The Mechanical Characteristics and Damage Model of Helan Mountain Rock based on Acoustic Emission

YANG Youzhen, ZHAO Yishen, MA Hailong, CHEN Wenwu, FENG Haiyan,JIN Menghua, LIN Qingqing, MA Wenguo,2

(1. School of Physics and Electronic and Electrical Engineering, Ningxia University, Yinchuan 750021, China; 2. Institute of Solid Mechanics, Ningxia University, Yinchuan 750021, China; 3. Institute of Ethnic Preparatory Education, Ningxia University, Yinchuan 750021, China; 4. College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China; 5.Shanghai Junnel Engineering Co. Ltd., Shanghai 200030, China)

Abstract: With the risk of disappearing for the rock paintings considering long-term exposure in Helan Mountain, the freeze-thaw(F-T) cycling experiments were carried out with 12-hour F-T cycling (0, 10, 20, 30, and 40 F-T cycles) under five kinds of confining pressures(5, 10, 20, 30, and 40 MPa). The acoustic emission (AE)detect technology was used to reveal the rock fracturing characteristics during the triaxial compression test whole process. The stress-strain relation changes along with different confining pressures and F-T cycles. Peak stress and residual stress changes along with different confining pressures and damages, and the variation of axial stress-AE ringing counts-time changes along with different confining pressures and F-T cycles. The damage variable with AE parameter under F-T and force coupling was defined for the first time, and the damages model was established. The experimental results show that the F-T cycles lead to the decrease of rock strength and the gradual transformation of compression failure mode from brittleness to plasticity. The confining pressure provides a certain ability to resist deformation and inhibit crack growth for rock samples after F-T cycles. The temporal and spatial evolution law of AE counts well corresponds to the loading and failure process of the rock samples. The AE 3D positioning technology can accurately capture the development position and direction of internal cracks and pores of rock, and the failure form is conical shear failure. The established damage model has a better fittingness between the theoretical calculation results and the test results, and is reasonable to be used in the future for protection of Helan Mountain rock painting.

Key words: helan mountain rock; freeze-thaw(F-T); acoustic emission (AE); damage model

1 Introduction

The Helan Mountain, located in Ningxia of Northwest China(Fig.1), is the largest and most concentrated rock paintings area in the world[1]. The rock paintings with high historical, cultural and scientific values, engraved early in the late period of the times old tools and mainly created in the times of new stone tools, are mainly distributed on the mountain rock walls at the eastern foot of Helan Mountain shown in Fig.2, as well as on the lonely rocks in the flooded fan desert grasslands outside the passes. The rock walls have undergone severe weathering and erosion from long-term exposure to the environment, and followed by the resulting diseases, such as crackles, powdery exfoliation and flaky exfoliation[2], which that will make the rock paintings in danger of disappearing, as shown in Fig.3. It can be seen that the exfoliation, cracks, collapse and carrier loss are the four main diseases, which severely destroy the integrity of rock, and cause rock paintings to disappear in the end. The main reasons are related to the current environment of Helan Mountain with the water content and temperature change circularly in monsoon climate zone, which are the external incentives of rock diseases occurrence and development. The Helan Mountain rock painting carrier sandstone has inside geological defects such as primary faults, joints, bedding, holes and fissures, as an uneven natural rock material. When the water enters the pores and fractures in the sandstone and is in the low-temperature environment, the water turns into ice and expands, resulting in tensile stress and accelerating the initiation and development of fractures and pores. When the temperature increases, the water melts and transfers in the rock, and the cement between rock’s grains is lost. The repeated freeze-thaw cycles lead to the damage of sandstone structure.Therefore, a study on the rock damage mechanism under the multiple factors is urgent, which helps to further study and predicts the type, location and deterioration of rock painting in Helan Mountain, and develops effective protection schemes.

Fig.1 Location of the Ningxia

Fig.2 Distribution of Helan Mountain rock paintings

Fig.3 Typical diseases of rock (a: exfoliation, b: cracks, c: collapse, and d: carrier loss)

The research on the damage mechanism of stone cultural relics has always been a difficult and important issue in the field of cultural relics protection[3]. Mikayamaa[4]et alfound that the collision and adhesion of sand particles driven by the wind into the cave were the candidate cause of deterioration on Mogao caves. Li[5]et alconsidered that salting liquids were the main cause of rock deterioration in Longyou Grottoes. Zong[6]et alfound that temperature difference could lead to serious spalling of the surface layer of Dazu rock carvings. Li[7]studied the formation mechanism and characteristics of breakage of Wofo sandstone in Dazu rock carvings. Zhang[8]et al. obtained the corresponding weathering model by studying the mechanism and characteristics of the spalling damage of the bricks of the walls of the Xi’an City. Liu[9]et alstudied the distribution and characteristics of different diseases in Yun gang Grottoes. Liao[10]et alpredicted the deterioration types and locations of sandstone relics through tests. The above-mentioned researchers only considered the influence of single or dual factors of water, temperature and force in the study of deterioration of stone cultural relics, and the study of the three fields coupling is relatively rare. In recent years, due to the advances of instrumentation and technology, existing non-destructive testing technology has been gradually applied to the field of stone cultural relics[11]. For example, scanning electron microscopy (SEM) was applied to observe the microscopic change characteristics of stone relics[12,13]. X-ray diffraction (XRD) was used to analyze mineral components of the weathering products of stone relics[14]. Ultrasonic CT technology was used to detect the internal cracks[15-17]. Moreover, acoustic emission (AE), as a nondestructive testing, can determine the deformation characteristics and damage degree of materials under the action of external factors[18], and is mostly used in the fields of ceramics[19], wood[20], rock[21], concrete[22], etc. AE can detect the acoustic signal produced by dislocation movement, crack propagation, fracture, temperature stress,etc.in the material and obtain the damage of materials in real time in the whole deformation process. Therefore, AE has been used to establish the damage constitutive model in recent years. Hang[23]et alexplored the thermal damage evaluation of rock by AE test system and established a thermal damage evolution model which considers both heating and cooling processes. Gu[24]et aldeveloped a statistical damage constitutive model of cement mortar of uniaxial compression and fitted the damage evolution equation with AE energy data. Yang[25]et alresearched the influence of different loading rates on the AE characteristics of rock and deduced the damage constitutive model of sandstone based on the cumulative acoustic emission count and the loading rate. But, the above researches only focused on the damage model of materials under the action of a single factor. For open-air stone cultural relics, theirs damage is caused by the multiple factors such as temperature, water and force, which are often closely related to the specific environment. Therefore, the establishment based on AE signal of damage constitutive model of Helan Mountain rock painting carrier rock under the multi factors will help to understand the damage of rock paintings and the cultural relics protection. To date, the damage mechanism of the rock in Helan Mountain is not fully understood.

In this paper, the AE was used to investigate deterioration of rock mechanical properties under the three fields of temperature, water and force. The mechanical properties of triaxial compression under freeze-thaw cycling were studied. Combing with AE, the rock failure process was located by three-dimensional positioning. The damage variable based on the cumulative bell count of AE and elastic modulus was defined for the first time, and the damage model was established finally. The results presented in this study can provide a deeper understanding of rock damage and reference for subsequent restoration and protection of rock paintings.

2 Experimental

2.1 Rock samples preparation

The rock blocks were taken from the area near the rock painting, which does not affect the safety of cultural relics in Helan Mountain. Rock and mineral identification was identified through microscope 2.5 mm × 10 mm in the laboratory of Geology and Mineral Resources Center of Ningxia, as shown in Fig.4. The test results implied that the rock is composed of quartz, feldspar (plagioclase, potassium feldspar), rock debris, calcite,etc. Angular subangular quartz (with particle size of 0.15-0.9 mm mainly), subangular feldspar (with particle size of 0.18-0.65 mm, with varying degrees of soil and sericitization) and rounded and plastic shaped rock debris (with particle size of 0.2-0.55 mm mainly) are slightly evenly distributed in the rock. A small amount of flake mica (with flake size of 0.2-0.8 mm) can be seen scattered in the gap. The cement is mainly calcite, with a small amount of unevenly distributed clay minerals and iron. Calcite is slightly evenly filled in the above debris gap, resulting in basal cementation of the rock, and some clay minerals have been metamorphosed into chlorite aggregates.

Fig.4 Microscopic view of rock sample

According to the method suggested by ISRM[26], the rock blocks were cut and polished into a cylinder with a diameter (D) of 50 mm and a height (H) of 100 mm. The angle between two adjacent surfaces was 90° ± 0.25°. For both ends of the sample, the inhomogeneity error was less than 0.05 mm and the parallelism was less than 0.1 mm. Then the rock samples wave velocity were measured to ensure the uniformity of physical properties under natural condition. Of 75 samples were prepared, shown in Fig.5, and were divided into 5 groups with 15 in each group. The 75 rock samples were marked as T-Aa1 - T-Ee3, where A-E are corresponding to the number of cycles with 0, 10, 20, 30, and 40 respectively and a - e re corresponding to the confining pressures with 5, 10, 20, 30, and 40 MPa, respectively. The grouped rock samples are shown in Fig.5. Their average wave velocity is 5 189 m/s, and the deviation is less than 2%.

Fig.5 Prepared rock samples (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

2.2 F-T treatment

The rock samples were saturated with vacuum saturation cylinder. The selected and grouped samples were placed in saturated bucket with water which was 2 cm above the surface of the samples. The sealing cover was closed with rubber ring to ensure that air does not enter at negative pressure. The air pump drew air through the connected rubber tube until the air pressure was -0.8 MPa and continued 30 min. When the air pressure in bucket was constant, the valve of rubber tube and the power supply of air pump were shut off. The samples were soaked for 12 h in this state. And the mass was measured again after the end of water saturation test.The F-T cycles were generated from a TMS9012 programmable high-low temperature device, as shown in Fig.6. The saturated sample was frozen in the F-T machine at −20 ℃ for 4 h, then the temperature was increased to 20 ℃ in 2 h and remained unchanged for 4 h, as shown in Fig.7. In this study, five groups of the samples were treated with F-T cycles for 0, 10, 20, 30, and 40, respectively. After the experiment, the saturated mass, dry mass and wave velocity of samples are measured.

Fig.6 TMS9012 programmable high-low temperature device

Fig.7 Temperature changes with time in one F-T cycle

2.3 Triaxial compression test

Here, three samples were prepared for each experimental group. The triaxial compression tests were conducted using the TFD-1000G rock mechanics system, including the axial loading system, pressure loading system, deformation measurement system, servo con trol system, data-processing system and other auxiliary systems，as shown in Fig.8. The testing machine adopts original German DOLI all-digital servo controller, which has high control precision, complete protection function and strong reliability. The maximum axial force of system is 1 000 kN with the accuracy of ±1%, and the maximum piston displacement of system is 140 mm with the accuracy of ±0.5%. The tested samples were treated with different F-T cycles. During the test, the sample was wrapped up by a shrink-fit hose for protecting the sensors from rock fragmentation and equipped with axial displacement sensor and radial displacement sensor. The triaxial compression test needed to load the confining pressure, and the set values of confining pressure were 5, 10, 20, 30, and 40 MPa. The axial pressure began to load after that the confining pressure was loaded to the target value, and the loading method was displacement loading at a rate of 0.002 mm/s to obtain the stable deformation variation data. The central computer recorded the strength and deformation information of rock.

Fig.8 Rock mechanics machinery

2.4 Acoustic emission test

The AE system was used to monitor the initiation, propagation and coalescence behavior of rock crack in the whole loading process. According to the AE principle (as plotted in Fig.9), we calculated the AE event rate (or cumulative number), AE energy, and established AE 3D positioning to reveal the influence of the previous F-T damage on the deterioration of rock samples in these experiments. The test carried out with PCI-2 AE system of American Physical Acoustics Company (PAC), had the sampling rate of up to 40MSPS and the resolution of 18 bits, to record AE signals. As shown in Fig.10, the AE monitoring system includes AE sensors, signal acquisition, preamplifier, recording, processing and display units. The AE count and AE energy were recorded during the deformation of rock samples. In the triaxial compression test, the AE sensors were fixed on the outer wall of the pressure chamber to avoid the damage to sensors by confining pressure. The sampling interval was set at 0.1 s, and the threshold was set at 40 dB. Before the loading, a pencil lead test would be conducted in each test to ensure that the sensors worked normally, the coupling between the sensors and the sample, and to check the accuracy of the parameters setting.

Fig.9 Scheme of the AE monitoring principle

Fig.10 Connection between AE equipment and sample

3 Results and analyses

3.1 Triaxial compression test

The carrier material of Helan Mountain rock painting is subject to complex stress conditions all the year round, which aggravates the weathering degree. Therefore, the triaxial compression test is used in the laboratory to explore the strength degradation process after the F-T cycling, which can lay the foundation for establishing the damage model of Helan Mountain rock painting carriers.

3.1.1 The evolution law of stress-strain curve

The triaxial compression test of grouped rock samples were carried out by using sandstone triaxial servo testing machine. The confining pressures of the test are 5, 10, 20, 30, and 40 MPa. There are three rock samples under each confining pressure and each number of F-T cycles with a total of 75 rock samples. Due to limited space, a single rock sample is selected for analysis under each group. The stress-strain curves of triaxial compression test are shown in Fig.11.

Fig.11 Triaxial stress-strain curves of Helan Mountain rock under different confining pressures after F-T cycles (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

It can be seen from the figure that the stress-strain curve of sample is greatly affected by confining pressures and F-T cycles. The deformation and failure process of rock sample is divided into compaction stage, elastic stage, plastic yield stage, destruction stage and residual deformation stage, which is in line with the typical rock triaxial deformation process.

Compaction stage: Under the confining pressure, the primary pores and cracks close rapidly. The compaction stage of rock sample is basically in the range of 0-0.0025, and the curve of compaction stage is concave. It can be seen from the figure that the slope of the curve increases with the confining pressure in (a)-(e), which indicates that the larger the initial confining pressure is, the faster the internal pores and cracks close, the denser the overall structure is and the higher the strength of the rock sample is.

Elastic stage: The secondary pores and cracks initiate and develop with the increase of the axial load. The elastic stage of rock sample increases linearly and extends with the increase of confining pressure. As the number of F-T cycles increases gradually, the slope of curve shows a decreasing trend. Due to the inhibition of confining pressure, secondary pores and cracks in rock sample expand slowly. Under the frost heaving effect caused by F-T cycles, the mutual cementation of internal grains is weakened, and the bearing capacity is decreased.

Plastic yield stage: The secondary pores and cracks gradually develop in the rock sample, and large cracks begin to appear. Under the confining pressure of 5MPa, it can be clearly seen that the stress curve of plastic yield stage has a step-by-step rise. With the increase of F-T cycles, the number of step-by-step rise becomes more, and the curve is not smooth. However, with the increase of confining pressure, the trend of stepped rising curve is weakened, which indicates that the confining pressure effectively inhibits the crack penetration phenomenon under load inside rock sample, the crack development is relatively stable, the mutual dislocation of grains is not easy to occur, and the overall stability of rock sample is effectively improved.

Destruction stage: The speed of internal crack development and propagation increases continuously with the axial load. The large cracks connect with each other which gradually forms surface macro cracks. It can be seen from the figure that the peak stress and peak strain of rock sample decrease with the F-T cycles, indicating that the F-T cycles are negatively correlated with the peak stress and peak strain. In addition, the peak stress increase with the confining pressure, which shows that applying confining pressure can effectively reduce the strength degradation of rock under F-T cycles.

Residual deformation stage: It is a typical phenomenon in triaxial compression test that the rock sample will not completely lose its stability and bearing capacity under confining pressure after failure. It can be seen from the Fig.11 that (a)-(e) all show this typical phenomenon. On the one hand, the number of F-T cycles is negatively correlated with the residual stress. On the other hand, the residual stress of rock sample increases significantly with the confining pressure. In the residual deformation stage, due to the inhibition action of confining pressure, the fragments and large fracture surfaces produced in the destruction stage continue to provide the bearing capacity.

3.1.2 Relationship between confining pressure and peak stress

The different F-T cycling number, confining pressures, residual stress and peak stress of rock samples are listed in Table 1. The changes of average peak stress and average residual stress in different F-T cycle number and confining pressures are drawn, as shown in Fig.12.

Fig.12 Changes of peak stress and residual stress (a: changes of peak stress and residual stress with F-T cycles; b: changes of peak stress and residual stress with confining pressure)

Table 1 Average peak stress and average residual stress

It can be seen from Fig.12(a) that the peak stress and residual stress of rock samples decrease with the increase of freeze-thaw cycles. The peak stress decreased 32.68 MPa in the first 20 cycles and decreased 26.49 MPa in the last 20 cycles, indicating that the deterioration of the peak stress gradually intensifies with the increase of the freeze-thaw cycles. The residual stress decreased 19.09 MPa in the first 20 cycles and decreased 6.53 MPa in the last 20 cycles, indicating that the previous freeze-thaw cycles have a great influence on the residual stress.It can be seen from Fig.12(b) that the peak stress and residual stress of rock samples increase gradually with the confining pressure. The peak stress under 30 MPa confining pressure is 65.54 MPa higher than that under 20 MPa confining pressure, which is the largest increase and indicates that 30 MPa confining pressure can most effectively put forward the peak stress of rock samples. In the range of 10 - 40 MPa confining pressure, the residual stress increases 49.55, 40.24, 20.4 MPa, respectively every 10 MPa, indicating that the residual stress is sensitive to the change of low confining pressure.

The fitting functions of peak stress, residual stress and F-T cycles, confining pressure are:

where, σpis the peak stress, σcis the residual stress,nis the number of F-T cycles, and σ3is the confining pressure.

In the Eqs.(1) and (2), the coefficients of -1.412 and -0.652 indicate that the peak stress and residual stress are negatively correlated with F-T cycles. In the Eq.(3) and Eq.(4), the coefficients of -1.412 and -0.652 indicate that the peak stress and residual stress are positively correlated with confining pressure. The all four correlation coefficients are more than 0.9, which signify that Eqs.(1)-(4) can reasonably deduce the variation characteristics of test results.

3.2 Analysis of acoustic emission characteristics

The triaxial compression test of Helan Mountain rock paintings carriers after F-T cycling is carried out. The probe is arranged on the outer wall of pressure chamber to monitor the deformation and failure process of the rock sample in real time, and the AE ringing count under triaxial compression is obtained. When crack initiates or friction is generated between the minerals during rock deformation, an AE event occurs due to the release of transient elastic waves in the rock. Changes in AE ringing count can reflect the crack initiation, propagation and coalescence characteristics. within rock in real-time. Through the changes of AE parameters, it is possible to infer the internal damage evolution law of the rock during the failure process[27]. Figs.13-17 illustrate the changes in AE ringing count of samples during the triaxial compression experiments.

The curve of ringing counts and energy can be obtained and the positions at which curve changes substantially are taken to be a segmented boundary. Based on this criterion, the variations in AE ringing count of samples, which have undergone F-T cycles, can be divided into five stages: initial growth stage, steady growth stage, rapid growth stage, steep increase stage and post-peak stage.

Initial growth stage: At this stage, the magnitude of AE ringing count remains small in a stable range of 0 to 5. This implies that there is no clear crack propagation during this period. This state is characterized by the closure of pores, microcracks, modifications or rearrangements of the rock grains, friction on the loading plates. Under confining pressure, the speed of pore closure is faster in the compaction stage, and the probability of mutual dislocation and collision between grains is small. So the acoustic signal received by AE probes is basically not generated, which indicates that the overall density of rock sample is high, and can be used as an index to evaluate the internal structural integrity of rock sample.

Steady growth stage: The cracks begin to develop slowly under different confining pressures. It can be seen from the figure that the AE ringing counts under different F-T cycles are dispersed with time interval in this stage. This stage is also the longest one in the AE count changing process, which further indicates that the initiation and development of internal secondary crack and pore are limited under the confining pressure. When the F-T cycle is zero, the AE ringing count is in the range of 10-209 under the confining pressure of 5 MPa, and in the range of 5-90 under the confining pressure of 40 MPa. When the F-T cycle is 40, the AE ringing count is in the range of 5-120 under the confining pressure of 5 MPa, and in the range of 5-63 under the confining pressure of 40 MPa. The growth rate of AE ringing counts decreases with the F-T cycles and confining pressure. It shows that confining pressure can effectively improve the internal structural strength of rock samples and inhibit the development of cracks.

Rapid growth stage: With the increase of load, the distribution of AE ringing counts begins to concentrate. Currently, the triaxial stress changes gradually into the plastic deformation stage with the propagation of internal pores and microcracks. When the F-T cycle is zero, the AE ringing count increases from 287 to 2 509 under the confining pressure of 5 MPa, and increases from 193 to 748 under the confining pressure of 40MPa. When the F-T cycle is 40, the AE ringing count increases from 248 to 10 898 under the confining pressure of 5 MPa, and increases from 206 to 635 under the confining pressure of 40 MPa. It can be seen that the cracks develop rapidly under load with the increase of F-T cycles. The confining pressure can effectively reduce the crack growth rate, so as to improve the stability of rock and reduce the weathering effect under F-T cycles.

Fig.13 Relationship between axial stress, AE ringing counts, and time for the tested rock samples with different F-T cycles under 5 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Fig.14 Relationship between axial stress, AE ringing counts, and time for the tested rock samples with different F-T cycles under 10 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Fig.15 Relationship between axial stress, AE ringing counts, and time for the tested rock samples with different F-T cycles under 20 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Fig.16 Relationship between axial stress, AE ringing counts, and time for the tested rock samples with different F-T cycles under 30 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Fig.17 Relationship between axial stress, AE ringing counts, and time for the tested rock samples with different F-T cycles under 40 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Steep increase stage: The secondary micro pores and cracks gradually develop from the inside to the surface, causing the formation of fracture surface which affects the structural stability. The sharp increase of AE ringing counts is synchronized with the peak stress. It can be seen from the figure that the stages of stress failure under different F-T cycles and confining pressures are one-to-one corresponding to the drastic changes of AE ringing counts. When the F-T cycle is zero, the AE ringing count increases from 272 to 22 053 under the confining pressure of 5 MPa, and increases from 270 to 24 189 under the confining pressure of 40MPa. When the F-T cycle is 40, the AE ringing count increases from 370 to 20 365 under the confining pressure of 5MPa, and increases from 375 to 23 065 under the confining pressure of 40 MPa. It shows that the growth rate of AE ringing counts increases with the confining pressure, which indicates that the crack distribution is more concentrated and the acoustic signal is easier to be collected under higher confining pressure.

Post-peak stage: After failure, the rock sample still maintain a certain bearing capacity under the action of confining pressure, and will not lose stability immediately. This phenomenon is reflected by the AE real-time monitoring, which is that the AE ringing counts still maintain a certain level after the peak and will not be completely reduced to 0. It can be seen from the figure that under the different F-T cycles and confining pressure, the AE ringing counts are consistent with the stress change curves and maintain in a certain range. When the F-T cycle is zero, the AE ringing count is maintained at about 1 478 under the confining pressure of 5 MPa, and maintained at about 1 059 under the confining pressure of 40 MPa. When the F-T cycle is 40, the AE ringing count maintained at about 495 under the confining pressure of 5 MPa, and maintained at about 607 under the confining pressure of 40 MPa. The results show that when the rock sample continues to be loaded after failure, the mutual dislocation and penetration of the fracture surface and macro cracks can be effectively captured by the AE probes, and the formation of large cracks, which affecting the rock integrity, can be judged by the phenomenon that the AE ringing count remains at a certain level.

3.3 Failure analysis

Fig.18 Triaxial failure diagram and 3D location of rock samples under 5 MPa confining pressure

Fig.19 Triaxial failure diagram and 3D location of rock samples under 10 MPa confining pressure

Fig.20 Triaxial failure diagram and 3D location of rock samples under 20 MPa confining pressure

Fig.21 Triaxial failure diagram and 3D location of rock samples under 30 MPa confining pressure

Fig.22 Triaxial failure diagram and 3D location of rock samples under 40 MPa confining pressure

To explore the expansion direction and status of the internal rock cracks, the distribution law of positioning points in the AE three-dimensional location are analyzed. The triaxial compression failure morphology maps, sketch maps and AE location maps of rock samples in different F-T cycles are corresponded one by one, as shown in Figs.18-22.

Figs.18-22 shows the failure diagram and AE 3D positioning diagram of triaxial compression of Helan mountain sandstone under different F-T cycles. It can be seen from the figure that with the increase of confining pressure, the distribution of 3D positioning points in the process of rock sample failure is concentrated and regular, and has a good corresponding relationship with the rock sample failure diagram. In the 3D positioning of the rock samples without F-T cycle, the number of positioning points is large, widely distributed and scattered. It can be seen from the rock sample failure diagram that in addition to forming the main fracture surface, there are many micro cracks distributed around the rock sample, accompanied by a certain degree of debris falling. This is because the internal structure of the rock sample without F-T cycles is relatively dense, the cementation between grains is high, and the dislocation between grains is frequent under load. After the number of F-T cycles which is greater than 20, the AE 3D location diagrams show that there are few 3D location points which are mainly distributed around the main fracture surface of the rock sample. And the rock sample failure diagrams reflect that there are few micro cracks on the rock sample surface, but only macro cracks which leading to samples’ complete instability. The failure forms of rock samples under triaxial compression are mainly conical shear failure with a certain cone angle. Under different F-T cycles, the surface macro crack and axis angle gradually increase with the confining pressure, and the cone angles of the two cones formed by the main fracture surface also increase with the confining pressure. Compared with the research results[28]under low confining pressure (less than 5 MPa), the surface cracks of rock samples under high confining pressure are less, and the failure is concentrated in one or two main cracks. The high confining pressure makes the rock samples still maintain a certain integrity after failure.

The damage characteristics of rock samples reflect the damage characteristics of rock paintings to a certain extent. In the process of freeze-thaw cycles, the water dissolution and frost heaving force increases the rock’ pores and reduces the rock’ strength. Under the crustal stress and rock mass self weight, the cracks and exfoliation appear on the surface of rock paintings, and gradually evolve into more serious damage such as collapse and material loss.

4 Damge model

4.1 Determination of damage variables

The change of elastic modulus is selected as the F-T cycle damage variableDn:

where,E0is the rock elastic modulus before the F-T cycle, andEnis the rock elastic modulus afternF-T cycling.

By selecting the cumulative AE ringing count and considering the critical damage variable, the AE damage variableDCis defined as:

where,Cis the accumulative AE ringing count generated when the particles inside the rock are completely broken,Cdis the cumulative ringing count when the strain is ε, σmis the peak stress, and σnis the stress corresponding to the maximum ringing countC.

The constitutive relationship of rock stress-strain based on AE damage variable is expressed as:

where, σ is the axial stress, and ε is the axial strain.

The constitutive relationship of rock stress-strain based on total damage variable is expressed as:

where,Dis the total damage variables.

In the past, when established damage variables, scholars only considered macroscopic variables[29-31], such as elastic modulus, longitudinal wave velocity, and elongation, or only mesoscopic variables[32,33], such as AE ringing count, AE cumulative energy, and AE energy probability density. The above-mentioned researches rarely combine the two to characterize the damage variables, which made it impossible to achieve a cross-scale definition of damage variables, this article combines Eqs.(6)-(9) to define the total damage variableDfor the first time:

The above formula shows that the damage of F-T rock increases nonlinearly with loading. The damage variable defined by the coupling of elastic modulus and cumulative ringing count not only contains the damage caused by the increase of pores and the loss of cement under the action of temperature-water cycles, but also contains the time evolution damage caused by the development and extending of internal rock cracks under loading condition. The two kinds of damage affect each other, and the coupling effect makes the total damage variableDweakly decrease.

According to Eq.(10), the curves of damage variable and stress change of the sandstones, which is the carrier material of Helankou rock art, can be obtained as shown in Figs.23-27. It can be seen from the figure that the damage variable curves in the rock failure process are mainly divided into four stages: initial growth stage, stable growth stage, accelerated growth stage and rapid growth stage.

Initial growth stage: When the rock sample does not experience the F-T cycles, it can be considered that the damage variable increases gradually from 0. With the increase of the load, the primary defects of the rock sample begin to fill and close with each other, and this stage ends when the axial strain is about 0.0006. The damage value of the initial stage increases gradually under the action of F-T cycles and confining pressure. It can be shown that there is a certain degree of damage in the rock sample which experiences the F-T cycles before the initial stage of loading, and the damage does not expand immediately but increases slowly under the action of confining pressure.

Stable growth stage: When the load increases gradually, the micro cracks between the rock joints and bedding, and the pores between the particles begin to stagger each other. The damage increases gradually but gently. At this stage, the damage variable increases from 0.00067 to 0.0014, and ends when the axial strain is about 0.0018. With the increase of F-T cycles and confining pressure, the damage change is relatively stable. The damage variable under confining pressure increases steadily based on F-T damage, but it is relatively weak. In this stage which corresponds to the elastic stage of stress change, the development of microcracks and micropores is relatively slow and stable, the irreversible damage is basically not produced, the overall strength and integrity of rock samples are high, and the deterioration degree of damage to rock is weak.

Fig.23 Damage variable curves with different F-T cycles under 5 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Fig.24 Damage variable curves with different F-T cycles under 10 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Fig.25 Damage variable curves with different F-T cycles under 20 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Fig.26 Damage variable curves of sandstone in different F-T cycles under 30 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Fig.27 Damage variable curves of with different F-T cycles under 40 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

Accelerated growth stage: With the increase of load, the secondary microcracks and pores in the rock sample develop to be complete, the pores and microcracks are gradually extended to form permanent cracks, the damage is gradually accelerated. In this stage, the damage variable increases by 79% from 0.0014 to 0.0025, and ends when the axial strain is about 0.0034. It indicates that the peak strength of rock sample under confining pressure does not mean that the damage reaches the maximum, and the damage will further increase when it enters the plastic flow stage. The confining pressure greatly improves the overall stability and integrity of rock samples.

Rapid growth stage: In the continuous application of load, the secondary cracks and pores of rock samples accumulate more and more, forming many permanent cracks and fissures. These cracks expand rapidly in the failure stage and post peak stage, forming large fracture surface, macro cracks and fragment spalling. In this stage, the damage variable increases by 500% from 0.0025 to 0.015 and the growth rate reaches the maximum in the damage variable curve, which is the last stage in the test process. The maximum value of damage variable increases gradually with the number of F-T cycles, which proves that the F-T cycle is an important cause of rock damage.

It can be seen from the four stages that the damage variable based on macro elastic modulus and meso AE parameters can well reflect the damage evolution process of Helan Mountain rock, and the Eq.(10) can be used as the characterization formula of rock damage variable.

4.2 Establishment and verification of damage model

For the triaxial constitutive model of rock, Yang Yongjie[34]introduced the lame constant and applied the generalized Hooke’s law of linear elasticity, obtaining the following formula :

where, λ andGare the lame constants, and μ is the Poisson’s ratio.

By substituting lame constant λ,Ginto Eq.(11), the triaxial constitutive equation of rock is obtained as follows:

where, μ is the Poisson’s ratio.

Eq.(10) is substituted into Eq.(12), and the theoretical stress-strain relationship of Helan Mountain sandstone is obtained. Fig.28 shows the comparison diagram between the test result curve and the theoretical calculation curve, taking 20 MPa confining pressure as typical examples. It can be seen from the Fig.28 that the theoretical results of stress-strain curves of Helan Mountain sandstone under different F-T cycles are basically consistent with the experimental results. From the theoretical results, the five stages under triaxial compression are reflected in the curves. It further shows that it is very effective to define the damage variable of Helan Mountain rock painting carriers based on the degradation of elastic modulus under F-T cycles and the change of AE cumulative ringing count under load. The three fields of temperature, water and force, realizes the leap from macro to micro, and the constitutive model can provide a very valuable reference for the subsequent study of the weathering damage of Helan Mountain rock paintings.

Fig.28 Comparison of stress-strain relationship between theory and test of Helan Mountain rock with different F-T cycles under 20 MPa confining pressure (a: F-T=0, b: F-T=10, c: F-T=20, d: F-T=30, and e: F-T=40)

5 Conclusions

a) The compression deformation process of rock is affected by F-T cycles and confining pressure. In the process of F-T cycles, the cementitious material in the rock sample is lost, and the bedding and inter joint fissures are increased. Meanwhile the pores among particles are expanded. The overall strength decreases and the fracture mode is gradually transformed from brittleness to plasticity. Taking 20 cycles as the boundary, the early F-T cycles weaken the peak stress and residual stress of rock samples more. The confining pressure provides rock samples with a certain ability to resist deformation and inhibit crack development and propagation. The peak stress and residual stress of rock samples increase gradually with the confining pressure. Specifically, the largest proportion of peak stress increase occurs near 30 MPa, and the largest proportion of residual stress increase occurs near 20 MPa.

b) The AE technology can be used as a monitoring way for the propagation and development of cracks in rock paintings. Specifically, the AE signals can effectively illustrate the deformation characteristics of rocks. The temporal and spatial evolution law of AE counts well corresponds to the loading and failure process of the rock samples. The AE ringing count increases greatly to the maximum in the failure stage of rock sample. The AE 3D positioning technology can accurately be used as capturing the development position and direction of internal cracks and pores of rock samples. It can be seen from the results of AE 3D positioning that the major failure form of rock samples under triaxial compression is conical shear failure.

c) The damage variable based on elastic modulus and AE cumulative ringing count, is defined to achieve the macro and micro quantities to characterize the damage. The established damage constitutive relationship has a better fittingness of coincidence between the theoretical calculation results and the test results, which shows that it is reasonable to use this equation to evolve the test process. It provides a scientific model reference for the protection of F-T damage of Helan Mountain rock paintings.