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}

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