A novel chevron notched short rod bend method for measuring the mode I fracture toughness of rocks
测量岩石 I 型断裂韧性的新型切口短棒弯曲法

Ming-Dong Wei, Feng Dai $^{*}$ , Nu-Wen Xu, Yi Liu, Tao ZhaoState Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu, Sichuan 610065, China
四川大学水利水电学院水力学与山河工程国家重点实验室，中国四川成都 610065

ARTICLE INFO 文章信息

Article history: 文章历史：

Received 30 September 2017
2017 年 9 月 30 日收到
Received in revised form 19 November 2017
2017 年 11 月 19 日收到修订稿
Accepted 29 November 2017
2017 年 11 月 29 日接受
Available online 2 December 2017
2017 年 12 月 2 日在线提供

Keywords: 关键词：

Fracture toughness  断裂韧性
Chevron notched short rod bend
雪佛龙槽口短杆弯管
Three-point bending  三点弯曲
Chevron bend  雪佛龙弯管
EMTSN criterion  EMTSN 标准

Abstract 摘要

A novel chevron notched short rod bend (CNSRB) method is developed for measuring the mode I fracture toughness ( $K_{Ic}$ ) of rocks. Similar to the conventional short rod (SR) method suggested by the International Society for Rock Mechanics (ISRM), the proposed CNSRB method can also measure the fracture toughness along axial directions of rock cores. Moreover, the CNSRB has obvious advantages compared with the SR method, such as simpler installing and testing procedure, lower requirement on testing machine, higher failure load and lower required amount of intact rock core. To assess the reliability of the CNSRB method, $K_{Ic}$ results of CNSRB specimens and ISRM-suggested chevron bend (CB) specimens are both experimentally and theoretically compared. Laboratory tests on two rock types indicate that the CNSRB method can produce $K_{Ic}$ values comparable to those measured using the CB method. Moreover, theoretical predictions based on an extended maximum tangential strain (EMTSN) criterion show that, $K_{Ic}$ of the CNSRB specimen is closer to that of the $C B$ specimen than that of the SR specimen, which inherently yields higher fracture toughness results than other ISRM-suggested specimens. Therefore, the CNSRB method can be reliably used to measure the mode I fracture toughness of rocks. Most importantly, accompanied with other test methods (e.g., CB), the proposed CNSRB method contributes to forming a complete set of methods for investigating fracture toughness anisotropy from a single rock core.
研究人员开发了一种新型的雪佛龙缺口短棒弯曲（CNSRB）方法，用于测量岩石的 I 型断裂韧性（ $K_{Ic}$ ）。与国际岩石力学学会（ISRM）建议的传统短棒（SR）方法类似，所提出的 CNSRB 方法也可以测量岩心沿轴向的断裂韧性。此外，与 SR 方法相比，CNSRB 具有明显的优势，如安装和测试程序更简单、对试验机的要求更低、失效载荷更大以及所需的完整岩芯量更少。为了评估 CNSRB 方法的可靠性， $K_{Ic}$ 对 CNSRB 试样和 ISRM 建议的楔形弯曲（CB）试样的结果进行了实验和理论比较。对两种岩石类型进行的实验室测试表明，CNSRB 方法可以产生与 CB 方法测量结果相当的 $K_{Ic}$ 值。此外，基于扩展最大切向应变（EMTSN）准则的理论预测表明，与 SR 试样相比，CNSRB 试样的 $K_{Ic}$ 更接近于 $C B$ 试样的 $K_{Ic}$ 值，而 SR 试样的 $C B$ 值本身就比其他 ISRM 建议的试样产生更高的断裂韧性结果。因此，CNSRB 方法可以可靠地用于测量岩石的 I 型断裂韧性。最重要的是，与其他测试方法（如 CB）配合使用，所提出的 CNSRB 方法有助于形成一套完整的方法来研究单个岩芯的断裂韧性各向异性。

1. Introduction 1.导言

The principles, methods and techniques of rock fracture mechanics have been widely applied to rock engineering activities associated with brittle breakage and fracture, such as hydraulic fracturing, rock drilling, tunnel boring, underground excavation and oil exploration [1,2]. In rock fracture mechanics, the most important parameter is the critical stress intensity factor (SIF), i.e., fracture toughness, which represents the resistance of the rock containing an initial crack against further crack extension. There are three fundamental loading types that a cracked rock can experience: tension/opening mode (mode I), in-plane shearing mode (mode II) and out-of-plane shearing mode (mode III). As rocks are relatively weak in tension, the mode I fracture toughness (

K_{Ic}

) of rocks has attracted considerable research interests.

K_{Ic}

can be applied as a basic mechanical property to the classification of rocks and to the stability assessment of rock structures; it can also be used as an input parameter to numerically simulate rock fracture processes [3].
岩石断裂力学的原理、方法和技术已被广泛应用于与脆性破碎和断裂有关的岩石工程活动中，如水力压裂、凿岩、隧道掘进、地下开挖和石油勘探等[1,2]。在岩石断裂力学中，最重要的参数是临界应力强度因子（SIF），即断裂韧性，它表示包含初始裂缝的岩石对裂缝进一步扩展的阻力。裂缝岩石可能经历三种基本加载类型：拉伸/张开模式（模式 I）、平面内剪切模式（模式 II）和平面外剪切模式（模式 III）。由于岩石在拉力作用下相对较弱，岩石的 I 模式断裂韧性（

K_{Ic}

）引起了相当大的研究兴趣。

K_{Ic}

可作为基本力学性能，用于岩石分类和岩石结构稳定性评估；也可作为数值模拟岩石断裂过程的输入参数[3]。

Nomenclature
\(a \quad\) crack length
\(a_{0} \quad\) initial crack length
\(a_{1} \quad\) final crack length
\(a_{\mathrm{c}} \quad\) critical crack length
\(B \quad\) biaxiality ratio
CB chevron bend
CCCD center-cracked circular disc
CCNBD cracked chevron notched Brazilian disc
CNSRB chevron notched short rod bend
\(D\) diameter of specimen
E Young's modulus
H height of specimen
ISRM International Society for Rock Mechanics
LEFM linear elastic fracture mechanics
\(K_{\mathrm{I}} \quad\) mode I stress intensity factor
\(K_{\text {Ic }} \quad\) mode I fracture toughness
\(K_{\text {Ic }}^{*} \quad K_{\text {Ic }}\) corresponding to zero T-stress
\(m, n \quad\) parameters related to Young's modulus and Poisson's ratio
EMTSN extended maximum tangential strain
\(P \quad\) load on specimen
\(P_{\text {max }}\) maximum load
\(r_{\mathrm{c}} \quad\) critical distance from the crack tip
\(S\) distances between bottom supports
SCB semi-circular bend
SIF stress intensity factor
SR short rod
\(T \quad\) T-stress
\(T_{\mathrm{c}} \quad\) T-stress at the onset of fracture
\(v \quad\) Poisson's ratio
\(Y^{*} \quad\) normalized stress intensity factor
\(Y_{\text {min }}^{*} \quad\) minimum normalized stress intensity factor
\(\alpha \quad\) normalized crack length
\(\alpha_{c} \quad\) critical normalized crack length
\(\beta \quad\) normalized critical distance
\(\sigma_{\text {rr }} \quad\) radial stress
\(\sigma_{\mathrm{t}} \quad\) tensile strength
\(\sigma_{\theta \theta} \quad\) tangential stress
\(\varepsilon_{\theta \theta} \quad\) tangential strain
\(\varepsilon_{\theta \theta c} \quad\) critical tangential strain
\(\theta_{0} \quad\) fracture initiation angle

As given in the literature, myriads of testing methods with different specimen geometries have been developed to measure

K_{Ic}

of rocks; some of these methods are reviewed in Table 1 [4-42]. Unfortunately, different testing methods usually yield inconsistent toughness results even for a given rock material [43-45]. For example, in the experiments by Khan and Al-Shayea [46] on a limestone,

K_{Ic}

measured using the SCB method was 1.62 times of that using the CCCD method. To achieve reliable

K_{Ic}

measurements, the International Society for Rock Mechanics (ISRM) has suggested four testing methods, including CB [4], SR [4], CCNBD [8] and SCB [1]. The four suggested specimens (Fig. 1) can be individually used to measure rock fracture toughness, and a selected combination of these specimens can also form a set of testing scheme (e.g., CB+SR+CCNBD or

CB + SR + SCB

) for a full fracture investigation along three orthotropic directions of cored rocks, because the fracture propagation material directions of the suggested specimens can be easily designed to be orthogonal to each other for the same rock core. This statement can be interpreted from Fig. 2. The notch plane of the CB specimen is parallel to cross sections of the rock core, and thus the CB specimen can measure the fracture toughness within the cross sections. The CCNBD and SCB specimens can measure the fracture toughness along radial directions of the rock core, and their crack planes are always perpendicular to cross sections of the rock core. Thus, the facture planes of CCNBD and SCB specimens are always perpendicular to that of the CB specimen if they are fabricated from the same rock core. As for the SR specimen, it can determine the fracture toughness along the axial direction of the rock core, different from the measurements using the other three ISRM-suggested specimens.
如文献所述，目前已开发出无数种具有不同试样几何形状的测试方法来测量岩石的

K_{Ic}

；表 1 综述了其中一些方法[4-42]。遗憾的是，即使是针对特定岩石材料，不同的测试方法通常也会得出不一致的韧性结果 [43-45]。例如，在 Khan 和 Al-Shayea [46] 对石灰石进行的实验中，使用 SCB 方法测得的

K_{Ic}

是使用 CCCD 方法测得的

K_{Ic}

的 1.62 倍。为了获得可靠的

K_{Ic}

测量值，国际岩石力学学会（ISRM）提出了四种测试方法，包括 CB [4]、SR [4]、CCNBD [8] 和 SCB [1]。这四种建议的试样（图 1）可以单独用于测量岩石的断裂韧性，而这些试样的选定组合也可以形成一套测试方案（如 CB+SR+CCNBD 或

CB + SR + SCB

），用于沿岩心岩石的三个正交方向进行全面的断裂研究，因为对于同一岩心，建议试样的断裂传播材料方向很容易设计成相互正交。这一说法可以从图 2 中得到解释。CB 试样的切口平面与岩心的横截面平行，因此 CB 试样可以测量横截面内的断裂韧性。CCNBD 和 SCB 试样可测量岩芯沿径向的断裂韧性，其裂纹平面始终垂直于岩芯的横截面。因此，如果 CCNBD 和 SCB 试样是由同一岩芯制作而成，它们的裂纹平面总是与 CB 试样的裂纹平面垂直。至于 SR 试样，它可以测定岩芯沿轴向的断裂韧性，这与使用 ISRM 建议的其他三种试样进行的测量有所不同。

Table 1 表 1
Some typical methods for measuring the mode I fracture toughness of rocks.
测量岩石 I 型断裂韧性的一些典型方法。

Test method 测试方法	Loading type 装载类型
Chevron bend (CB) method [4,5] 雪佛龙弯曲（CB）法 [4,5]	Three-point bending 三点弯曲
Chevron-notched short rod (SR) method [4,6,7] 雪佛龙缺口短杆（SR）法 [4,6,7]	Direct tension 直接拉力
Cracked chevron notched Brazilian disc (CCNBD) method [8-15] 裂纹楔形缺口巴西圆盘（CCNBD）法 [8-15]	Brazilian-type compression 巴西式压缩
Center-cracked circular disc (CCCD) method [16-18] 中心裂纹圆盘（CCCD）法 [16-18]	Brazilian-type compression 巴西式压缩
Modified ring method [19] 改良环形法 [19]	Brazilian-type compression 巴西式压缩
Hollow center cracked disc method [20,21] 中空中心开裂圆盘法 [20,21]	Brazilian-type compression 巴西式压缩
Cracked chevron notched semi-circular bend method [22-26] 裂纹雪佛龙缺口半圆弯曲法 [22-26]	Three-point bending 三点弯曲
Straight-crack semi-circular bend (SCB) method [1,27-34] 直裂半圆弯（SCB）法 [1,27-34]	Three-point bending 三点弯曲
Flattened Brazilian disc method [35] 巴西扁平圆盘法 [35]	Brazilian-type compression 巴西式压缩
Edge-cracked triangular method [36] 边缘裂纹三角法 [36]	Three-point bending 三点弯曲
Straight notched disc bend method [37-39] 直缺口圆盘弯曲法 [37-39]	Three-point bending 三点弯曲
Edge-cracked bend method [40-42] 边缘裂纹弯曲法 [40-42]	Three-point or four-point bending 三点或四点弯曲

Fig. 1. Schematics of four ISRM-suggested specimens for measuring

K_{Ic}

of rocks.
图 1.四种用于测量岩石

K_{Ic}

的国际空间遥感模型建议试样示意图。
Indeed, after the CB and SR methods were suggested by ISRM in 1988, more consistent

K_{Ic}

results can be obtained using the two methods. However, a difference of

\sim 20 - 30 %

was still reported between the

K_{Ic}

values measured by the two methods [ 9,10 ]. Moreover, many experimental data indicate that the SR method usually produces higher

K_{Ic}

measurements than other
事实上，1988 年 ISRM 提出 CB 和 SR 方法后，使用这两种方法可以获得更加一致的

K_{Ic}

结果。然而，仍有报告称这两种方法测得的

K_{Ic}

值之间存在

\sim 20 - 30 %

的差异 [ 9,10 ]。此外，许多实验数据表明，SR 方法通常比其他方法测得的

K_{Ic}

值更高。

Fig. 2. Schematics of different crack growth directions in ISRM-suggested specimens prepared from a single rock core.
图 2.从单个岩芯制备的 ISRM 建议试样中不同裂缝生长方向的示意图。

ISRM-suggested methods [45,47]. For example, in a recent study by Aliha et al. [45], the fracture toughness of Harsin marble measured by the SR method was

1.89 MPa m^{0.5}

, while the fracture toughness values measured with the CB, CCNBD and SCB methods were only

1.39, 0.95

and

1.34 MPa m^{0.5}

, respectively. Aliha et al. [45] reasonably interpreted the discrepancies among these

K_{Ic}

results using an extended maximum tangential strain (EMTSN) fracture criterion proposed by Ayatollahi and Abbasi [48]; the sign and magnitude of T-stress [49] are believed to be key factors affecting the measuring inconsistency [45,48,50,51].
ISRM建议的方法[45,47]。例如，在 Aliha 等人[45] 最近的一项研究中，用 SR 方法测得的哈辛大理石断裂韧性为

1.89 MPa m^{0.5}

，而用 CB、CCNBD 和 SCB 方法测得的断裂韧性值分别只有

1.39, 0.95

和

1.34 MPa m^{0.5}

。Aliha等人[45]使用Ayatollahi和Abbasi[48]提出的扩展最大切向应变（EMTSN）断裂准则合理地解释了这些

K_{Ic}

结果之间的差异；T应力的符号和大小[49]被认为是影响测量不一致的关键因素[45,48,50,51]。

In addition to the measuring accuracy, the practicability of a test method is also a major concern for widespread applications. A convenient

K_{Ic}

testing method is expected to have some merits such as easy specimen preparation, simple installing and testing procedure, and low requirement on testing machine. However, as pointed out in some studies [23,52], the preparation and the testing procedure of the SR specimen are rather complicated. Moreover, as shown in Fig. 1, the mode I loading condition for the SR specimen is achieved by directly applying a tensile load to the initial notched plane. However, directly applying tensile loads on rock specimens can often encounter many practical difficulties, and thus it is not favored for fracture tests on brittle rocks [19,35,37]. For example, to apply the tensile load on the SR specimen, special fixtures (e.g., grip jaws and a suitable linkage system for eliminating bending and torsional stresses in the specimen) are needed and bonding failures may occur at the contact region between the specimen and the fixtures [19,37]. In addition, the failure load of the SR specimen is much lower than that of the other suggested specimens made of the same rock. According to linear elastic fracture mechanics (LEFM), the failure load of the SR specimen is only

43.4 %

and

8.8 %

of that of standard CB and CCNBD specimens, respectively

[4, 8]

. Thus, the

S R

method has a higher requirement on the measuring precision of testing machine due to its lower failure load especially for soft rocks [8].
除了测量精度，测试方法的实用性也是广泛应用的一个主要问题。一种方便的

K_{Ic}

测试方法应具有一些优点，如试样制备容易、安装和测试程序简单、对试验机要求低等。然而，正如一些研究[23,52]所指出的，SR 试样的制备和测试步骤相当复杂。此外，如图 1 所示，SR 试样的模式 I 加载条件是通过在初始缺口平面上直接施加拉伸载荷来实现的。然而，在岩石试样上直接施加拉伸载荷往往会遇到很多实际困难，因此在脆性岩石的断裂试验中并不适用[19,35,37]。例如，要在 SR 试样上施加拉伸载荷，需要特殊的夹具（如夹钳和消除试样弯曲和扭转应力的合适连接系统），而且试样与夹具的接触区域可能会发生粘结失效 [19,37]。此外，SR 试样的破坏载荷远低于其他由相同岩石制成的建议试样。根据线性弹性断裂力学（LEFM），SR 试样的破坏载荷分别仅为标准 CB 试样和 CCNBD 试样的

43.4 %

和

8.8 %

[4, 8]

。因此，

S R

方法对试验机的测量精度要求较高，因为其破坏载荷较低，尤其是在软岩中[8]。

In this study, we propose a novel chevron notched short rod bend (CNSRB) method to measure the mode I fracture toughness of rocks. This CNSRB fracture test is easier to perform and has a higher failure load than the traditional SR test. The remainder of this paper is organized as follows. In Section 2, the specimen geometry and testing principle of the CNSRB method are presented, and the critical normalized SIF required for

K_{Ic}

determination is numerically calibrated. In Section 3, laboratory tests using CNSRB and CB specimens are conducted on two rock types, and the measured results are compared. Section 4 theoretically assesses the measuring consistency between the CNSRB and CB methods using the EMTSN criterion, and the ratios between

K_{Ic}

results via the two different methods are predicted. A comprehensive discussion on the theoretical and experimental results is presented in Section 5. Section 6 summarizes the study.
在这项研究中，我们提出了一种新型的切口短棒弯曲（CNSRB）方法来测量岩石的 I 型断裂韧性。与传统的 SR 试验相比，这种 CNSRB 断裂试验更容易进行，破坏载荷也更高。本文的其余部分安排如下。第 2 节介绍了 CNSRB 方法的试样几何形状和测试原理，并对测定

K_{Ic}

所需的临界归一化 SIF 进行了数值校准。第 3 部分使用 CNSRB 和 CB 试样对两种岩石类型进行了实验室测试，并对测量结果进行了比较。第 4 节利用 EMTSN 准则从理论上评估了 CNSRB 和 CB 方法的测量一致性，并预测了两种不同方法得出的

K_{Ic}

结果之间的比率。第 5 节对理论和实验结果进行了全面讨论。第 6 节对本研究进行总结。

2. The CNSRB method
2.CNSRB 方法

2.1. Specimen geometry and testing principle
2.1.试样几何形状和测试原理

A schematic of the CNSRB specimen geometry is shown in Fig. 3. The CNSRB specimen is core-based with a similar shape of the SR specimen, and it is also chevron notched along the axial direction of a rock core. However, different from the traditional SR specimen, the CNSRB is loaded by three-point bending, not by direct tension. In a CNSRB test, the specimen should be placed in a three-point bending fixture; the two bottom support rollers should be symmetrically located at both sides of the notch plane (Fig. 3); and the top loading roller should be aligned with the notched plane.
图 3 是 CNSRB 试样的几何示意图。CNSRB 试样以岩心为基础，形状与 SR 试样相似，也是沿岩心轴向切口。但与传统的 SR 试样不同的是，CNSRB 采用三点弯曲加载，而非直接拉伸。在 CNSRB 试验中，试样应放置在三点弯曲夹具中；两个底部支撑辊应对称位于缺口平面的两侧（图 3）；顶部加载辊应与缺口平面对齐。

Fig. 3. Schematic of the CNSRB method proposed in this study.
图 3.本研究提出的 CNSRB 方法示意图。

Geometric parameters of the CNSRB specimen used in this investigation are tabulated in Table 2. The CNSRB specimen can be fabricated according to the procedures illustrated in Fig. 4. First, a short rod is hold in a fixture, a circular diamond saw with a radius of 50 mm is moved to touch the edge of the rod (Step I in Fig. 4), and the circular saw should be located in the designed chevron notch plane. Then the saw is moved downwards 20 mm along the axial direction of the short rod, shown in Step II (Fig. 4). In Step III, the first cut is made by moving the rotating diamond saw into the rock with a horizontal displacement of 35 mm . In Step IV, the second cut is done in a similar way. A desired CNSRB specimen can thus be prepared.
表 2 列出了本次研究中使用的 CNSRB 试样的几何参数。CNSRB 试样的制作过程如图 4 所示。首先，将一根短杆固定在夹具上，然后移动半径为 50 毫米的金刚石圆锯接触短杆边缘（图 4 中的步骤 I），圆锯应位于设计的楔形缺口平面内。然后将圆锯沿短杆的轴向向下移动 20 毫米，如步骤 II 所示（图 4）。在步骤 III 中，将旋转金刚石锯移入岩石中，水平位移 35 毫米，进行第一次切割。在步骤 IV 中，以类似方式进行第二次切割。这样就可以制备出所需的 CNSRB 试样。

During a CNSRB test, the onset of crack growth occurs at the notch tip, and then the crack propagates within the chevron notched ligament in a stable fashion until reaching a certain critical length. The crack length at the onset of unstable crack growth is usually called the critical crack length, denoted as

a_{c}

. At this critical crack length, the loading force reaches its maximum. Thus, the mode I fracture toughness can be determined as follows:
在 CNSRB 试验中，裂纹开始在凹口尖端生长，然后裂纹以稳定的方式在楔形凹口韧带内扩展，直至达到一定的临界长度。不稳定裂纹开始增长时的裂纹长度通常称为临界裂纹长度，用

a_{c}

表示。在此临界裂纹长度处，加载力达到最大。因此，模式 I 断裂韧度可按以下方式确定：

K_{Ic} = \frac{P_{max}}{H \sqrt{D}} Y_{min}^{*}

where

P_{max}

is the failure load, and

Y_{min}^{*}

is the minimum normalized SIF. The value of

Y_{min}^{*}

is dependent on specimen geometries, i.e.,

H, D, a_{1}

and

a_{0}

, and it can be determined by numerical tools.
其中，

P_{max}

为破坏载荷，

Y_{min}^{*}

为最小归一化 SIF。

Y_{min}^{*}

的值取决于试样几何形状，即

H, D, a_{1}

和

a_{0}

，可通过数值工具确定。

2.2. Determination of $Y_{min}^{}$
2.2. $Y_{min}^{}$ 的确定

A sub-modeling technique in finite element modeling, also known as the cut boundary displacement method, is used to calibrate

Y_{min}^{*}

of the CNSRB specimen. A full model with finite element meshes is first built to obtain full-field displacements
有限元建模中的子建模技术（也称为切割边界位移法）用于校准 CNSRB 试样的

Y_{min}^{*}

。首先建立带有有限元网格的完整模型，以获得全场位移

Table 2 表 2
Geometric parameters of the CNSRB specimen used in this study.
本研究中使用的 CNSRB 试样的几何参数。

Geometric parameter 几何参数	Suggested value 建议值
$D$	50 mm 50 毫米
$H$	$40 mm (H / D = 0.8)$
$a_{0}$	$10 mm (a_{0} / D = 0.2)$
$a_{1}$	$27.7 mm (a_{1} / D = 0.554)$
$S$	$40 mm (S / D = 0.8)$

Fig. 4. Schematic of the preparation process of the CNSRB specimen.
图 4.CNSRB 试样的制备过程示意图。
of the specimen, and then a sub-model with a more elaborate mesh is detached from the full model to re-analyze the region adjacent to the crack tip.
然后从完整模型中分离出一个网格更精细的子模型，对裂纹尖端附近区域进行重新分析。

Fig. 5 shows a set of full model and sub-model for the CNSRB specimen. Geometric parameters of the models are the same as those in Table 2. Due to symmetry, a quarter of CNSRB specimen is emulated in the full model, which consists of 61,498 ten-node quadratic tetrahedron elements and 88,303 nodes. A semi-cylinder enclosing the crack front is created for submodeling with 21,600 twenty-node quadratic brick elements and 94,341 nodes. The mesh in the sub-model is refined near the crack tip with high stress concentration (Fig. 5). In addition, given the stress singularity of

r^{- 1 / 2}

at the crack tip (where

r

is the distance from the crack tip), singular elements (i.e., quarter point elements) are used in the region around the crack front.
图 5 显示了 CNSRB 试样的全模型和子模型。模型的几何参数与表 2 中的参数相同。由于对称性，全模型中模拟了 CNSRB 试样的四分之一，由 61 498 个十节点四面体元素和 88 303 个节点组成。子模型由 21,600 个 20 节点的二次方砖元素和 94,341 个节点组成，用于创建包围裂缝前沿的半圆柱体。子模型中的网格在应力集中的裂纹尖端附近进行了细化（图 5）。此外，考虑到裂纹尖端的应力奇异性

r^{- 1 / 2}

（其中

r

为距离裂纹尖端的距离），在裂纹前沿周围区域使用了奇异元素（即四分之一点元素）。

Using the sub-modeling technique, the mode I SIFs (

K_{I}

) of CNSRB specimens at different crack lengths can be numerically obtained. Then the SIFs can be normalized using the following formula.
利用子建模技术，可数值求得不同裂缝长度 CNSRB 试样的模 I SIF (

K_{I}

) 。然后可使用以下公式对 SIF 进行归一化处理。

Y^{*} = \frac{K_{I} H \sqrt{D}}{P}

where

Y^{*}

is the normalized SIF; for a given specimen geometry,

Y^{*}

only depends on the crack length. The variation of

Y^{*}

versus normalized crack length

α (= a / H)

for the CNSRB specimen is shown in Fig. 6 . It can be observed that

Y^{*}

decreases first and increases thereafter with the increase of the crack length. The critical/minimum normalized SIF

(Y_{min}^{*})

can thus be determined as 3.03 for the CNSRB specimen and the corresponding critical crack length is about 0.46 times of the specimen height. In addition, similar to

K_{l}

, the magnitude of T-stress (denoted as

T

) can also be directly obtained by finite element analysis using ABAQUS. For the CNSRB specimen with the critical crack length,

T / K_{I}

is determined as

0.445 m^{- 0.5}

.
其中

Y^{*}

为归一化 SIF；对于给定的试样几何形状，

Y^{*}

仅取决于裂纹长度。CNSRB 试样的

Y^{*}

随归一化裂纹长度

α (= a / H)

的变化如图 6 所示。可以看出，随着裂纹长度的增加，

Y^{*}

首先减小，随后增大。因此可以确定 CNSRB 试样的临界/最小归一化 SIF

(Y_{min}^{*})

为 3.03，相应的临界裂纹长度约为试样高度的 0.46 倍。此外，与

K_{l}

类似，T 应力的大小（表示为

T

）也可以通过使用 ABAQUS 进行有限元分析直接获得。对于具有临界裂缝长度的 CNSRB 试样，

T / K_{I}

可确定为

0.445 m^{- 0.5}

。

3. Experimental results 3.实验结果

To examine the reliability of the CNSRB method,

K_{Ic}

of the CNSRB specimen will be compared with that of the ISRMsuggested CB specimen, which has already received wide acceptance. Two kinds of rocks (a granite and a sandstone), which are relatively isotropic and homogeneous, were tested. The granite was quarried from Changtai County, Fujian Province,
为了检验 CNSRB 方法的可靠性，CNSRB 试样的

K_{Ic}

将与 ISRM 建议的 CB 试样进行比较，后者已被广泛接受。我们测试了两种各向同性且相对均匀的岩石（花岗岩和砂岩）。花岗岩采自福建省长泰县、

Fig. 5. Numerical model for determining the normalized SIF of the CNSRB specimen.
图 5.确定 CNSRB 试样归一化 SIF 的数值模型。

Fig. 6. Variation of the normalized SIF with the normalized crack length for the CNSRB specimen.
图 6.CNSRB 试样的归一化 SIF 随归一化裂纹长度的变化。

China, and the sandstone was from Longchang County, Sichuan Province, China. The Poisson’s ratio

v

and tensile strength

σ_{t}

of the granite are 0.21 and 13.2 MPa [53,54], respectively, and those of the sandstone are 0.19 and 5.4 MPa [32], respectively. A total of 24 specimens were prepared (Fig. 7). The CNSRB specimens were fabricated according to the procedures described in Section 2, and their geometric dimensions are the same as those listed in Table 2. The CB specimens were prepared according to ISRM-suggested method [4], the specimen diameter is 50 mm and all other geometric parameters are the same as those of the ISRM-suggested standard CB specimen.
砂岩产自中国四川省隆昌县。花岗岩的泊松比

v

和抗拉强度

σ_{t}

分别为 0.21 和 13.2 MPa [53,54]，砂岩的泊松比

v

和抗拉强度

σ_{t}

分别为 0.19 和 5.4 MPa [32]。共制备了 24 个试样（图 7）。CNSRB 试样按照第 2 节所述程序制作，其几何尺寸与表 2 所列尺寸相同。CB 试样根据 ISRM 建议的方法 [4] 制作，试样直径为 50 毫米，所有其他几何参数与 ISRM 建议的标准 CB 试样相同。

Experiments were conducted using a MTS815 Rock Mechanics Test System (Fig. 8). The compressive loads were applied on the specimens with a constant displacement-controlled rate of

0.05 mm / \min

, and the load point displacements were recorded by a linear variable differential transformer. Data logging was controlled by a computer, and the variation of applied force versus load point displacement can be displayed in real time.
实验使用 MTS815 岩石力学测试系统进行（图 8）。以

0.05 mm / \min

的恒定位移控制速率对试样施加压缩荷载，并通过线性可变差分变压器记录荷载点位移。数据记录由计算机控制，可实时显示加载力与加载点位移的变化。

Fig. 7. Typical photos of the prepared CNSRB and CB specimens.
图 7.制备的 CNSRB 和 CB 试样的典型照片。

Fig. 8. Loading configuration of the CNSRB and CB fracture tests in the MTS machine.
图 8.在 MTS 设备中进行 CNSRB 和 CB 断裂试验的加载配置。

Fig. 9. Four typical force-displacement curves recorded during the tests.
图 9.试验过程中记录的四条典型的力-位移曲线。

Fig. 9 shows four typical force-displacement curves recorded during the tests. After an initial nonlinear stage, all curves are nearly linear until the peaks are reached and then drop suddenly, indicating a brittle fracture. Fig. 10 shows the fracture surfaces and the side view of some typical recovered specimens; there is no local damage close to the load points for both the CNSRB and CB specimens. The whole experimental data including the failure loads and corresponding fracture toughness are summarized in Table 3, and the fracture toughness values of the CNSRB and CB specimens are compared in Fig. 11. Note that

K_{Ic}

values of the CB specimens are calculated using the critical normalized/dimensionless SIF (i.e., 9.167) calibrated in our recent study [7], instead of the value 10.42 given in the document [4], which has been proved by Matsuki et al. [55] to be about

10 %

higher than its true value. The fracture toughness values of the granite measured with the CNSRB and CB methods are 2.304 and

2.228 MPa m^{0.5}

, respectively, and those of the sandstone are 0.741 and

0.712 MPa m^{0.5}

, respectively. It is evident that the measured fracture toughness values with the CNSRB and CB tests are comparable, with a discrepancy of only about

3 %

for the granite and

4 %

for the sandstone. In addition, standard deviations of

K_{Ic}

results of the CNSRB specimens ( 0.051 and

0.038 MPa m^{0.5}

for the granite and sandstone, respectively) are lower than those of the CB specimens ( 0.069 and

0.074 MPa m^{0.5}

, accordingly), indicating that the CNSRB method yields smaller data scatter compared with the CB method. It is also noteworthy that the mean failure load of the CNSRB specimens is about 2.5 times that of the CB specimens for both granite and sandstone rock materials.
图 9 显示了试验过程中记录的四条典型的力-位移曲线。经过最初的非线性阶段后，所有曲线都接近线性，直到达到峰值后突然下降，表明发生了脆性断裂。图 10 显示了一些典型的复原试样的断裂面和侧视图；CNSRB 和 CB 试样的加载点附近都没有局部损伤。表 3 总结了整个实验数据，包括破坏载荷和相应的断裂韧性，图 11 比较了 CNSRB 和 CB 试样的断裂韧性值。需要注意的是，CB 试样的

K_{Ic}

值是用我们最近的研究[7]中标定的临界归一化/无尺寸 SIF（即 9.167）计算的，而不是文件[4]中给出的 10.42，后者已被 Matsuki 等人[55]证实比其真实值高约

10 %

。用 CNSRB 和 CB 方法测量的花岗岩断裂韧性值分别为 2.304 和

2.228 MPa m^{0.5}

，砂岩的断裂韧性值分别为 0.741 和

0.712 MPa m^{0.5}

。由此可见，CNSRB 和 CB 试验测得的断裂韧性值相当，花岗岩的差异仅约为

3 %

，砂岩的差异约为

4 %

。此外，CNSRB 试样的

K_{Ic}

结果的标准偏差（花岗岩和砂岩分别为 0.051 和

0.038 MPa m^{0.5}

）低于 CB 试样的标准偏差（分别为 0.069 和

0.074 MPa m^{0.5}

），这表明 CNSRB 方法产生的数据散差小于 CB 方法。值得注意的是，对于花岗岩和砂岩两种岩石材料，CNSRB 试样的平均破坏荷载都是 CB 试样的 2.5 倍左右。

4. Theoretical assessment
4.理论评估

In this study, the extended maximum tangential strain (EMTSN) criterion [48] is utilized to theoretically assess the consistency between

K_{Ic}

results of CNSRB and CB methods, as this criterion has been proved reliable by Aliha et al. [45,50] for predicting the geometry dependence of

K_{Ic}

measured by different methods.
在本研究中，利用扩展最大切向应变 (EMTSN) 准则 [48] 从理论上评估 CNSRB 和 CB 方法的

K_{Ic}

结果之间的一致性，因为 Aliha 等人 [45,50] 已证明该准则在预测不同方法测得的

K_{Ic}

的几何依赖性方面是可靠的。

In the EMTSN criterion, the onset of a crack is considered to occur after the maximum tangential strain

ε_{θ θ}

at a critical distance

r_{c}

from the crack tip reaches a critical value

ε_{θ θ c}

, where

r_{c}

and

ε_{θ θ c}

are constant material properties

[45, 48, 50, 51]

.
在 EMTSN 准则中，裂纹的产生被认为是在距裂纹尖端临界距离

r_{c}

处的最大切向应变

ε_{θ θ}

达到临界值

ε_{θ θ c}

之后发生的，其中

r_{c}

和

ε_{θ θ c}

是恒定的材料属性

[45, 48, 50, 51]

。

Fig. 10. The fracture surfaces and the side view of typical recovered samples.
图 10.典型回收样品的断裂面和侧视图。

Table 3 表 3
Summary of experimental results.
实验结果摘要。

Rock specimen 岩石标本	$P_{max} (kN)$	Average of $P_{max} (kN)$ $P_{max} (kN)$ 的平均数	$K_{Ic} (MPa \cdot m^{0.5})$	Average of $K_{Ic} (MPa \cdot m^{0.5})$ $K_{Ic} (MPa \cdot m^{0.5})$ 的平均数	$T_{c} (MPa)$	Average of $T_{c}$ ( MPa ) $T_{c}$ 的平均值 ( 兆帕 )
CNSRB (granite) CNSRB （花岗岩）	7.061	$6.801 \pm 0.151$	2.392	$2.304 \pm 0.051$	1.064	$1.025 \pm 0.023$
	6.815		2.308		1.027
	6.643		2.250		1.001
	6.912		2.285		1.042
	6.744		2.341		1.017
	6.634		2.247		1.000
CB (granite) CB (花岗岩)	2.798	$2.717 \pm 0.084$	2.294	$2.228 \pm 0.069$	-2.562	$- 2.489 \pm 0.077$
	2.744		2.250		-2.513
	2.589		2.123		-2.371
	2.810		2.304		-2.574
	2.620		2.149		-2.400
	2.743		2.249		-2.512
CNSRB (sandstone) CNSRB （砂岩）	2.276	$2.187 \pm 0.114$	0.771	$0.741 \pm 0.038$	0.343	$0.330 \pm 0.017$
	2.005		0.679		0.302
	2.309		0.782		0.348
	2.146		0.727		0.324
	2.294		0.777		0.346
	2.094		0.709		0.316
CB (sandstone) CB（砂岩）	0.993	$0.868 \pm 0.090$		$0.712 \pm 0.074$		$- 0.795 \pm 0.083$
	0.774		0.635		$- 0.709$
	0.968		0.793		-0.886
	0.761		0.624		-0.697
	0.896		0.735		-0.821
	0.817		0.670		-0.749

where some values are given as mean

\pm SD

(standard deviation).
其中一些值以平均值

\pm SD

（标准偏差）表示。

In the conventional crack-tip polar coordinate system, the in-plane stresses in a cracked body subjected to mode I loading can be described using the following formulas.
在传统的裂纹尖端极坐标系中，受模式 I 荷载作用的裂纹体的平面内应力可以用下面的公式来描述。

\begin{aligned} σ_{rr} = \frac{1}{\sqrt{2 π r}} K_{I} \cos \frac{θ}{2} (1 + \sin^{2} \frac{θ}{2}) + T \cos^{2} θ + O (r^{1 / 2}) \\ σ_{θ θ} = \frac{1}{\sqrt{2 π r}} K_{I} \cos^{3} \frac{θ}{2} \cos \frac{θ}{2} + T \sin^{2} θ + O (r^{1 / 2}) \\ σ_{r θ} = \frac{1}{2 \sqrt{2 π r}} K_{I} \sin \frac{θ}{2} \cos \frac{θ}{2} - T \sin θ \cos θ + O (r^{1 / 2}) \end{aligned}

Fig. 11. The mode I fracture toughness results of the (a) granite and (b) sandstone.
图 11.(a) 花岗岩和 (b) 砂岩的模式 I 断裂韧性结果。
where

T

(usually called T-stress) is a non-singular stress term, which is not affected by the distance from the crack tip.

O (r^{1 / 2})

denotes the remaining terms of the series expansion and can usually be left out near the crack tip. According to Hook’s law, the tangential strain can be written as follows:
其中，

T

（通常称为 T 应力）是非奇异应力项，不受裂缝尖端距离的影响。

O (r^{1 / 2})

表示序列展开的其余项，通常可以在裂纹尖端附近省略。根据胡克定律，切向应变可写成以下形式：

ε_{θ θ} = m σ_{θ θ} + n σ_{rr}

where 其中

m = \frac{1}{E}, n = - \frac{v}{E}, for plane stress

m = \frac{1 - v^{2}}{E}, n = - \frac{v + v^{2}}{E}, for plane strain

Combining Eqs. (3), (4) and (6), and neglecting

O (r^{1 / 2})

, the tangential strain can be rewritten as:
结合公式 (3)、(4) 和 (6)，并忽略

O (r^{1 / 2})

，切向应变可重写为

ε_{θ θ} = m [\frac{1}{\sqrt{2 π r}} K_{I} \cos^{3} \frac{θ}{2} + T \sin^{2} θ] + n [\frac{1}{\sqrt{2 π r}} K_{I} \cos \frac{θ}{2} (1 + \sin^{2} \frac{θ}{2}) + T \cos^{2} θ]

In the conventional mode I fracture tests where mode I fracture occurs, the fracture initiation angle

θ_{0} = 0^{\circ}

. Thus, at the onset of mode I fracture, the following equation can be obtained.
在发生模式 I 断裂的传统模式 I 断裂试验中，断裂起始角

θ_{0} = 0^{\circ}

。因此，在模式 I 断裂开始时，可以得到以下方程。

ε_{θ θ c} = (m + n) \frac{1}{\sqrt{2 π r_{c}}} K_{Ic} + n T_{c}

where

T_{c}

is the T-stress at the onset of fracture.

T_{c}

and

r_{c}

are normalized as follows:
其中

T_{c}

是断裂开始时的 T 应力。

T_{c}

和

r_{c}

归一化如下：

\begin{aligned} B = \frac{T_{c} \sqrt{π a_{c}}}{K_{Ic}} \\ β = \sqrt{\frac{2 r_{c}}{a_{c}}} \end{aligned}

Subsequently, Eq. (10) can be rewritten as:
随后，公式 (10) 可改写为

ε_{θ θ c} \sqrt{2 π r_{c}} = (1 + \frac{n}{m + n} B β) (m + n) K_{Ic}

In the MSTN criterion,

ε_{θ θ c}, r_{c}, m

and

n

are constants for a given rock material. Thus, the ratio between the fracture toughness results of two different test methods, e.g., methods A and B, can be determined as follows:
在 MSTN 标准中，

ε_{θ θ c}, r_{c}, m

和

n

是特定岩石材料的常数。因此，两种不同测试方法（如 A 和 B 方法）得出的断裂韧性结果之间的比值可按下式确定：

\frac{{(K_{Ic})}_{A}}{{(K_{Ic})}_{B}} = \frac{1 + \frac{n}{m + n} (B β)_{B}}{1 + \frac{n}{m + n} (B β)_{A}}

This equation indicates that the discrepancies between

K_{Ic}

results of different test methods are induced by their inconsistent

B β

values, or essentially, by their inconsistent ratios of T-stress to

K_{l}

, because of

B β = \frac{T_{c} \sqrt{2 π r_{c}}}{K_{l}} = \frac{T \sqrt{2 π r_{c}}}{K_{l}}

. According to the numerical analysis in the previous section, the value of

T / K_{I}

can be calculated as

0.445 m^{- 0.5}

for the CNSRB test. For the CB test using the

50 - mm

-diameter specimen,

T / K_{I}

can be determined as

- 1.117 m^{- 0.5}

[45]. Thus, if the critical distance

r_{c}

and the Poisson’s ratio

v

of a rock are known, the ratio between the fracture toughness results of the CNSRB and CB methods can be estimated.
该等式表明，不同试验方法的

K_{Ic}

结果之间的差异是由它们的

B β

值不一致引起的，或者说本质上是由它们的 T 应力与

K_{l}

的比率不一致引起的，因为

B β = \frac{T_{c} \sqrt{2 π r_{c}}}{K_{l}} = \frac{T \sqrt{2 π r_{c}}}{K_{l}}

。根据上一节的数值分析，对于 CNSRB 试验，

T / K_{I}

的值可计算为

0.445 m^{- 0.5}

。对于使用

50 - mm

直径试样的 CB 试验，

T / K_{I}

可确定为

- 1.117 m^{- 0.5}

[45]。因此，如果已知岩石的临界距离

r_{c}

和泊松比

v

，就可以估算出 CNSRB 方法和 CB 方法的断裂韧性结果之间的比率。

Note that the critical distance

r_{c}

is usually estimated as the size of fracture process zone for rock materials, and it can be determined using Eq. (15) [56-62].
需要注意的是，临界距离

r_{c}

通常被估算为岩石材料断裂加工区的大小，可通过公式 (15) [56-62] 确定。

r_{c} = \frac{1}{2 π} {(\frac{K_{Ic}}{σ_{t}})}^{2}

By substituting

K_{Ic}

and

σ_{t}

measured using ISRM-suggested CB specimens and Brazilian disc specimens (i.e.,

2.228 MPa m^{0.5}

and 13.2 MPa for the granite, and

0.712 MPa m^{0.5}

and 5.4 MPa for the sandstone),

r_{c}

is estimated as 4.5 and 2.8 mm for the granite and sandstone, respectively. Subsequently, the value of

B β

can be determined as 0.075 and -0.188 for the CNSRB granite specimens and the CB granite specimens, respectively; as for the sandstone specimens,

B β

is 0.059 and -0.148 for CNSRB and CB, respectively. Thus, according to Eq. (14), the ratio of

K_{Ic}

of the CNSRB specimen to that of the CB specimen is estimated as 1.09 for the granite and 1.06 for the sandstone, indicating that the CNSRB and CB tests can theoretically produce relatively close values of

K_{Ic}

with the difference less than

10 %

.
将使用 ISRM 建议的 CB 试样和巴西圆盘试样测得的

K_{Ic}

和

σ_{t}

值（即花岗岩的

2.228 MPa m^{0.5}

和 13.2 兆帕，砂岩的

0.712 MPa m^{0.5}

和 5.4 兆帕）代入后，估计花岗岩和砂岩的

r_{c}

值分别为 4.5 毫米和 2.8 毫米。随后，可确定 CNSRB 花岗岩试样和 CB 花岗岩试样的

B β

值分别为 0.075 和 -0.188；至于砂岩试样，CNSRB 和 CB 的

B β

值分别为 0.059 和 -0.148。因此，根据公式 (14)，CNSRB 试样的

K_{Ic}

与 CB 试样的

K_{Ic}

之比估计花岗岩为 1.09，砂岩为 1.06，这表明 CNSRB 和 CB 试验理论上可以产生相对接近的

K_{Ic}

值，其差值小于

10 %

。

5. Discussion 5.讨论

The traditional SR method is an important ISRM-suggested method for measuring

K_{Ic}

of rocks, because the SR specimen facilitates the formation of a set of testing scheme (i.e.,

C B + S R + C C N B D

C B + S R + S C B

) for a full fracture investigation along three orthotropic directions of a single rock core. Unfortunately, the measured

K_{Ic}

results by the SR specimen are inherently higher than those of other suggested specimens [45]. Moreover, the direct application of a tensile load on a rock specimen is not easy, and the failure load of the SR specimen is too low, leading to a relatively high requirement on the precision of the testing machine. It is thus meaningful to develop a modified method that has the same advantages as the SR method and can also avoid its disadvantages. In this study, a chevron notched short rod bend (CNSRB) method is proposed. Similar to the conventional SR method, the CNSRB method can measure the fracture toughness along axial directions of rock cores. Moreover, the CNSRB specimen is loaded by three-point bending, which is easier to apply than direct tension in rock fracture tests.
传统的 SR 方法是测量岩石

K_{Ic}

的一种重要的 ISRM 建议方法，因为 SR 试样便于形成一套测试方案（即

C B + S R + C C N B D

或

C B + S R + S C B

），用于沿单个岩芯的三个正交方向进行全面断裂研究。遗憾的是，SR 试样测得的

K_{Ic}

结果本身就高于其他建议的试样 [45]。此外，在岩石试样上直接施加拉伸载荷并不容易，而且 SR 试样的破坏载荷太低，导致对试验机精度的要求相对较高。因此，开发一种既具有 SR 方法的优点，又能避免其缺点的改进方法是很有意义的。本研究提出了一种雪佛龙缺口短杆弯曲（CNSRB）方法。与传统的 SR 方法类似，CNSRB 方法可以测量岩心沿轴向的断裂韧性。此外，CNSRB 试样通过三点弯曲加载，比直接拉伸更容易应用于岩石断裂测试。

On the other hand, the CNSRB specimen has a chevron notch, instead of a straight-through notch. For the specimen using a straight-through notch, the notch tip should be machined sharp enough (or the notch width should be narrow enough), and thus the notch can be regarded as the thin crack for determining

K_{Ic}

. Otherwise, directly applying theories of traditional fracture mechanics to the notched specimens may cause some errors [63,64], and pre-cracking the notch tip (e.g., by cyclic fatigue loading) is required to create the thin crack, especially for fine-grained rocks. However, fabricating a sharp notch tip is difficult for hard rocks, and pre-cracking is tedious, time-consuming, and hard to guarantee the desired geometry (e.g., uniform crack lengths for the prepared specimens). In the

K_{Ic}

measurements using chevron notched specimens including the CNSRB specimen, a main crack can be generated from the tip of the chevron-notched ligament due to stress concentration, before the maximum load is reached. Then the critical crack related to

K_{Ic}

determination can be formed after the main crack extends along the ligament up to a certain distance. Therefore, the difficulties in preparing a sharp notch tip can be avoided.
另一方面，CNSRB 试样采用的是雪佛龙切口，而不是直通切口。对于使用直通切口的试样，切口尖端应加工得足够锋利（或切口宽度应足够窄），因此可将切口视为用于确定

K_{Ic}

的细裂纹。否则，直接将传统断裂力学理论应用于缺口试样可能会造成一些误差 [63,64]，而且需要预先使缺口尖端开裂（例如通过循环疲劳加载）才能产生细裂纹，特别是对于细粒岩石。然而，对于坚硬的岩石来说，制造尖锐的切口尖端是很困难的，而且预裂纹是繁琐、耗时的，也很难保证所需的几何形状（例如，所制备试样的裂纹长度一致）。在使用包括 CNSRB 试样在内的切口试样进行

K_{Ic}

测量时，由于应力集中，在达到最大荷载之前，切口韧带顶端可能会产生主裂纹。然后，在主裂纹沿韧带延伸到一定距离后，就会形成与

K_{Ic}

测定有关的临界裂纹。因此，可以避免在准备尖锐缺口尖端时遇到的困难。

Laboratory tests on a granite and a sandstone indicate that the CNSRB specimen can produce the fracture toughness comparable to that measured using the ISRM-suggested CB specimen, which has already received wide acceptance. Moreover, the mean failure load of the CNSRB specimens is about 2.5 times that of the CB specimens for both rock types. If CNSRB and SR specimens can obtain the same

K_{Ic}

results, the failure load of the CNSRB specimen is, in theory, about 6.3 times that of the SR specimen. Therefore, the CNSRB specimen has a much higher failure load than CB and SR specimens and thus has a lower requirement on the precision of the testing machine. Our experiments also reveal that the CNSRB specimens yield a lower scatter of testing results than the CB specimens. One may argue that the ratio of the support span

(S)

to specimen height

(H)

in the CNSRB test is relatively small, this can cause a considerable vertical compression between the upper and bottom supports, and the considerable compressive stresses may affect the accuracy of the fracture toughness measurement. In fact,

S / H

used in the CNSRB test is the same as those in the SCB test with

S / D = 0.5

(i.e.,

S / R = 1

), which is within the ISRM-suggested valid range of support span for the SCB specimens. As

S / R = 1

is valid for the SCB test,

S / H = 1

can also be suitable for the CNSRB test. For a given rock specimen, although a higher compressive load may make a larger amount of energy consumed due to compression-induced micro-damage, however, for conventional fracture tests, the load is not high enough to severely damage the rock material away from the crack-tip region. In other words, the support span

(S / H = 1)

used in the CNSRB test has a negligible influence on the accuracy of the fracture toughness measurement. Indeed, no unexpected local damage can be observed in the recovered samples after testing. Moreover, as indicated by the experiments, the CNSRB test can produce the fracture toughness close to that measured from the CB test, which also shows the reliability of the CNSRB method.
在花岗岩和砂岩上进行的实验室测试表明，CNSRB 试样的断裂韧性可与使用 ISRM 建议的 CB 试样测量的断裂韧性相媲美。此外，对于两种岩石类型，CNSRB 试样的平均破坏载荷约为 CB 试样的 2.5 倍。如果 CNSRB 和 SR 试样能获得相同的

K_{Ic}

结果，那么 CNSRB 试样的破坏荷载理论上约为 SR 试样的 6.3 倍。因此，CNSRB 试样的破坏载荷远高于 CB 和 SR 试样，从而对试验机的精度要求较低。我们的实验还显示，CNSRB 试样的测试结果散布度低于 CB 试样。有人可能会说，CNSRB 试验中的支撑跨度

(S)

与试样高度

(H)

之比相对较小，这会导致上下支撑之间产生相当大的垂直压缩应力，而相当大的压缩应力可能会影响断裂韧性测量的准确性。事实上，CNSRB 试验中使用的

S / H

与 SCB 试验中使用的

S / D = 0.5

相同（即

S / R = 1

），都在 ISRM 建议的 SCB 试样支撑跨度有效范围内。由于

S / R = 1

适用于 SCB 试验，因此

S / H = 1

也适用于 CNSRB 试验。对于给定的岩石试样，虽然较高的压缩载荷可能会因压缩引起的微破坏而消耗较大的能量，但对于传统的断裂试验，载荷不足以严重破坏裂缝尖端区域以外的岩石材料。换句话说，CNSRB 试验中使用的支撑跨度

(S / H = 1)

对断裂韧性测量精度的影响可以忽略不计。事实上，测试后复原的样品不会出现意外的局部损伤。此外，实验还表明，CNSRB 试验所产生的断裂韧性与 CB 试验所测得的断裂韧性接近，这也说明 CNSRB 方法是可靠的。

Table 4 表 4
Theoretical assessments of the CNSRB method using the EMTSN criterion for some rocks studied in the literature.
使用 EMTSN 标准对文献中研究的一些岩石进行 CNSRB 方法的理论评估。

Rock Material 岩石材料	$r_{c} (mm)$	$v$	$(B β)_{CNSRB}$	$(B β)_{CB}$	${(K_{Ic})}_{CNSRB} / {(K_{Ic})}_{CB}$
Saudi Arabian limestone [46] 沙特阿拉伯石灰石 [46]	$5.2 [58, 61]$	$0.2 [65]$	0.080	-0.202	1.10	${(K_{Ic})}_{CNSRB} / K_{IC}^{*}$
Italian light marble [66] 意大利轻质大理石 [66］	$0.6 [58, 61]$	$0.3 [61]$	0.027	-0.069	1.07	1.03
Guiting limestone [67] 桂亭石灰岩 [67]	$2.3 [67]$	$0.2 [65]$	0.053	-0.134	1.06	1.02
Neyriz marble [43] 内里兹大理石[43]	$3.6 [43]$	$0.18 [68]$	0.067	-0.168	1.07	1.02
Harsin marble [60] 哈辛大理石 [60]	$3 [60]$	$0.2 [45]$	0.061	-0.153	1.07	1.02
Keochang granite [23] 柯昌花岗岩 [23]	$0.8 [58]$	$0.21 [23]$	0.032	-0.079	1.04	1.02

Recently, using the EMTSN criterion, Aliha et al. [45] have reasonably interpreted the measuring discrepancies between

K_{Ic}

results of four ISRM-suggested specimens. Inconsistent

K_{Ic}

results of the specimens were reported to be caused by their different ratios of T-stress to

K_{I}

, and the value of

T / K_{I}

could be determined as

3.019, - 1.117, - 13.172

and

0.838 m^{- 0.5}

for the

50 - mm

-diameter

SR, CB, CCNBD

and SCB specimens, respectively. The larger

T / K_{I}

of the SR specimen was found to be responsible for its higher

K_{Ic}

than the CB, CCNBD and SCB specimens [45]. In this study,

T / K_{I}

of the CNSRB specimen with a diameter of 50 mm is determined as

0.445 m^{- 0.5}

, closer to

T / K_{I}

C B

and SCB specimens than that of the SR specimen. Thus, the CNSRB specimen is expected to produce more comparable

K_{IC}

C B

, and SCB specimens than the SR specimen. Indeed, according to our laboratory tests and theoretical prediction, the measuring discrepancy between CNSRB and CB methods is not significant, obviously less than the discrepancies between

S R

and

C B

methods often reported in the literature. For example, a variation

\sim 20 - 30 %

is often observed between

K_{Ic}

values of CB and SR specimens [9,10]. Therefore, the CNSRB is more suitable than the SR method for the formation of a complete set of testing methods (e.g., CNSRB+CB+SCB) that can determine the fracture toughness anisotropy in orthogonal directions of a single rock core.
最近，Aliha 等人[45] 使用 EMTSN 标准合理解释了四个 ISRM 建议试样的

K_{Ic}

测量结果之间的差异。据报道，试样的

K_{Ic}

结果不一致是由于它们的 T 应力与

K_{I}

的比率不同造成的，而

T / K_{I}

的值可分别确定为

3.019, - 1.117, - 13.172

和

0.838 m^{- 0.5}

适用于直径为

50 - mm

的

SR, CB, CCNBD

和 SCB 试样。与 CB、CCNBD 和 SCB 试样相比，SR 试样的

T / K_{I}

较大，这是其

K_{Ic}

较高的原因 [45]。在本研究中，直径为 50 mm 的 CNSRB 试样的

T / K_{I}

被确定为

0.445 m^{- 0.5}

，比 SR 试样更接近

C B

和 SCB 试样的

T / K_{I}

。因此，与 SR 试样相比，CNSRB 试样产生的

K_{IC}

与

C B

和 SCB 试样更具有可比性。事实上，根据我们的实验室测试和理论预测，CNSRB 和 CB 方法之间的测量差异并不大，明显小于文献中经常报道的

S R

和

C B

方法之间的差异。例如，CB 和 SR 试样的

K_{Ic}

值之间经常出现

\sim 20 - 30 %

差异 [9,10]。因此，CNSRB 比 SR 方法更适合于形成一套完整的测试方法（如 CNSRB+CB+SCB），以确定单个岩芯在正交方向上的断裂韧性各向异性。

Classical LEFM theory normally neglects all the higher-order terms in Williams expansion (e.g., Eqs. (3)-(5)) except for the singular term, resulting in a single-parameter description of the near-tip fields (e.g., stress, strain and energy). However, for rocks, the critical distance

r_{c}

can be as large as several millimeters [56], and the contribution of T-stress to the stress, strain and energy at the critical distance may be significant. As indicated by Aliha et al. [45], the T-stresses in the ISRM-suggested specimens have certain influences on

K_{Ic}

measurements, according to the EMTSN criterion. Thus,

K_{Ic}

values measured in the laboratory are usually not strictly the fracture toughness expected in the classical LEFM (denoted as

K_{Ic}^{*}

in the following), which is not affected by the T-stress. Note that the absolute value of

T / K_{I}

of the CNSRB specimen is smaller than that of the SR, CB, CCNBD and SCB specimen configurations studied in Ref. [45], and thus, the effect of T-stress on mode I fracture is smaller in the CNSRB fracture test. Consequently,

K_{Ic}

measured via CNSRB specimens can be used to approximate

K_{Ic}^{*}

. For example, for both the granite and sandstone tested in this study, the discrepancy between

K_{Ic}

of the CNSRB specimen and

K_{Ic}^{*}

can be estimated as only

\sim 2 - 3 %

according to Eq. (14).
经典的 LEFM 理论通常忽略 Williams 展开（如公式 (3)-(5)）中除奇异项之外的所有高阶项，从而对近端场（如应力、应变和能量）进行单参数描述。然而，对于岩石而言，临界距离

r_{c}

可大至几毫米[56]，T应力对临界距离上的应力、应变和能量的贡献可能很大。如 Aliha 等人[45]所述，根据 EMTSN 准则，ISRM 建议试样中的 T 应力对

K_{Ic}

测量值有一定影响。因此，实验室测得的

K_{Ic}

值通常与经典 LEFM 中预期的断裂韧性（下文中表示为

K_{Ic}^{*}

）并不严格，后者不受 T 应力的影响。请注意，CNSRB 试样的

T / K_{I}

的绝对值小于参考文献 [45] 中研究的 SR、CB、CCNBD 和 SCB 试样配置的

T / K_{I}

的绝对值。[因此，在 CNSRB 断裂试验中，T 应力对模式 I 断裂的影响较小。因此，通过 CNSRB 试样测得的

K_{Ic}

可用于近似

K_{Ic}^{*}

。例如，对于本研究中测试的花岗岩和砂岩，CNSRB 试样的

K_{Ic}

与

K_{Ic}^{*}

之间的差异可根据公式 (14) 仅估算为

\sim 2 - 3 %

。

In addition, as indicated by Eq. (14), the differences between

K_{Ic}

results of different testing methods are related to the Poisson’s ratio

v

and the critical distance

r_{c}

of the rock. To further assess the CNSRB method, some typical rocks studied in the literature (Table 4) with different Poisson’s ratios and critical distances are theoretically analyzed using the EMSTN criterion. Shown in Table 4,

K_{Ic}

results of CNSRB and CB specimens made of these rocks are compared, as well as

K_{Ic}

of the CNSRB specimen with

K_{Ic}^{*}

. It can be observed that the discrepancies between

K_{Ic}

values of CNSRB and CB specimens are, in theory, not significant, and

K_{Ic}

of CNSRB specimens can approximate the fracture toughness corresponding to

T_{c} =

0 . Given these theoretical predictions and our experimental results, the CNSRB method is reliable to measure

K_{Ic}

of rocks. Combined with other testing methods (e.g., CB), the CNSRB method can also contribute to forming a complete set of testing methods for determining the fracture toughness anisotropy in orthogonal directions of a single rock core.
此外，如公式 (14) 所示，不同测试方法的

K_{Ic}

结果之间的差异与岩石的泊松比

v

和临界距离

r_{c}

有关。为了进一步评估 CNSRB 方法，我们使用 EMSTN 准则对文献中研究的一些具有不同泊松比和临界距离的典型岩石（表 4）进行了理论分析。如表 4 所示，比较了由这些岩石制成的 CNSRB 和 CB 试样的

K_{Ic}

结果，以及 CNSRB 试样的

K_{Ic}

与

K_{Ic}^{*}

的结果。可以看出，CNSRB 和 CB 试样的

K_{Ic}

值之间的差异在理论上并不显著，而且 CNSRB 试样的

K_{Ic}

可以近似达到

T_{c} =

0 所对应的断裂韧性。鉴于这些理论预测和我们的实验结果，CNSRB 方法在测量岩石的

K_{Ic}

方面是可靠的。与其他测试方法（如 CB）相结合，CNSRB 方法还有助于形成一套完整的测试方法，用于确定单个岩芯在正交方向上的断裂韧性各向异性。

Note that the CNSRB specimen consumes a relatively large volume of rock material, compared with the CCNBD and SCB specimens. Fortunately, a smaller volume of material is required for a CNSRB specimen than for SR and CB specimens. Given the advantages elaborated above, the CNSRB specimen can be regarded as a modified SR specimen. Although the CCNBD or SCB specimen consumes a smaller volume of rock material, they cannot measure the fracture toughness along axial directions of rock cores, especially when only the rock cores along a single material direction are available. It should also be noted that although this study has proposed the CNSRB method and associated features for fracture toughness testing of rocks, further investigations (e.g., conducting a parametric study on the geometry of the CNSRB specimen, testing more CNSRB specimens with different geometries, and comparing the CNSRB method with more test methods) may still be needed in the future to fully explore our proposed CNSRB method.
请注意，与 CCNBD 和 SCB 试样相比，CNSRB 试样消耗的岩石材料相对较多。幸运的是，与 SR 和 CB 试样相比，CNSRB 试样所需的材料量较少。鉴于上述优点，CNSRB 试样可视为改进的 SR 试样。虽然 CCNBD 或 SCB 试样消耗的岩石材料较少，但它们无法测量岩心沿轴向的断裂韧性，尤其是当只有沿单一材料方向的岩心时。还需注意的是，尽管本研究提出了用于岩石断裂韧性测试的 CNSRB 方法及相关特性，但未来可能仍需进一步研究（例如对 CNSRB 试样的几何形状进行参数研究、测试更多不同几何形状的 CNSRB 试样，以及将 CNSRB 方法与更多测试方法进行比较），以充分探索我们提出的 CNSRB 方法。

6. Summarization 6.总结

A novel chevron notched short rod specimen was developed for measuring the mode I fracture toughness of rocks. Similar to the ISRM-suggested SR specimen, the proposed CNSRB specimen can also measure the fracture toughness along the axial direction of rock cores. Moreover, the CNSRB method has obvious merits compared with some other test methods, such as simple installing and testing procedure, low requirement on testing machine, higher failure load and lower required amount of intact rock core than

S R

and

C B

methods.
为测量岩石的 I 型断裂韧性，开发了一种新型切口短棒试样。与 ISRM 建议的 SR 试样类似，所提出的 CNSRB 试样也可以测量岩心沿轴向的断裂韧性。此外，与其他一些测试方法相比，CNSRB 方法具有明显的优点，如安装和测试程序简单、对试验机要求低、失效载荷高，以及与

S R

和

C B

方法相比，所需的完整岩芯量较少。

The reliability of the CNSRB method was both experimentally and theoretically assessed by comparing

K_{Ic}

measurements with those of the CB method. Laboratory fracture tests on a granite and a sandstone indicate that the CNSRB method can yield

K_{Ic}

results close to those of the CB method, while the data scatter is slightly lower. A theoretical comparison based on the EMTSN fracture criterion also suggests that the discrepancy between

K_{Ic}

values of CNSRB and CB specimens is not significant, and the CNSRB method gives

K_{Ic}

values closer to the CB method than the SR method. In addition, it is demonstrated that

K_{IC}

of the CNSRB specimen can be used to approximate the fracture toughness corresponding to

T_{c} = 0

, due to the relatively small ratio of T-stress to

K_{I}

in the CNSRB method. Therefore, the CNSRB method can be reliably used to measure the mode I fracture toughness of rocks. Most importantly, combined with other testing methods (e.g., CB), our proposed CNSRB method contributes to forming a complete set of methods for investigating fracture toughness anisotropy from a single rock core.
通过比较

K_{Ic}

测量值与 CB 方法的测量值，对 CNSRB 方法的可靠性进行了实验和理论评估。在花岗岩和砂岩上进行的实验室断裂测试表明，CNSRB 方法得出的

K_{Ic}

结果与 CB 方法的结果接近，而数据散度则略低。基于 EMTSN 断裂准则的理论比较也表明，CNSRB 和 CB 试样的

K_{Ic}

值差异不大，而且 CNSRB 方法的

K_{Ic}

值比 SR 方法更接近 CB 方法。此外，由于 CNSRB 方法中 T 应力与

K_{I}

的比值相对较小，因此 CNSRB 试样的

K_{IC}

可用来近似计算与

T_{c} = 0

相对应的断裂韧性。因此，CNSRB 方法可用于可靠地测量岩石的 I 型断裂韧性。最重要的是，结合其他测试方法（如 CB），我们提出的 CNSRB 方法有助于形成一套完整的方法来研究单个岩芯的断裂韧性各向异性。

Acknowledgments 致谢

The authors are grateful for the financial support from the National Program on Key Basic Research Project (No. 2015CB057903) and the National Natural Science Foundation of China (No. 51779164 and No. 51679158).
作者感谢国家重点基础研究发展规划项目（编号：2015CB057903）和国家自然科学基金项目（编号：51779164和51679158）的资助。

References 参考资料

[1] Kuruppu MD, Obara Y, Ayatollahi MR, Chong KP, Funatsu T. ISRM-Suggested method for determining the mode I static fracture toughness using semicircular bend specimen. Rock Mech Rock Eng 2014;47:267-74.
[1] Kuruppu MD，Obara Y，Ayatollahi MR，Chong KP，Funatsu T.利用半圆形弯曲试样确定模式I静态断裂韧性的ISRM建议方法.Rock Mech Rock Eng 2014;47:267-74.
[2] Liu Y, Dai F, Fan PX, Xu NW, Dong L. Experimental investigation of the influence of joint geometric configurations on the mechanical properties of intermittent jointed rock models under cyclic uniaxial compression. Rock Mech Rock Eng 2017;50(6):1453-71.
[2] Liu Y, Dai F, Fan PX, Xu NW, Dong L. 循环单轴压缩条件下节理几何构型对间歇节理岩石模型力学性能影响的实验研究.Rock Mech Rock Eng 2017;50(6):1453-71.
[3] Kuruppu MD, Chong KP. Fracture toughness testing of brittle materials using semi-circular bend (SCB) specimen. Eng Fract Mech 2012;91:133-50.
[3] Kuruppu MD, Chong KP.使用半圆弯曲（SCB）试样测试脆性材料的断裂韧性。Eng Fract Mech 2012;91:133-50.
[4] Ouchterlony F. ISRM commission on testing methods. Suggested methods for determining fracture toughness of rock. Int J Rock Mech Min Sci Geomech Abstr 1988;25:71-96.
[4] Ouchterlony F. ISRM 测试方法委员会。岩石断裂韧性的建议测定方法.Int J Rock Mech Min Sci Geomech Abstr 1988;25:71-96.
[5] Funatsu T, Shimizu N, Kuruppu M, Matsui K. Evaluation of mode I fracture toughness assisted by the numerical determination of K-resistance. Rock Mech Rock Eng 2015;48:143-57.
[5] Funatsu T, Shimizu N, Kuruppu M, Matsui K. 通过数值测定K抗力来评估模式I断裂韧性。Rock Mech Rock Eng 2015;48:143-57.
[6] Mostafavi M, McDonald SA, Mummery PM, Marrow TJ. Observation and quantification of three-dimensional crack propagation in poly-granular graphite. Eng Fract Mech 2013;110:410-20.
[6] Mostafavi M, McDonald SA, Mummery PM, Marrow TJ.多粒状石墨中三维裂纹扩展的观察与量化。Eng Fract Mech 2013;110:410-20.
[7] Wei MD, Dai F, Xu NW, Zhao T. Stress intensity factors and fracture process zones of ISRM-suggested chevron notched specimens for mode I fracture toughness testing of rocks. Eng Fract Mech 2016;168:174-89.
[7] Wei MD，Dai F，Xu NW，Zhao T. 岩石I型断裂韧性测试中ISRM-suggested chevron notched specimens的应力强度因子和断裂过程区.Eng Fract Mech 2016;168:174-89.
[8] Fowell RJ. ISRM commission on testing methods. Suggested method for determining mode I fracture toughness using cracked chevron notched Brazilian disc (CCNBD) specimens. Int J Rock Mech Min Sci Geomech Abstr 1995;32(1):57-64.
[8] Fowell RJ.ISRM 测试方法委员会。使用裂纹切口巴西圆盘（CCNBD）试样确定模式I断裂韧性的建议方法。Int J Rock Mech Min Sci Geomech Abstr 1995;32(1):57-64.
[9] Iqbal MJ, Mohanty B. Experimental calibration of stress intensity factors of the ISRM suggested cracked chevron-notched Brazilian disc specimen used for determination of mode-I fracture toughness. Int J Rock Mech Min Sci 2006;43:1270-6.
[9] Iqbal MJ, Mohanty B.用于确定I型断裂韧性的ISRM建议裂纹楔形缺口巴西圆盘试样应力强度因子的实验校准.Int J Rock Mech Min Sci 2006;43:1270-6.
[10] Iqbal MJ, Mohanty B. Experimental calibration of ISRM suggested fracture toughness measurement techniques in selected brittle rocks. Rock Mech Rock Eng 2007;40(5):453-75.
[10] Iqbal MJ, Mohanty B.选定脆性岩中ISRM建议的断裂韧性测量技术的实验校准。Rock Mech Rock Eng 2007;40(5):453-75.
[11] Wei MD, Dai F, Xu NW, Zhao T. Experimental and numerical investigation of cracked chevron notched Brazilian disc specimen for fracture toughness testing of rock. Fatigue Fract Eng Mater Struct 2017:1-15. doi: https://doi.org/10.1111/ffe.12672.
[11] Wei MD, Dai F, Xu NW, Zhao T. 岩石断裂韧性测试中裂纹切口巴西圆盘试样的实验与数值研究.Fatigue Fract Eng Mater Struct 2017:1-15. doi: https://doi.org/10.1111/ffe.12672.
[12] Dai F, Wei MD, Xu NW, Ma Y, Yang DS. Numerical assessment of the progressive rock fracture mechanism of cracked chevron notched Brazilian disc specimens. Rock Mech Rock Eng 2015;48(2):463-79.
[12] Dai F, Wei MD, Xu NW, Ma Y, Yang DS.裂缝楔形缺口巴西圆盘试样的渐进式岩石断裂机理数值评估.Rock Mech Rock Eng 2015;48(2):463-79.
[13] Dai F, Xu Y, Zhao T, Xu NW, Liu Y. Loading-rate-dependent progressive fracturing of cracked chevron-notched Brazilian disc specimens in split Hopkinson pressure bar tests. Int J Rock Mech Min Sci 2016;88:49-60.
[13] Dai F, Xu Y, Zhao T, Xu NW, Liu Y.裂纹楔形缺口巴西圆盘试样在分体式霍普金森压力棒试验中的加载速率依赖性渐进断裂.Int J Rock Mech Min Sci 2016;88:49-60.
[14] Xu NW, Dai F, Wei MD, Xu Y, Zhao T. Numerical observation of three-dimensional wing cracking of cracked chevron notched Brazilian disc rock specimen subjected to mixed mode loading. Rock Mech Rock Eng 2016;49(1):79-96.
[14] Xu NW，Dai F，Wei MD，Xu Y，Zhao T. 混合模态加载下裂缝切口巴西圆盘岩试件三维翼状裂纹的数值观测.Rock Mech Rock Eng 2016;49(1):79-96.
[15] Xu Y, Dai F, Zhao T, Xu NW, Liu Y. Fracture toughness determination of cracked chevron notched Brazilian disc rock specimen via Griffith energy criterion incorporating realistic fracture profiles. Rock Mech Rock Eng 2016;49(8):3083-93.
[15] Xu Y, Dai F, Zhao T, Xu NW, Liu Y.通过格里菲斯能量准则结合实际断裂剖面确定开裂楔形缺口巴西圆盘岩石试样的断裂韧性.Rock Mech Rock Eng 2016;49(8):3083-93.
[16] Akbardoost J, Ayatollahi MR, Aliha MRM, Pavier MJ, Smith DJ. Size-dependent fracture behavior of Guiting limestone under mixed mode loading. Int J Rock Mech Min Sci 2014;71:369-80.
[16] Akbardoost J, Ayatollahi MR, Aliha MRM, Pavier MJ, Smith DJ.混合模式加载下桂亭石灰岩的尺寸依赖性断裂行为。Int J Rock Mech Min Sci 2014; 71:369-80.
[17] Ayatollahi MR, Zakeri M. An improved definition for mode I and mode II crack problems. Eng Fract Mech 2017;175:235-46.
[17] Ayatollahi MR, Zakeri M. 模式I和模式II裂缝问题的改进定义。Eng Fract Mech 2017;175:235-46.
[18] Aliha MRM, Ayatollahi MR. Brittle fracture evaluation of a fine grain cement mortar in combined tensile-shear deformation. Fatigue Fract Eng Mater Struct 2009;32:987-94.
[18] Aliha MRM, Ayatollahi MR.细粒水泥砂浆在拉伸剪切联合变形下的脆性断裂评估。疲劳断裂工程材料结构，2009；32：987-94.
[19] Tutluoglu L, Keles C. Effects of geometric factors on mode I fracture toughness for modified ring tests. Int J Rock Mech Min Sci 2012;51:149-61.
[19] Tutluoglu L, Keles C. 改良环形试验中几何因素对模式I断裂韧性的影响。Int J Rock Mech Min Sci 2012;51:149-61.
[20] Amrollahi H, Baghbanan A, Hashemolhosseini H. Measuring fracture toughness of crystalline marbles under modes I and II and mixed mode I-II loading conditions using CCNBD and HCCD specimens. Int J Rock Mech Min Sci 2011;48(7):1123-34.
[20] Amrollahi H, Baghbanan A, Hashemolhosseini H.使用 CCNBD 和 HCCD 试样测量模式 I、II 和混合模式 I-II 加载条件下结晶大理石的断裂韧性。Int J Rock Mech Min Sci 2011; 48(7):1123-34.
[21] Aliha MRM, Ayatollahi MR, Pakzad R. Brittle fracture analysis using a ring-shape specimen containing two angled cracks. Int J Fract 2008;153(1):63-8.
[21] Aliha MRM, Ayatollahi MR, Pakzad R. 使用包含两个角度裂缝的环形试样进行脆性断裂分析。Int J Fract 2008;153(1):63-8.
[22] Kuruppu MD. Fracture toughness measurement using chevron notched semi-circular bend specimen. Int J Fract 1997;86(4):33-8.
[22] Kuruppu MD.使用切口半圆形弯曲试样测量断裂韧性。Int J Fract 1997;86(4):33-8.
[23] Chang SH, Lee CI, Jeon S. Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Eng Geol 2002;66:79-97.
[23] Chang SH, Lee CI, Jeon S. 使用圆盘型试样测量模式 I 和模式 II 以及混合模式条件下的岩石断裂韧性.Eng Geol 2002;66:79-97.
[24] Ayatollahi MR, Mahdavi E, Alborzi MJ, Obara Y. Stress intensity factors of semi-circular bend specimens with straight-through and chevron notches. Rock Mech Rock Eng 2016;49(4):1161-72.
[24] Ayatollahi MR，Mahdavi E，Alborzi MJ，Obara Y. 带直通缺口和楔形缺口的半圆形弯曲试样的应力强度因子。Rock Mech Rock Eng 2016；49（4）：1161-72.
[25] Wei MD, Dai F, Xu NW, Xu Y, Xia K. Three-dimensional numerical evaluation of the progressive fracture mechanism of cracked chevron notched semicircular bend rock specimens. Eng Fract Mech 2015;134:286-303.
[25] Wei MD，Dai F，Xu NW，Xu Y，Xia K. 裂纹切口半圆形弯曲岩石试件渐进断裂机制的三维数值评估.Eng Fract Mech 2015;134:286-303.
[26] Wei MD, Dai F, Xu NW, Liu JF, Xu Y. Experimental and numerical study on the cracked chevron notched semi-circular bend method for characterizing the mode I fracture toughness of rocks. Rock Mech Rock Eng 2016;49:1595-609.
[26] Wei MD, Dai F, Xu NW, Liu JF, Xu Y.裂纹雪佛龙缺口半圆弯曲法表征岩石模态I断裂韧性的实验与数值研究.Rock Mech Rock Eng 2016;49:1595-609.
[27] Chong KP, Kuruppu MD. New specimens for mixed mode fracture investigations of geomaterials. Eng Fract Mech 1988;30:701-12.
[27] Chong KP, Kuruppu MD.土工材料混合模态断裂研究的新试样.Eng Fract Mech，1988；30：701-12.
[28] Ayatollahi MR, Akbardoost J, Berto F. Size effects on mixed-mode fracture behavior of polygranular graphite. Carbon 2016;103:394-403.
[28] Ayatollahi MR, Akbardoost J, Berto F. 多晶粒石墨混合模式断裂行为的尺寸效应。碳2016；103：394-403.
[29] Ayatollahi MR, Aliha MRM. Analysis of a new specimen for mixed mode fracture tests on brittle materials. Eng Fract Mech 2009;76(11):1563-73.
[29] Ayatollahi MR，Aliha MRM。脆性材料混合模式断裂试验新试样分析。Eng Fract Mech 2009；76（11）：1563-73.
[30] Funatsu T, Kuruppu MD, Matsui K. Effects of temperature and confining pressure on mixed-mode (I-II) and mode II fracture toughness of Kimachi sandstone. Int J Rock Mech Min Sci 2014;67:1-8.
[30] Funatsu T, Kuruppu MD, Matsui K. 温度和约束压力对Kimachi砂岩混合模式（I-II）和模式II断裂韧性的影响.Int J Rock Mech Min Sci 2014; 67:1-8.
[31] Dai F, Wei MD, Xu NW, Zhao T, Xu Y. Numerical investigation of the progressive fracture mechanisms of four ISRM-suggested specimens for determining the mode I fracture toughness of rocks. Comput Geotech 2015;69:424-41.
[31] Dai F, Wei MD, Xu NW, Zhao T, Xu Y.用于确定岩石I型断裂韧性的四种ISRM建议试样渐进断裂机制的数值研究.Comput Geotech 2015;69:424-41.
[32] Wei MD, Dai F, Xu NW, Liu Y, Zhao T. Fracture prediction of rocks under mode I and mode II loading using the generalized maximum tangential strain criterion. Eng Fract Mech 2017;186:21-38.
[32] Wei MD, Dai F, Xu NW, Liu Y, Zhao T. 使用广义最大切向应变准则预测模态I和模态II加载下的岩石断裂。Eng Fract Mech 2017;186:21-38.
[33] Xu Y, Dai F, Xu NW, Zhao T. Numerical investigation of dynamic rock fracture toughness determination using a semi-circular bend specimen in split Hopkinson pressure bar testing. Rock Mech Rock Eng 2016;49(3):731-45.
[33] Xu Y，Dai F，Xu NW，Zhao T. 分体式霍普金森压杆试验中使用半圆形弯曲试样测定岩石动态断裂韧性的数值研究.Rock Mech Rock Eng 2016；49（3）：731-45.
[34] Wei MD, Dai F, Xu NW, Zhao T, Xia KW. Experimental and numerical study on the fracture process zone and fracture toughness determination for ISRM-suggested semi-circular bend rock specimen. Eng Fract Mech 2016;154:43-56.
[34] Wei MD, Dai F, Xu NW, Zhao T, Xia KW.ISRM建议的半圆形弯曲岩石试样断裂过程区及断裂韧性测定的实验与数值研究.Eng Fract Mech 2016;154:43-56.
[35] Keles C, Tutluoglu L. Investigation of proper specimen geometry for mode I fracture toughness testing with flattened Brazilian disc method. Int J Fract 2011;169(1):61-75.
[35] Keles C, Tutluoglu L. 用扁平巴西圆盘法进行模式 I 断裂韧度测试的适当试样几何形状研究。Int J Fract 2011;169(1):61-75.
[36] Aliha MRM, Hosseinpour GR, Ayatollahi MR. Application of cracked triangular specimen subjected to three-point bending for investigating fracture behavior of rock materials. Rock Mech Rock Eng 2013;46(5):1023-34
[36] Aliha MRM, Hosseinpour GR, Ayatollahi MR.应用受三点弯曲的开裂三角形试样研究岩石材料的断裂行为。Rock Mech Rock Eng 2013;46(5):1023-34
[37] Tutluoglu L, Keles C. Mode I fracture toughness determination with straight notched disk bending method. Int J Rock Mech Min Sci 2011;48 (8):1248-61.
[37] Tutluoglu L, Keles C. 用直缺口圆盘弯曲法测定模式I断裂韧性。Int J Rock Mech Min Sci 2011;48 (8):1248-61.
[38] Aliha MRM, Bahmani A. Rock fracture toughness study under mixed mode I/III loading. Rock Mech Rock Eng 2017;50(7):1739-51.
[38] Aliha MRM, Bahmani A.I/III 混合模式加载下的岩石断裂韧性研究。Rock Mech Rock Eng 2017;50(7):1739-51.
[39] Aliha MRM, Bahmani A, Akhondi S. Determination of mode III fracture toughness for different materials using a new designed test configuration. Mater Des 2015;86:863-71.
[39] Aliha MRM, Bahmani A, Akhondi S. 使用新设计的测试配置确定不同材料的模式III断裂韧性。Mater Des 2015;86:863-71.
[40] Ingraffea AR. Mixed-mode fracture initiation in Indiana limestone and Westerly granite. In: The 22nd US Symposium on Rock Mechanics (USRMS). American Rock Mechanics Association; 1981.
[40] Ingraffea AR.印第安纳石灰岩和威斯特利花岗岩中的混合模式断裂起始。In：第22届美国岩石力学研讨会（USRMS）。美国岩石力学协会；1981 年。
[41] Razavi SMJ, Aliha MRM, Berto F. Application of an average strain energy density criterion to obtain the mixed mode fracture load of granite rock tested with the cracked asymmetric four-point bend specimens. Theor Appl Fract Mech 2017. doi: https://doi.org/10.1016/j.tafmec.2017.07.004.
[41] Razavi SMJ, Aliha MRM, Berto F. 应用平均应变能量密度准则获取花岗岩石混合模式断裂荷载，用开裂的非对称四点弯曲试样进行测试。Theor Appl Fract Mech 2017. doi: https://doi.org/10.1016/j.tafmec.2017.07.004.
[42] Fakhri M, Amoosoltani E, Aliha MRM. Crack behavior analysis of roller compacted concrete mixtures containing reclaimed asphalt pavement and crumb rubber. Eng Fract Mech 2017;180:43-59.
[42] Fakhri M，Amoosoltani E，Aliha MRM。含有再生沥青路面和橡胶屑的碾压混凝土混合物的裂缝行为分析。Eng Fract Mech 2017;180:43-59.
[43] Ayatollahi MR, Akbardoost J. Size and geometry effects on rock fracture toughness: mode I fracture. Rock Mech Rock Eng 2014;47:677-87.
[43] Ayatollahi MR, Akbardoost J. 岩石断裂韧性的尺寸和几何效应：I型断裂。Rock Mech Rock Eng 2014;47:677-87.
[44] Aliha MRM, Sistaninia M, Smith DJ, Pavier MJ, Ayatollahi MR. Geometry effects and statistical analysis of mode I fracture in guiting limestone. Int J Rock Mech Min Sci 2012;51:128-35.
[44] Aliha MRM, Sistaninia M, Smith DJ, Pavier MJ, Ayatollahi MR.guiting石灰岩中I型断裂的几何效应和统计分析。Int J Rock Mech Min Sci 2012;51:128-35.
[45] Aliha MRM, Mahdavi E, Ayatollahi MR. The influence of specimen type on tensile fracture toughness of rock materials. Pure Appl Geophys 2017;3 (174):1237-53.
[45] Aliha MRM, Mahdavi E, Ayatollahi MR.试样类型对岩石材料拉伸断裂韧性的影响。Pure Appl Geophys 2017;3 (174):1237-53.
[46] Khan K, Al-Shayea NA. 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 2000;33(3):179-206.
[46] Khan K, Al-Shayea NA.试样几何形状和测试方法对沙特阿拉伯石灰岩I-II混合模式断裂韧性的影响。Rock Mech Rock Eng，2000；33（3）：179-206.
[47] Cui ZD, Liu DA, An GM, Sun B, Zhou M, Cao FQ. A comparison of two ISRM suggested chevron notched specimens for testing mode-I rock fracture toughness. Int J Rock Mech Min Sci 2010;47:871-6.
[47] Cui ZD, Liu DA, An GM, Sun B, Zhou M, Cao FQ.两种ISRM建议的用于测试I型岩石断裂韧性的楔形缺口试样的比较.Int J Rock Mech Min Sci 2010;47:871-6.
[48] Ayatollahi MR, Abbasi H. Prediction of fracture using a strain based mechanism of crack growth. Build Res J 2001;49(3):167-80.
[48] Ayatollahi MR, Abbasi H.使用基于应变的裂纹生长机制预测断裂。Build Res J 2001; 49(3):167-80.
[49] Ayatollahi MR, Saboori B. T-stress effects in mixed mode I/II/III brittle fracture. Eng Fract Mech 2015;144:32-45.
[49] Ayatollahi MR, Saboori B. I/II/III混合模式脆性断裂中的T应力效应。Eng Fract Mech 2015;144:32-45.
[50] Aliha MRM, Sarbijan MJ, Bahmani A. Fracture toughness determination of modified HMA mixtures with two novel disc shape configurations. Constr Build Mater 2017;155:789-99.
[50] Aliha MRM, Sarbijan MJ, Bahmani A.采用两种新型圆盘形状配置的改性 HMA 混合物的断裂韧性测定。Constr Build Mater 2017;155:789-99.
[51] Mirsayar MM, Berto F, Aliha MRM, Park P. Strain-based criteria for mixed-mode fracture of polycrystalline graphite. Eng Fract Mech 2016;156:114-23.
[51] Mirsayar MM, Berto F, Aliha MRM, Park P. 基于应变的多晶石墨混合模式断裂标准。Eng Fract Mech 2016;156:114-23.
[52] Lim IL, Johnston IW, Choi SK. Assessment of mixed mode fracture toughness testing methods for rock. Int J Rock Mech Min Sci Geomech Abstr 1994;31 (3):265-72.
[52] Lim IL, Johnston IW, Choi SK.岩石混合模式断裂韧性测试方法评估。Int J Rock Mech Min Sci Geomech Abstr 1994;31 (3):265-72.
[53] Wei MD, Dai F, Xu NW, Zhao T, Liu Y. An experimental and theoretical assessment of semi-circular bend specimens with chevron and straight-through notches for mode I fracture toughness testing of rocks. Int J Rock Mech Min Sci 2017;99:28-38.
[53] Wei MD, Dai F, Xu NW, Zhao T, Liu Y.岩石I型断裂韧性测试中带有楔形缺口和直通缺口的半圆形弯曲试样的实验和理论评估.Int J Rock Mech Min Sci 2017;99:28-38.
[54] Wei MD, Dai F, Liu Y, Xu NW, Zhao T. An experimental and theoretical comparison of CCNBD and CCNSCB specimens for determining mode I fracture toughness of rocks. Fatigue Fract Eng Mater Struct 2017:1-17. doi: https://doi.org/10.1111/ffe.12747.
[54] Wei MD, Dai F, Liu Y, Xu NW, Zhao T. CCNBD和CCNSCB试样测定岩石模I断裂韧性的实验与理论比较.Fatigue Fract Eng Mater Struct 2017:1-17. doi: https://doi.org/10.1111/ffe.12747.
[55] Matsuki K, Matsune S, Takahashi H. Boundary element analysis for standard specimen configurations in the ISRM suggested methods for determining fracture toughness of rock. Int J Rock Mech Min Sci Geomech Abstr 1991;28(5):355-63.
[55] Matsuki K, Matsune S, Takahashi H. ISRM 中标准试样配置的边界元分析建议的岩石断裂韧性测定方法.Int J Rock Mech Min Sci Geomech Abstr 1991;28(5):355-63.
[56] Aliha MRM, Ayatollahi MR. Two-parameter fracture analysis of SCB rock specimen under mixed mode loading. Eng Fract Mech 2013;103:115-23.
[56] Aliha MRM, Ayatollahi MR.混合模式加载下SCB岩石试样的双参数断裂分析。Eng Fract Mech 2013;103:115-23.
[57] Ayatollahi MR, Aliha MRM. Fracture toughness study for a brittle rock subjected to mixed mode I/II loading. Int J Rock Mech Min Sci 2007;44:617-24.
[57] Ayatollahi MR, Aliha MRM.承受 I/II 混合模式加载的脆性岩石的断裂韧性研究。Int J Rock Mech Min Sci 2007;44:617-24.
[58] Ayatollahi MR, Aliha MRM. On the use of Brazilian disc specimen for calculating mixed mode I-II fracture toughness of rock materials. Eng Fract Mech 2008;75:4631-41.
[58] Ayatollahi MR, Aliha MRM.使用巴西圆盘试样计算岩石材料的混合模式I-II断裂韧性。Eng Fract Mech 2008;75:4631-41.
[59] Akbardoost J, Ayatollahi MR. Experimental analysis of mixed mode crack propagation in brittle rocks: the effect of non-singular terms. Eng Fract Mech 2014;129:77-89.
[59] Akbardoost J, Ayatollahi MR.脆性岩石中混合模式裂纹扩展的实验分析：非星形项的影响。Eng Fract Mech 2014;129:77-89.
[60] Aliha MRM, Ayatollahi MR, Akbardoost J. Typical upper bound-lower bound mixed mode fracture resistance envelopes for rock material. Rock Mech Rock Eng 2012;45:65-74.
[60] Aliha MRM, Ayatollahi MR, Akbardoost J. 岩石材料的典型上限-下限混合模式断裂阻力包络。Rock Mech Rock Eng 2012;45:65-74.
[61] Ayatollahi MR, Sistaninia M. Mode II fracture study of rocks using Brazilian disk specimens. Int J Rock Mech Min Sci 2011;48:819-26.
[61] Ayatollahi MR, Sistaninia M. 利用巴西圆盘试样进行的岩石模态II断裂研究.Int J Rock Mech Min Sci 2011;48:819-26.
[62] Sistaninia M, Ayatollahi MR, Sistaninia M. On fracture analysis of cracked graphite components under mixed mode loading. Mech Adv Mater Struct 2014;21(10):781-91.
[62] Sistaninia M, Ayatollahi MR, Sistaninia M. On fracture analysis of cracked graphite components under mixed mode loading.Mech Adv Mater Struct 2014;21(10):781-91.
[63] Ayatollahi MR, Berto F, Campagnolo A, Gallo P, Tang K. Review of local strain energy density theory for the fracture assessment of V-notches under mixed mode loading. Eng Solid Mech 2017;5(2):113-32.
[63] Ayatollahi MR, Berto F, Campagnolo A, Gallo P, Tang K. Review of local strain energy density theory for the fracture assessment of V-notches under mixed mode loading.Eng Solid Mech 2017;5(2):113-32.
[64] Berto F, Gomez G. Notched plates in mixed mode loading (I+II): a review based on the local strain energy density and the cohesive zone mode. Eng Solid Mech 2017;5(1):1-8.
[64] Berto F, Gomez G. 混合模式加载（I+II）中的缺口板：基于局部应变能密度和内聚区模式的综述。Eng Solid Mech 2017;5(1):1-8.
[65] Al-Shayea NA. Effects of testing methods and conditions on the elastic properties of limestone rock. Eng Geol 2004;74:139-56.
[65] Al-Shayea NA.测试方法和条件对石灰岩弹性特性的影响。Eng Geol 2004;74:139-56.
[66] Awaji H, Sato S. Combined mode fracture toughness measurement by the disk test. J Eng Mater Tech 1978;100:175-82.
[66] Awaji H, Sato S. 通过圆盘试验测量组合模式断裂韧性。J Eng Mater Tech 1978;100:175-82.
[67] Aliha MRM, Ayatollahi MR, Smith DJ, Pavier MJ. Geometry and size effects on fracture trajectory in a limestone rock under mixed mode loading. Eng Fract Mech 2010;77(11):2200-12.
[67] Aliha MRM, Ayatollahi MR, Smith DJ, Pavier MJ.混合模式加载下石灰岩断裂轨迹的几何形状和尺寸效应。Eng Fract Mech 2010; 77(11):2200-12.
[68] Doghozlou HM, Goodarzi M, Renani HR, Salmi EF. Analysis of spalling failure in marble rock slope: a case study of Neyriz marble mine, Iran. Environ Earth Sci 2016;75:1478.
[68] Doghozlou HM, Goodarzi M, Renani HR, Salmi EF.大理岩边坡剥落破坏分析：伊朗Neyriz大理石矿的案例研究。Environ Earth Sci 2016;75:1478.

- Corresponding author. 通讯作者：
E-mail address: fengdai@scu.edu.cn (F. Dai).
电子邮件地址：fengdai@scu.edu.cn (F. Dai)。

A novel chevron notched short rod bend method for measuring the mode I fracture toughness of rocks 测量岩石 I 型断裂韧性的新型切口短棒弯曲法

ARTICLE INFO 文章信息

Article history: 文章历史：

Keywords: 关键词：

Abstract 摘要

1. Introduction 1.导言

2. The CNSRB method2.CNSRB 方法

2.1. Specimen geometry and testing principle2.1.试样几何形状和测试原理

2.2. Determination of Y min ∗ Y min ∗ Y_(min)^(**)Y_{\min }^{*}2.2. Y min ∗ Y min ∗ Y_(min)^(**)Y_{\min }^{*} 的确定

3. Experimental results 3.实验结果

4. Theoretical assessment4.理论评估

5. Discussion 5.讨论

6. Summarization 6.总结

Acknowledgments 致谢

References 参考资料

A novel chevron notched short rod bend method for measuring the mode I fracture toughness of rocks
测量岩石 I 型断裂韧性的新型切口短棒弯曲法

2. The CNSRB method
2.CNSRB 方法

2.1. Specimen geometry and testing principle
2.1.试样几何形状和测试原理

2.2. Determination of $Y_{min}^{}$
2.2. $Y_{min}^{}$ 的确定

4. Theoretical assessment
4.理论评估