Peiyang Yu, Peng-Zhi Pan,c,*, Guangliang Feng, Zhenhua Wu, Shankun Zhao

a State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences, Wuhan,430071,China

b University of Chinese Academy of Sciences, Beijing,100049, China

c Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang,110819, China

d State Key Laboratory of Coal Resources High Efficient Mining and Clean Utilization, China Coal Research Institute, Beijing,100013, China

ABSTRACT Understanding rock mechanical behaviors after thermal shock is critically important for practical engineering application. In this context, physico-mechanical properties of Beishan granite, Gansu Province,China after cyclic thermal shock were studied using digital image correlation (DIC), acoustic emission(AE) monitoring, and microscopic observation. The results show that the peak strength and elastic modulus decreased gradually with increase in thermal shock cycle. However, the above two parameters showed no further changes after 10 thermal shock cycles. The loading stress ratio (i.e. the ratio of the current loading stress level to the peak stress in this state)corresponding to the occurrence of the uneven principal strain field and the local strain concentration zone on the surface of the granite specimen decreased with increase in thermal shock cycle. Three transformation forms of the standard deviation curves of the surface principal strain were found. For granite with fewer thermal shock cycles (e.g. no more than 2 cycles), the standard deviation curves exhibited approximately exponential growth in exponential form. With increase in thermal shock cycle, the S-shaped curve was dominant. After 10 thermal shock cycles, an approximate ladder-shaped curve was observed.It is displayed that AE activity was mainly concentrated around the peak strength zone of the granite specimen when the rock samples underwent fewer thermal shock cycles. With increase in thermal shock cycle, AE activity could occur at low loading stress levels.Microscopic observation further confirmed these scenarios,which showed that more microcracks were induced with increase in thermal shock cycle. The number of induced microcracks at the edge location of the granite specimen was significantly larger than that at the interior location. Finally, a continuum damage model was proposed to describe the damage evolution of the granite specimen after cyclic thermal shock during loading.

Keywords:Beishan granite Cyclic thermal shock Digital image correlation (DIC)Acoustic emission (AE)Physico-mechanical properties Continuum damage model

1. Introduction

For rock mass engineering projects in areas where the environmental temperature changes quickly, such as in geothermal energy extraction,deep mining of coal resources and underground storage of high-level nuclear waste, cyclic thermal shock is an important temperature indicator affecting rock mechanical behaviors. In the drilling process of geothermal energy extraction,surrounding rocks at a temperature above 200°C continuously encounter cooling effects of drilling fluid and water injection. The temperature of the surrounding rocks in case of underground fires can reach 1000°C,and the rock mass will be cooled by continuous firefighting cool water when combating the fire disasters (Hajpál,2002). This also will cause the surrounding rocks to undergo cyclic thermal shock. Due to the temperature gradient generated in the high-temperature surrounding rocks when it is cooled down,thermal stress inside rocks is generated and it will cause crack initiation and expansion inside the rock and finally affect the physico-mechanical properties of the rocks (Zhang et al., 2017).Therefore, the physico-mechanical properties of the rocks after cyclic thermal shock will change significantly compared with that of intact rocks. A correct understanding of the impact of cyclic thermal shock on rock properties is of great significance for longterm stability of rock mass engineering.

Many experiments have been carried out to investigate the effects of thermal shock on the physico-mechanical properties of rock(Freire-Lista et al., 2016; Wang et al., 2017; Rossi et al., 2018; Zhu et al.,2018;Han et al.,2019).The physico-mechanical properties of rocks after thermal shock include permeability (Lima et al., 2019),P-wave velocity, elastic modulus, static compressive strength(Freire-Lista et al., 2016; Yin et al., 2016; Su et al., 2017a; Jin et al.,2018; Rossi et al., 2018; Zhang et al., 2018a; Zhu et al., 2018; Han et al., 2019) and dynamic compressive strength (Wang et al.,2016, 2017). These parameters usually present a decreasing trend with increasing applied temperature.Rock specimens are generally subjected to two thermal shock treatments, i.e. slow cooling after heating and rapid cooling after heating.The first scenario refers to as the natural cooling method. After heating, specimens are generally exposed to an open area in a form of slow cooling.In the second scenario,the cooling method mainly includes water cooling and rapid air cooling. It reveals that rapid cooling has a greater impact on rock performance than slow cooling (Kim et al., 2014;Shao et al., 2014; Eren Sar?c?, 2016). However, most of the above researches are mainly focused on the physico-mechanical properties of rock after thermal shocks at different temperatures. The performances of rocks after cyclic thermal shock treatment under the same heating temperature conditions are not maturely understood studied yet, which needs to be further investigated.

With the advancement of experimental monitoring technology,it facilitates our understanding of rock failure process gradually from macroscopic to microscopic. The digital image correlation(DIC) technology has made it possible to continuously track the entire process of rock failure (Peters and Ranson, 1982). The DIC method is a non-contact, high-precision full-field observation technology based on digital image technology.Its basic principle is to track the geometrical points of random digitalized speckle images on the surface between the undeformed and deformed states to determine the deformations of an object, thus achieving continuous track of the entire process of rock failure. At present,this technology has been widely used in rock experiments. Many studies have used DIC technology to explore evolution of surface deformation fields (Zhang et al., 2013; Munoz et al., 2016; Tang et al., 2019), expansion of pre-existing cracks (Nguyen et al.,2011; Li et al., 2017; Miao et al., 2018), and fracture process (Lin and Labuz, 2013; Lin et al., 2014; Zhang et al., 2015, 2018b; Ji et al., 2016; Su et al., 2017b; Miao et al., 2020) of rocks during compression.

In this study,the effects of cyclic thermal shock on the physicomechanical properties of granite were studied experimentally.The uniaxial compression tests were carried out on granite specimens after cyclic thermal shock treatment under the same heating temperature conditions,and the evolution of the deformation field on the granite surface and the internal fractures were tracked in real time by DIC and acoustic emission (AE) monitoring systems during the entire test process.Meanwhile,the microstructure of the granite specimens after cyclic thermal shock was observed microscopically with thin sections of the specimen.A continuum damage model was finally introduced to describe the damage evolution of granite after cyclic thermal shock during loading.

2. Experimental programme

2.1. Specimen preparation

The rock used was granite collected from the Jijicao quarry of Beishan located in Gansu Province, China.This granite is mediumfine grained with grain size of approximately 0.2—4 mm, and is mainly composed of quartz (35%), feldspar (55%—65%) and a small amount of mica. The specimens were cut and polished to form rectangles with a length of 50 mm,a width of 50 mm and a height of 100 mm, which had a length deviation of less than 0.5 mm in three directions.

To ensure the consistency of the experimental specimens,all the specimens were taken from a granite block with the dimensions of 1.2 m × 0.5 m × 0.6 m, as shown in Fig.1. This granite block was sampled by non-drill-and-blast method (i.e. drill holes around the rock block to be taken, then cut shallow grooves along the periphery, and finally add proppant into the holes to make the rock blocks crack along the direction of shallow grooves,thus separating the rock block from the rock mass),thus the damage to the rock is slight and its relative integrity is good. Weighing and ultrasonic testing were used to remove discrete fragments. The granite specimens after screening have a density of 2.601—2.620 g/cm3and a P-wave velocity of 2.714—2.906 km/s, with the average values of 2.611 g/cm3and 2.802 km/s, respectively. The scheme for testing the specimens for this study is shown in Table 1.

2.2. Procedure of cyclic thermal shock tests

It shows that the physico-mechanical properties of Beishan granite will change obviously when the temperature exceeds 200°C (Chen et al., 2017a). However, when the temperature of granite reaches 573°C, the internal quartz composition will undergo a phase transition, which will increase its volume by more than 5%, leading to an obvious increase in the number of internal cracks (Xi,1994; Chen et al., 2017b). Based on the above two considerations,to understand the influence of cyclic thermal shock on rocks and avoid the impact of excessive temperature, the temperature selected in this study is assumed 300°C,which is used as the upper limit temperature of cyclic thermal shock.

The specimens were first heated to a prescribed temperature at a heating rate of 3°C/min during the process of thermal shock and then rapidly water-cooled with different cycles,as shown in Fig.2.The heating device was using a box-type resistance furnace QSH-1400M-2030T, which is composed of a control box and a furnace.Its maximum operating temperature is 1400°C, with generation power of 6 kW. When the target temperature reached, the temperature was kept constant for 2 h to ensure temperature uniformity inside rocks (Jin et al., 2018). Then, the specimens were removed from the furnace and immersed in a container filled with tap water for at least 1 h. During the process of immersion, the water was constantly stirred to ensure that the specimens could be quickly cooled down to room temperature of(8—11)°C.The granite specimens were heated to the prescribed temperature and then rapidly cooled down to room temperature of(8—11)°C,which was regarded as one thermal shock. In this experiment, the granite specimens were treated with 0(i.e.no thermal treatment),1,2,4,6,8,10,14 and 20 cycles of thermal shock,respectively.The number of rock specimens contained in each type of thermal shock treatment is shown in Table 1.

Fig.1. Beishan granite block.

Table 1Testing scheme foe the specimens prepared in this study.

After cyclic thermal shock,the specimens showed differences in water saturation(Kim et al.,2014).To eliminate the influence of this factor on the subsequent experimental results, the above specimens were placed in a dry oven at 105°C for 48 h to obtain dried specimens.

2.3. Experimental set-up

Fig. 2. Cyclic thermal shock treatment.

Fig. 3. Schematic diagram of the testing system.

The experimental facility is composed of loading system, AE monitoring system and digital image acquisition system,as shown in Fig.3.The loading system is the servo-controlled rock mechanics testing system RMT-150C with a maximum loading capacity of 1000 kN and a maximum vertical stroke of 50 mm, whose force sensor accuracy and stroke sensor accuracy are 5‰ and 3‰ of full scale, respectively. An axial linear variable differential transformer(LVDT) was used to measure axial deformation during loading process. Axial displacement-controlled loading was used to apply axial stress incrementally at a rate of 0.002 mm/s (Jin et al., 2018;Miao et al., 2018). AE activities were recorded using the AE detection function of the Sensor Highway II series manufactured by American Physical Acoustic Corporation, sampling accuracy of which is 18 bits.The pre-amplification of the AE signals was set to 40 dB, and the threshold level for AE recording was set to 45 dB(Miao et al.,2018).DIC technology was used to capture the variation of the in-plane deformation field of the specimen surface in real time during loading process. The digital image acquisition system contains a CCD camera with 3376 × 2704 effective square pixels,two digital white lights and a computer with image acquisition software(Ji et al.,2016;Miao et al.,2018).The computer-controlled image acquisition was five frames per second.

3. Results

3.1. Mechanical properties

3.1.1. Stress—strain relationship

Fig.4 presents the axial stress—strain relationship of the granite specimens after cyclic thermal shock.The results reveal that cyclic thermal shock has a significant influence on the stress—strain relations. With the increase in thermal shock cycle, the slope of the stress—strain curves at the initial deformation stage of the granite specimen decreased gradually (i.e. the crack compaction stage of the specimens during loading became more and more obvious).The reason is that cyclic thermal shock causes more microcracks inside the granite,and these microcracks gradually close under the action of external force at the early stage of loading, resulting in greater irreversible deformation at the stage of crack compaction(Rong et al., 2018).

Fig.4. Stress—strain relationship of the granite specimens after cyclic thermal shock(0 cycle(RD-0-3),1 cycle(RD-1-3),2cycles(RD-2-3),4 cycles(RD-4-3),6 cycles(RD-6-3),8 cycles (RD-8-3),10 cycles (RD-10-3),14cycles (RD-14-3), and 20 cycles (RD-20-3)).

Cyclic thermal shock also affects the post-peak stress—strain behavior of the granite. When the granite underwent less cyclic thermal shock treatment, the post-peak stress—strain relationship exhibited brittle behavior(i.e.the strength decreases quickly when the peak stress is reached). With the increase in thermal shock cycle,the post-peak stress—strain curves showed a certain ductility behavior.

3.1.2. Strength and deformation behavior

Fig. 5 presents the variations of the peak strength, peak strain,and elastic modulus of the granite specimens with the number of cyclic thermal shocks. The peak strength of the granite specimens generally decreased with increase in thermal shock cycle. After 4 thermal shock cycles, the peak strength of the granite specimen decreased by 13.26% compared with those of specimens without thermal shock. The peak strength of the granite specimen decreased by 18.48%after 10 thermal shock cycles and decreased by 18.98% after 20 thermal shock cycles.

With the increase in thermal shock cycle,the peak strain of the granite specimens gradually increased, and the elastic modulus gradually decreased. After 4 thermal shock cycles, the peak strain and elastic modulus of the granite specimen were 1.18 times and 0.82 times greater than those of specimens without thermal shock,respectively. The peak strain and elastic modulus were 1.24 times and 0.71 times greater than that after 10 thermal shock cycles and 1.26 times and 0.68 times greater than that after 20 thermal shock cycles, respectively.

Based on the above analysis,it suggests that the peak strength,peak strain and elastic modulus of the granite specimens show no obvious changes after 10 thermal shock cycles. It can also be seen that the specimen damage caused by cyclic thermal shock is limited a certain degree. Above this limit, cyclic thermal shock will not significantly impact the mechanical properties of the rock.

3.2. Deformation evolution on the surface of granite and AE characteristics

3.2.1. Principal strainfield on the surface of granite

The surface deformation field of granite specimens after cyclic thermal shock was monitored during the uniaxial compression process.It is difficult to capture the evolution of the deformation field at the post-peak stage due to the speckle stripping on the rock surface.Therefore,this paper only explored the evolution of the deformation field on the surface of granite at the pre-peak stage.Fig.6 presents the contour map of the principal strain field on the surface of the granite specimens at the pre-peak and peak stages after 0, 4, 10 and 20 thermal shock cycles. In addition, to discuss the effects of different stress levels on the surface strain evolution after cyclic thermal shock,the loading stress ratio (LR) (Kao et al., 2016) (i.e. the ratio of the current loading stress level to the peak stress in this state) is used rather than stress value to describe the strain field distribution on the surface of granite under different loading stress levels.

Fig. 5. Variations of the peak strength, peak strain and elastic modulus with the number of cyclic thermal shocks for the granite specimens.

Fig.6a presents the evolution of the principal strain field on the surface of the granite specimen without thermal shock under different loading stress levels.The distribution of principal strain on the granite surface was relatively uniform at the initial stage of loading.When increasing to 70%of the peak strength,the principal strain field on the granite surface was unevenly distributed.When stress reached 80% of the peak strength, the local strain concentration zone appeared in the middle and lower parts of the granite surface, indicating microcrack nucleation on the surface of the granite specimen. Then, the surface strain increased continuously with increasing loading stress, and its distribution was gradually concentrated on the local strain zone. The local strain zone was continuously extended and interconnected,and showed a tendency to develop towards both ends of the specimen, indicating that the surface microcracks propagated continuously after nucleation with the increment of the loading stress.When the stress reached 95%of the peak strength, the surface of the specimen gradually formed a strain zone extending throughout the entire plane, and became more significant when the loading stress reached the peak strength. A similar phenomenon of principal strain field evolution can also be found in granite after 4,10 and 20 thermal shock cycles during loading process (Fig. 6b—d). Meanwhile, this phenomenon can be found in published literature (Song et al., 2013).

Fig.6b—d presents the evolution of the principal strain field on the surface of the granite specimens after 4,10 and 20 thermal shock cycles.It can be observed that the uneven distribution of the principal strain field and local strain concentration zone on the surface of the granite specimen after 10 and 20 thermal shock cycles appeared earlier compared with that after 4 thermal shock cycles.

Table 2 presents the stress levels corresponding to the occurrence of the uneven principal strain field and local strain concentration zone on the surface of the granite specimens after cyclic thermal shock.It shows that the stress levels corresponding to the uneven principal strain field and local strain concentration zone decreased gradually as the number of thermal shocks increased and no obvious changes were observed after 10 thermal shock cycles.Using loading stress ratio to reflect the occurrence of the uneven principal strain field and local strain concentration zone can indirectly reflect cyclic thermal shock accelerating the formation of uneven principal strain field and local strain concentration zone.

3.2.2. Degree of dispersion of the surface principal strain and AE count

The differences in the principal strain distribution on the rock surface were caused by rock surface damage (Song et al., 2013). In this context,the principal strain at any time during the rock loading process was extracted,and the

standard deviationSof the principal strain at that time was used to measure the degree of principal strain dispersion on the granite specimen surface, which also reflected the damage to the rock surface.

The standard deviationSof the principal strain field at a certain moment can be expressed as (Song et al., 2013):

whereXkis the principal strain at pointk;Nis the total number of data points in field;andis the average value ofXk,which can be expressed as

Fig.6. Contour map of the principal strain field on the surface of the granite specimens at the pre-peak and peak stages after cyclic thermal shock:(a)0 cycle(RD-0-3);(b)4 cycles(RD-4-3); (c) 10 cycles (RD-10-3); and (d) 20 cycles (RD-20-3).

Table 2Loading stress ratio corresponding to the occurrence of the uneven principal strain field and the local strain concentration zone.

Table 3Stress level corresponding to the turning points of the standard deviation curve of the surface principal strain.

The degree of dispersion of the surface principal strain and AE count are shown in Fig.7.It can be observed that as the number of cyclic thermal shocks increased,the form of the standard deviation curve of the surface principal strain became more diverse,and the AE activity became more drastic. Upon few cyclic thermal shocks,the standard deviation curve of the surface principal strain of the granite specimen exhibited approximately exponential growth with the strain. With further increase in thermal shock cycle, the standard deviation curve presented an S-shaped curve approximately(i.e.the rate of the standard deviation curve showed a trend of first increasing and then decreasing).When the granite specimen were treated with more thermal shock cycles, the standard deviation curve exhibited an increasing tendency of approximate laddershaped, and this phenomenon became more obvious with increasing thermal shock cycles. According to the variation between the slope of the standard deviation curve of the surface principal strain and the AE count, the standard deviation curve of the surface principal strain of the granite specimens after cyclic thermal shock can be divided into various stages. The turning points of each stage are denoted asA,B,CandD, and the corresponding stress levels are recorded in Table 2.

(1) The first stage(O-A)is the initial stage of rock loading and is accompanied by original crack compaction,and the principal strain field was evenly distributed(Fig.6).It can be seen that the compaction of the original crack has little impact on the lateral deformation of the rock. At this stage, the standard deviation of the principal strain first grew slowly and then changed slightly, showing a tendency becoming almost parallel to the strain axis. The AE count was almost none,indicating that there were few microcracks at this stage.

(2) The second stage (A-B) is the stage of stable crack propagation in the rock, and the principal strain field appeared unevenly(Fig.6).The standard deviation of the principal strain increased with the strain at an approximately constant rate in this stage. Meanwhile, the AE count appeared in a relatively stable trend.

(3) In the third stage (B-peak orB—C), a local strain concentration zone appeared in the middle and lower parts of the rock surface. With increase in loading stress, the principal strain increased continuously, and its distribution gradually concentrated on the local strain zones.The local strain zones continuously developed and intersected each other, and finally a “vertical”strain concentration zone appeared on the rock surface (Fig. 6). The standard deviation of the principal strain increased significantly in this stage and presented two different forms relative to strain. When the number of thermal shocks did not exceed 4, the standard deviation of the principal strain presented approximately exponential growth.When the number of thermal shocks exceeded 4,the standard deviation curve of the surface principal strain presented an approximately logarithmic form(i.e.the rate of the standard deviation curve showed a trend of increasing first and then decreasing).The number of AE counts increased at this stage, indicating that AE activity became more significant.

(4) When the number of thermal shocks exceeded 8, the standard deviation curve contained the fourth and fifth stages in addition to the above three stages.In the fourth stage(C-D),with the increase in loading stress, a “platform segment”appeared on the standard deviation curve, i.e. the principal strain changed slightly at this stage.Meanwhile,the AE count was relatively reduced,indicating that the AE activity was at a weakened state, and the rate of microcrack generation inside the rock was reduced. At the fifth stage (D-peak), the“vertical” strain concentration zone became more obvious,and other local strain concentration zones appeared around it(Fig.6).The standard deviation of the principal strain again presented approximately exponential growth with the increase in strain, and the AE activity became more drastic.

(5) With the increase in thermal shock cycle, the standard deviation of the surface principal strain at the peak stage presented an overall increasing trend, and the AE activity became more drastic, indicating that cyclic thermal shock causes increasing damage to the rocks, which makes the uniformity worse.

(6) Fig. 8 presents the relation between the stress level of the principal strain field (Table 2) and the stress level corresponding to the turning points of the standard deviation curve(Table 3).It can be observed that the variation trends of the above two stress levels with the number of cyclic thermal shocks were roughly the same, but the former was higher than that of the latter. The reason is likely that the uneven strain field and strain concentration zone were observed artificially,while the turning points were determined by the slope of the standard deviation curve and the AE count,thus the latter was more accurate than the former. Therefore, it can be considered that the stress level of the turning pointAwas the stress level when the uneven strain field occured,and the stress level of the turning pointBwas the stress level when the local strain concentration zone appeared.

Fig.7. Standard deviation curve of the surface principal strain and AE count of the granite specimen after cyclic thermal shock during loading:(a)0 cycle(RD-0-3);(b)1 cycle(RD-1-3); (c) 2 cycles (RD-2-3); (d) 4 cycles (RD-4-3); (e) 6 cycles (RD-6-3); (f) 8 cycles (RD-8-3); (g) 10 cycles (RD-10-3); (h) 14 cycles (RD-14-3); and (i) 20 cycles (RD-20-3).

Fig.8. Relationship between the stress level of the principal strain field and the stress level corresponding to the turning points of the standard deviation curve.

3.3. Microscopic characteristics

In order to understand the effect of cyclic thermal shock on the granite microstructures, optical microscopic thin sections are directly observed in the granite specimen after various thermal shock treatments.The microscopic test herein was divided into two parts. The first part was the microscopic characteristics on the surface of the granite specimen after cyclic thermal shock prior to the mechanical tests, and the second part was the microstructure characteristics of the edge and interior of the cross-section of the granite specimen before the mechanical tests. The extraction position of examined thin sections is shown in Fig.9,in whichA—Ais the cross-section of the granite specimen.

Fig. 10 presents the surface microstructures of the granite specimens subjected to 0, 1, 4, 10 and 20 thermal shock cycles.The results reveal that cyclic thermal shock can affect rock mineral grains with respect to the microcrack behaviors. More thermal-shock microcracks were induced with increase in thermal shock cycle. When the granite specimen experienced no or fewer thermal shock treatments, very few microcracks were observed, and the grains were well bonded with each other.Thermal shock-induced microcracks were distributed along the grain boundaries and extend into the interior of the grain in the granite specimen subjected to 4 thermal shock cycles. The length of the microcracks increased with the increase in thermal shock cycle, gradually forming microcrack networks surrounding the grains. When the number of thermal shocks exceeded 10, a large number of intragranular cracks and some transgranular cracks could be observed inside the grains,suggesting that the degree of development of intragranular cracks can be used as an important index to evaluate the damage induced by cyclic thermal shock(Rong et al., 2018).

Fig. 9. Microscopic observation of thin sections of the granite specimens.

The thin sections on the edge and interior of the cross-section of the granite specimen after 14 thermal shock cycles prior to the mechanical tests are shown in Fig. 11. The results reveal that the microstructure of the edge and interior of the specimens presented different characteristics prior to the mechanical tests.As shown in Fig.11a and b, a large number of grain boundary cracks and intragranular cracks were observed at the edge of the granite specimen,while the microcracks inside the rock were relatively small. It can be seen that the effect of cyclic thermal shock on the edge microstructure of the granite specimen is greater than that on the internal microstructure.

3.4. Damage model for granite after cyclic thermal shock

3.4.1. Damage model

To quantitatively describe the damage evolution of the pre-peak stage of granite after cyclic thermal shock during uniaxial loading,the uniaxial continuum damage model can be expressed as (Kao et al., 2016):

where σ is the axial stress, ε is the axial strain, and φ is the Kasyanov’s continuity parameter. The damage parameterDis an internal variable, which is defined by the continuity parameter:

The axial stress acting on this model can be considered to act on a composite material with two phases that are connected in series with the same axial strain,and the sum of the partial stresses acting on the two phases is the total stress(Kao et al., 2016):

wherea,bandmare the material parameters.Eq.(6)represents the first phase,which is equivalent to a non-damaged component that presents a linearly elastic stress—strain relation. Eq. (7) represents the second phase which is equivalent to a damaged component.This damaged component is used to capture the damage process during loading and follows a nonlinearly irreversible damage law,as described by the continuity function in Eq. (8).

Fig. 7 shows that no uneven principal strain or AE count occurred until the loading stress ratio reached stress level of turning pointA. Thus, it can be conclusive that the damage parameter starts when the axial strain reaches a critical strain εAat turning pointA. The continuity function in Eq. (8) and damage evolution function in Eq. (4) can be rewritten as

where continuity parameter φ = 1 means that the material has no damage when ε ≤ εA, and the damage starts during loading when ε ≥ εA.

Fig.10. Thin sections of the surface of the granite specimen after cyclic thermal shock:(a)0 cycle(RD-0-4);(b)1 cycle(RD-1-4);(c)4 cycles(RD-4-4);(d)10 cycles(RD-10-4);and(e) 20 cycles (RD-20-4).1 - Grain boundary cracks, 2 - Intragranular cracks, Qtz - Quartz, Kfs - K-feldspar, Pl - Plagioclase, Bt - Biotite, Mu - Muscovite.

The parameters of uniaxial continuum damage model are determined in two steps:(1)When the strain is less than the critical strain, the rock is in the elastic stage. The least squares method is adopted to fit the stress—strain curve of the rock, and the parametersaandbare obtained; and (2) When the strain reaches the critical value, the rock is damaged. At this point, the least squares method with known parameters (aandb) is adopted to fit the stress—strain curve to obtain the parameterm. Fig. 7 presents an overall good match between the continuum damage model and the axial stress—strain relation.The fitted parameters are presented in Table 4.It can be observed that the parametersa,bandmpresent a downward trend globally with increase in thermal shock cycle because of the increased initial damage to the rock after cyclic thermal shock.

3.4.2. Evaluation of damage parameter

To evaluate the applicability of the above damage parameter, a method is proposed here to determine the damage parameter using the inelastic strain of the surface deformation field from DIC (Kao et al., 2016). Fig. 12a presents a contour map of the principal strain field on the surface of the granite after 4 thermal shock cycles(LR= 0.8). By taking an average along they-direction (Kao et al.,2016), an average value of the principal strain (εl) is obtained in Eq.(11).A profile of the average value of the principal strain(εl)in thex-direction is shown in Fig.12b.

Fig.13a presents the profile of the average value of the principal strain at different stress levels for granite after 4 thermal shock cycles. It can be observed that the profile of the average value of the principal strain remained relatively uniform when the loading stress level was low (LR≤0.5), suggesting that the rock exhibits elastic behavior without damage. However, the distribution of the average value of the principal strain was not uniform, with the middle part of the rock displaying a higher level of principal strain than other parts of the rock, when the loading stress level was relatively high, indicating that damage was developed on the surface of the rock. This part where the principal strain was significantly higher than the other parts of the rock was called the damaged zone in this study.The extent of the damaged zone gradually increased with increment of the stress, while in the region away from the damaged zone, the distribution of the average principal strain was uniform without damage. Such region was called the undamaged zone. Therefore,the surface of the rock could be divided into two regions during the loading process:the undamaged zone and the damaged zone.Note that the average principal strain is composed of the elastic and inelastic principal strains (Fig. 13b). The elastic principal strain can be estimated using the average principal strain from the undamaged zone by fitting and extrapolating a linear trend during loading (Kao et al., 2016) (Fig. 13c). Once the elastic principal strain at different stress levels is determined, the inelastic principal strain can be calculated by subtracting the elastic principal strain from the profile of the average value of the principal strain (Fig.13d).

Fig.11. Thin sections of the edge and interior of the cross-section of the granite specimen after 14 thermal shock cycles (RD-14-4): (a) Edge location;and (b) Interior location.1-Grain boundary cracks, 2 - Intragranular cracks, Qtz - Quartz, Kfs - K-feldspar, Pl - Plagioclase, Bt - Biotite, Mu - Muscovite.

Table 4Fitted parameters of the continuum damage model.

The inelastic principal strain is averaged,and the current state of damage is represented by the average inelastic principal strain,which is related to the damage parameter at different stress levels,thus the damage parameter 〈D〉 determined from DIC can be expressed as (Kao et al., 2016):

where 〈D〉is the damage parameter determined fromis the average inelastic principal strain; and εDis the damage saturation strain, which represents the critical principal strain at the peak stage. Note that a fracture was visible when the loading stress reached the peak strength, and the value of the damage parameterDin the above damage model is equivalent to the damage parameter〈D〉determined from DIC.

Fig.12. (a)Principal strain contour map and(b)the corresponding profile of the average value of the principal strain in the x-direction at LR=0.8 for granite after 4 thermal shock cycles (RD-4-3).

Fig.13. (a)Profile of the average value of the principal strain at different stress levels;(b)Illustration of the elastic and inelastic principal strains at LR=0.8;(c)Determination of the elastic principal strain and (d) determination of the inelastic principal strain at LR = 0.8 for granite after 4 thermal shock cycles (RD-4-3).

Table 5 presents the damage level, average inelastic principal strain,and damage saturation strain of rock at the peak stage.It can be observed that the damage level, average inelastic principal strain, and damage saturation strain increased gradually with increase in thermal shock cycle. This suggests that cyclic thermal shock causes initial damage to the rock,and the initial damage can be intensified with increase in thermal shock cycle,as shown by the results from microscopic observations in Fig.10.

The process of damage evolution in the continuum damage model is compared with that in DIC, displaying overall reasonable consistency in Fig.14 and suggesting that this continuum damage model can be used to well simulate the damage evolution law of the granite after cyclic thermal shock during loading. However, after more than 8 thermal shocks, there is slight deviation between the damage evolution curves of the damage model and DIC.The reason may be that the thermal shock causes the surface microcracks of the rock to be much greater in number than the internal microcracks(Fig.11a and b).Therefore,when the loading stress reached the peak strength,the inelastic deformation of the rock surface was much larger than that inside rock. This indirectly leads to the increase of the damage saturation strain (i.e. the denominator of Eq.(12) is increased), resulting in the damage value under the same stress level during loading being less than that in the continuum damage model,appearing as a shift in the damage evolution curve of DIC to the right of the damage evolution curve of the continuum damage model.

4. Discussion

Previous studies have shown that thermally induced microcrack is dependent upon applied temperature, thermal expansion mismatch, temperature gradient and initial porosity (Fredrich and Wong,1986; Jin et al., 2018). In this study, since only one type of rock is used and the upper limit temperature of thermal shock is the same, it can be considered that the initial porosity and appliedtemperature are not the main causes of thermally induced microcracks.In addition,it shows that the effect of temperature gradient on the physico-mechanical properties of rocks is much greater than that of thermal expansion mismatch(Shao et al.,2014;Eren Sar?c?,2016). Therefore, it can be considered that the temperature gradient is the most important factor causing thermal cracking in this study.When immersed in tap water,the specimens entered the water with temperature (8—11)°C from a high-temperature of 300°C, generating a great temperature gradient, which formed large thermal stresses. Therefore, repeated thermal shocks will cause further damage to the granite specimen.

Table 5Damage level,average inelastic principal strain and damage saturation strain of rock at the peak stage.

Fig.14. Damage evolution from the continuum damage model and DIC:(a)0 cycle(RD-0-3);(b)1 cycle(RD-1-3);(c)2 cycles(RD-2-3);(d)4 cycles(RD-4-3);(e)6 cycles(RD-6-3);(f) 8 cycles (RD-8-3); (g) 10 cycles (RD-10-3); (h) 14 cycles (RD-14-3); and (i) 20 cycles (RD-20-3).

Fig. 15. Normalized physico-mechanical parameters of the granite specimen after cyclic thermal shock.

The physico-mechanical parameters of the granite specimens after cyclic thermal shock are normalized in Fig.15. With the increase in thermal shock cycle, the peak strength and elastic modulus decreased gradually, and the peak strain presented an upward trend. However, the above parameters of the granite specimen showed no obvious changes after 10 thermal shock cycles.It can be seen that cyclic thermal shock leads to deterioration of the physico-mechanical properties of rock, but with limited effect. Beyond this limit, cyclic thermal shock will not have a significant impact on the physico-mechanical properties of rock.

A large number of microcracks were observed at the edge of the granite specimen, while the microcracks inside the rock were relatively fewer(Fig.11a and b).The reason is likely that rock after thermal shock treatment will generate large tensile stresses near its surface,and compressive stresses will be generated inside the rock(Kim et al., 2014; Shao et al., 2014), and the compressive limit of rock is far greater than the tensile limit. Therefore, it can be considered that the microcracks on the surface will be more complicated than those in the interior with increase in thermal shock cycle. Meanwhile, this also explains that the physicomechanical parameters tended to be stable with the increase in thermal shock cycle, and the form of the standard deviation curve of the surface principal strain became more complicated.

Crack propagation is related to the density of microcracks, and only when the density of the microcracks reaches a critical value,the crack will propagate. More thermal-shock microcracks were induced on the surface of the granite specimen with increase in thermal shock cycle(Fig.10).For rocks subjected to fewer thermal shocks,there were no or fewer microcracks on the surface.During the loading process, microcracks were continuously generated.When the loading stress approached the peak strength, the microcracks reached the critical density,and the cracks propagated,leading to an increase in the dispersion of surface principal strain,and a sharp rise in the standard deviation curve, which were accompanied by a large number of AE signals.When the number of cyclic thermal shocks further increased, the number of surface microcracks also increased. The corresponding stress level would decrease when the microcracks reached the critical density,and the crack would propagate in advance. However, in the process of cracks interconnecting each other,some cracks would be unloaded,resulting in weakening of the growth trend of the surface principal strain. After the number of cyclic thermal shocks reached 10, the microcracks on the surface would further increase. When the critical density was reached, the corresponding stress level would further decrease.However,in the process of cracks interconnecting each other, the phenomenon of crack unloading became more violent, resulting in a “platform section” in the standard deviation curve, and the AE count dropped significantly. When the loading stress approached the peak strength, the surface cracks and the internal cracks connected with each other.The degree of dispersion of the surface principal strain increased,and the standard deviation curve rose sharply,accompanied by a large number of AE activities.

5. Conclusions

In this study, the main purpose is to investigate the effect of cyclic thermal shock on the physico-mechanical properties of granite.The main conclusions are summarized as follows:

(1) The peak strength and elastic modulus decreased gradually with increase in thermal shock cycle, and the peak strain presented an upward trend.However,the above parameters of the granite specimen showed no obvious changes after 10 thermal shock cycles.It can be seen that cyclic thermal shock leads to deterioration of the physico-mechanical properties of rock, but this effect has a certain limit. Beyond this limit,cyclic thermal shock will not significantly affect the physicomechanical properties of rock.

(2) The surface damage to the granite specimen first exhibited a diffusion distribution at the initial stage of loading,and then concentrated on the local area, with corresponding strain value significantly larger than those of other areas.With the increase in loading stress, the damage to the surface of the granite specimen concentrated further, resulting in surface microcrack nucleation. When the loading stress was further increased, the microcracks continuously propagated, connected,and tended to develop towards both end faces of the granite specimen.

(3) The form of the standard deviation curve of the surface principal strain became more diverse with increase in thermal shock cycle, appearing to develop from the exponential form through S-shaped to ladder-shaped,and the AE activity became more drastic. The stress levels of turning pointsAandBon the standard deviation curve are the stress levels corresponding to the occurrence of an uneven strain field and a local strain concentration zone, respectively. They decreased gradually with increase in thermal shock cycle.The results obtained from microscopic observations showed that the effect of cyclic thermal shock on the edge microstructure of the granite specimen is greater than that on the internal microstructure,and more microcracks were induced with increase in thermal shock cycle.

(4) The continuum damage model was used to describe the damage evolution of granite after cyclic thermal shock during loading. The results showed that the stress—strain relation could be well modeled by this damage model.To verify the applicability of the continuum damage model, damage evolution from this damage model was compared with that from DIC, displaying an overall consistency.

Declaration of Competing Interest

The authors wish to confirm that there are no known conflicts of interests 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 supported by the State Key Research Development Program of China (Grant No. 2017YFC0804203), National Nature Science Foundation of China (Grant No. 51621006), Key Research Program of Frontier Sciences, Chinese Academy of Sciences (Grant No.QYZDB-SSW-DQC029).