# The Loading Rate Effect on the Fracture Toughness of Marble Using Semicircular Bend Specimens

Authors: Fengqiang Gong; Yunliang Wang; Shanyong Wang
Journal: Geofluids
 (2020)

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## Abstract

A series of dynamic fracture experiments on semicircular bend (SCB) marble specimens were conducted to characterize the loading rate effect using the INSTRON testing machine and the modified SHPB testing system. The fracture toughness of the marble specimens was measured from a low loading rate to a high loading rate (10-3~106 MPa·m1/2s-1). The results show that the fracture toughness will increase with the loading rate. Since the fracture toughness at a magnitude of 10-3 MPa·m1/2s-1 is regarded as the static fracture toughness, the specific value of DIFf (the dynamic increase factor of fracture toughness) can be obtained at the other loading magnitudes from dynamic fracture tests. To describe the variation in DIFf from low to high loading rates, a new continuous model of DIFf was put forward to express the quantitative relation between the loading rate and rock dynamic fracture toughness. It is shown that the new DIFf model can accurately describe the loading rate effect on the dynamic fracture testing data for rock materials.


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## Body

## 1. Introduction

The failure of rock or rock mass, such as rock cutting, hydraulic fracturing, rock burst, and spalling, is closely related to the initiation and propagation of internal cracks under complex stress [1–6]. This phenomenon has been observed in many laboratory tests or engineering sites [7–10]. Fracture mechanics plays a crucial role in geophysical processes and engineering applications involving rock or underground engineering [11–14]. Rock fracture toughness, as an intrinsic material property of rocks, is considered an important factor for resisting crack initiation and propagation and thus has been widely investigated in the rock mechanics community [15–17]. Accurate measurement of fracture toughness is critical for understanding rock fracture mechanisms and solving engineering problems.Many methods have been proposed to measure the fracture toughness of brittle materials such as rocks. For example, short rod (SR) and chevron bending (CB) tests in 1988 [18], the cracked chevron-notched Brazilian disc (CCNBD) testing in 1995 [19], and chevron-notched SCB testing [20]. The Brazilian disc test [21] was proposed by the International Society for Rock Mechanics (ISRM) as a method for rock fracture testing. In addition, cracked straight through Brazilian disk (CSTBD) tests [22, 23] are also often used to measure fracture toughness. In addition, the semicircular bend (SCB) was proposed in 1984 by Chong and Kuruppu [24], and the geometry of the sample is convenient for sample processing (directly from rock cores) and experimentation. The ISRM recently adopted the semicircular bend (SCB) method for characterizing the static and dynamic mode I fracture toughness of rocks [25, 26]. Thus, this method is used for static and dynamic fracture experiments, and a large number of experimental results have shown that the loading rate has an important influence on the mechanical properties of rock [27, 28].Existing attempts to measure rock fracture toughness were mostly conducted under limited dynamic loading by using a split Hopkinson pressure bar (SHPB) [29, 30]. Limited attempts have been made to study the static and dynamic fracture toughness of rocks simultaneously. Zhang and Zhao [31] studied the effect of the loading rate on the fracture toughness and failure micromechanisms in marble. Zhang et al. [27] measured the fracture toughness of marble for a wide range of loading rates from10-2 MPa·m1/2s-1 to 106 MPa·m1/2s-1. Backersa et al. [32] investigated the influence of the loading rate on the fracture toughness of sandstone samples subjected to mode I loading and noted that at low velocities, the fracture toughness remains approximately constant. By exceeding a fracture velocity threshold, the fracture toughness increases significantly. All the above research shows the following conditions: (1) Although many people have studied SCB testing, most only consider the dynamic loading range and obtain the rate effect of fracture toughness. Therefore, the wide ranges of loading rates have not been investigated by SCB testing at present. (2) There is no description of the normalized model for the dynamic increase factor available from low loading rates to high loading rates.In this study, the semicircular bend (SCB) test was conducted on marble using a servo-hydraulic testing machine and a modified SHPB system. Then, the change rule of the fracture toughness of rock with the loading rate was obtained. Based on this result, at a wide range of low to high loading rates, a new continuous model ofDIFf for the rock dynamic fracture was put forward to express the quantitative relation between the loading rate and dynamic fracture toughness.


## 2. SCB Specimen Preparation

To investigate the fracture properties of the rock material, marble [15, 16] extracted from Leiyang in Hunan Province was chosen as the experiment material. According to the suggested method of specimen processing for SCB testing [24], marble cores with 50 mm diameters were first drilled from a rock block and then sliced to obtain discs with thickness of 20 mm. The surface roughness of all the specimens is less than 0.5% of the thickness. Then, SCB specimens were machined from the obtained discs by radial cutting; i.e., a notch with a width of approximately 1 mm was subsequently processed with a 0.5 mm thick hacksaw blade by radial cutting from the centre of the disc. In addition, sufficient crack tip sharpness is necessary for accurately measuring the fracture initiation toughness. The specimen processing is shown in Figure 1.Figure 1
(a) Cored rock and (b) SCB specimen.
(a)(b)


## 3. Experimental Setup

The INSTRON testing system (Figure2) from the modern analysis and testing centre of Central South University was employed to perform the static measurement of fracture toughness. The test results were collected by a computer. The INSTRON testing system is shown in Figure 2(a). In addition, the SCB specimen geometry and schematic loading arrangement are shown in Figure 2(b), in which the specimen was loaded by a three-point bending fixture.Figure 2
(a) INSTRON testing system and (b) SCB specimen geometry and schematic loading arrangement.
(a)(b)A constant loading rate of 4 kN/min was applied by displacement control during the experiments. The specimen was loaded at the constant rate until it was totally fractured. This relatively low loading rate is not only conducive to the stable development of a surface crack and nonlinear fracture process zone at a crack tip but also convenient to measure the fracture toughness. The SHPB system is often used to test various dynamic parameters of rock materials [33–36]. In these tests, the dynamic fracture testing of the rock was conducted using a 50 mm diameter SHPB system (see Figure 3), which has been used in many dynamic tests [37, 38]. c mm; this apparatus can simulate pulse waveforms to reduce high-frequency vibration and minimize the dispersion degree of the test results. This system also includes an incident bar 2000 mm in length and a transmission bar 1500 mm in length. To carry out the tests, the specimens were clamped between the incident and transmitted bars. In addition, before the tests, the longitudinal wave velocity of the marble was measured, and its minimum value is 3.09 km/s.Figure 3
(a) SHPB system for rock dynamic fracture testing. (b) Schematics of the semicircular bend (SCB) specimen in the split Hopkinson pressure bar (SHPB) system.
(a)(b)


## 4. Formula

The initiation fracture toughnessKIC of SCB is determined by the ISRM-suggested method [24]:
(1)KIC=Pmaxπa2RBY′,(2)Y′=−1.297+9.516S2R−0.47+16.457s2Rβ+1.071+34.401s2Rβ2,(3)β=aR,where Pmax is the measured peak load at specimen failure; R and B are the radius and the thickness of the specimen, respectively; A is the prefabricated crack length; Y′ is the minimum dimensionless stress intensity factor; and S is the length of the load end. In this work, the standard sizes mentioned in the literature are used as follows [25]:S/2R=0.667 and a/R=0.2. According to the principle of SHPB in dynamic fracture testing, the specific value of the dynamic fracture toughness can be calculated by substituting Pmax into Equation (1).In the test of the SCB mode I fracture toughness, the loading rate of the test was defined as the rate of change in the stress intensity factor at the crack tip (K̇Id) because K̇Id can accurately respond to the rate of change in the stress field during the loading process. The dynamic loading rate can be calculated according to the following equation:
(4)K⋅Id=KIdtf,where KId refers to the dynamic fracture toughness and tf is the time needed for the main crack to run through the entire specimen.


## 5. Experimental Results

### 5.1. Quasistatic Fracture Test Results at a Low Loading Rate

To analyse the relationship between the load and the displacement of the fracture under a quasistatic load, the load-displacement curves of the specimens under different loading rates during the fracture tests are shown in Figure4. From Figure 4, it appears that the fracture process of a specimen can be divided into two distinct phases, i.e., the steady increase phase and the sharp increase phase. Between these two phases, a turning point is clearly observed. After entering the sharp increase phase, the specimen is in a relatively stable loading process; near the fracture point, the specimen suddenly brakes, leading to destruction. During the initial loading stage, some microcracks existing in the rock closed, which caused larger deformation under a lower load, i.e., the steady increase phase. With the load increasing, the new microcracks developed and extended, which led to the sharp increase phase. The fracture test results of rock specimens under the quasistatic load are summarized in Table 1.Figure 4
Load-displacement curves during the fracture tests for different loading rates.Table 1
Fracture test results of rock specimens under the quasistatic load.
No.Control rate (mm/min)t (s)Log timeFracture toughness (MPa·m0.5)Loading rate (MPa·m0.5s-1)Log loading rateS-20.015662.75280.54070.0010-3.0199S-30.017812.89270.53790.0007-3.1620S-60.017452.87220.57480.0008-3.1126S-70.12122.32630.64340.0030-2.5178S-80.11312.11730.72280.0055-2.2582S-90.1881.94450.68200.0077-2.1107S-1016.000.77820.64650.1078-0.9676S-1115.750.75970.65330.1136-0.9446S-118.910.94980.66250.0744-1.1286S-13101.16560.06650.68100.5842-0.2334S-14101.23460.09150.75750.6135-0.2122S-15101.33260.12470.71050.5332-0.2731S-161000.1426-0.84590.87886.16270.7898S-171000.1836-0.73610.93175.07460.7054S-181000.1706-0.76800.81084.75260.6769The results show that the fracture toughness of rock changes with the increase in the loading rate under quasistatic loading. The fracture toughness increases with the loading rate, and the maximum value of the fracture toughness is 0.9317 MPa·m0.5, which is an increase of approximately 72% compared with the minimum fracture toughness of 0.5407 MPa·m0.5. According to the traditional method, a linear function was used to express the relationship between the loading rate and fracture toughness.


### 5.2. Fracture Test Results of Rock under Dynamic Loading

The relationship between the load and the displacement of the fracture under a dynamic load in the SHPB test, as illustrated in Figure5, is analysed here. The results of the rock dynamic fracture tests are also shown in Table 2.Figure 5
Load-displacement curves of different loading rates.Table 2
The results of the rock dynamic fracture tests under dynamic loading.
No.t (s)Log timeFracture toughness (MPa·m0.5)Loading rate (MPa·m0.5s-1)Loading rate logD-10.000092-4.03622.291624908.95864.3964D-30.000100-4.00002.385823858.19674.3776D-50.000103-3.98722.318822513.09074.3524D-60.000101-3.99573.498634639.44624.5396D-80.000102-3.99142.449424014.11154.3805D-90.000089-4.05061.826820525.44664.3123B-10.0000441.643457.1655162853.9645.2117B-20.0000471.672093.097165896.61404.8188B-30.0000431.6334686.1324142614.6635.1541B-40.0000501.698974.306686133.50324.9351B-60.0000721.857333.665250906.32574.7067It appears that the load-displacement curve of the specimen during the dynamic fracture process has a similar change rule to that in the static state; the fracture process can clearly be divided into a steady increase phase and a sharp increase phase. Relative to the curve under a static state, the steady increase phase during the dynamic fracture process is shorter and the proportion of the sharp increase phase tends to increase. This may be caused by the different dynamic responses of the rock materials under impact loading. In addition, the failure displacement of a specimen under dynamic loading is clearly smaller than that of a specimen under static loading. This is because the dynamic impact speed is very high, causing the crack to rapidly develop through the specimen, and the failure of a specimen produces relatively less deformation.The relationship between the fracture toughness and logarithm of the loading rate is shown in Figure6. From the scatter diagrams in Figure 6, it appears that the marble fracture toughness under dynamic loading was significantly higher than that under a static condition and follows a similar increasing trend with the increase in the loading rate [28]. The maximum value of the fracture toughness is 7.1655 MPa·m0.5, which is an increase of approximately 292% from the minimum fracture toughness of 1.8268 MPa·m0.5; the loading rate effect is much more apparent.Figure 6
Comparison of the fracture toughness under dynamic and static loading.Based on the above research results, it can be concluded that a clear loading rate effect exists for the fracture toughness of marble. Both the fracture toughness at a low loading rate and a high loading rate are logarithmically increased as the loading rate increases, but their growth trends are different. It can be seen from Figure7 that at low loading rates, the marble fracture toughness and the loading rate (logarithmic) show a linear but relatively slow increase. At high loading rates, the marble fracture toughness increases rapidly with the increase in the loading rate (logarithmic), and these parameters are related. Their relationships can be expressed by the following equations:
(i)
Low loading rate:(5)KId=0.0681logσdf.+0.7763,(ii)
High loading rate:(6)KId=0.0002logσsf.6.2835,where KId is the fracture toughness under different loading rates, σdf. is the loading rate of the dynamic fracture, and σsf. is the static fracture loading rate.Figure 7
Comparison of the fitting formula and measured data.


## 5.1. Quasistatic Fracture Test Results at a Low Loading Rate

To analyse the relationship between the load and the displacement of the fracture under a quasistatic load, the load-displacement curves of the specimens under different loading rates during the fracture tests are shown in Figure4. From Figure 4, it appears that the fracture process of a specimen can be divided into two distinct phases, i.e., the steady increase phase and the sharp increase phase. Between these two phases, a turning point is clearly observed. After entering the sharp increase phase, the specimen is in a relatively stable loading process; near the fracture point, the specimen suddenly brakes, leading to destruction. During the initial loading stage, some microcracks existing in the rock closed, which caused larger deformation under a lower load, i.e., the steady increase phase. With the load increasing, the new microcracks developed and extended, which led to the sharp increase phase. The fracture test results of rock specimens under the quasistatic load are summarized in Table 1.Figure 4
Load-displacement curves during the fracture tests for different loading rates.Table 1
Fracture test results of rock specimens under the quasistatic load.
No.Control rate (mm/min)t (s)Log timeFracture toughness (MPa·m0.5)Loading rate (MPa·m0.5s-1)Log loading rateS-20.015662.75280.54070.0010-3.0199S-30.017812.89270.53790.0007-3.1620S-60.017452.87220.57480.0008-3.1126S-70.12122.32630.64340.0030-2.5178S-80.11312.11730.72280.0055-2.2582S-90.1881.94450.68200.0077-2.1107S-1016.000.77820.64650.1078-0.9676S-1115.750.75970.65330.1136-0.9446S-118.910.94980.66250.0744-1.1286S-13101.16560.06650.68100.5842-0.2334S-14101.23460.09150.75750.6135-0.2122S-15101.33260.12470.71050.5332-0.2731S-161000.1426-0.84590.87886.16270.7898S-171000.1836-0.73610.93175.07460.7054S-181000.1706-0.76800.81084.75260.6769The results show that the fracture toughness of rock changes with the increase in the loading rate under quasistatic loading. The fracture toughness increases with the loading rate, and the maximum value of the fracture toughness is 0.9317 MPa·m0.5, which is an increase of approximately 72% compared with the minimum fracture toughness of 0.5407 MPa·m0.5. According to the traditional method, a linear function was used to express the relationship between the loading rate and fracture toughness.


## 5.2. Fracture Test Results of Rock under Dynamic Loading

The relationship between the load and the displacement of the fracture under a dynamic load in the SHPB test, as illustrated in Figure5, is analysed here. The results of the rock dynamic fracture tests are also shown in Table 2.Figure 5
Load-displacement curves of different loading rates.Table 2
The results of the rock dynamic fracture tests under dynamic loading.
No.t (s)Log timeFracture toughness (MPa·m0.5)Loading rate (MPa·m0.5s-1)Loading rate logD-10.000092-4.03622.291624908.95864.3964D-30.000100-4.00002.385823858.19674.3776D-50.000103-3.98722.318822513.09074.3524D-60.000101-3.99573.498634639.44624.5396D-80.000102-3.99142.449424014.11154.3805D-90.000089-4.05061.826820525.44664.3123B-10.0000441.643457.1655162853.9645.2117B-20.0000471.672093.097165896.61404.8188B-30.0000431.6334686.1324142614.6635.1541B-40.0000501.698974.306686133.50324.9351B-60.0000721.857333.665250906.32574.7067It appears that the load-displacement curve of the specimen during the dynamic fracture process has a similar change rule to that in the static state; the fracture process can clearly be divided into a steady increase phase and a sharp increase phase. Relative to the curve under a static state, the steady increase phase during the dynamic fracture process is shorter and the proportion of the sharp increase phase tends to increase. This may be caused by the different dynamic responses of the rock materials under impact loading. In addition, the failure displacement of a specimen under dynamic loading is clearly smaller than that of a specimen under static loading. This is because the dynamic impact speed is very high, causing the crack to rapidly develop through the specimen, and the failure of a specimen produces relatively less deformation.The relationship between the fracture toughness and logarithm of the loading rate is shown in Figure6. From the scatter diagrams in Figure 6, it appears that the marble fracture toughness under dynamic loading was significantly higher than that under a static condition and follows a similar increasing trend with the increase in the loading rate [28]. The maximum value of the fracture toughness is 7.1655 MPa·m0.5, which is an increase of approximately 292% from the minimum fracture toughness of 1.8268 MPa·m0.5; the loading rate effect is much more apparent.Figure 6
Comparison of the fracture toughness under dynamic and static loading.Based on the above research results, it can be concluded that a clear loading rate effect exists for the fracture toughness of marble. Both the fracture toughness at a low loading rate and a high loading rate are logarithmically increased as the loading rate increases, but their growth trends are different. It can be seen from Figure7 that at low loading rates, the marble fracture toughness and the loading rate (logarithmic) show a linear but relatively slow increase. At high loading rates, the marble fracture toughness increases rapidly with the increase in the loading rate (logarithmic), and these parameters are related. Their relationships can be expressed by the following equations:
(i)
Low loading rate:(5)KId=0.0681logσdf.+0.7763,(ii)
High loading rate:(6)KId=0.0002logσsf.6.2835,where KId is the fracture toughness under different loading rates, σdf. is the loading rate of the dynamic fracture, and σsf. is the static fracture loading rate.Figure 7
Comparison of the fitting formula and measured data.


## 6. Discussion

Most of the existing studies on the normalized model of the dynamic increase factor have focused on compression or tensile tests. Additionally, the dynamic increase factor is usually divided: one function is used for low loading rates and another function is used for high loading rates. Over the past decades, the normalized dynamic uniaxial compressive strength has been obtained as a function of the strain rate [39, 40], and the normalized dynamic tensile strength [41, 42] has been obtained as a function of the loading rate. Thus, the accurate determination of the normalized model of the dynamic increase factor is crucial for understanding the fracture mechanisms over a wide range of loading rates and is beneficial for engineering applications.TheDIFf (DIF of fracture toughness) was proposed to compare the trend of the change in the fracture toughness for a range of low and high loading rates. In addition, DIFf can be expressed as follows:
(7)DIFf=KIdKIs,where KId is the fracture toughness under different loading rates and KIs is the fracture toughness under the minimum loading rate. The DIFf test results under medium-low loading rates and high loading rates are plotted in Figure 8.Figure 8
Variation in the rock fracture toughness at low to high loading rates.As mentioned previously, traditional analysis methods are generally used to separate the results of the medium-low loading rates and high loading rates. Therefore, according to the latest research results from Gong and Zhao [43], a unified expression for the relationship between the fracture toughness and the loading rate of rock for all the loading rates is presented. The expression of the fitting function is as follows:
(8)DIFf=αα−logσ̇df/σ̇sf1−βlogσ̇df/σ̇sf/α,where σ̇df is the loading rate of the dynamic fracture and σ̇sf is the static fracture loading rate. In this work, the dynamic loading rate with the slowest order of magnitude is equal to the static loading rate, and α and β are constants. According to the obtained experimental data and Equation (4), α and β are 8.18 and 0.45, respectively. Based on the fitting formula, Figure 7 gives a comparison of the fitting curve and experimental data, and it can be seen that the fit is good; the fitting curve is very close to the measured data.


## 7. Conclusions

In this study, the fracture toughness of SCB specimens was measured at different loading rates with an INSTRON testing system and a modified SHPB system, and the mechanical properties of the marble obtained from the quasistatic and dynamic fracture tests were qualitatively and quantitatively analysed. In addition, the main results are as follows:(1)
The load-displacement curves at different loading rates were obtained, and it was noted that both the static and dynamic fracture processes can be divided into two distinct stages; the fracture forms near the inflection point between these two stages. However, owing to the high speed of the dynamic impact, the dynamic fracture displacement is clearly smaller than the displacement of the static fracture in the rock(2)
The fracture toughness under pure static conditions and a pure dynamic state was measured. It is found that both the static fracture toughness and dynamic fracture toughness are influenced by the loading rate, and the effect of the loading rate under the static load is obviously smaller than that under the dynamic one. It is also proven that the fracture toughness in a pure static state (logarithmic) increases linearly with the loading rate; hence, it can be concluded that the fracture toughness of the rock increases exponentially with the increasing loading rate(3)
A continuous function was proposed to express the relation between the rock fracture toughness and the loading rate and can better characterize the fracture toughness of rock under low to high loading rates



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