Study of Energy Evolution Law and Damage Characteristics during Uniaxial Cyclic Loading and Unloading of Sandstone

Abstract

Using the rock mechanics test (RMT) and acoustic emission acquisition system (DS9), based on the energy principle, uniaxial compression, uniaxial cyclic loading, and unloading tests are used to study the energy transformation characteristics of the process of sandstone absorbing axial strain energy, accumulating and releasing elastic strain energy, plastic deformation, and crack extension dissipation energy. The study results show the increase of loading rate, rock fracture surface increases, number of fragments increases, and size of fragments decreases; the sandstone damage process causes: shear damage to tensile shear damage and then splitting damage for change; the input energy and elastic energy increase nonlinearly with an increase of stress, and the dissipative energy is larger at the beginning of loading. After a small decrease, it enters the nonlinear growth stage, and the input energy density grows the fastest. The elastic energy density is the second fastest, and the dissipative energy density grows the slowest; with an increase of loading rate, in any deformation stage, the elastic energy density and dissipation energy density are increased, proportion of elastic energy density is decreased, and proportion of dissipation energy density is increased. Near the peak stress stage, the proportion of elastic energy decreases, and the proportion of dissipative energy increases; the damage variable stress curve of sandstone is “weakly concave”, which is consistent with the logistic function, and the damage evolution process has chaotic dynamics properties; the acoustic emission energy of sandstone in cyclic loading and unloading test has a similar variation with the theoretical calculation of dissipated energy. The cumulative energy curve shows a step-up law, and the stress corresponding to the step point is near the historical maximum stress.

1. Introduction

Rocks are a product of geological history, with non-uniform internal structure with joints, fractures, and microdefects. The strength and deformation characteristics of rocks are the basis of the theoretical calculation and design of underground engineering, water conservancy engineering, petroleum engineering, mine, geological disaster, etc. [1,2,3,4,5]. The strength and deformation characteristics of rocks are the key to establishing rock intrinsic relationships and analyzing rock deformation damage. Rock stress–strain behavior describes the inherent physical and mechanical properties of rock. However, for a particular kind of rock, the stress–strain relationship is actually a macroscopic reflection of the energy evolution within the rock. The energy evolution process inside the rock is the most essential feature of the whole process of rock destruction. From the laws of thermodynamics, it is clear that the destruction of any material is a destabilization phenomenon driven by energy, and rock destruction is no exception. In fact, in rock engineering activities, the action of external loads is the process of energy input. Part of the energy is accumulated as elastic strain energy of the rock and released at the time of damage; the other part of the energy leads to the expansion of rock cracks and produces plastic strain, which is dissipated in the form of energy, such as rock fracture surface, friction, heat, and electromagnetic radiation energies.

Energy perspectives regarding rock destruction is gradually gaining popularity among scholars and engineers. Xie Heping [6] discussed the intrinsic connection between energy dissipation, energy release, and rock strength during rock damage and pointed out that rock damage is the combined result of energy dissipation and release. Tarasov et al. [7] believe that brittleness is the ability of rock to self-sustain macroscopic failure in the post-peak stage. Based on the energy balance after the peak, Tarasov et al. proposed the ratio of the increment of fracture energy after the peak to the increment of elastic energy released after the peak as the index to evaluate the brittleness of rock. Manoj N. Bagde. et al. [8] showed that increasing the stress level of cyclic load would increase the input, dissipated, and elastic energies of rock in a single cycle. Zhou [9] studied the anisotropy of rocks, based on the strain energy density factor, through the traditional triaxial test. Q. B. Meng et al. [10] investigated the energy evolution and distribution pattern of granite during uniaxial cyclic loading and unloading. Gong Fengqiang et al. [11] obtained the linear energy storage law, on the basis of considering the energy consumption characteristics in the whole process of rock loading. Li Jiangteng et al. [12] studied the fatigue deformation, damage characteristics, and energy evolution law of red sandstone under low frequency uniaxial cyclic loading and unloading. Based on triaxial cyclic loading and unloading tests and energy principles under different surrounding pressures, Li Ziyun et al. [13] explored the energy evolution law during the loading of shale and established a strength failure criterion, based on energy mutation. From the energy perspective, Xu Jiang et al. [14] analyzed the deformation damage process of sandstone under the action of circulating pore water pressure and explored the evolution of energy absorption and release in this deformation damage process. Filimonov et al. [15] studied the effects of different loading rates on the acoustic emission characteristics of salt rocks. Zhang et al. [16] used the BPM model to analyze the acoustic emission characteristics of rocks with different compression rates. B Mahanta et al. [17] studied the effect of different strain rates on fracture toughness and the energy release rate of shale. Meng Qingbin and other scholars [18,19] explored the size effect of rock energy evolution. By rock tri-axial cyclic loading and unloading tests, Peng Ruidong [20] et al. analyzed the damage evolution behavior of coal rocks under different surrounding pressure and concluded that the energy-defined damage variables can better describe the damage degree of the rocks. From the perspective of energy dissipation, Jin Fengnian et al. [21] defined the damage variables of rocks and provided the theoretical formula for calculating the damage variables and determination of the damage threshold. Through the relationship between dissipation energy and input energy, Zhu Weishen et al. [22] defined the damage variable of the rock. By conducting triaxial cyclic loading and unloading tests on granite, Miao Shengjun et al. [23] investigated the evolution characteristics of rock dissipation, frictional dissipation, and crushing dissipation energies under cyclic loading and proposed rock damage variables, based on crushing dissipation energy. X. S. Liu et al. [24] developed a rock damage intrinsic model, in terms of energy loss.

The above studies have greatly enriched the content of rock energy theory; however, as a whole, the loading rate effects of rock energy evolution in different types of rocks and different loading rate intervals are very different. In this paper, we study the energy evolution and distribution characteristics of sandstone during uniaxial cyclic loading and unloading at three different loading rates, define the damage variables from the perspective of dissipative energy, and explore the nonlinear characteristics of the damage evolution process. The research results enrich the content of rock energy theory and provide a reference for rock damage prediction and engineering practice.

2. Materials and Methods

2.1. Calculation Method of Energy in the Test

The deformation of the rock unit under the action of external forces can be approximated as a closed-loop system, assuming that there is no heat exchange between the mechanical system and external environment. Due to the diverse forms of dissipative energy, it is difficult to monitor all of the energy. Considering the irreversibility of dissipative energy and reversibility of elastic energy, only external input, elastic, and dissipative energies are examined in this study. The following relationship can be obtained from the first law of thermodynamics:

$$U^0=U^{\mathrm{e}}+U^{\mathrm{d}}$$

(1)

$U^0$ is the input energy density, which refers to the outside transfers energy to the rock; $U^{\mathrm{e}}$ is the elastic energy density, which refers to the elastic energy stored inside the rock; $U^{\mathrm{d}}$ is the dissipated energy density, which refers to the energy dissipated by the rock during the loading process.

Corresponding to the loading and unloading stress–strain curve in the $i$-direction up to ${\sigma }^{\prime }$, as shown in the Figure 1, the dissipated energy density in this direction can be determined by the area between the loading and unloading curves, and the stored elastic energy density is determined by the area between the unloading curve and horizontal axis; the formula is as follows:

$$u_i^{\mathrm{d}}={\displaystyle {\int }_0^{{\epsilon }^{\prime }}{\sigma }_i}\mathrm{d}{\epsilon }_i-{\displaystyle {\int }_{{\epsilon }^{″}}^{{\epsilon }^{\prime }}{\sigma }_i}\mathrm{d}{\epsilon }_i$$

(2)

$$u_i^{\mathrm{e}}={\displaystyle {\int }_{{\epsilon }^{″}}^{{\epsilon }^{\prime }}{\sigma }_i}\mathrm{d}{\epsilon }_i$$

(3)

2.2. Test Apparatus and Test Program

The rock samples used in the test are from the slope of G210 national highway project. A total of 16 platforms were excavated on the slope. The rock samples are from the 13th level platform. In order to ensure the homogeneity of the rock samples, all the test samples were taken from the same large sandstone. Sandstone is gray and formed by sand cementation, fine-grained structure, and layered structure. The mineral composition is mainly quartz and feldspar. The sample has good integrity. Rock strength is closely related to longitudinal wave velocity [25], with measured average longitudinal wave speed of 6.4 km/s and average density of 2.5 × 103 kg/m. According to the recommended method of the International Society of Rock mechanics, the specimen was processed into a diameter of 50 mm and height of 100 mm. In order to ensure the homogeneity of the specimen rock samples, this test rock samples are taken out on the same large block of sandstone.

In order to explore the energy evolution law and damage characteristics of the whole process of rock damage, uniaxial compression tests and cyclic loading and unloading tests were conducted. This test uses the RMT-150C rock mechanics test system to load the specimens, as shown in Figure 2. This mechanical system can load the specimen automatically; after input, the rock information and loading command, the loading is performed automatically. The specimen is preloaded first, and the press will stop automatically when it comes into contact with the rock sample, which makes it easy to record the real loading time and avoids the rock being crushed before the test; after the specimen is destroyed, the press can unload automatically and return to the initial state, which improves the test efficiency. The test adopts stress control mode, and the loading system has 12 loading rates, in the range of 0.001 to 100 KN/S. In this study, three loading rates of 0.2 KN/S, 0.5 KN/S, and 1 KN/S were selected for uniaxial compression, as well as uniaxial acceleration and deceleration cycle, tests. The loading path is shown in Figure 3.

The DS9 acoustic emission meter is small and light, so it is easy to carry around. It does not need to be connected to a computer. Just plug in to work. It can be easily placed in vulnerable parts to monitor, and two channels can excite and trigger the signal, respectively.

3. Experimental Results and Analysis

3.1. Uniaxial Compression Test

At three different loading rates, the uniaxial compressive stress–strain curves and post-damage morphology of sandstone are shown in Figure 4 and Figure 5. The physical and mechanical parameters are shown in

Table 1. Table of mechanical parameters of uniaxial compression test.

Loading Rate (KN/S)	Peak Stress (MPa)	Peak Strain	Modulus of Elasticity (GPa)	Type of Damage
0.2	41.17	0.0129	35.42	Shear Damage
0.5	42.27	0.00787	37.44	Tension shear damage
1	44.45	0.00566	46.19	Splitting damage

. It can be seen that, although the stress–strain curves obtained at different loading rates have slightly different patterns, the sandstone undergoes four processes, from intact to damaged: first it is compacted, then it undergoes elastic and plastic deformation. The rock was finally destroyed. When the test was conducted with a loading rate of 0.2 KN/S, the average value of peak stress of sandstone was 41.17 MPa; the average value of peak strain was 0.0129, and the average value of elastic modulus was 35.42 GPa. The damage form was shear damage, with two oblique shear surfaces through the specimen, a small number of large-scale rock pieces along the shear surface, accompanied by a continuous “friction” sound. When the loading rate was increased from 0.2 to 0.5 KN/S, the average value of peak stress in sandstone was 42.27 MPa, with an increase of 0.25%; the average value of peak strain was 0.00787, with a decrease of 38.99%, and the average value of elastic modulus was 37.44 GPa. The damage form was mainly in tension shear, accompanied by splitting damage, multiple penetrating fracture surfaces, and along the fracture surface. The damage form is mainly in tension shear, accompanied by splitting damage, with multiple penetration fracture surfaces; along the fracture surface appears a large number of medium-scale rock blocks, which are accompanied by continuous bursting sound. The damage form is mainly in tension shear, accompanied by splitting damage, multiple penetrating rupture surfaces, and multiple medium-scale rock blocks along the rupture surface, which are accompanied by a continuous bursting sound. When the loading rate was increased from 0.5 to 1 KN/S, the average value of peak stress of sandstone was 44.45 MPa, with an increase of 5.16. The average value of peak strain was 0.00556, with a decrease of 29.35%, the average value of elastic modulus was 46.19 GPa, and the damage form was mainly in the form of splitting blocks, with multiple penetrating splitting surfaces; a large number of small rock blocks formed along the splitting surface. The damage is mainly in the form of splitting blocks, with multiple penetrating cleavage surfaces, and a large number of small rock pieces formed along the cleavage surface, with a continuous bursting sound, loud noise, and small amount of rock chips flying out during the damage.

From the results, it can be seen that there is a great difference in the response of the same rock at different loading rates. As the loading rate increases the strength of sandstone increases, the elastic modulus becomes larger, and the peak strain is reduced. As the loading rate increases, the number of fragments increases, scale becomes smaller, and sound is more violent after the sandstone is broken. The damage form is “shear damage tensile shear damage → splitting damage”.

3.2. Cyclic Loading and Unloading Energy Evolution Law

Rocks are natural geological materials containing joints and microcracks, which cause non-ideal elastic deformation. As the rock specimen is loaded, a plastic hysteresis occurs in the stress–strain curve, and the envelope area of the curve is equal to the dissipated energy. As the cyclic loading and unloading proceeds, the stress corresponding to the unloading point gradually increases. Correspondingly, the envelope area of the plastic hysteresis loop also increases, and the dissipated energy is used for the development and expansion of cracks, which causes damage to the structure and leads to a decrease in the ability of the rock to store elastic energy. With the increase of stress at the unloading point, the energy absorbed by the rock gradually reaches the limit of the stored elastic energy, and the rock will be damaged.

Using the above energy calculation method, the rock energy densities for different stages of the cyclic loading and unloading test were obtained. The energy density obtained from the calculation is plotted in Figure 6. It can be seen from the graph that, under the three loading rates, before the peak stress, the input and elastic energy densities all grow nonlinearly with the increase of stress, and the input energy density grows rapidly, followed by the elastic energy density. The dissipative energy density is high in the first cycle and decreases slightly in the second cycle; then, this trend shows a non-linear pattern, and the growth rate is the slowest among the three energies. The rocks exhibit different deformation characteristics under different stress states, and similarly, their energy evolution law has different characteristics in different deformation stages. In the initial loading stage, corresponding to the rock compacting stage, the accumulated elastic energy in the rock increases slowly, and the dissipation energy caused by the friction of the closure of the original microcracks and misalignment between the rock masses is larger than the accumulated elastic energy, and the growth of all three types of energy is slow in this stage. In the elastic deformation stage, the rock specimen continuously absorbs energy, and most of the input energy is converted into elastic energy. A small part is used for the formation and expansion of microcracks, and the energy grows approximately linearly. In the plastic stage, more and more input energy is dissipated in the formation, expansion, and nucleation of microcracks inside the rock. The accumulated elastic energy is dissipated in the form of crack surface energy, acoustic emission, and other energy. Near the peak stress stage, the three energy densities show an accelerated growth trend. The elastic energy of the rock specimen reaches its maximum at the end of the plastic stage, which is the peak stress stage. The energy evolution directly reflects the inherent properties of the rock; it can be inferred that the elastic energy is the main driving force for the destruction of the rock specimen. The variation of the elastic energy density indicates that the rock has a certain energy storage limit, which is related to the lithology of the rock and its stress state. The number, size, and distribution of the initial microcracks are the main factors affecting the energy storage limit of the rock.

Rocks exhibit different forms of damage at different loading rates. In essence, the different stress states determine how the rock energy is distributed and transformed. At the same time, the difference in energy state drives the different deformation and damage modes of the rock. It can be seen that the ratio of elastic energy and dissipative energy-to-input energy affects the damage mode of the rock. Figure 7 shows when the loading rate is 0.5 KN/s. The variation of elastic energy ratio and dissipation energy ratio with axial stress of the rock. From the figure, it can be seen that the elastic energy ratio and dissipation energy ratio vary nonlinearly with the axial stress. The elastic energy ratio increases at the beginning of loading, and then increases when the loading reaches about 48% of the peak stress; it reaches the maximum when the axial stress is loaded to about 73% of the peak stress. The dissipation energy ratio exceeds the elastic energy ratio at the early stage of loading. This shows that the closed friction of microcracks and dislocation between rock blocks dissipate a lot of energy in the compaction stage, thus resulting in the proportion of dissipated energy greater than the proportion of elastic energy. As the loading continues, the dissipated energy ratio gradually decreases. From the initial loading to the loading of about 73% of the peak stress, the dissipation energy ratio decreases by about 30%. After that, the dissipative energy ratio increases. With the continuous expansion of microcracks, the specimens are continuously damaged, and the microcracks are connected from random and disorderly sprouting to orderly nucleation, thus forming macroscopic cracks and destabilizing extension. The ability of the rock to store elastic energy decreases, percentage of elastic energy decreases, and percentage of dissipated energy increases. This can be used as a precursor to destabilization damage in rock specimens.

It has been shown that rock energy evolution has a strong rate correlation [26,27,28,29]. However, there is no uniform understanding of the variation of various forms of energy with the loading rate, so it is necessary to make an analysis of the loading rate effect of that. As Figure 8 and Figure 9 represent, at different loading rates, the evolution law of elastic energy and dissipative energy with axial stress is similar, which has been analyzed in the previous section. The elastic energy increases with the increase of the loading rate. The increase of elastic energy with stress at the beginning of loading is not obvious. With the increase of axial stress, the loading rate effect of elastic energy is gradually amplified. Dissipation energy also increases with the increase of loading rate. This is because the increase of loading rate will lead to the increase of peak stress; that is, the rock strength will increase, so the input energy will also increase, and the energy allocated to elastic energy and dissipation energy will be increased accordingly. The distribution ratio of elastic energy and dissipated energy under different loading rates is shown in Figure 10 and Figure 11. In different deformation stages, the percentage of elastic energy decreases with the increase of loading rate; the percentage of dissipated energy increases with the increase of loading rate. The more energy that is dissipated during the deformation of the rock, the more microcracks will be produced and fracture surfaces will be formed; at the same time, more fragments appear, and the fragment sizes become smaller. This explains the phenomenon that, the larger the loading rate, the more fracture surfaces, more fragments, and smaller the scale after damage.

3.3. Damage Evolution Analysis of Sandstone under Cyclic Loading and Unloading

The process of rock deformation and damage is essentially a process of rock energy dissipation, which can also be described as a process of rock damage evolution. In the process of rock loading damage, as the input energy increases, the primary microcracks within the rock expand, and new cracks are created, accompanied by the continuous dissipation of energy, thus resulting in the continuous weakening of the mechanical properties of the rock. The dissipation energy reflects the degree of rock damage, so the dissipation energy is used to define the rock damage variable as:

$$D=\frac{U_{\mathrm{d}}^{\epsilon }}{U_{\mathrm{t}}}$$

(4)

where $D$ is the damage variable; $U_{\mathrm{d}}^{\epsilon }$ denotes the accumulated dissipation energy at a given stress state of $\epsilon$; $U_{\mathrm{t}}$ is the sum of the dissipated energy generated during the whole deformation process. $U_{\mathrm{t}}$ is to be 0.154, 0.246, and 0.350 MJ.m⁻³ at 0.2, 0.5, and 1 KN/s loading rates, respectively (The peak point dissipation energy is calculated by the equation $U^{\mathrm{d}}=U^0-{\sigma }^2/2E$).

The aforementioned results show that the rock energy evolution law exhibits a significant nonlinearity. The rock system is a complex and dissipative system, which has completely different properties from the closed system. Zheng Zaisheng [30] pointed out that there are three interactions in the main types of deformation mechanisms (strain hardening and strain softening) during rock deformation: (1) the interaction between rock elastic energy and dissipative energy refers to the fact that an active accumulation of the former provides more energy for the latter to dissipate, while an active dissipative action also inhibits the accumulation of the former. (2) The interaction between micromechanisms in two major types of deformation mechanisms, i.e., the enhancement of one micromechanism will inhibit the action of other mechanisms. For rock systems, this micromechanism cannot be enhanced indefinitely, and other mechanisms are not yet, which, in turn, will inhibit the enhancement of this mechanism. (3) Spatial interaction of different mechanisms, i.e., the more energy consumed by one mechanism in one region, the more it will inhibit the action of this mechanism in other regions, as well as the action of other mechanisms in this region. In summary, there is a competition among the macroscopic mechanisms of rock energy evolution, as well as competition among microscopic mechanisms and different mechanisms in space. Such an energy evolution law with internal constraints is similar to biological population evolution law. This has been discussed by scholars [31], and it is shown that the variation of any energy evolution process in rocks with stress level can be expressed by a logistic equation; then, the damage variable defined by dissipative energy in this paper can be written in the following form:

$$D=\frac{k}{{1+\mathrm{e}}^{a-r\sigma }},0 < k<1$$

(5)

$D$ is the damage variable, $\sigma$ is the axial stress, $a$ is the parameter reflecting the initial damage level, $r$ is the endogenous growth rate of damage evolution, and $k$ is the maximum damage level of 1.

Under the three loading rates, the variation of damage variables with axial stress of the rock in cyclic loading and unloading tests can be calculated, as shown in Figure 12. Under the action of cyclic loading and unloading, the damage variation curve of the rock is the “weakly concave” type, with the increase of axial stress, the damage variation gradually increases, and the rate of change also gradually increased. When near failure, the change rate of the damage variable with stress is the largest, but there is no obvious mutation.

The data from the test were fitted with Equation (5). Its fitting curve and equation can be acquired. As can be seen from Figure 13, the fitting equation has high accuracy, and the equation can well-describe the rock damage evolution process in the prepeak deformation stage of the rock. It can be seen that, with the increase of loading rate, the initial damage parameter a keeps decreasing; that is, the initial damage degree $D_0$ keeps increasing. As shown in Figure 14, the endogenous growth rate of damage evolution r shows a good linear relationship with the initial damage parameter $a$.

From the derivation of the damage evolution Equation (5), the differential form of the damage evolution equation can be obtained:

$$\frac{\mathrm{d}D}{\mathrm{d}\sigma }=rD(1-\frac{D}{k})$$

(6)

If the value of $D$ is measured once every stress interval by increasing $\Delta \sigma$, and the nth value is denoted by $D_n$, the original continuous variables $D\left(\sigma \right)$ and $\sigma$ become discrete variables ($D_0, D_1,D_2,\dots$) and $\left(n=0,1,2,\dots \right)$, and the continuous differential Equation (6) becomes a discrete difference equation, as follows:

$$D_{n+1}=(r\Delta \sigma +1)D_n(1-\frac{r\Delta \sigma }{(r\Delta \sigma +1)k}{D}_n)$$

(7)

Define $\mu =r\Delta \sigma +1$, $D_n^{\prime }=\frac{r\Delta \sigma }{(r\Delta \sigma +1)k}{D}_n$; $D_n^{\prime }$ is the generalized damage variable, and the above equation can be simplified as:

$$D_{n+1}^{\prime }=\mu D_n^{\prime }(1-D_n^{\prime })=f(D_n^{\prime })$$

(8)

where $n=0,1,2,3\dots$, and the mapping f is a logistic mapping reflecting the generalized damage variable $D_n^{\prime }$.

Equation (8) is the classical one-dimensional logistic mapping, which has very complicated properties. If the control parameter $\mu \in [0,4]$, the mapping f acts to still map any value $D_n^{\prime }\in [0,1]$ to that interval, i.e., the interval $[0,1]$ is an invariant interval for the mapping $f$. In the invariant interval, the sequence of $D_n^{\prime }$ shows very different orbits with $\mu$. As the control parameter μ increases, the mapping $f$ will go through the stable region ($\mu <3$), multiplicative period bifurcation region ($3\le \mu <3.5699$), and finally enter the chaotic state ($3.5699\le \mu <4$). The values of each bifurcation point are shown in the following

Table 2. Logistic mapping bifurcation point value.

Bifurcation Situation	$\mathbf{Control} \mathbf{Parameters} {\mathit{\mu }}_{\mathit{n}}$	$\left({\mathit{\mu }}_{\mathit{n}}-{\mathit{\mu }}_{\mathit{n}-1}\right)/\left({\mathit{\mu }}_{\mathit{n}+1}-{\mathit{\mu }}_{\mathit{n}}\right)$
$2^0\to 2^1$	3	/
$2^1\to 2^2$	3.44949	4.751466
$2^2\to 2^3$	3.54409	4.656251
$2^3\to 2^4$	3.56441	4.688242
$2^4\to 2^5$	3.56876	4.66874
$2^5\to 2^6$	3.56969	4.6691
$2^6\to 2^7$	3.56989	4.669
$2^7\to 2^8$	3.56993	4.669
…	…	…
Periodic → Chaos	3.569946	…

:

$0<D_n\le k\le 1$, so $D_n^{\prime }=\frac{r\Delta \sigma }{(r\Delta \sigma +1)k}{D}_n\in (0,1)$, and $\mu =r\Delta \sigma +1>1$.

The rock damage evolution process will have the nature of multiplicative period bifurcation or chaotic dynamics. The rock damage evolution process is analyzed at a loading rate of 0.5 KN/s, and the damage evolution equation is

$$D=\frac{1}{{1+\mathrm{e}}^{4.338-0.134\sigma }}$$

(9)

The discrete equation for the generalized damage is:

$$D_{n+1}^{\prime }=\mu D_n^{\prime }(1-D_n^{\prime })$$

so $\mu =0.134\Delta \sigma +1, D_n^{\prime }=\frac{0.134\Delta \sigma }{0.134\Delta \sigma +1}{D}_n$.

When $0<\Delta \sigma <14.93 \mathrm{MPa}$, there is $1<\mu <3$; that is, when stress is increasing in this interval, the internal energy of the rock keeps accumulating, deformation keeps accumulating, accompanied by continuous energy dissipation, and the rock keeps being damaged; the generalized damage process will eventually enter a stable state, that is, the fixed point $D_n^{\prime }=1-1/\mu$ formed after the contraction of phase space. When $14.93 \mathrm{MPa}<\Delta \sigma <19.18 \mathrm{MPa}$, there is $3<\mu <3.5699$. The generalized damage process enters the multiplicative period bifurcation zone with the increase of stress, and a series of bifurcation points and corresponding limit rings appear; at first $D_n^{\prime }$ oscillates between two values, and, with the increase of stress, the bifurcation performance oscillates between four values. The distance ratio between adjacent multiplicative bifurcation points, $({\mu }_n-{\mu }_{n-1})/({\mu }_{n+1}-{\mu }_n)\approx 4.669$, the famous Feigenbaum constant, which indicates that the damage evolution from multiplicative bifurcation into chaotic states show some regularity in their quantitative relationship. When $19.18 \mathrm{MPa} <\Delta \sigma <22.39 \mathrm{MPa}$, the generalized damage process enters the chaotic state from arbitrary damage. The generalized damage variables will run through all the states from 0 to 1. When $19.18 \mathrm{MPa} <\Delta \sigma <22.39 \mathrm{MPa}$, the generalized damage process enters the chaotic state, and starts to evolve from any damage states. If the two initial damage states are very close to each other, and the track in the phase space may be completely different after many iterations, i.e., the damage is completely different under the same stress state, which can be said as “the difference between a hair and a thousand miles”.

3.4. Acoustic Emission Characteristics

Deformation damage of rocks is always accompanied by the release of strain energy, in the form of elastic waves, i.e., acoustic emission phenomenon (AE). The acoustic emission of rocks under load can be divided into two categories: the first category is the acoustic emission caused by violent elastic vibration, due to elastic strain changes; the second category is the acoustic emission caused by microcrack development, friction, and misalignment. Before rock damage, acoustic emission is dominated by the second type of acoustic emission. After rock damage acoustic emission is dominated by the first type, during the energy evolution of rocks, acoustic emission, as a form of dissipative energy, is released in the form of crack expansion before rock destruction. The phenomenon of acoustic emission contains the rich physical and mechanical properties of rocks, especially describing the stress state and dissipative energy characteristics during rock deformation and damage, which provides the possibility for revealing the physical and mechanical processes of rock deformation and damage, as well as forecasting rock damage.

Figure 15 shows the axial stress, energy, and accumulated energy, with time under 0.2, 0.5, and 1 KN/s cyclic loading and unloading conditions.

The test results show that there is a relatively active acoustic emission signal at the beginning of loading, after which the acoustic emission signal is somewhat weakened, compared with the beginning of loading. With the continuation of cyclic loading and unloading, the stress at the unloading point increases, and the redeveloped microcracks keep appearing. The acoustic emission signal tends to be active, and the acoustic emission energy increases. This means an increase in dissipated energy, which is consistent with the aforementioned dissipation energy calculation results. When the load approaches the peak stress, acoustic emission is significantly active, and energy appears to increase more significantly, at which time, the microcrack penetrates and the specimen is damaged. In the post-peak fracture stage, the specimen loses the load carrying capacity, and the stored elastic energy is dissipated mainly by the slip friction of the microcrack; the acoustic emission phenomenon disappears at the same time. When the loading stress did not exceed the previous maximum loading stress, there was almost no acoustic emission phenomenon; when the loading stress exceeded the previous maximum loading stress, the acoustic emission phenomenon reappeared and showed a step increase in the cumulative energy curve, indicating that this specimen has Kaiser effect.

4. Conclusions

Based on the cyclic loading and unloading tests of sandstone specimens, the following conclusions are drawn:

• In the cyclic loading and unloading test, in the initial stage of loading, most of the input energy is converted into dissipated energy, and a small part is converted into stored elastic energy. The corresponding acoustic emission phenomenon is active in the early stage of loading. The influence of initial damage on rock failure should be studied further.
• In the cyclic loading and unloading test, the three kinds of energy density increased nonlinearly with an increase of loading stress; the input energy density increased fast, the elastic energy density followed, and the dissipated energy density increased slowly. Near the peak stress stage, the proportion of elastic energy decreases and proportion of dissipated energy increases, which can be used as a precursor of rock failure and provides a basis for the prediction and early warning of rock failure.
• Rock energy evolution is closely related to the loading rate. With the increase of the loading rate, the elastic energy density and dissipated energy density increase; the proportion of elastic energy density decreases and proportion of dissipated energy density increases. This indicates that the greater the loading rate, the more intense the friction slip of microcracks inside the rock, and more energy is converted into dissipated energy. This provides a theoretical reference for explaining the loading rate effect of rock failure morphology.
• The damage variable defined by dissipative energy can well-describe the damage process of rock, and the damage evolution process satisfies the logistic equation. The damage evolution of the loaded rock has bifurcation and chaotic properties. This shows that the study of the rock damage process from the perspective of energy is more helpful for understanding the complex physical and mechanical behavior of rocks.