Investigating rock mutation characteristics and damage state warning model based on energy conversion

Rock engineering achieves the secondary stress balance through rock mass structure adjustment, where energy conversion is throughout and associated closely with rock deformation and damage. In this study, a series of triaxial compression tests were conducted on red sandstone to investigate these features. The results showed that the damage state of red sandstone specimens presented five stages under different confining pressure, corresponding to the multistage evolution characteristic of the energy conversion. In the case of the dissipation energy conversion ratio (η), it showed five stages: a gradual increase, decreasing gradually and reaching a minimum value, increasing gradually, increasing with growth rate, and accelerated growth, therein the strong nonlinearity reflected the stability and instability of the internal structure of the rock and had the basic characteristics of the mutation theory, therefore the damage state warning model was established on just that. The relation between the η and time fitted by a four‐rank potential function had a fitting parameter (R2) larger than 0.9, and the bifurcation set of the η calculated by the damage state warning model had twice stages less than 0. The second stage, which occurred near the minimum value of the η and run through the plastic deformation stage, could be used to predict rock damage and fracture, and it was proven feasible by acoustic emission (AE) precursor and better than AE warning. This research can enrich the methods for identifying rock damage state and provide reference for revealing the occurrence and development mechanism of various rock instability disasters.

of energy-related hazard sources, the energy or energy carrier, which may injure people and damage property due to accidental release or conversion, is regarded as the class Ⅰ hazard source, and the various factors that may cause constraint failure are regarded as the class Ⅱ hazard source. 2,3In underground space, tunnel tunneling, mountain road construction, and another rock engineering, the objects of construction are all kinds of rocks.Before being affected by construction activities, the rock mass, which is in a state of three-dimensional equal stress constraint, is the energy carrier, and the internal accumulated energy is in a state of equilibrium.After the construction activities, the constraint state of the rock changes, and the energy in the rock accumulates, dissipates, and releases. 4,5Then people take various support methods to limit the accidental release and conversion of energy and make the rock stable again, while there will be varying degrees of disaster accidents when it exceeds the artificial control.Herein rock instability process contains both the class Ⅰ hazard source and the class Ⅱ hazard source.Therefore, studying the physical mechanism of rock deformation and damage from the standpoint of energy, can effectually identify the root which causes the disasters and put forward reasonable precautions for engineering design and construction.
Currently, most researchers mainly focus on energy evolution characteristics and influenced factors by conducting uniaxial and triaxial static or dynamic loading tests.][8][9][10] Generally, the energy curves show a prominent nonlinear evolution characteristic: because of the low stress at the initial loading stage, the imported energy is small, the elastic strain energy and the dissipated energy gently change; then the elastic strain energy increases in harmony with the imported energy until reaching the yield point, that is, most of the imported energy is mainly stored in the form of elastic energy, the dissipated energy increases with a lower growth rate; the more microcrack growth, the more energy consuming, and the rate of energy dissipation gradually increases with the loading until it sharply increases near the peak stress, and most of the elastic strain energy releases. 11,124][15][16][17][18][19] And using the evolution characteristics, scholars have explained the occurrence and development mechanism of various rock instability disasters.For example, rock burst, which occurs in the water resources and hydraulic tunnels, the underground space and the well roadway engineering, is a dynamic fracture process where the elastic strain energy rapidly and violently releases 20,21 ; coal bumps is a nonlinear dynamic process of energy stable accumulating and unsteady releasing when coal mass which is in specific geological environment deforms and fractures under mining activities 22,23 ; coal and gas outburst is an speedy crushing which is caused by the sudden release of the potential deformation energy of coal mass near the working face and the internal energy of gas. 24n general, the work of external force on the rock will be converted into elastic strain energy and dissipated energy, so rock deformation and damage are closely related to the evolution characteristics of elastic strain energy and dissipated energy.However, it is difficult to intuitively predict when rock enters the damage state only from the evolution characteristics of the energy curves.Taking rock under load as a system, some scholars have described the evolution of energy by using catastrophe theory and explored the secret behind the energy sequence. 25Cai et al. introduced catastrophe theory into identifying rock engineering instability, and achieved determining the possibility of stability of the rock engineering by using the finite element program 26 ; Yang et al. derived the bifurcation equation of energy in rock engineering by introducing cusp catastrophe model, and the value of the bifurcation set was using to determine whether the energy was is stable, critical, or unstable, further the stope state could be judged according to the energy state 27 ; Zhang et al. and Wang et al. concluded that the energy release process of the rock specimens was a typical catastrophe process, and the cusp catastrophe model could be used to fit the elastic energy and analyze its sudden change 28,29 ; Zhou et al. used the catastrophe theory to analyze the total energy of the goaf support system, and concluded the necessary and sufficient condition for system instability 30 ; Guo et al. considered the plastic strain energy as an index to investigate the progressive failure phenomena of the tunnel roof, and derived the criterions of collapsed and spalling for shallow tunnel roof 31 ; Wang et al. constructed the potential function based on the accumulative energy of acoustic emission (AE) parameters and predicted the hollow cylinder rock failure by using the cusp catastrophe model. 32However, these research achievements mainly considered sudden changes in energy evolution and rarely involved jumps, discontinuities, and sudden qualitative changes in energy conversion.
The primary objective of this study is to take the energy conversion in the triaxial compression test of red sandstone as an entry point.Below, the energy conversion is quantified by the ratio of the dissipated energy or the elastic strain energy to the imported energy at any stress level, the evolution process and the mutation characteristics of the energy conversion are investigated, the damage state warning model based on energy conversion is built by using the cusp mutation theory, the unstable state of energy conversion ratio is identified to predict rock damage state.Additionally, the precursor information of AE characteristic parameters with high recognition, such as the ringing count rate and energy, are used to verify the rationality of the early warning information obtained from the energy conversion warning model.The results can further enrich energy analysis methods for rock mechanical properties and provide a new idea and practice for the stability analysis and disaster evaluation of engineering rock mass.

| Experiments
In the present study, red sandstone was chosen for the experiment due to its homogeneous and isotropic nature.The experiment specimens were machined from the same red sandstone block exhibiting fine consistency and integrity with no obvious joints and cracks.According to the ISRM suggested method, all specimens were carefully produced standard specimens with 50 mm in diameter and 100 mm in height, ground to produce flat parallel surfaces with ±0.02 mm parallelism.As shown in Figure 1A, four groups of specimens (Groups A, B, C, and D, three specimens in each group) were assigned to test with the confining pressures of 5, 10, 20, and 30 MPa, respectively.
The RLJW-2000 triaxial shear-creep testing system (as shown in Figure 1B), with a 2000 kN maximum axial loading capacity and a 50 MPa maximum confining pressure, was used to implement the triaxial compression tests.The axial extensometer and the chain-type lateral extensometer (see Figure 1C) were used to collect the axial deformation and the circumferential deformation of the rock specimens, respectively.The AE monitoring system (as shown in Figure 1D, manufactured by Vallen System) recorded the AE parameters (counts, accumulative counts, energy, and so on) by AE sensors during rock deformation and damage, as shown in Figure 1B two AE sensors were arranged with 150°angle on the head of the press.The AE monitoring system preamplifier gain was set to 40 dB, the threshold value was set to 50 dB, and the sampling rate was 10 MHz.First, rock specimen was fixed with 2-3 MPa axial pressure.Then, the confining pressure was loaded to the set value and remained constant, and the axial and radial displacements were cleared.Finally, the displacement control method was adopted with 0.005 mm/s loading rate, and the axial pressure was loaded until the rock specimen broken, and the stress-strain curves and AE information were recorded.
The stress-strain curves reflect the progressive processes of pore compaction, microcrack closure, initiation, propagation, coalescence, and instability in the rock.The σ 1 -ε 1 and the σ 1 -ε 3 curves of red sandstone specimens under four different confining pressures were collected in Figure 2. As shown in Figure 2, the peak stress and the residual stress of each specimen increased with the increase of confining pressure, the overall stress-strain curve showed an obvious nonlinearity, and each damage stage had specific physical and mechanical properties.Using the volumetric strain method, the crack volumetric strain method, the lateral strain method, the lateral strain response method, or the AE method, the damage state of rock is normally divided into five stages, including crack closure stage (I), linear-elastic deformation stage (II), crack growth stable stage (III), crack growth unstable stage (IV), and postpeak damage stage (V), and corresponding to the crack closure stress (σ cc ), the crack initiation stress (σ ci ), the crack damage stress (σ cd ), and the peak stress (σ p ). [33][34][35][36][37] As shown in Figure 3, the lateral strain method (drawing a tangent line [red dotted line in Figure 3] on the curve of axial stress-lateral strain, and the points where the lateral strain deviated from linearity are the σ cc and the σ ci , respectively) and the volumetric strain method (marking the turning point on the curve of volumetric strain-axial strain, and the corresponding stress value is the σ cd ) were chosen to determine the characteristic stress, all characteristic stress were counted in Table 1 and shown in Figure 4.As can be seen from Table 1, the σ p , the σ cd , and the σ ci increased with the increase of confining pressure, indicating a positive correlation with the increase of confining pressure.
While the σ cc decreased with increasing confining pressure and was negatively correlated with confining pressure.The σ cd /σ p and the σ ci /σ p had little change with confining pressure; the range of the σ cd /σ p varied from 76.44% to 80.76%, with an average value of 78.76%; the σ ci /σ p varied from 46.75% to 58.50%, with an average of 51.80%.Due to the degree of original cracks closure increasing with increasing confining pressure, the σ cc /σ p decreased with increasing confining pressure.As shown in Figure 4, the different stages had different characteristics for the stress-strain relationship, and it manifested an upward-concave trend at the crack closure stage, an approximately linear at the linear-elastic deformation stage.While it gradually deviated from a linear direction at the crack growth stable stage and showed a clear upward-convex trend at the crack growth unstable stage.

| Energy conversion characteristics
The whole deformation and damage process of the rock is the internal defects evolution, propagation, and connection driven by energy. 6,38Compared with the deformation parameters (E, ν) and the strength parameters, the energy analysis method provides a new way to study the rock mechanical properties and has been widely used in recent years.It is assumed that no thermal transmission occurs between the rock specimen and the external environment in the triaxial compression test, and according to the First Law of Thermodynamics, the relationship of energy parameters for per unit volume of the rock specimen can be described as follows 25 : where W im , W el , and W di are the imported energy, the strain energy for the elastic deformation, and the F I G U R E 3 Evaluation damage evolution state.
dissipation energy for the plastic deformation and the internal defects evolution, propagation, and connection, respectively.
In the literature, the W im and the W el are always determined by loading and unloading stress-strain curve, respectively, and the formula can be described as follows 39 : where σ 1 and σ 3 are the axial stress and the confining pressure during the loading and unloading process, respectively; ε 1 and ε 3 are the axial strain and the radial strain during the loading process, respectively; ε 1 el and ε 2 el are the recoverable axial strain and the recoverable radial strain during the unloading process, respectively.Generally, the unloading process at any stress level is assumed to approximately unloads with the unloading elastic modulus as the slope, and the integrals to W el can be expressed as follows 39 : where E u and ν u are the unloading elastic modulus and the unloading Poisson's ratio, respectively.The development of energy conversion at any stress level could reflect the degree of elastic deformation and plastic damage in the rock.Therefore, the energy conversion ratio is proposed to evaluate the ability to T A B L E 1 Characteristic stress under different confining stress.

Confining pressure (MPa) Characteristic stress (MPa)
The ratio of characteristic stress to σ p (%)  | 1225 store elastic energy or the degree of plastic damage of a rock and can be calculated as follows: where λ and η are the conversion ratio of elastic strain energy and the conversion ratio of dissipation energy, respectively.Further, the energy parameters (W im , W el , and W di ) and the energy conversion ratio (λ and η) of red sandstone specimens under four different confining pressures were calculated and were plotted in Figure 4. Figure 4, in terms of the damage state, showed that the energy parameters presented a nonlinear evolution under four different confining pressures.The W im increased with increasing load in the whole damage process.During the crack closure stage and the linear-elastic deformation stage, the most of W im converted to the W el , and the increase of the W el was synchronized with the W im .At the same time, the W di presented an approximately smooth increase with the slight plastic deformation.After that, the trend of the W el which deviated from the W im became apparent, and the W di gave an increasing tendency.With the unstable growth of crack, the growing rate for the W el became decreasing and reached the max value at peak stress, and the growing rate for the W di increased.After the peak stress, the W el was released for the crack connection, and the most of W im converted to the W di ; the W di increased significantly, and the W el decreased rapidly.
The energy conversion ratio can enrich the energy analysis method to understand the deformation and damage in the rock.Equation (5) showed that λ + η = 1, and the η would be used for analyzing the energy conversion.A more significant nonlinearity could be obtained in Figure 4.The internal defects closure would dissipate some of the energy at the crack closure stage, and it led the η to a gradual increase.However, this condition decreased with the increasing confining pressure, and the W im mainly converted into the W el (i.e., λ > η).With entering the linear-elastic deformation stage, the η decreased gradually and reached a minimum value near the σ ci .Thereafter, the η increased gradually with the new crack initiation, and its growth rate increased with the damage development in the rock.The loading stress exceeded the peak stress, the fracture of the rock specimens dissipated most of the energy, the η exhibited an accelerated growth, and the W im was mainly converted into the W di , that is, λ < η.Therefore, the energy conversion ratio nonlinearly developed in parallel with the deformation and damage in the rock.

| Mutation characteristics in energy conversion
The strong nonlinearity of energy conversion was the result of competition between energy storage and energy dissipation, and it reflected the stability and instability of the internal structure of the rock, that is, rock deformation and damage process had the essential characteristics of the mutation theory. 40,41As described as follows: (1) Energy conversion is closely related to rock damage and instability.The energy conversion dominated by elastic strain energy predicts the existence of stable structure in rock, and the energy conversion dominated by dissipated energy predicts the loss of stable structure.Thus, energy conversion can cause rock to appear two different states: stable state or unstable state, that is, rock deformation and damage process have multimodal properties.(2) In the process of energy conversion, energy conversion dominated by elastic strain energy can be transformed to be conquered by dissipated energy, which means that rock changes from a stable structure to an unstable system, and the internal units in rock constantly break and maintain unsteady balance.But this balance cannot really achieve, that is, rock deformation and damage process have inaccessible properties.(3) Energy conversion in rock deformation and damage process has obvious sudden transitions; small changes of the imported energy can make energy conversion being dominated by elastic strain energy or by dissipated energy, that is, from one stable state to another steady state.The apparent manifestation of this feature is that the energy conversion has multiple stages, corresponding to the multistage characteristic of the stress-strain curve, that is, rock deformation and damage process have sudden transitions.(4) Near the extremum of energy conversion, even a tiny perturbation can reverse the energy conversion.When rock enters the crack growth unstable stage, a tiny perturbation can also cause a polarization of the evolution rate of energy conversion.It indicates that rock deformation and damage process have divergence.(5) The previous study indicates that any physical system will exhibit hysteresis when it cannot strictly repeat a process of change in reverse.Rock, as a specific dissipative structure, is no exception, and the irreversibility of its internal energy conversion is bound to cause the irreversibility of macro deformation and damage.Thus, rock deformation and damage process have hysteresis.Because the deformation and damage process of rock have prominent mutation forms, such as multimodality, inaccessibility, sudden transitions, divergence, and hysteresis.Thus, it is reasonable to use the cusp catastrophe theory to describe the deformation and damage process of rock.

| Cusp catastrophe model
The cusp catastrophe model is one of the seven elementary catastrophes, and it is commonly used to analyze engineering stability. 42It has two control variables (u and v), and one state variables (x).Its potential function V(x) can be expressed as follows: In the cusp catastrophe model, the equilibrium surface equation follows V′(x) = 0, that is, the first derivative of the potential function V(x); the singularity set equation follows V x ″( ) = 0, that is, the second derivative of the potential function V(x).The bifurcation set of cusp catastrophe model should meet V′(x) = 0 and V x ″( ) = 0; based on this, the bifurcation set equation can be expressed as follows: The equilibrium surface, bifurcation set, and singularity set of the cusp catastrophe model can be described in Figure 5. Figure 5 shows that the bifurcation set is the projection of the singularity set on the control variables space in the equilibrium surface.The equilibrium surface has a top, middle, and bottom page, representing three different equilibrium positions, that is, the top page and bottom page are stable equilibrium position, and the middle page is unstable equilibrium position.When u > 0, a phase point moves from the top page to the bottom page (or from the bottom page to the top page), the changes of u and v almost always cause a smooth change of x.When u < 0, the u and v satisfy the bifurcation set equation (i.e., Δ = 0), and the phase point falls on the edge of the middle page, and the system is in a critical state between stability and instability; while Δ > 0, the system is in a stable state; and Δ < 0, the system is in an unstable state.

| Damage state warning method based on energy conversion
Based on the above analysis, the energy conversion curves were applied for the state variable to estimate the rock sample failure.Firstly, the function of λ-time and η-time is established, respectively.Then, it is transformed to meet the potential function (i.e., V(x)) of the cusp mutation model, and the u and v are obtained.Finally, the bifurcation set of energy conversion in time series is calculated to evaluate the state of rock deformation and damage.The detailed steps follow: Due to the displacement control method in the test, the deformation and time satisfy a specific relation, so energy conversion which is in strain space can be converted into time-space, and the mapping relationship between energy conversion and time is expressed as f(t).Then, Taylor's formula is used to expand f(t), and the order of the Taylor polynomial is determined by precision analysis. 32,43,44The mapping relationship between energy conversion and time is expressed by a four-rank Taylor polynomial as follows: where η(t) is the mapping relationship between the elastic strain energy conversion ratio and time; λ(t) is the mapping relationship between the dissipation energy conversion ratio and time; t and r are time, and index of expanded form, respectively.
F I G U R E 5 Equilibrium surface, bifurcation set, and singularity set of the cusp catastrophe model.
, and Equation ( 8) can be converted as follows: where m j and n j are time-related undetermined coefficients, which are determined by the least-square fitting method; j = 0, 1, Further, set t = x-m 3 /4m 4 in η(t), and t = x-n 3 /4n 4 in λ (t), and Equation ( 9) can be converted into a standard potential function form similar to Equation (6).
According to Equation (10), the control variables (u and v) of the standard potential function established by energy conversion are expressed as follows: Finally, according to Equation (7). the bifurcation set expression of rock damage state warning model based on energy conversion can be derived as follows: ) ( )

| Bifurcation set of dissipation energy conversion ratio
Section 2.3 analysis showed that the λ and η characterized rock deformation and damage process in the same way, so only the η was taken as the research object to carry out the mutational prediction and the mutational characteristic analysis in this section.According to the loading rate, the evolution process of the η under different confining pressures in Figure 4 was converted into time-space, see Figure 6.Then Equation ( 9) was used to fit the η-time curves by using Origin software, and the fitting results were summarized in Table 2, and the R 2 indicated that Equation ( 9) could be used to express the evolution process of the η in time series.Next, according to the deduction of the bifurcation set, the equal data interval (i.e., 100 data points, corresponding to the time interval was 14.5, 13.5, 14.5, and 19 s for the confining pressure 5, 10, 20, and 30 MPa, respectively) was chosen to carry out the least squares fitting for dissipation energy conversion ratio at different moments, and the n j was obtained.Then, Equations ( 11) and ( 12) were used to calculate the bifurcation set (Δ λ ) of the dissipated energy conversion ratio at different moments.Moreover, in view of the extensive variation range in time series, the Δ λ was performed logarithmic conversion by Equation ( 13) with maintaining sign.The Δ′-time curves of the dissipation energy conversion ratio under different confining pressures were plotted in Figure 6.
The damage state warning aimed at the prepeak stress stage, it could be observed for all the specimens that the Δ′ of dissipation energy conversion ratio had two jumps to a negative value (i.e., area A and area B in Figure 6), which was directly related to the strong nonlinear evolution of energy conversion.The first jump appeared at the initial stage where the η started to decrease gradually, which could be considered as a false instability and was not worthy of our attention.While the second jump occurred near the minimum value of the η, and it continued until the η exhibits an accelerated growth, where the energy conversion which was dominated by elastic strain energy was broken, and the condition of the energy conversion dominated by dissipated energy indicated that the specimen entered the unstable status.Moreover, the duration of two jumps had confining pressure effect, especially the second jump, its duration increased with the increase of confining pressure, and it was 59, 94, 115, and 139 s for the confining pressures of 5, 10, 20, and 30 MPa, respectively.

| Relation with damage state
Rock, because of microcrack closure, initiation, propagation, and coalescence, tend to show multistage evolution, which implies that the Δ′ jumps of the η in the previous section are related to the rock damage process.As shown in Figure 6, due to microcrack closure and stable solid skeleton formation, there was a discontinuous change between the approximately linear decline and the gradual decline of the η.So, the first jump generally occurred at the early of linear-elastic deformation stage for all the specimens, but it inclined to crack closure stage under the low confining pressure.It could be regarded as the formation of stable energy storage structure in the rock, that is, entering the linear-elastic deformation stage.
When σ 3 = 10, 20, and 30 MPa, the second jump of the specimens began at the early of crack growth stable stage, crossed the crack growth unstable stage, and ended at the peak stress σ p .While σ 3 = 5 MPa, there was a difference for the beginning of the second jump, it began at the crack growth unstable stage.Currently, the external load had exceeded the σ ci of the specimens, and the new crack in the rock specimens began to develop, it indicated that the stable internal energy storage structure lost balance and the stable internal dissipated structure was formed.So, there was another discontinuous change between the gradual decline and the gradual increase of the η.That could be considered as the beginning of the rock instability, and its duration implied the progression of damage rock damage.

| Verification with AE
In addition, the process of rock deformation and damage is accompanied by a series of physical parameters, such as sound, electricity, and heat, which show the characteristics of fluctuation across scales, critical sensitivity, and power law singularity.AE is a phenomenon where rock elastic strain energy is released in the deformation and damage process, and it is a powerful method to detect microcracks before macroscopic failure and track crack propagation. 45The AE count rate, which is times in unit time that the signals exceed the present threshold, can reflect the fracture developing in rock.And its quiet period or sudden increase, which appears at the plastic deformation stage, usually is taken as a precursor characteristic. 46,47Figure 7 showed the curves of the AE count rate for the specimens with σ 3 = 5 and 30 MPa.At the initial loading stage, the crack in the rock were closed, and the AE count rate fluctuated, where the Δ′ of the η was less than 0 for the first time.Then, the AE count rate kept a relatively minor variation before the peak stress.Near the peak stress, the AE count rate appeared a clearly decreasing, that is, quiet period which was marked in Figure 7.In the quiet period, the Δ′ was less than 0 for the second time and turned from negative to positive, and it indicated the formation of stable dissipative structure.Moreover, the starting time of the second less than 0 of the Δ′ was ahead of the quiet period, that was to say the second less than 0 of the Δ′ was better than the quiet period of the AE count rate for predicting rock damage.AE energy, which is the total absolute energy of all events received by the AE sensors, reflects the elastic energy release when the cracks in rock are generated and propagated, and the step change of the cumulative AE energy rate usually predicts rock fracture. 48As shown in Figure 8, for the rock specimens with σ 3 = 5 MPa and 30 MPa, the step change of the cumulative AE energy rate corresponded to the Δ′ turn from negative to positive.Similarly, the starting time of the second less than 0 of the Δ′ was ahead of the step change of the cumulative AE energy rate, and the second less than 0 of the Δ′ was better than the step change of the cumulative AE energy rate for predicting rock damage.
Combining the above observations, the damage state warning model revealed that there were two mutations in the nonlinear evolution process of energy conversion, and those had agreeable with the early linear-elastic deformation stage and the plastic stage of the rock, respectively.The second mutation implied the beginning of the rock damage, and it could earlier predict the rock damage than the AE precursor.Therefore, the damage state warning model, which revealed the mutations of the energy conversion in rock, could be used for rock damage warning.Moreover, it only needed to monitor the rock deformation and stress.

| CONCLUSIONS
(1) The strong nonlinear evolution of energy conversion in rock was found under different confining stress, which showed five stages for the dissipation energy conversion ratio: a gradual increase at the crack closure stage, decreasing gradually and reaching a minimum value at the linear-elastic deformation stage, increasing gradually at crack growth stable stage, increased with growth rate at crack growth unstable stage, and an accelerated growth exceeding the peak stress.The process of energy conversion had the essential characteristics of the mutation theory: multimodal properties, inaccessible properties, sudden transitions, divergence, and hysteresis.(2) The bifurcation set of the dissipated energy conversion ratio was calculated by using the damage state warning model which was deduced based on the cusp catastrophe model.It showed that there were two jumps to a negative value; the first jump appeared at the initial stage where the η started to decrease gradually; the second jump occurred near the minimum value of the η and continued until the η exhibited an accelerated growth; the duration of two jumps had confining pressure effect, especially the second jump, its duration increased with the increase of confining pressure.(3) Compare the rock damage state, the first jump of the bifurcation set of the η occurred at the early of linearelastic deformation stage, while the second jump occurred at the plastic deformation stage, where the external load had exceeded the σ ci , so it could be considered as the beginning of rock instability.In addition, the AE precursors, such as the quiet period of AE count rate and the step change of cumulative AE energy rate, were found appearing in the process of the second jump and being behind the starting time of the second jump, those could verify that the result of the damage state warning model was feasible.