A waveform modification method for testing dynamic properties of rock under high temperature

Tubing Yin, Chao Wang, You Wu, Bingqiang Wu

School of Resources and Safety Engineering, Central South University, Changsha, 410083, China

Keywords:High temperature Split Hopkinson pressure bar (SHPB)Dynamic loading Rock materials Mechanical properties

ABSTRACT In this study, a waveform modification method was proposed using a self-designed heating device combined with the split Hopkinson pressure bar (SHPB) technique for determination of dynamic behaviors of rock at high temperature. Firstly, the temperature gradient distribution on the incident bar was measured according to the variation of elastic modulus of the bar with temperature, and the relationship between the longitudinal wave velocity and temperature of the bar was obtained based on onedimensional stress wave theory. The incident bar with a temperature gradient was divided into a series of microelements, and then the transmission coefficient of the whole incident bar was obtained. Finally,the stress wave was modified by the transmission coefficient from 25 °C to 600 °C.This method was used to study the dynamic properties of rock at high temperature, which not only preserves a classical SHPB device, but also effectively ensures the accuracy of the experimental results. A dynamic Brazilian disc experiment was carried out to explore the influences of loading rate and temperature on dynamic tensile strength of sandstone at high temperature using the proposed waveform modification method.

1. Introduction

Temperature plays an important role in many rock engineering projects,such as deep mining of mineral resources(Cai and Brown,2017;Wagner,2019),borehole drilling in tunnel excavation(Fomin et al., 2005), exploitation and utilisation of geothermal resources(Zhao,1994), storage of deep nuclear waste (Dwivedi et al., 2008),and fire hazard(Ozguven and Ozcelik,2013;Mambou et al.,2015).The mechanical properties of rock change under the influence of high temperature (Heuze, 1983; Xu and Karakus, 2018). As the temperature increases, the porosity and volume of rock increase,the longitudinal wave velocity and density decrease (Chaki et al.,2008; Ozguven and Ozcelik, 2014; Zhang et al., 2016; Yin et al.,2020), the conductivity and thermal diffusivity decrease (Dwivedi et al., 2008; Sun et al., 2016a), and the uniaxial compressive/tensile strength, fracture toughness, and elastic modulus decrease(Ding et al.,2016;Yin et al.,2019;Yu et al.,2020).Moreover,cracks appear in the rock due to the fact that the thermal expansion coefficients of various minerals are different and the mineral composition is transformed under high temperature. Ding et al.(2016) used a scanning electron microscope (SEM) and found that the microcracks inside the sandstone increase when the thermal treatment temperature exceeds 400°C.Yang et al.(2017)utilised Xray computed tomography (CT) and found that the damage of granite sample increases with increasing temperature. In a word,the influence of temperature on rock properties directly jeopardises the stability and safety of rock engineering projects.

In addition, it is also a hot research point that safety of rock engineering is threatened by dynamic loading (such as blasting,rockburst, and seismic activity) (Dang et al., 2020). Due to the characteristics of transient and high strain rate,dynamic properties of rock are quite different from static ones(Zhou et al.,2012;Dang et al., 2016; Peng et al., 2019). Extensive studies have been carried out on various dynamic properties of rocks after thermal treatment,such as dynamic compression (Huang and Xia, 2015; Fan et al.,2017), dynamic tension (Mardoukhi et al., 2017), and dynamic fracture(Yin et al.,2012).However,previous studies found that the mechanical properties of rocks under high temperature vary widely from those after thermal treatment in both the static and dynamic experiments(Liu and Xu,2015;Yin et al.,2016a).The advantage of studying the mechanical characteristics of rock under high temperature is mainly reflected in the fact that it can restore the high temperature state of the engineering site.

With the limitations of measurement techniques, research on the effect of temperature on dynamic properties of rock is mainly confined to the state after thermal treatment. There are two methods to eliminate the influence of temperature on elastic bar parameters. The first method is to design a new dynamic test system at high temperature based on the original split Hopkinson pressure bar(SHPB).For example,Nemat-Nasser and Isaacs(1997)proposed a synchronous assembly system to avoid temperature gradient effect, and Chen et al. (2015) developed a microwaveheating and automatic time-controlled SHPB to understand the effects of temperature and high strain rate on normal concrete.The second method is to retain the classical SHPB and modify the stress wave affected by temperature. For example, Xia et al. (1998) used one-dimensional (1D) stress wave propagation theory and heat transfer theory to correct the effect of temperature gradient field on waveform measurement,and Shang et al.(2010)heated the elastic bar separately and modified the waveform at high temperature by difference schemes.

Up to now,there are few studies on the dynamic characteristics of rock at elevated temperature. Liu and Xu (2013) found that the failure pattern of the sample is related to the impact velocity at the same temperature. Wang et al. (2018) discussed the coupling influences of temperature and dynamic load on the compression behavior of granite, and established the damage constitutive models based on Weibull distribution. Wong et al. (2017) investigated the Carrara marble in the four heating states under dynamic uniaxial compression. Yin et al. (2016b) stated that with the increasing temperature, there are many new cracks growing and penetrating inside the rock. Dynamic fracture toughness of Fangshan gabbro and Fangshan marble at high temperatures (100°C-330°C) was measured in laboratory test (Zhang et al., 2001). Yin et al. (2018a) analyzed the fracture initiation time and fracture initiation toughness of straight-through notch Brazilian disc(CSTBD) samples at high temperatures using the discrete element particle flow code.

It is acknowledged that the tensile strength of rock is much less than its compressive strength and the tensile failure is the most common failure mode in nature.Therefore,it is necessary to study the dynamic tensile characteristics of rock at different elevated temperatures. In this study, custom-designed furnace was used to study the effect of high temperature on the stress wave, and the transmission coefficient was obtained. Based on this waveform modification method, dynamic tension test of sandstone at high temperature was carried out. The relationship of tensile strength with loading rate and temperature was analyzed, and the failure modes of the samples were monitored by a high-speed camera.

2. Experimental apparatus

The experimental device system used in this study is composed of dynamic impact system, heating system and high-speed photography system,as shown in Fig.1.

Fig.1. Schematic diagram of experimental equipment system (In.: Incident; Re.: Reflected; Tr.: Transmitted; Ab.: Absorption; SG: Strain gauge).

2.1. Split Hopkinson pressure bar (SHPB) system

The impact device used in the experiment is an improved SHPB system designed by Li et al. (2000), which can realise slow-rising half-sine repeated loading (Zhou et al., 2011) and determine the dynamic properties of brittle rocks.The system is mainly composed of a nitrogen-driven device,spindle-shaped striker(Li et al.,2005),incident bar, transmitted bar, absorption bar, and energy trap device, as shown in Fig. 1. All bars and striker are made of highstrength 40Cr alloy steel, and the basic parameters of the bars are listed in Table 1.

Table 1 Basic parameters of the bars.

During the experiment, the spindle-shaped striker was driven by high-pressure nitrogen and hit the front end of the incident bar at high speed. The generated half-sine incident wave propagated along the incident bar to the sample and the strain gauge on the incident bar recorded an incident voltage signal.When the incident wave reached the contact area between the bar and the sample, a part of the wave was reflected due to the change in impedance.The voltage signal of the reflected wave was recorded by the strain gauge on the incident bar.The remaining wave propagated further to the transmitted bar through the sample, and the voltage signal was recorded by the strain gauges on the transmitted bar. The transmission wave was eventually absorbed by the energy capture device. The voltage signals were amplified by a dynamic strain indicator,and displayed on the oscilloscope in the form of waveform signals. Based on the 1D stress wave theory (Kolsky, 1949) and considering the loading time t, the force history at the two end surfaces of the sample and bars was obtained(Kolsky,1964;Zhang and Zhao, 2013; Yao and Xia, 2019):

where P1and P2are the forces on the contact areas between the incident bar/sample and sample/transmitted bar, respectively; Aband Ebare the cross-sectional area and elastic modulus of the bar,respectively; and εI, εRand εTare the incident, reflected and transmitted strains, respectively.

2.2. Heating system

The self-designed heating system is composed of a heating furnace and a temperature controller, as shown in Fig. 1. The maximum operating temperature of heating device is 1200°C.The high-temperature furnace mainly consists of four parts:metal shell,outer insulation layer,inner insulation layer,and furnace chamber,as plotted in Fig.2.The bar-hole is slightly larger than the diameter of the bar and connects with the furnace chamber.The length of the square furnace body is 450 mm along the direction of the bar-hole,the width is 470 mm, and the height is 400 mm. The insulation materials of the outer and inner insulation layers are asbestos fibre and perlite, respectively. The furnace chamber size is 260 mm×200 mm×140 mm(length×width×height),and the upper and lower sides of the furnace chamber are arranged symmetrically with high temperature-resistant Ni-Cr alloy coil, for a total of 12 coils.

Fig. 2. Top view of heating furnace with SHPB assembly.

In order to use a high-speed camera to capture the crack initiation mode of the sample, and guarantee the accuracy of dynamic experimental results, a 100 mm × 80 mm (length × width) visual window is arranged on the furnace door. The material of the window is high temperature-resistant thick quartz glass of sufficient strength to ensure that it will not be damaged by rock fragments during the test.

3. Stress wave modification

There are two test methods used to study the dynamic properties of materials under high temperature using SHPB. First, special experimental devices are used to limit the influence of temperature on the bar, such as using materials that are not affected by temperature to manufacture the bars,or the sample can be heated separately, and then the impact of the docking bar and sample can be completed quickly (Liu and Xu, 2013). In this case,the cold contact time (CCT) should be less than 50 ms (Apostol et al., 2003). Zhang et al. (2017) found that when the CCT is less than 10 ms, the temperature drop of the samples does not exceed 50°C. Second, the bar is heated together with the sample (Zhang et al., 2001; Wang et al., 2018). Hence, the bar parameters will change with temperature.

Although the first approach does not need to consider the influence of temperature on the bar and directly uses the data recorded by the strain gauges for calculations, it makes the experimental device complex in structure, difficult to synchronise docking and impact,and ensures a short operation time.When the CCT is long, the sample loses plenty of heat, which makes the experimental results inaccurate. Moreover, it is difficult to obtain reliable results for dynamic splitting tensile or dynamic fracture experiments (Lindholm and Yeakley,1968). In the second method,there is a temperature gradient on the bar that makes it impossible to use the data obtained directly by the strain gauges to describe the mechanical properties of rock,but the SHPB device with simple structure is retained.

3.1. Variation of bar parameters with temperature

During testing,it is necessary to guarantee that the temperature gradient generated by the heating furnace will not affect the strain gauge. The highest experimental temperature in the study is 600°C.Therefore,it should be ensured that the room temperature(25°C) is maintained around the strain gauge even when the furnace operates at 600°C. The lengths of the incident and transmitted bars are 2 m each; the strain gauges are attached to the centre of the bars; the heating furnace and furnace chamber are 450 mm and 200 mm long,respectively;and the overall thickness of the insulation layer is 125 mm.

When the Brazilian disc(BD)sample is sandwiched between the incident and transmitted bars,the lengths of the bar in the furnace chamber and in the insulation layer are 75 mm and 125 mm,respectively. The strain gauge is installed 800 mm away from the heating furnace. When the temperature reaches 600°C and the dwelling time is sufficient,the temperature in the furnace chamber is considered to be 600°C. The junction between the furnace chamber and the inner insulation layer is taken as the coordinate origin, the temperature data point on the bar is measured by thermocouple, and the temperature distribution on the incident bar is illustrated in Fig. 3. By fitting the measured data, the temperature on the bar presents a negative exponential distribution,and a functional relationship between temperature and distance with good correlation is obtained:

where T is the temperature of the bar, and x is the distance to the coordinate origin, as depicted in Fig. 3.

As can be seen from Fig.3,the temperature at the position 60 cm away from the coordinate origin is close to the room temperature.The room temperature around the strain gauge is always maintained during the test.Since the incident and transmitted bars are symmetric about the centre of the furnace chamber, the distribution curve on the transmitted bar is also obtained.

When the bar and rock are heated together,the elastic modulus of the bar changes under the influence of temperature, thus affecting the waveform.In this study,the waveguide bar is made of 40Cr alloy steel,and its elastic modulus changes with temperature(Editorial Board of Practical Manual of Engineering Materials,2002), as plotted in Fig. 4. The function of the elastic modulus with temperature is

where E is the elastic modulus of the bar.

As can be seen from Fig.4,with the increase in temperature,the elastic modulus gradually decreases, maintaining a quadratic polynomial relationship with temperature.When the temperature reaches 600°C, the elastic modulus of the bar is 165 GPa, which decreases by 21.8%compared with that at room temperature.It can be seen that the temperature has a significant influence on bar parameters. Thus it is necessary to modify the waveform.

Fig. 3. Temperature distribution on the incident bar at 600 °C.

Fig. 4. Variation of elastic modulus of the bar with temperature.

Fig. 5. Element division of the incident bar with a temperature gradient. Bi is the boundary of two neighbouring elements, and ρi is the density of the element.

3.2. Stress wave propagation and transmission coefficient

In order to modify the waveform affected by temperature,Francis (1966) studied the relationship between elastic modulus and temperature, as well as the distribution of temperature gradient on the bar.Lindholm and Yeakley(1968)adopted a unified modified formula based on Francis’s theory. The influence of temperature gradient on elastic wave was investigated in terms of continuous variation of P-wave velocity and amplitude. Since the distance between the strain gauge and the sample is equal and the temperature distribution is symmetrical, the propagation time of the wave from the sample to each strain gauge is the same,and the change in P-wave velocity does not cause a time error in the recorded signal (Lindholm and Yeakley,1968).

The incident bar at the experimental temperature of 600°C is taken as an example.As shown in Fig.5,the bar with a temperature gradient is divided into n elements. The time interval Δt of each element is consistent with the acquisition time of oscilloscope,which is set to 1 μs.Then the length of the element i is CiΔt(Ciis the P-wave velocity on the i element).When the incident wave reaches the boundary of each of two units, one part of the wave will be transmitted and another part will be reflected.

Fig.6 presents the dynamic stress equilibrium curve obtained by the empty impact (there is no sample sandwiched between the incident and transmitted bars)at 600°C in the furnace.We can see from Fig.6 that although the signal measured by the strain gauge is not modified, the equilibrium of the waveform acquired can be maintained.

The amplitude of the reflected wave in Fig. 6 is minimal compared with that of the transmitted wave, thus the reflected wave between every two neighbouring elements is ignored and only the transmitted wave is considered. At this point, the transmission coefficient λTibetween the two elements i-1 and i can be expressed as

Fig. 6. Dynamic stress equilibrium obtained by empty impact at 600 °C (without modification).

The transmission coefficient λTof the incident bar can be written as

Then the relationship between the measured waveform and the actual waveform is

where ωMand ωAare the measured waveform and the actual waveform, respectively.

It is considered that the temperature is distributed along the axial direction of the bar,and the same temperature is maintained in the radial direction of the bar. Based on the 1D stress wave theory, we have

where Cbis the P-wave velocity of the bar.

The density of the bar varies negligibly with temperature,thus it can be considered as a constant.According to Eqs.(4)and(8),as the temperature increases, the elastic modulus of the bar and the Pwave velocity decrease, and the length of the element division becomes smaller in Fig. 5.

According to Eqs.(3)-(5)and(8),the transmission coefficient λTat 600°C in Eq.(6)is 0.938.Taking incident wave as an example,the modified waveform is shown in Fig.7.It can be seen that the actual waveform amplitude under high temperature is smaller than that recorded by the strain gauge.

Similarly,the transmission coefficients at each temperature can be obtained (see Table 2). As the temperature increases, the transmission coefficient decreases, suggesting that the amplitude of reflection increases. If there is no modification, the calculated result at 600°C is about 6% larger than the real value.

4. Laboratory test

4.1. Sample preparation

The rock material used in the experiment is the fine-grained homogeneous red sandstone without visible cracks or defects sampled from Chuxiong Yi Autonomous Prefecture, Yunnan Province,China.The recommended testing method of the International Society for Rock Mechanics and Rock Engineering (ISRM) was followed (Zhou et al., 2012). All samples were extracted from the same sandstone block.The sample preparation process is shown in Fig.8.First,the core was removed from the rock block and then cut with clean water to make a pre-sample of the BD.Then the surface of the sample was polished to make the surface roughness less than 0.02 mm and the end surface perpendicular to its axis was less than 0.001 rad.Finally,the sample was made into a disc with a nominal diameter of 50 mm and a length/diameter ratio of 0.5.Fig.9 shows the surface and size of the sandstone sample after production.

The static test was conducted with Instron 1342 (capacity 2.5-250 kN) and Instron 1346 (capacity 250-2000 kN) testing machines.Some basic physico-mechanical properties of the sandstone are listed in Table 3. The mineral composition of sandstone measured by X-ray diffraction(XRD)is depicted in Fig.10.The main components are quartz (59.57%), feldspar (18.45%), calcite (8.79%),chlorite(5.62%),mica(4.57%),and a small amount of other mineral components.

Fig. 7. Incident waves before and after modification at 600 °C.

Fig. 8. Sample preparation process.

Fig.10. XRD analysis of sandstone.

Table 2 Transmission coefficients at various temperatures.

4.2. Experimental method

Dynamic tension test methods for rocks are divided into direct tension and indirect tension types (Xia and Yao, 2015). Several indirect tension test methods such as BD, bending, and spalling experiments have been widely used to overcome the shortcomings of direct tension methods. The BD experiment is commonly applied due to the advantages of simple sample preparation, universal experimental equipment, and convenient experimental protocol,and it has been recommended by the ISRM as a suggested specification for determination of the dynamic tensile strength of rock(Zhou et al., 2012). The dynamic BD test is adopted in this study.

The dynamic Brazilian splitting test evolved from the static Brazilian splitting test (Bieniawski and Hawkes, 1978). Thus,according to Eqs.(1)and(2),the average load P（t）for the two ends of the sample is

When the dynamic stress equilibrium is reached by using halfsine wave loading, the following equation is obtained:

Then, the dynamic tensile strengths calculated as (Zhou et al., 2012):

where P（t）maxis the maximum load during the test.

Fig.9. Surface and size of sandstone sample.D and B are the diameter and thickness of the sample, respectively.

4.3. Dynamic stress equilibrium

Fig.11a shows the dynamic stress equilibrium of sample before modification at 600°C. Although there is no modification, the superposition of incident and reflected stresses also reaches a balance and is coincident with the transmitted stress. The graph indicates that all the reflected waves generated at the cell boundary under the influence of temperature do not distort the whole waveform,which further indicates that the influence of reflection on waveform is negligible.Fig.11b shows the dynamic stress equilibrium of sample after modification at 600°C, which is obtained by multiplying the incident,reflected and transmitted stresses of Fig.11a by a correction coefficient of 0.938 according to Eq. (7). The only difference between Fig. 11a and b is the amplitude of the stress waveform. The data mentioned in following sections under high temperature have been modified.

4.4. Determination of loading rate and tensile strength

As rock is typically a brittle material, its yield strength and failure mode follow certain rules with respect to the increase in strain rate,which is also referred to as the rate effect(Zhao and Li,2000;Cho et al.,2003).In fact,taking the strain rate as the variable to investigate the rate effect is generally only applicable in compression tests. For fracture and splitting tests, it is generally impossible to measure the strain,but it is more appropriate to use loading rate as the variable to study the rate effect (Gong et al.,2018). Therefore, the loading rate is selected as the variable in this test(Zhou et al., 2012).

According to the measurement method of rock dynamic tensile properties recommended by ISRM, the loading rate of BD sample can be determined by the time-history curve of dynamic tensile stress(Zhou et al.,2012).When the sample reaches dynamic stress equilibrium, the change in tensile stress with time is obtained(Fig.12). It can be seen from Fig.12 that there is an obvious linear increase stage before sample failure, and the slope of this linear part can be defined as the loading rate of the sample.According to Eq. (11), the highest point of the curve corresponds to the tensile strength of the sample.In this study,the loading rate and dynamic tensile strength of all samples are determined by this method.

5. Test results and analysis

In this test, the BD samples were divided into seven groups based on the impact performed at 25°C (room temperature),100°C,200°C,300°C,400°C,500°C and 600°C,respectively,and there were no fewer than six samples in each group. The heating rate was 10°C/min.When the temperature in the furnace reached the assigned value, it was kept constant for 1 h to ensure that the whole sample attained the same temperature field. The real-time heating status of the sample is depicted in Fig.1. When the holding time was sufficient, the dynamic splitting test was carried out under high temperature. After impact, the furnace door was opened to start cooling in the air. The sample fragments were collected and prepared for the next test after the temperature in the furnace chamber dropped to room temperature.

5.1. Dynamic tensile strength of sandstone subjected to high temperature

After verifying dynamic stress equilibrium and crack initiation for each sample,the experimental data of sandstone samples were obtained.During the test,some samples did not reach equilibrium or cracked at the centre, thus these data were discarded. Finally,dynamic tensile strength data for 45 samples under high temperature were obtained and are tabulated in Table 4.

Table 3 Basic physico-mechanical properties of sandstone.

Table 4 Parameters and experimental results of sandstone samples.

Fig.13 shows the relationship between dynamic tensile strength and loading rate of sandstone at different temperatures. One can see that the loading rate-dependence of sandstone exists at high temperature, i.e. for the loading rate ranging from 140 GPa/s to 400 GPa/s, the dynamic tensile strength increases approximately linearly with the increase in loading rate for each temperature group. Therefore, it must be noted that even under high temperatures,the strength of sandstone at high loading rate is greater than that at low one. In addition, from the data at 25°C, it can be seen that the dynamic tensile strength of the sample at the minimum loading rate of 144 GPa/s is 8.57 MPa,which is 2.23 times the static tensile strength of 3.84 MPa, indicating that the dynamic tensile strength of the sample at 25°C is far greater than the static tensile strength. The dynamic tensile strength of the sample at high temperature as a function of loading rate is consistent with the dynamic compressive strength (Liu and Xu, 2015; Wang et al., 2018)and dynamic fracture toughness (Zhang et al., 2001; Yin et al.,2018a), which indicates that the effect of loading rate on rock materials will not change under high temperatures.

Since the dynamic tensile strength has a linear relationship with the loading rate in each temperature group, the following formula can be introduced:

In addition to considering the effect of loading rate on dynamic tensile strength, temperature is also a key factor that cannot be ignored. Data points for the relationship between dynamic tensile strength and temperature with roughly the same loading rate(such as 200 GPa/s, 260 GPa/s and 300 GPa/s) are found from each temperature group. These comparable data points are connected, as shown in Fig.14.

Fig.14. Dynamic tensile strengths of sandstone at various temperatures.

Combining Figs.13 and 14,it can be seen that with the increase in temperature, the dynamic tensile strength of the sample generally shows a trend of first increasing and then decreasing.From 25°C to 200°C, the dynamic tensile strength keeps increasing. However, the dynamic tensile strength at 300°C is basically unchanged from that at 200°C. When the temperature reaches 400°C, the dynamic tensile strength of the sample is reduced compared with that at 300°C. Nevertheless, the dynamic tensile strength of the sample does not always decrease, for example,the strengths at 400°C and 500°C are roughly the same.The strengths at 400°C-500°C are still greater than that at 25°C.The dynamic tensile strength of the sample decreases significantly when the temperature increases from 500°C to 600°C,which is the most obvious stage at the constant temperature gradient.It can be seen from Fig.13 that the tensile strength at 600°C is smaller than that at 25°C.

5.2. Determination of failure mode of BD samples under high temperature

The dynamic BD test is effective only when the sample cracks first near the centre of the disc along the loading direction (Dai et al., 2010; Xia and Yao, 2015). In order to verify the accuracy of the results for each sample, a high-speed camera was used to monitor the initiation and propagation of cracks.As shown in Fig.1,the camera was connected to and triggered by the oscilloscope,and it could record the destruction of the sample during impact. The camera was set to 72,000 frames per second (FPS) and the resolution was 256 × 240 pixels.

As shown in Fig.15, effective dynamic tensile failure modes of sandstone at high temperature can be divided into two types:type-I (Fig.15a) and type-II (Fig.15b). In Type-I failure mode, a tensile crack is generated at disc sample centre,and the sample along the centre line is fragmented into two large flaps and small triangular shear zone that break around the bar/sample contact areas. In Type-II failure mode,the secondary cracks form at the two ends of the sample and eventually extend to the disc centre, and finally coalesce to form a crushing zone. The triangular shear area consisting of X-shaped secondary cracks increases. The sample is fragmented into several small blocks.

Seven samples with a loading rate of around 200 GPa/s were tested to understand the influence of temperature on the failure mode.Fig.16a shows the failure mode of these seven samples failed at different temperatures.One can see that when the loading rate is low, the disc failure mode is type-I, and only the size of the two edge wedges is different.Fig.16b depicts the shapes of each sample broken at 600°C after splicing. At the same temperature, failure mode transforms from type-I to type-II with the increase in loading rate.More attention is paid to the last sample U66 in Fig.16b,where an area marked by a white dotted line and labelled “exfoliation”,because this debris, unlike other fragments that span the entire thickness of the sample, peels off directly from the surface of the disc.The results show that the loading rate plays a dominant role in the tensile failure mode of the sample compared with the effect of temperature.

Fig.11. Dynamic stress equilibrium diagrams of sample: (a) Before and (b) After modification.

Fig.12. History of dynamic tensile stress of sample. ˙σ is the loading rate.

5.3. Experimental phenomena and causes

As mentioned previously in Section 5.1, the dynamic tensile strength of sandstone changes with temperature, which is quite different from previous studies of rock after exposure to high temperature.For example,the dynamic tensile strength of Longyou sandstone(LS)decreased with the increase in temperature(except at 450°C) after heat treatment (Yao et al., 2016). In addition, we conducted dynamic BD tests on Laurentian granite (LG) after thermal treatment, and the results showed that except for 100°C,the dynamic tensile strength at other temperatures was lower than that at room temperature (Yin et al., 2015). A comparison of the results of Yin et al. (2015) and Yao et al. (2016) after thermal treatment with this experiment is plotted in Fig.17, in which the tensile strength is normalised. According to Fig. 17, although the rocks are different,the dynamic tensile strength of LS and LG after thermal treatment tends to change with temperature,and basically decreases with increasing temperature.

Furthermore, as depicted in Fig.18a, the dynamic compressive strength of sandstone under high temperature is greater than that at room temperature except for 600°C,and the overall compressive strength first increases and then decreases (Liu and Xu, 2015).Fig.18b shows a comparison of the results of this study with Liu and Xu(2015)after the strength is normalised.It can be seen from the figure that the strength variation trend of the two studies is basically the same.The results of this study show that variations in the mechanical properties of rocks with increasing temperature are indeed different from those after heat treatment.

Fig. 13. Variations of dynamic tensile strength of sandstone with loading rate at different temperatures.

The dynamic tensile strength of sandstone at 100°C,200°C and 300°C is larger than that at room temperature in this study. The main reasons are the evaporation of free water and decomposition of mineral combination water,and the closure of microcracks after thermal expansion of minerals. At this time, temperature plays an enhanced role. However, at 400°C and 500°C, due to different thermal expansion coefficients of minerals, the rock strength decreases, indicating that the enhancement effect of temperature gradually fades,and the deterioration effect of temperature on the sample dominates. Because of the high quartz content (59.57%) in the sandstone used in this study,the transformation of quartz from α phase to β phase occurs at 573°C (Glover et al., 1995; Nasseri et al., 2007), and new microcracks are generated, leading to a significant decrease in tensile strength at 600°C.

Fig.15. Two typical effective failure modes of sandstone under high temperature:(a)Type-I,and(b)Type-II.The incident bar is on the left and the transmitted bar is on the right.Note that the time in the selected picture is relative to crack initiation and development, and not to the time zero of the experiment.

It is worth noting that within the high temperature range(100°C-600°C) used in this study, except 600°C, the dynamic tensile strength of samples is greater than that at room temperature. Because the dynamic tensile strength of samples at 600°C is distinctly lower than that at 25°C, we consider 600°C as the threshold temperature.

It is well known that the mineral composition of sandstone determines its colour(Pettijohn et al.,1972). The sandstone sample in this study has a reddish surface at room temperature (Fig. 9). As shown in Fig.16a, the colour gradually darkens as the temperature rises from 25°C to 300°C. This may be due to the chlorite in the sample which turns brownish at 300°C (Hajpál and T?r?k, 2004).However, when the temperature exceeds 300°C, the colour of sandstone turns from dark to bright. At 600°C,it appears brick red with yellowish, and is brighter than the colour of sample at room temperature. There are three possible reasons: (1) When the temperature exceeds 300°C,chlorite will be decomposed by heat(Hajpál and T?r?k,2004);(2)Ferrous ions containing hydroxyls in sandstone turns into ferric ions at temperatures above 300°C(Chakrabarti et al.,1996;Hajpál and T?r?k,2004;Dong et al.,2019);and(3)The ferromagnetic substance (Fe3O4) will react with the oxygen in the air to generate hematite(Fe2O3),which turns the sandstone into red(Sun et al., 2016b; Zhu et al., 2016; Dong et al., 2019). Moreover, the colour variance caused by this chemical reaction is irreversible and cannot be changed due to cooling or other factors.

Fig. 17. Comparison of variations in tensile strength of rocks as a function of temperature. σT and σ25 are the tensile strengths of rock at temperature T and 25 °C,respectively.

Fig. 18. (a) Dynamic compressive strength of sandstone (Liu and Xu, 2015); and (b)Comparison of compression with tension under high temperature.

Using thismethod of waveformmodificationto study the dynamic properties of rocks at high temperature can retain the classical SHPB and has universal applicability, but the experimental temperature should not be too high. The elastic modulus of 40Cr alloy steel will decrease from 211 GPa at room temperature to 165 GPa at 600°C,indicating that the strength of elastic bar decreases and the deformation will be easier.Moreover,when the experimental temperature is too high,the assumption of ignoring reflected wave in Section 3.2 is invalid.Therefore,it is not recommended to test the dynamic properties of hard materials at very high temperatures.

Fig.16. Recovered BD samples: (a) Failure mode of samples at various temperatures; and (b) Failure mode of samples at 600 °C.

6. Conclusions

In this study,a waveform modification method was proposed by using a SHPB system and self-designed heating device. The dynamic tensile strength and failure mode of sandstone at high temperature were preliminarily explored based on this method.The main conclusions are drawn as follows:

(1) The bar is divided into several elements to analyse the influence of temperature on waveform.Based on the 1D stress wave theory, the transmission coefficients corresponding to each of six temperature groups from 100°C to 600°C are 0.996, 0.989, 0.98, 0.969, 0.955 and 0.938, respectively,which means that without waveform modification at 600°C,the calculated result would be 6%larger than the real value.

(2) The loading rate-dependence of sandstone still exists at high temperature, and with the increase in loading rate, the dynamic tensile strength increases approximately linearly.Meanwhile, the tensile strength first increases and then decreases with a rise in temperature. Except at 600°C, the strength at other temperatures is higher than that at room temperature.Therefore,600°C is considered as the threshold temperature. The variation in rock strength under high temperature is indeed different from those after thermal treatment.

(3) The effective tensile failure modes of BD sample under high temperature can be divided into two types. The rock failure mode is basically unaffected by temperature,but the increase in loading rate causes type-I to type-II transformation.Due to the presence of chlorite and iron minerals in sandstone,with the increase in temperature, the sandstone sample turns from reddish to dark and then to bright,and it appears brick red with yellowish at 600°C.

Declaration of competing interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication, and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China(Grant Nos.41972283 and 51774325).