspeed camera with LED lighting.
Results and Discussion
Size Effect on the Elastic Properties
Figure 5a and b shows the typical stress–strain curves of the diskshaped slate samples of Type N and Type P, respectively. The \(\varepsilon_{x} {{\pi Dt} \mathord{\left/ {\vphantom {{\pi Dt} P}} \right. \kern0pt} P}\), \(\varepsilon_{y} {{\pi Dt} \mathord{\left/ {\vphantom {{\pi Dt} P}} \right. \kern0pt} P}\), and \(\gamma_{xy} {{\pi Dt} \mathord{\left/ {\vphantom {{\pi Dt} P}} \right. \kern0pt} P}\) used in Sect. 2.2 to compute \(E^{\prime}\), \(\nu^{\prime}\), and \(G^{\prime}\) correspond to the secant values at 50% peak stress as depicted by the green lines in Fig. 5.
Theoretically, only two diskshaped samples are needed to determine the five elastic constants of transversely isotropic rocks by the Brazilian tests. To ensure accuracy, in this study, more than two specimens were employed to measure the elastic constants of the slate for each size, as listed in Table 1. The \(E\), \(E^{\prime}\), \(G^{\prime}\), \(\nu\), and \(\nu^{\prime}\) results for each size slate specimen are listed in Table 1 and plotted in Fig. 6. Parameter \(E\) dramatically increases as the sample diameter increases from 25 to 38 mm, then fluctuates when the sample diameter is in the range of 38–75 mm, and finally increases again until the sample diameter reaches 100 mm. The total trend of \(E\) increases with specimen size, which is similar to that observed in the Blanco Mera granite (Quiñones et al. 2017) and Stanstead granite (Walton 2017). This is because \(E\) and \(\nu\) are obtained by Type N samples, in which the foliation planes have little influence on the deformability of transversely rocks under the Brazilian tests. At this point, the slate of Type N and the two granites are treated as isotropic rocks, irrespective of rock anisotropy. The ascending size effect on \(E\) can be attributed to the nearsurface damage during sample preparation. The larger specimens have a lower surface areatovolume ratio (\({8 \mathord{\left/ {\vphantom {8 d}} \right. \kern0pt} d}\) for the Brazilian disks), so the larger specimens can be expected to be more rigid with less damage densities. In contrast, \(E^{\prime}\), \(\nu^{\prime}\), and \(G^{\prime}\) were determined by Type P samples, in which the foliation planes have a great influence on the deformability of transversely isotropic rocks under the Brazilian tests. Moreover, the stiffness of the foliation plane is far smaller than that for the rock matrix. The increasing number or volume of foliation planes per unit volume in the loading direction with increasing sample diameter reduces the rigidity of the sample, which can explain the decrease with the sample diameter as shown in Fig. 6a for \(E^{\prime}\) and \(G^{\prime}\). The singularities of \(E^{\prime}\) for the 38mmdiameter sample and of \(G^{\prime}\) for the 50mmdiameter sample may be induced by the greater effect of the surface damage in sample preparation than that of the foliation plane. Additionally, \(\nu\) and \(\nu^{\prime}\) do not present an evident size effect, as depicted in Fig. 6b.
The shear modulus \(G^{\prime}\) is often obtained (Togashi et al. 2017) using Saint–Venant’s empirical relation (Saint–Venant 1863) as follows:
Cho et al. (2012) investigated the validity of the empirical equation and found that the equation did not agree well with the experimental data obtained from Asan gneiss, Boryeong shale, and Yeoncheon Schist. With respect to this issue, the results of \(G^{\prime}\) obtained from experiment and theory for the slate of various diameters are compared and shown in Fig. 7. The results demonstrate that with increasing sample diameter, the difference between \(G^{\prime}\) measured by experiment and that predicted by theory decreases gradually, almost equal to zero when D = 100 mm.
Size Effect on the Tensile Strength
As mentioned in the introduction section, the indirect tensile strength (\(\sigma_{\text{t}}\)) is well approximated as the principal tension at the disk center, expressed as follows (Claesson and Bohloli 2002):
where \(b = \frac{{\sqrt {EE^{\prime}} }}{2}\left( {\frac{1}{{G^{\prime}}}  \frac{{2\nu^{\prime}}}{{E^{\prime}}}} \right)\).
For each condition (a prescribed size and \(\beta\)), there are 3–5 slate samples being tested. The tensile strength results for each of the samples tested are plotted in Fig. 8. The results indicate that the variation in \(\sigma_{\text{t}}\) with \(\beta\) for specimens of different sizes mostly exhibits trend III: \(\sigma_{\text{t}}\) increasing systematically with \(\beta\). Sometimes, the minimum \(\sigma_{\text{t}}\) occurs at 15° or 30°, which is consistent with the results observed in other transversely isotropic rocks (Khanlari et al. 2015; Mighani et al. 2016).
Assessment of Tensile Failure Criteria
To assess the performance of existing typical tensile failure criteria, as mentioned in Sect. 2.3, three different assessment indicators are employed. They are the maximum absolute relative error (MARE), the average absolute relative error (AARE), and the standard error (SE), defined by the following formulae:
The three assessment indicators can reflect the reliability or misfit of each failure criterion, and the higher MARE, AARE, and SE indicate the lower reliability and higher misfit, and vice versa. The order of the failure criteria is sorted according to the magnitude of the MARE, AARE, and SE when the four failure criteria are evaluated. Note that the SE is recommended as the final assessment indicator if the order of the MARE, AARE, and SE is not the same for a certain failure criterion (Ma et al. 2017).
The comparisons between the predicted and test results are shown in Fig. 8, and the evaluation results using the MARE, AARE, and SE are listed in Table 2. The results indicate that the orders of the reliability of the failure criteria are as follows: (1) N–Z > L–P > H–B > SPW for the 25, 38 and 50mmdiameter specimens, (2) N–Z > HB > L–P > SPW for the 63 and 100mmdiameter specimens, and (3) L–P > SPW > N–Z > H–B for the 75mmdiameter specimens. Except for the 75mmdiameter specimens, the predicted results by the N–Z criterion are the most in line with those from the experiments. Therefore, the N–Z criterion is recommended to describe the anisotropy of the tensile strength of the slate.
Anisotropic Size Effect of the Tensile Strength
Based on the existing sizeeffect models referred to in Sect. 2.4 and the experimental data of this study as shown in Fig. 9, three principles defining the sizeeffect response are as follows:

1.
The relationship between the tensile strength and specimen size depends on the mechanical properties of the material.

2.
The variation in the tensile strength with specimen size shows both ascending and descending trends and is correlated with the loadingfoliation angle.

3.
For samples of a prescribed shape, the tensile strength has upper and lower boundaries with varying sample sizes.
Thus, the SEL and FFSEL are recommended to describe the size dependence of the tensile strength of the slate. This combination has the advantages of capturing increasing and decreasing trends with the common parameters (\(\lambda d_{0}\)).
Based on the SEL, when the upper and lower boundaries are considered, the sizeeffect relation is transformed as follows:
where \(T_{1} (d)\) is the tensile strength of the specimen with a diameter of \(d\); \(\eta { = }\lambda d_{0}\) is the unit of length; and \(\sigma_{0}\) and \(\sigma_{\text{M}}\) are the tensile strengths when \(d \to 0\) and \(d \to \infty\), respectively.
Correspondingly, the FFSEL is transformed as follows:
where \(T_{2} (d)\) has the same meaning as \(T_{1} (d)\); \(d_{f}\) represents the fractal dimension; and \(\bar{\sigma }_{0}\) is the tensile strength when \(d \to 0\).
The curvefitted results of Eqs. (21) and (22) are shown in Fig. 9, and the fitting parameters (\(\sigma_{0}\), \(\sigma_{M}\), \(\eta\), \(\bar{\sigma }_{0}\), and \(d_{\text{f}}\)) of the equations in every loadingfoliation direction are listed in Table 3. The fitted curves are in agreement with the experimental data and the correlation coefficients of Eqs. (21) and (22) are greater than 0.73. The results indicate that with increasing specimen diameter, the tensile strength continues decreasing when β ranges from 0° to 45°, and the tensile strength increases first and then decreases when β is in the range of 60°–90° or the specimen belongs to Type N. The transition occurs when β ranges from 45° to 60°, which may be attributed to the change in the failure modes in this range. The results are elaborated in the following. When β ranges from 0° to 45°, the tensile strength of slate is mainly dependent on the strength of the foliation planes, displaying the failure mostly along the foliation plane. When β ranges from 60° to 90° or the specimen belongs to Type N, the tensile strength of the slate is primarily determined by the strength of the rock matrix, exhibiting the failure mostly across the rock matrix. The typical descending size effect can be attributed to Fairhust’s theory (Fairhurst 1971), originating from elastic energy principles, that the product of the length of the critical flaw and the square of the strength of the tested sample is constant. The length of the critical flaw is proportional to the specimen size. Accordingly, the strength decreases with increasing specimen diameter. On the other hand, the reverse size effect observed in the slate specimens can be explained by the smaller specimens having lower strength because of the higher surface damage density or the developing cracks inclining more to intersect the free sample surface in the smaller specimens (Quiñones et al. 2017).
Referring to Table 3, \(\sigma_{0}\) and \(\sigma_{\text{M}}\) exhibit an almost increasing trend with β; \(\bar{\sigma }_{0}\) also continues increasing as β increases from 60° to 90°; however, the parameters \(\eta\) and \(d_{\text{f}}\) are basically constant. It is worth noting that the curve fitting parameters for the Type N specimen are greatly different from those obtained for the Type P specimens, except for \(\eta\). The results also suggest that the loading direction plays an important role in the sizeeffect response.
In addition, to compare the anisotropy of the size effect on the tensile strength of the slate, the derivative of Eq. (21) is deduced as follows:
where \(T^{\prime}_{1} (d)\) is the derivative of \(T_{1} (d)\). Values of \(T^{\prime}_{1} (d)\) are negative, since \(\sigma_{0}  \sigma_{\text{M}}\) and \(\eta\) are positive, as listed in Table 3. The values of \(T^{\prime}_{1} (d)\) are comparable when β = 0°–30°, reaching the maximum value. \(T^{\prime}_{1} (d)\) is the minimum value when β = 90°. This implies that the downward size effect on the tensile strength of the slate is strongly increasing with β, reaching the strongest value when β = 90°.
Anisotropy of the Tensile Strength
Darlington et al. (2011) found that the results of specimens 300 mm in diameter, at which asymptotic strength is met, are of crucial importance to largescale design. Accordingly, the tensile strength of the 300mmdiameter specimen is recommended as the tensile strength of the representative elementary volume (REV), which is termed \(T_{\text{REV}}\). The predicted \(T_{REV}\) by Eq. (21) are listed in Table 4, which increases steadily with increasing β. Furthermore, the experimental data fitted by Eq. (8) are plotted in Fig. 10, and the curve fitting parameters \(T_{\text{b}}\) and \(T_{\text{m}}\) are listed in Table 5. \(T_{\text{b}}\), characterizing the tensile strength of weak planes, decreases gradually, and \(T_{\text{m}}\), characterizing the tensile strength of the rock matrix, increases first and then decreases with increasing specimen size. The variation trends correspond to the two kinds of sizeeffect responses as depicted in the previous section (see Fig. 9). It further illustrates that the transition of the sizeeffect trends is closely related to the change in the failure modes.
The curvefitted results agree well with the experimental data, with correlation coefficients > 0.82 for the various specimen diameters (25–300 mm). In summary, the N–Z criterion is capable of describing the anisotropy of the tensile strength of the slate irrespective of the sample size.
Universal Equations for Size Effect and Strength Anisotropy
A unified sizeeffect relation capturing the relationship among the tensile strength, sample size, and loading direction in transversely isotropic rocks is vital for the correct estimation of the rock strength of a certain specimen size in a given loading direction. As \(\sigma_{\text{M}}\) and \(\sigma_{0}\) of Eq. (21) and \(\bar{\sigma }_{0}\) of Eq. (22) can be replaced by Eq. (8), two universal equations describing both the size effect and anisotropy in transversely isotropic rocks are proposed in the following forms:
where \(T_{1} (d,\beta )\) is the tensile strength of the specimen with a diameter d at a loadingfoliation angle β; \(T_{{{\text{b}}0}}\) and \(T_{{{\text{m}}0}}\) are the tensile strength of the weak plane and the rock matrix, respectively, when \(d \to 0\); \(T_{\text{bM}}\) and \(T_{\text{mM}}\) represent the tensile strength of the weak plane and the rock matrix, respectively, when \(d \to \infty\); and \(\eta\), as mentioned in Eq. (21), denotes a material constant with unit of length (mm).
where \(T_{2} (d,\beta )\) has the same meaning as \(T_{1} (d,\beta )\); \(\bar{T}_{{{\text{b}}0}}\) and \(\bar{T}_{{{\text{m}}0}}\) represent the tensile strength of the weak plane and the rock matrix, respectively, when \(d \to 0\); \(\eta\) is the same as that in Eq. (24); and \(d_{\text{f}}\), as mentioned in Eq. (22), is the dimensionless fractal dimension.
If both ascending and descending trends are observed in the experimental data, let \(T_{1} (d_{i} ,\beta ) = T_{2} (d_{i} ,\beta )\), and the specimen diameter (\(d_{i}\)) at which the maximum tensile strength is reached can be determined. According to the experimental data observed in the slate, Eqs. (24) and (25) are fitted as Eqs. (26) and (27) with R^{2} > 0.93, and the two fitted surfaces are plotted in Fig. 11:
Combining Eqs. (26) and (27), \(d_{i}\) for \(\beta\) = 60°, 75°, and 90° are 41.0 mm, 35.2 mm, and 34.1 mm, respectively. The predicted results are comparable to the experimental results (Fig. 9b), presenting a downward trend with increasing \(\beta\). The corresponding maximum tensile strengths of the slate for \(\beta\) = 60°, 75°, and 90° are 13.32 MPa, 18.35 MPa, and 21.75 MPa, respectively.
As seen from the discussion, Eqs. (24) and (25) provide a method to obtain the tensile strength of slate with varying sample sizes and anisotropic angles. Moreover, the characteristic properties of the strength of the representative elementary volume, the upward and downward sizeeffect trends, and the specimen diameter corresponding to the maximum strength are all combined to enable a systematic description of the behavior to be assessed. Nevertheless, the two equations suffer from the problem that specimens of different sizes and anisotropic angles are needed to determine the parameters in the equations for a certain material. It is also highlighted that more research into this issue is needed to demonstrate the capability of the proposed equations for other transversely isotropic rocks.
Size Effect on the Tensile Failure Mechanism
Fracture Pattern
The failure pattern of a slate specimen under the Brazilian test is nearly twodimensional, mostly being similar on two flat surfaces. Thus, only the failure pattern on one surface is displayed. Figure 12 shows the representative fracture patterns of specimens of different sizes at different loadingfoliation angles after testing. The sketches of the fracture patterns are also depicted in Fig. 13. The failure pattern is closely related to the loadingfoliation angle and specimen size. In total, three types of failure patterns are presented: layer activation failure (type I), mixed failure (type II), and nonlayer activation failure (type III).
As shown in Fig. 13, when β ranges from 0° to 30°, the disk specimen fails mainly in the type I mode; when β ranges from 45° to 60°, the failure pattern of the specimen covers all three types; and when β ranges from 75° to 90°, the failure of the specimen is mostly manifested by the type III failure mode. This result is consistent with those observed by other researchers (Dan et al. 2013; Tavallali and Vervoort 2010; 2013). The Type N specimen fails by single or multiple cracks along the loading direction accompanying the layer activation, as shown in Fig. 13h.
Furthermore, considering the size effect, the percentages of the three failure types of specimens at different loadingfoliation angles are listed in Table 6. For β equal to 0° or 90°, the specimen size has little influence on the failure patterns of the specimens. For β varying from 15° to 30°, the failure patterns of the specimens include types I and II, and the percentage of type II increases first and then decreases and then stabilizes with the specimen size. For β increasing from 45° to 60°, the failure patterns of the specimens include all three types, and the percentage of type III increases first and then decreases with increasing specimen size. For β reaching 75°, the failure patterns of the specimens include type II and III, and the percentage of type III increases first and then decreases and then stabilizes with the specimen size.
Generally, the tensile strength of the specimen failed by type III is the highest, that by type II is secondary, and that by type I is the lowest. The variation in the failure pattern with the specimen size can account for variation in the tensile strength with the specimen size, as discussed in Sect. 4.2. The percentage of the fracture type is also similar to the relative fracture length used in the previous study (Tavallali and Vervoort 2010, 2013).
Transverse Strain
The maximum transverse strain is defined as the transverse strain measured by the horizontal strain gage of the strain rosette at failure. As shown in Fig. 14, the specimen size impacts the maximum transverse strains of the slate specimens under diametral loading. The lowest value of the maximum transverse strain occurs at β = 0° when the specimen diameter is smaller than or equal to 50 mm, while it occurs at β = 15° when the specimen diameter is larger than 50 mm. In general, the maximum transverse strains, regardless of the loading direction, decrease with the specimen diameter, reaching the minimum value at d = 100 mm. Accordingly, based on the maximum extension strain criterion (Li and Wong 2013), the result implies that the slate samples exhibit an increased brittle failure as the specimen size increases, with Type N samples of large size failing by multiple cracks (Fig. 12). This is consistent with the observations in various isotropic rocks, including sandstone, andesite, granite, basalt, and monzonite (Bahaaddini et al. 2019; Masoumi et al. 2017, 2018; Serati et al. 2017). They stated that the increased brittleness of rocks resulting from the increasing sample size in a Brazilian test would cause the tensile breakage to change from single cracking to multiple cracking, which may undermine the validity of the Brazilian test for brittle rocks. Meanwhile, similar to the size effect on the tensile strength, the size effect on the maximum transverse strain is stronger at high loadingfoliation angles than at low loadingfoliation angles, reaching the strongest effect at β of 90°.
In addition, it has been found that the contact angle plays an important role in the failure mechanism in the Brazilian test (Bahaaddini et al. 2019). The critical contact angle for the transition from multiple cracking to single cracking is material dependent, which can be estimated by the ratio of the uniaxial compressive strength (UCS) to the tensile strength. As shown in Fig. 15, the UCStotensile strength ratios of the slate samples do not present an evident size effect but vary with loadingfoliation angle. At β of 45°–90°, the UCStotensile strength ratios of the slate samples are comparable and in the range of 5 to 10, while at β of 0°–30°, they range from 15 to 30. According to the recommendation of Bahaaddini et al. (2019), the minimum contact angles required for the slate samples at β of 0°–30° and 45° to 90° are 14° and 22°, respectively. In this study, the flat loading platen was adopted to neglect the influence of the contact angle. As an alternative, the curved loading jaw with different radii of curvature regarding samples of different sizes can be used to ensure a constant contact angle (ISRM 1978; Lin et al. 2015). Note that the UCS data of the slate samples are provided in the reference (Li et al. 2019).
It can be concluded that the specimen size, loadingfoliation angle, and loading configuration together control the failure mechanism of transversely isotropic rocks in the Brazilian test. Due to the presence of weak planes, in the Brazilian test, transversely isotropic rocks are mostly not failed by a single tensile crack initiated at the center. Accordingly, considering the influences of the specimen size, loadingfoliation angle, and loading configuration, a more reliable solution for the tensile strength of transversely isotropic rocks can be derived in future work.
Conclusions
Slate, as a transversely isotropic rock, was employed to investigate the size effect and anisotropy on its deformation, tensile strength, and failure mechanism. Diskshaped slate samples of six sizes ranging from 25 to 100 mm were cored parallel to (Type P) and normal to (Type N) the foliation planes. A series of Brazilian tests were performed on the slate samples at loadingfoliation angles of 0–90° at intervals of 15°. The main findings are summarized as follows:

1.
The Young’s modulus (\(E\)) in the transversely isotropic plane presents an increasing size effect, while the Young’s modulus (\(E^{\prime}\)) and shear modulus (\(G^{\prime}\)) perpendicular to the transversely isotropic plane exhibit a decreasing size effect. The Poisson’s ratios parallel to (\(\nu\)) and normal to (\(\nu^{\prime}\)) the transversely isotropic plane do not show an evident size effect. Moreover, the difference between \(G^{\prime}\) obtained from the experiment and Saint–Venant’s empirical equation decreases with specimen size. The size effects on these elastic properties of slate can be attributed to the combined influence of foliation planes and nearsurface damage during sample preparation.

2.
Irrespective of the sample size, the variation in the tensile strength of slate with loadingfoliation angle presents an increasing trend, captured well by the N–Z criterion. At loadingfoliation angles of 0°–45°, the tensile strength of the slate presents a typical descending size effect. In contrast, at loadingfoliation angles of 60°–90°, the tensile strength of the slate presents a first ascending and then descending size effect. The transition of the sizeeffect trends is closely related to the failure mechanism. A unified sizeeffect relation including two equations is proposed and validated against the experimental data on slate. This is the first time that the relationship among the tensile strength, specimen size, and loadingfoliation angle has been comprehensively captured for the transversely isotropic rock.

3.
With increasing specimen size, the fracture pattern of the specimen at the loadingfoliation angle of 0° or 90° does not vary, but at other loadingfoliation angles, the percentage of the fracture type with a higher strength increases first and then decreases. Regardless of the loading direction, the maximum transverse strain of the slate under the Brazilian test shows a descending size effect. This implies that the slate samples exhibit an increased brittle failure as the specimen size increases, which is consistent with the observations in various isotropic rocks. It is also found that the specimen size, loadingfoliation angle, and loading configuration together control the failure mechanism of transversely isotropic rocks in the Brazilian test. Taking these influences into account, a more reliable solution for the tensile strength of transversely isotropic rocks should be derived.
Abbreviations
 \(\beta\) :

Angle between the loading direction and the transversely isotropic plane (°)
 \(\sigma_{x} ,\;\sigma_{y} \;{\text{and}}\;\tau_{xy}\) :

Stresses in global coordinate system (MPa)
 \(\varepsilon_{x} ,\;\varepsilon_{y} ,\;{\text{and}}\;\gamma_{xy}\) :

Strains in global coordinate system
 \(q_{xx} ,\;q_{yy} ,\;{\text{and}}\;q_{xy}\) :

Stress concentration factors
 \(a_{ij}\) :

Compliance matrix
 \(E\;{\text{and}}\;E^{\prime}\) :

Elastic moduli parallel to and perpendicular to the plane of transverse isotropy, respectively (GPa)
 \(\nu \;{\text{and}}\;\nu^{\prime}\) :

Poisson’s ratios parallel to and perpendicular to the plane of transverse isotropy, respectively
 \(G^{\prime}\) :

Shear modulus normal to the transversely isotropic plane (MPa)
 \(T(\beta )\) :

Tensile strength of a specimen at \(\beta\) (MPa)
 \(T_{\text{m}}\) and \(T_{\text{b}}\) :

Tensile strength of rock matrix and weak plane, respectively (MPa)
 \(\sigma_{\text{t}}\) :

Tensile strength (MPa)
 \(d\) :

Sample diameter (mm)
 \(k\) :

Positive constant
 \(\sigma_{{{\text{t}}50}}\) :

Tensile strength obtained from a specimen of 50 mm in diameter (MPa)
 \(B\;{\text{and}}\;\lambda\) :

Dimensionless material constants
 \(f_{\text{t}} \;{\text{and}}\;\sigma_{0}\) :

Strength of a specimen with an infinitesimal size (MPa)
 \(f_{\text{c}}\) :

Strength of a specimen with an infinite size (MPa)
 \(d_{0}\) :

Maximum aggregate size (mm)
 \(l\) :

Material constant (mm)
 \(d_{\text{f}}\) :

Fractal dimension
 \(v_{\hbox{max} }\), \(v_{\hbox{min} }\) and \(v_{\text{mean}}\) :

Maximum, minimum, and average ultrasonic wave velocities, respectively (m/s)
 \(T_{1} (d)\) and \(T_{2} (d)\) :

Tensile strength of a specimen with the diameter of \(d\) (MPa)
 \(\sigma_{0}\) and \(\sigma_{\text{M}}\) :

Tensile strength when \(d \to 0\) and \(d \to \infty\), respectively (MPa)
 \(\bar{\sigma }_{0}\) :

Tensile strength when \(d \to 0\) (MPa)
 \(T_{1} (d,\beta )\) and \(T_{2} (d,\beta )\) :

Tensile strength of a specimen with d at β (MPa)
 \(T_{{{\text{b}}0}}\) and \(T_{{{\text{m}}0}}\) :

Tensile strength of the weak plane and the rock matrix, respectively, when \(d \to 0\) (MPa)
 \(T_{\text{bM}}\) and \(T_{\text{mM}}\) :

Tensile strength of the weak plane and the rock matrix, respectively, when \(d \to \infty\) (MPa)
 \(d_{i}\) :

Specimen diameter at which the maximum tensile strength reaches (mm)
 VA:

Velocity anisotropy index (%)
 MARE:

Maximum absolute relative error
 AARE:

Average absolute relative error
 SE:

Standard error
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Acknowledgements
The authors would like to thank Dr. H. Masoumi for his precious suggestions and Mr. R. Leung for his assistance during the experiments. The work in this paper is financially supported by the Hong Kong Polytechnic University (account RUF4), National Natural Science Foundation of China (Grant No. 51778313), and Cooperative Innovation Center of Engineering Construction and Safety in Shandong Blue Economic Zone.
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Li, K., Cheng, Y., Yin, ZY. et al. Size Effects in a Transversely Isotropic Rock Under Brazilian Tests: Laboratory Testing. Rock Mech Rock Eng 53, 2623–2642 (2020). https://doi.org/10.1007/s00603020020587
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Keywords
 Slate
 Transversely isotropic rock
 Brazilian test
 Tensile strength
 Size effect
 Anisotropy