Quantitative description of infrared radiation characteristics of preflawed sandstone during fracturing process

Ke Zhang, Xianghua Liu, Yulong Chen, Heming Cheng

a Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming, 650500, China

b Faculty of Civil and Architectural Engineering, Kunming University of Science and Technology, Kunming, 650500, China

c School of Energy and Mining Engineering, China University of Mining and Technology, Beijing,100083, China

Keywords: Remote sensing rock mechanics Infrared radiation characteristics Statistics Precursor point Verhulst inverse model

ABSTRACT Previous studies show that infrared radiation temperature (IRT) abnormalities are always accompanied by the crack development in rocks under external loads.In this context,experiments were conducted on preflawed sandstone to investigate the infrared radiation characteristics during failure process. Two indicators were defined herein,i.e.coefficient of variation of IRT(CVIRT)and skewness of IRT(SIRT).The regression analysis shows that the IRT probability distributions during loading process fit the Gaussian model. The variations in the CVIRT are characterized by four stages: primary stage, steady stage, accelerating stage and post-peak stage. Besides, the variations in the SIRT are divided into three stages:primary stage, steady stage and failure and post-peak stage. The precursor point for preflawed rock failure is identified based on the CVIRT-time curve, with average precursor point of 83% of the peak stress.Compared with other IRT indicators,the proposed two IRT indicators have higher sensitivity to IRT abnormalities during failure process. Furthermore, the connection between the IRT indicators and the rock fracturing was investigated to interpret the IRT indicator abnormalities. Based on the Verhulst inverse function, a new quantitative model was presented to describe the primary stage, steady stage and accelerating stage of the CVIRT-time curve.The results obtained in this study can provide early-warning information for rock failure prediction.

1. Introduction

Use of advanced monitoring technologies has become a common practice for understanding the fracturing process of rock masses in laboratory tests and rock engineering projects. The acoustic emission (AE) and microseismic (MS) monitoring techniques have been widely used to detect crack developments in rocks.The distribution of AE/MS events can be used to indicate the location where new cracks develop (Xiao et al., 2016; Ishida et al.,2017; Feng et al., 2019; Li et al., 2019; Sun et al., 2020). Besides,the digital image correlation(DIC)technique has been employed to reconstruct the displacement and strain fields in the rock surfaces(Munoz and Taheri, 2017; Hassan et al., 2018; Miao et al., 2018;Zhou et al., 2018).

Rocks and rock-like materials under loading could produce detectable electromagnetic phenomena, including infrared radiation with wavelengths of 700 nm-1 mm. Infrared thermography has been applied to the field of rock mechanics as a nondestructive,noncontact and real-time monitoring technique, which provides a new perspective for understanding the fracturing behavior of rocks(Zhao and Jiang, 2010; Sun et al., 2017; Ma et al., 2019). In field application, thermal infrared imaging data from remote sensing satellites recorded thermal anomalies prior to major earthquakes(Genzano et al., 2007; Rawat et al., 2011). This technique was also used for detecting potentially weakened zones of unstable rock slopes and cliffs,including karst cave openings,fractures,moisture and material inhomogeneities(Teza et al.,2012;Baroˇn et al.,2014;Mineo et al., 2015; Guerin et al., 2019). Pappalardo et al. (2016)defined the cooling rate index in the remote survey of rock masses and found a relation linking the infrared data to the geomechanical parameters (rock quality designation and volumetric joint count). In laboratory experiments, various studies were reported on the infrared radiation characteristics from rock materials under different loading conditions, including uniaxial and biaxial compression tests(Wu and Wang,1998;Wu et al.,2006;Zhao and Jiang, 2010; Ma et al., 2016, 2018, 2019; Wang et al., 2016; Huang et al., 2018; Lou and He, 2018), the Brazilian split test (Salami et al., 2017), the shear-box test (Wu et al., 2006), the rockburst test (Sun et al., 2017) and the model test (He, 2011; Gong et al.,2015; Wang et al., 2018; Sun et al., 2020). Abnormalities in infrared radiation can be detected during the loading process.Based on their experimental observations, some researchers (e.g.Wu and Wang, 1998; Zhao and Jiang, 2010; Wang et al., 2016)suggested different stress levels as the precursor points for intact coal and rock failure.

Previous studies mainly focused on rock specimens in terms of infrared radiation characteristics, by using the average infrared radiation temperature (AIRT) as an indicator. However, the AIRT is known as a measure of central tendency and has the disadvantage of insensitivity to local variations, i.e. localization of temperature anomalies at the prefailure stage in rocks under compression has been found but may induce little change in the AIRT, as argued by Ma et al. (2016, 2018). In this study, two IRT indicators, i.e. the coefficient of variation of IRT (CVIRT) and skewness of IRT (SIRT),were defined to quantify the IRT evolution characteristics of preflawed sandstone specimens during rock failure process. The Verhulst inverse model was used to quantify growth in CVIRT,and the failure times were predicted. The IRT indicators and infrared thermal image abnormalities prior to failure were detected, and associated infrared precursors were proposed.

2. Specimen preparation and testing

2.1. Specimen preparation

Red sandstone used in this study due to its homogeneous and isotropic nature (Miao et al., 2018) was sampled from Chuxiong Prefecture, Yunnan Province, China, as shown in Fig.1. The experimental material is fine-grained red quartz sandstone of the Upper Cretaceous Zhaojiadian Formation,with layer thickness up to 35 m.The rock specimens were cut from one rock block revealing no apparent cracks or fracture. The tested sandstone has a crystalline and blocky structure, and is hard and compact with density of 2492 kg/m3, uniaxial compressive strength of 78.63 MPa, tensile strength of 4.22 MPa, and elastic modulus of 7.87 GPa. The sandstone is mainly composed of feldspar and quartz, with both emissivity values close to 0.8 as measured by Buettner and Kern(1963),indicating that the sandstone is essentially homogeneous in terms of emissivity.

All the specimens,with dimensions of 60 mm×25 mm×120 mm(length × width × height), were cut by a circular saw. A water-jet cutter was used to produce inclined flaws with a length of 20 mm(2l = 20 mm) and an aperture of 2 mm. Three different flaw inclinations (β) were selected: 30°, 45°and 60°. Mica sheets were tightly inserted through the thickness of the specimens to represent filled flaws.For each flaw inclination,three specimens were tested to ensure repeatability of results.Fig.2 and Table 1 show the geometries of sandstone specimens with different flaw inclinations and some of the specimens to be tested.

Fig.1. Sampling location of the red sandstone.

Fig. 2. Geometry of the specimen.

2.2. Experimental setup and procedure

An experimental system (see Fig. 3) consisting of a loading system,an infrared thermography instrument and a camcorder was used. The experimental procedure is described as follows:

(1) The servo-hydraulic universal testing system machine WAW1000 (manufactured by Changchun Research Institute of Testing Machine Co., Ltd., China) was applied for uniaxial compression tests with a maximum capacity of 1000 kN and a loading precision of 0.5%. The displacement-controlled axial loading was applied at a rate of 0.3 mm/min for all the tests.

(2) A high-sensitive and top-performance infrared thermography InfReC H2640 instrument (manufactured by Nippon Avionics Co., Ltd., Japan) was placed 1 m away from the specimen front face to continuously detect the temperature changes, as empirically suggested by previous researchers(e.g. Zhao and Jiang, 2010; Wang et al., 2016; Salami et al.,2017; Ma et al., 2018, 2019). Infrared thermal image was captured automatically per one second. The InfReC H2640 instrument has a temperature resolution of 0.03°C,a graphic resolution of 640× 480 pixels and an instantaneous field of view(IFOV)of 0.6 mrad.Fig.4 shows the measuring distance and measuring field of view for infrared camera.According to the given measuring distance and IFOV value,the horizontal and vertical ranges of the infrared thermal image are 0.38 m and 0.25 m, respectively, with minimum detectable resolution of 0.6 mm×0.6 mm.Therefore,the number of captured temperature data points can be calculated as 200 × 100(20,000)for the specimen front face,in order to ensure a high accuracy of the infrared testing results.

(3) A high-definition camcorder was also placed 1 m away from the specimen front face to record the fracturing process during loading, as suggested by Wu et al. (2020). The camcorder has a graphic resolution of 1920 ×1080 pixels, a rate of 30 frames per one second and an IFOV value of 2 mrad.

Table 1 Geometries of preflawed sandstone specimens.

Fig. 3. Layout of loading and monitoring systems.

In order to reduce the heat transfer effect and environmental radiation effect, the following experimental measures were adopted. For 24 h prior to test, all instruments and specimens were placed at room temperature. All the doors and windows were closed, the curtains were put down, the lights were switched off and no walk was permitted during testing. The room temperature during testing was approximately 17°C-18°C and the humidity was 54%. The paperboard with thermal insulating coating was inserted into the contact area between the specimen and the loading platform. The test of any specimen was completed in less than 10 min. Before test, the times of the different monitoring systems were synchronized. In addition, a platinum resistance temperature detector was attached to a specimen surface to record the reference temperature under the same experimental condition(Huang et al.,2018;Guerin et al.,2019),and then the emissivity was calibrated to find the reference temperature on the infrared thermal image (Guerin et al., 2019). The emissivity of the tested sandstone specimen is 0.83.

Fig. 4. Measuring distance and measuring field of view for infrared camera.

Fig. 5. Flowchart of the proposed method for processing infrared thermal images.

3. Quantitative indicators of infrared thermal image and MATLAB implementation

A valid area, represented by a white rectangle, was selected along the profile of the tested front face of the specimen on the infrared thermal images (see Fig. 5). The infrared thermal image time-series were acquired by the infrared thermography InfReC H2640 instrument, and N + 1 infrared thermal images were obtained at times t0, t1, …, tN, where t0is the initial time that equals zero. Note that the affiliated software InfReC analyzer NS9500 provides only the functions to calculate the maximum, minimum and average values. A method aimed at calculating other quantitative indicators was proposed herein with MATLAB (Version R2017a) implementation. The flowchart is shown in Fig. 5.

The IRT point of an infrared thermal image has a corresponding relation of the pixel point. These infrared thermal images of valid areas were exported into N + 1.CSV format files using the InfReC analyzer NS9500 software and then converted into a series of NR× NCmatrices T(tk) (k = 0, 1, …, N) with MATLAB software,representing the temporo-spatial evolution of thermal behaviors,where NRand NCrepresent the number of rows and columns of the matrices,respectively.Let Tij(tk)be the matrix element representing the temperature at time tkcorresponding to the ij-th pixel (i and j are the row and column indices,respectively,i=1,2,…,NR,j=1,2,…, NC).

Based on the mathematical statistics, five IRT indicators of the reference area at time tkwere defined to quantify the IRT characteristics, i.e. the maximum IRT (MAXIRT, unit:°C), minimum IRT(MINIRT, unit:°C), average IRT (AIRT, units:°C), CVIRT (dimensionless) and SIRT (dimensionless).

The MAXIRT, MINIRT and AIRT have commonly been used in previous studies (Wu et al., 2006; Zhao and Jiang, 2010; He, 2011;Gong et al., 2015; Wang et al., 2016, 2018; Salami et al., 2017; Sun et al., 2017), which are defined as follows:

In this study,the CVIRT and SIRT were proposed to characterize the localization of IRT anomalies with respect to the loading time and spatial distribution characteristics. The CVIRT was defined as the ratio of the standard deviation of IRT to the AIRT to measure the dispersion of an IRT probability distribution of testing surface.The higher the CVIRT,the greater the level of dispersion.The CVIRT(tk)is written as follows:

The SIRT was defined as the third standardized moment of IRT to measure the asymmetry of an IRT probability distribution of testing surface (Joanes and Gill,1998). Fig. 6 illustrates the graphical representation of the skewness.The positive SIRT value indicates that the IRT probability distribution is right-skewed, and the negative indicates that the IRT probability distribution is left-skewed.A zero value of the SIRT is the case for a symmetric distribution. The SIRT(tk) is expressed as follows:

Fig. 6. Graphical representation of the skewness.

Fig. 7. Evolution of five IRT indicators during loading: (a) MAXIRT, MINIRT and AIRT,(b) CVIRT, and (c) SIRT.

4. Results

4.1. Stress-time curve and IRT-time curves

A typical record(Specimen No.45-1)was chosen for analysis.The loading history is shown in Fig.7a.Based on the characteristics of the stress-time curves and crack developments during uniaxial tests,the typical failure process of the sandstone specimen can be divided into four stages(see Fig.7a),as suggested by Jaeger et al.(2007):original crack closure (stage I-S), elastic deformation (stage II-S), plastic deformation (stage III-S) and post-peak failure (stage IV-S). Five typical stress levels indicated as blue dots are identified and labeled with capital letters A (at the stage I-S, σ/σc= 0.01, where σcis the uniaxial compressive strength), B (at the stage II-S, σ/σc= 0.56), C(crack initiation stress,σ/σc= 0.84), D (peak stress,σ/σc= 1) and E(ultimate failure,σ/σc=0)in the stress-time curve.Fig.7 shows the plots of the five IRT indicators and stress as a function of time.Fig.8 illustrates the crack developments and corresponding infrared thermal images recorded during loading.The colorin the infrared thermal images represents IRT:the cooler the color,the lower the IRT.Though the stress and IRT levels are not the same at each stage for these specimens with different flaw inclinations under compression,almost all the stress-time curves and IRT-time curves have gone through the four stages as described above.The results show that the failure pattern obtained from this experiment is in reasonable agreement with the results reported by Li et al. (2005), Wong and Einstein (2009), and Miao et al. (2018). The results reported in the following were discussed with respect to four stages of the failure process:

Fig. 8. Evolution of cracks and corresponding infrared thermal images during loading. Crack developments at (a) point A, (b) point B, (c) point C, (d) point D, and (e) point E.Infrared thermal images at (f) point A, (g) point B, (h) point C, (i) point D, and (j) point E.

(1) Initial crack closure stage(stage I-S):An initial portion exhibits concaved upward behavior due to the closing of the microstructures under compression. With increasing compression loadings,the MAXIRT,MINIRT and AIRT fluctuate drastically. At the same time, the CVIRT and SIRT present fluctuating growth trends and then tend to stabilize in a narrow range.

(2) Elastic deformation stage(stage II-S):The curve has a portion that is essentially linear. The MAXIRT, MINIRT and AIRT continue to fluctuate as loading increases. The CVIRT and SIRT remain basically constant with fluctuations in a narrow range throughout this stage.

(3) Plastic deformation stage (stage III-S): Macroscopic tensile cracks are observed to initiate from the flaw tips(see Fig.8c),starting from a stress level (σ = 53.9 MPa) of approximately 84% of the peak stress (Point C in Fig. 7) and propagating for short distances.With further increase in the load, the propagation of tensile cracks stops as a result of the release of tensile force;then,shear cracks appear(see Fig.8d),which leads to an increase in the CVIRT fluctuation. A trough (stress decrease and increase)is observed on the stress-time curve associated with shearing along the shear cracks.When the stress reaches its peak (Point D in Fig. 7), the MAXIRT, CVIRT and SIRT increase abruptly, with values of 3.14°C, 4.05 ×10-3and 1.58,respectively. As specimen fails, a local high IRT abnormality belt is observed along the fracturing plane(see Fig.8i).

(4) Post-peak failure stage (stage IV-S): This stage is characterized by abrupt stress drop and IRT decrease after the peak stress has been reached.

Fig. 9. IRT field distributions and Gaussian fitting results under different load levels:(a) point A, (b) point B, (c) point C, and (d) point D.

4.2. Spatial distribution characteristics of IRT fields

To investigate the spatial distribution characteristics of the IRT fields,statistical analysis of the IRT fields under different load levels was carried out. Frequency histograms of Points A, B, C and D are shown in Fig. 9. Fig. 9 and Table 2 show that the IRT probability distributions fit a Gaussian model,given by the following equation,with the correlation coefficients R2more than 0.85:

where A,y0,w and xcare the coefficients determined by regression analysis. The results presented in Fig. 9 and Table 2 are consistent with that by Lou and He (2018) who performed uniaxial compression tests on concrete. The shape of Gaussian distribution changes along with the loading time,which will be discussed in the following sections.

Table 2 Coefficients for Eq. (6) under different load levels.

Fig.10. Idealized IRT indicator-time curves: (a) CVIRT, and (b) SIRT.

5. Comparison of five IRT indicators

5.1. Evolution models of new IRT indicator during failure process

5.1.1. Variation of CVIRT

Fig. 7b depicts the connection between the CVIRT-time curve and stress-time curve of a typical record (Specimen No. 45-1).These measured data collected from the present experimental study lead to construction of the idealized one-dimensional (1D)CVIRT-time curve,as shown in Fig.10a.The variations in the CVIRT can be divided into four stages:

(1) Primary stage(I-T1):The CVIRT increases at a decreasing rate(i.e.the curve is convex upward),corresponding to the initial half of the original crack closure stage in the stress-time curve.

(2) Steady stage(II-T1):During this stage,the CVIRT is basically constant,corresponding to the final half of the original crack closure stage and the whole elastic deformation stage in the stress-time curve.

Table 3 Classification of the skewness (Bulmer,1979).

(3) Accelerating stage (III-T1): This stage is characterized by a rapid increasing CVIRT till failure,corresponding to the crack propagation stage in the stress-time curve.

(4) Post-peak stage (IV-T1): The CVIRT eventually decreases,corresponding to the post-peak failure stage in the stresstime curve.

5.1.2. Variation of SIRT

Typical SIRT-time curve (Specimen No. 45-1) is plotted in Fig.7c.Fig.10b shows the idealized 1D SIRT-time curve,which can be divided into three stages:

(1) Primary stage (I-T2): In our experiments, the SIRT before loading is in the range of -1.01 to -0.58, in which the IRT probability distribution appears to be skewed to the left(see Fig. 9a). The curve has an initial portion which is convex upward,corresponding to the original crack closure stage in the stress-time curve.

(2) Steady stage (II-T2): A negative constant SIRT is basically maintained (approximately -0.5), whose IRT probability distribution is still left-skewed, corresponding to the elastic deformation stage and the crack propagation stage in the stress-time curve.

(3) Failure and post-peak stage (III-T2): The SIRT suddenly increases to exceed 1 at the point of peak stress (i.e. rightskewed IRT probability distribution), and then decreases,corresponding to the post-peak failure stage in the stresstime curve.

The classification of the skewness,suggested by Bulmer(1979),is presented in Table 3. It is found that as the applied loading increases,the skewness level of the IRT probability distribution shifts from moderately left-skewed towards approximately symmetric and then to highly right-skewed.

5.2. Anomalies of IRT indicators and precursor points before failure

As shown in Fig.7a,the MAXIRT,MINIRT and AIRT are observed to decrease rapidly and increase again before rock failure in the initial half of the crack propagation stage, which is less obvious than that in intact rocks,as reported by Wu et al.(2006),Zhao and Jiang(2010),and Wang et al.(2016).In addition,similar trends are observed in stages I-S and II-S (see Fig. 7a). In other words, using the change information of the MAXIRT, MINIRT and AIRT is very likely to result in erroneous judgment.

It is worthwhile to note that the measured CVIRT versus time abnormality takes the form of the accelerating stage (III-T1) as described in Section 5.1.1, wherein accelerating CVIRT can provide early-warning for imminent instability. Specifically, the turning point of the CVIRT from the steady stage to the accelerating stage can be recognized as the precursor point for preflawed rock failure(see Figs.7b and 10a),associated with initiation and propagation of the additional macroscopic cracks from the tips of the pre-existing flaw (see Fig. 8c). Compared with other IRT indicators, the monitoring of the CVIRTs gives a better indication of the propensity of a preflawed rock to fail. In our experiment, CVIRT abnormality precursors were observed in all sandstone specimens tested. Table 4 summarizes the precursor points of preflawed sandstone based on the evolution characteristics of the CVIRT (see Fig. 10a). The stress ratio, which is defined as the stress associated with the occurrence of a precursor point divided by the peak stress, offers quantitative information of how close the occurrence of the precursor point is to the specimen failure stress. It is found that the flaw inclination has little influence on the stress ratio of the precursor point in this experiment, ranging from 76% to 90% with average value of 83%.Wu and Wang(1998)suggested that a stress level of approximately 0.79σccan be regarded as the failure precursor for normal coal and sandstone. The stress ratio (σ/σc) of preflawed rock associated with a failure precursor is larger than that of intact sandstones. This difference occurs because the presence of a pre-existing flaw serves as a stress concentrator, leading to abrupt brittle failure.

Table 4 Precursor points of preflawed sandstone before failure.

5.3. Anomalies of IRT indicators at failure

The average increase rate before failure (AIRBF) of an IRT indicator was defined as follows:

where P is the time step corresponding to the peak stress, X is one of five IRT indicators.

The increase rate of an IRT indicator at failure(IRAF)was defined as follows:

The rate of mutation (RM) was defined as the ratio of IRAF to AIRBF:

The increase rates of IRT indicators before and at failure are presented in Table 5. The rates of mutation are listed in Table 6.Fig.7 and Tables 5 and 6 reveal that there is no significant increase of the MINIRT and AIRT,and their ratios of the IRAF to the AIRBF are in the ranges of 0.64-3.5 and 0.49-8.5, respectively. For the MAXIRT, CVIRT and SIRT, sudden increases can be observed after the peak stresses prior to specimen failure, and their ratios of the IRAF to the AIRBF are in the ranges of 27-261.85,11.57-164.97 and 39.95-483.65,respectively.Fig.8i shows that the infrared thermal image abnormality,characterized by local high IRT belt,is detected from our experiment at rock failure. Thus, the two new IRT indicators have successfully captured the IRT abnormalities at failure.

6. Quantitative model between the CVIRT and time

6.1. Verhulst inverse model

The logistic function proposed by Verhulst (Baca?r, 2011) is a nonlinear population growth model with a characteristic S-shaped curve(see Fig.11),which is applied in a range of fields from ecology to geosciences (Tong, 2009; Federico et al.,2012).

The Verhulst function is the solution of the following ordinary differential equation:

Table 5 Increase rates of IRT indicators before and at failure.

where x is the independent variable, a and b are the coefficients determined by regression analysis.

With the initial condition x = x1, the Verhulst inverse function(see Fig.11) is given by Federico et al. (2012):

Due to the shape similarity, the Verhulst inverse function is applied to modeling the CVIRT growth.As illustrated in Fig.11,the time and CVIRT are the independent and dependent variables,respectively.Thus,x and f(x)can be replaced by the time and CVIRT,respectively, leading to the following mathematical model:

This study found that the variations of IRT indicators are characterized by fluctuations, in reasonable agreement with the previous results(Wu et al.,2006;Zhao and Jiang,2010;He,2011;Gong et al., 2015; Ma et al., 2016, 2018, 2019; Wang et al., 2016, 2018;Huang et al., 2018; Lou and He, 2018), which may be due to the presence of noise from the surrounding environment and electric components of the test instruments, as argued by He (2011) and Gong et al.(2015).To better interpret the infrared data,the adjacent averaging method(Bregman and Irwin,2007)was used to smooth the CVIRT(tk) data points before failure (k = 0,1, …, P-1), and the smoothed curves are shown in Fig.12.

Table 6 Rates of mutation of IRT indicators.

6.2. Fitted results and failure time prediction

In this study,the initial time t1was selected as 1 s,and the fitted results based on the least square method are shown in Fig.12 and Table 7. The correlation coefficients R2of the fitted curves are all more than 0.7, suggesting that moderately or highly strong correlations (Lefsky, 2010) are found between the CVIRT and time. The comparison of the fitted curves and test results shows that the primary stage, steady stage and accelerating stage of the CVIRTtime curve can be more clearly described by this model with a simple and unified expression.

If the mutation of the CVIRT occurs, it can be considered as a catastrophic failure point.The time at the catastrophic failure point is the failure time.According to Eq.(12),the predicted failure timewas given as follows:

The relative difference ε between the predicted and measured failure time is defined as follows:

Fig.11. Schematic diagram of Verhulst inverse model.

where Tfis the measured failure time. The results of comparison between the predicted and measured failure times are showed in Table 7, and their relative differences are less than 1%. Therefore,the proposed model provides a reasonable estimate of failure time.

Fig. 12. Fitted curves based on Verhulst inverse model: (a) specimen No. 30-1, (b)specimen No. 45-1, and (c) specimen No. 60-1.

7. Theoretical interpretation of IRT indicator abnormalities

The IRT of rock surfaces is a comprehensive effect of rock deformation and fracturing (Wu and Wang,1998;Wu et al., 2006;Zhao and Jiang, 2010; He, 2011; Gong et al., 2015; Ma et al., 2016,2018, 2019; Wang et al., 2016, 2018; Salami et al., 2017; Sun et al.,2017; Lou and He, 2018). The pore gas desorbing-escaping effect,thermo-elastic effect,fracture effect,and friction-thermal effect are suggested as the four main mechanisms, resulting in the IRT increments which are respectively ΔT1, ΔT2, ΔT3and ΔT4(Wu et al.,2006; He, 2011; Gong et al., 2015; Wang et al., 2016; Sun et al.,2017). The heat transfer and environmental radiation can be nearly neglected due to the short test time (less than 10 min) and well-controlled experimental conditions, as described in Section 2.2. The IRT increment ΔT can be given by

7.1. Variation mechanism of the CVIRT

(1) Primary stage (I-T1)

Sandstones have complex and heterogeneous pore structures(Bernabé and Revil, 2013), leading to a heterogeneous IRT field during the early stage of the loading process, in which the IRTs around the pores decrease(ΔT1<0)due to the pore gas desorbingescaping effect.At the same time,the IRTs increase(ΔT2>0)due to the thermo-elastic effect. These two effects facilitate IRT differentiation, resulting in increased CVIRT.

(2) Steady stage (II-T1)

As the specimen is further loaded, the thermo-elastic effect is dominant,which means that the stresses and IRTs in the specimen increase simultaneously (Δσ1>0 and ΔT2> 0), but the extent of dispersion of the IRTs(i.e.CVIRT)remains approximately constant.

(3) Accelerating stage (III-T1)

The fractureeffectandfriction-thermaleffect aredominantduring the later stage of loading. The initiation and propagation of tensile crackscausethe IRTsaroundthecracktipsto decrease(ΔT3<0)dueto the fracture effect.On the other hand,the friction-thermal effect can increase the IRTs around the shear cracks (ΔT4> 0). Therefore, IRT localization occurs with an increase in CVIRT.

(4) Post-peak stage (IV-T1)

The CVIRT gradually decreases due to the disappearance of the IRT anomalies caused by a decrease in the stress level to zero.

7.2. Variation mechanism of the SIRT

(1) Primary stage (I-T2)

The initialIRTdistribution isfoundto beleft-skewedwithnegative SIRT. With increase in applied loading, the local IRTs are found to decrease because of the pore gas desorbing-escaping effect. Therefore,the IRT probability distribution tends to have a higher frequency of relatively low IRT values,and causes smaller tail on the left side of the distribution curve(see Fig.9b),thereby increasing the SIRT.

(2) Steady stage (II-T2)

Similar to II-T1 stage, the thermo-elastic effect is dominant at this stage, thus the asymmetry of the IRT probability distribution(i.e.the SIRT)becomes approximately constant.On the other hand,this IRT indicator is found to be insensitive to the fracture effect occurring at the plastic deformation stage.

(3) Failure and post-peak stage (III-T2)

Table 7 Coefficients for Eq. (12) and failure times.

Due to the friction-thermal effect,occurrence of shearing along the coalescence cracks leads to an abrupt rise of local IRTs at the peak of the stress-time curve.This phenomenon causes the tail on the right side of the IRT probability distribution to be longer than the left side (see Fig. 9d), which then makes the IRT probability distribution to be right-skewed with positive SIRT. Then, the SIRT decreases in the post-peak portion of the stress-time curve for the same reason as discussed at IV-T1 stage.

8. Discussion

8.1. Advantages and limitations of infrared technology

The AE, MS, DIC, and infrared methods were all applied as nondestructive techniques to estimate crack growth in rocks. The infrared technique has following advantages over the other nondestructive methods(Xiao et al.,2016;Ishida et al.,2017;Feng et al.,2019;Li et al.,2019):(1)It does not require installing sensors and preparing a contrasting random speckle pattern in specimen surfaces;and(2)Data processing is simpler without AE/MS source location and digital image correlation calculation. It should be noted that the main limitation of infrared technique is monitoring of the surface IRTs, rather than the interior IRTs.

8.2. Key characteristics and advantages of new IRT indicators

The failure patterns of sandstone specimens with three different flaw inclinations are all produced by the coalescence of shear cracks, as observed in our studies. The variation of IRT is closely related to the rock fracturing and thus an acceptable agreement is found among the thermal behaviors for these specimens. The key characteristics and advantages of the proposed IRT indicators(CVIRT and SIRT) revealed from this study can be summarized as follows:

(1) The MAXIRT, MINIRT and AIRT show drastic fluctuations during loading, and their abnormalities prior to failure are not detected (see Fig. 7a). In addition, an abrupt increase of the MAXIRT is a result of ultimate failure,not a precursor to failure (see Tables 5 and 6). Thus, the MAXIRT, MINIRT and AIRT, commonly used IRT indicators in previous researches(Wu et al.,2006;Zhao and Jiang,2010;He,2011;Gong et al.,2015; Wang et al.,2016, 2018; Salami et al.,2017; Sun et al.,2017), are found to be inadequate to describe the evolution process of IRT field for preflawed specimens.

(2) As noted in Section 5.2, the CVIRT-time curve has an accelerating stage after the occurrence of tensile cracks from the pre-existing flaw tips, which is the abnormal sign preceding the development of the coalescence cracks. The smoothing process and Verhulst inverse function were further used to identify the variation law of CVIRT and failure time quantitatively (see Section 6). Federico et al. (2012)stated that the initiation of tensile cracks is usually the first sign that a slope is approaching an unstable condition.Based on acquisition of time-series of the measured CVIRTs, the precursor point proposed in this study could thus be referred to as an early-warning precursor of rock failure. When the CVIRT of a rock mass exceeds the precursor point, effective measures should be adopted in a timely manner, such as reducing the working load and reinforcing unstable rock masses. Under further loading, a sharp and sudden increase of the CVIRT is observed at the peak of the stress-time curve,which can be identified as the failure sign, as described in Section 5.3.

(3) There is no visible SIRT sign preceding the developments of coalescence cracks, indicating that SIRT is insensitive to the crack initiation behavior. However, the SIRT is found as the most sensitive indicator of failure as the feedback signal,having the highest rate of mutation at failure within the range from 39.95 to 483.65(see Section 5.3).According to the classification of the skewness(Bulmer,1979),a SIRT value of 1 corresponding to highly right-skewed IRT probability distribution can be taken as the threshold to identify whether or not failure occurs in the rock.

(4) Compared with previous IRT indicators having dimensions of temperature, the proposed indicators are dimensionless.

Combining the above observations, it is conclusive that the CVIRT and SIRT show obvious multistage characteristics, and are able to capture the IRT abnormalities during failure process via making full use of the whole IRT field information.

8.3. Future works

The rock type, mineral component, and flaw geometry could have influences on the fracturing and thermal behaviors. Uniaxial compression experiments on coal, ironstone, sandstone, marble,limestone, granite, granodiorite, gabbro and gneiss have been performed using infrared thermography(e.g.Wu and Wang,1998;Wu et al.,2006;Zhao and Jiang,2010;Ma et al.,2016,2019;Wang et al.,2016).These experimental observations support the fact that thermal infrared anomalies are very sensitive to the brittle failure,thus they can be detected in the rock fracturing process (Wu and Wang, 1998; Wu et al., 2006; Zhao and Jiang, 2010; Ma et al.,2016, 2019; Wang et al., 2016). Further studies with different rock types, mineral components and flaw geometries are needed to validate the results provided in this study.

The natural fracturing is always three-dimensional (3D), hence further experimental studies on 3D crack growth are needed using infrared thermography.

9. Conclusions

This paper presented two IRT indicators and developed a MATLAB program for recognition of IRT abnormalities based on the processing of a series of infrared thermal images. The following conclusions can be drawn:

(1) The regression analysis shows that the IRT probability distributions under different load levels fit the Gaussian function,and the shape of Gaussian distribution changes with the loading time.

(2) The IRT indicators change with the loading time.The CVIRTtime curve can be divided into four stages: primary stage,steady stage,accelerating stage and post-peak stage.Besides,the SIRT-time curve can be classified into three stages: primary stage,steady stage and failure and post-peak stage.The associated skewness levels of the IRT probability distributions are moderately left-skewed, approximately symmetric and highly right-skewed.

(3) The variations in CVIRT and SIRT are closely related to the stress state, original microdefect closure, and macroscopic crack initiation, propagation and coalescence. The CVIRT is found to increase at an increasing rate after the initiation of tensile cracks from the inner flaw tips,starting from a stress level of about 83% of the uniaxial compressive strength,which can be regarded as an early-warning precursor of rock failure. The abrupt occurrence of coalescence cracks is associated with sharp and sudden increases in the CVIRT-time and SIRT-time curves, which can be regarded as the feedback signals of rock failure.

(4) A quantitative model is proposed to quantify growth in the CVIRT based on the Verhulst inverse function. The comparison of predicted curves and test results shows that the whole process of the CVIRT growth can be described by this model.

The proposed methodology and IRT indicators are expected to provide an efficient tool for the qualitative analysis of infrared thermal image time-series,as well as the monitoring of instabilities and failure for rock masses.

Declaration of competing interest

We 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

The research was funded by the National Natural Science Foundation of China (Grant No.11902128), and the Applied Basic Research Foundation of Yunnan Province (Grant Nos. 2019FI012 and 2018FB093).

List of notations