Energy evolution varies during the whole process of rock deformation, and mechanical parameters are markedly altered under cyclic loading and unloading. In order to investigate the effects of confining pressure on energy evolution and mechanical parameters, cyclic loading and unloading experiments were performed for granite under six different confining pressures. The experiment revealed the confining pressure effect on variation and allocation pattern of energy and mechanical characteristics. Four characteristic energy parameters, namely, storage energy rock, storage energy limit, energy storage ratio, and energy dissipation ratio, were proposed to describe energy storage and dissipation properties of rock. Elastic modulus and dissipation ratio presented a downward “U” and “U”-shaped trends, respectively, with loading and unloading cycles, while Poisson’s ratio increased linearly at the same time. Elastic energy was accumulated mainly before peak stress, while the energy dissipation and release were dominant after the peak strength. As the confining pressure increased, efficiency of energy accumulation and storage limit improved. An exponential function was proposed to express the relationship between the energy storage limit and confining pressure. Dissipation energy increased nonlinearly with the strain, and the volume dilatancy point defined the turning point from a relatively slow growth to an accelerated growth of dissipation energy. The dilatancy point can be used as an important indication for the rapid development of dissipation energy.
Many engineering projects have shown that rock is not in a stable stress environment but is subjected to a cyclic loading-unloading stress environment. Deformation and failure of granite under engineering disturbance is a very complicated damage evolution process, and failure is governed by energy storing and dissipation phenomenon [
The influence of loading and unloading cycles on energy variation and mechanical properties has been studied by domestic and foreign researchers. Wang et al. [
However, at present, the energy characteristics under the special loading level such as peak stress are paid more attention, while the real-time evolution and distribution of energy throughout the whole process of failure are insufficiently studied in the process of research. The purpose of this paper is to perform laboratory tests to explore quantitative relationships between energy and mechanical parameters throughout the whole process of failure. The results are expected to provide reference for rock mechanics tests and dynamic disaster prediction.
All rock samples used in the tests were granite from a depth of −1243∼−1245 m and were collected from the geological borehole of the new main shaft in Xincheng gold mine, Shandong province. Rocks are medium-fine grain structure. The analysis of a series of rock slices by optical microscopy shows that the main minerals are as follows: quartz, plagioclase, alkaline feldspar, and biotite. According to the QAP classification method, it can be named porphyritic granodiorite. Rock samples were processed into cylinders of 50 mm × 100 mm, according to the ISRM method recommended by international rock mechanics testing. A NM-4B ultrasonic testing instrument was used to detect wave velocity of samples. In order to avoid testing error caused by the variability of samples, samples with similar wave velocity were selected for triaxial testing. Basic physical parameters of rock samples are shown in Table
Basic physical parameters of rock samples.
Number | Diameter (mm) | Height (mm) | Wave velocity (m·s−1) | Density (kg·m−3) |
---|---|---|---|---|
X-H-1 | 50.01 | 100.03 | 4034 | 2676 |
X-H-2 | 49.97 | 99.99 | 4004 | 2689 |
X-H-3 | 49.98 | 99.99 | 4100 | 2685 |
X-H-4 | 50.00 | 100.00 | 4012 | 2683 |
X-H-5 | 50.01 | 100.01 | 3988 | 2695 |
X-H-6 | 49.98 | 100.02 | 4029 | 2658 |
A series of cyclic loading and unloading tests were conducted by an MTS815 rock mechanics testing machine. The maximum axial loading of the testing machine is 2700 kN, and the maximum confining pressure can be applied to 140 MPa. Axial and circumferential extensometers were used to measure axial and circumferential deformations in real time by placing extensometers in the middle of rock samples, as shown in Figure
Sample installation.
X-H-3 cyclic loading-unloading path.
Stress-strain curves under 45 MPa confining pressure and failure modes under different confining pressures are shown in Figure The outer envelopes of cyclic loading and unloading under different confining pressures are similar to conventional loading, and samples have undergone four distinct stages: compaction stage, elastic deformation stage, unsteady fracture development stage, and postpeak failure stage. Due to the closure of initial cracks caused by high confining pressure, the compaction stage was not obvious for larger confining pressure. The peak strength and peak strain were improved significantly with the larger confining pressures. Large confining pressure increased crack initiation stress and limited the development of circumferential deformation. Strength and rigidity were enhanced at the same time and had obvious confining pressure effect.
(a) Stress-strain curves and (b) failure modes.
Mechanical parameters of samples.
Confining pressure (MPa) | Dilatancy stress (MPa) | Peak stress (MPa) | Stress level (%) | Axial strain (%) | Hoop strain (%) |
---|---|---|---|---|---|
1 | 158.11 | 204.07 | 0.77 | 0.40 | 0.21 |
10 | 219.52 | 295.13 | 0.74 | 0.50 | 0.30 |
20 | 234.82 | 343.21 | 0.68 | 0.55 | 0.37 |
30 | 285.15 | 380.15 | 0.75 | 0.69 | 0.41 |
40 | 337.01 | 408.32 | 0.83 | 0.78 | 0.49 |
45 | 389.00 | 506.39 | 0.77 | 0.83 | 0.54 |
Under 1 MPa and 10 MPa confining pressures, the local region failed and small pieces fell off from the surface, and there were many small fragments beside the main crack, indicating local tensile failure. Under 20 MPa and 30 MPa confining pressures, small-scale failure existed on the surface of rock mass, and shear failure was the main failure mode. Under 40 MPa and 45 MPa confining pressures, rock samples were divided into two blocks separated by the main crack. Failure occurred with almost no surface fracture and accompanied by loud noise. With an increase in confining pressure, the number of surface fractures gradually decreased and failure modes changed from tension and shear coexistence to shear failure. Larger confining pressure restrained the development of surface cracks, and energy consumption was insufficient, so it was easy to produce through fracture surface. Sudden release of large amount of elastic energy was the intrinsic driving force for rock failure, and the failure mode was closely related to the characteristics of an internal structure.
Elastic energy is mainly stored in rock mass in the form of elastic strain, and it is reversible. Dissipation energy includes plastic deformation energy, surface damage energy, thermal energy, and radiation energy, and it is irreversible. The energy released after unloading is the elastic energy accumulated at a certain stress level. The decrease value relative to the total energy is the dissipation energy at this stress level. It is unrealistic to monitor each energy in real time during the dynamic loading. The energy evolution is mainly manifested by the dynamic balance of input energy, elastic strain energy, and dissipation energy [
Different forms of energy differ not only in quantity but also in quality. Elastic energy belonging to high-quality energy can be converted into other forms of energy. Surface fracture energy, radiation energy, and thermal energy are low-quality energy. The energy evolution processing of rock under loading is shown in Figure
Energy conversion relationship.
The relationship between internal structure and energy development is shown in Figure
Interaction between internal structure and energy.
The ratio of elastic strain energy to energy storage limit under current stress is defined as the rock energy level. The variation in the rock energy level with stress is shown in Figure
Evolution of containing energy level.
Storing energy through deformation from testing machines is an inherent property of hard rocks such as granite, which is defined as storage energy rock. Furthermore, the elastic strain energy stored at peak strength is defined as the energy storage limit. To describe the evolution of energy allocation, the ratio of elastic strain energy density to input energy density is defined as storage energy ratio, which is used to characterize the elastic energy storage features under different stress levels. The ratio of dissipation energy density to the input energy density is defined as dissipation energy ratio, which is used to characterize energy consumption and reflect the change in internal structure under different stress levels. The ratio of dissipation energy density to elastic energy density is defined as energy allocation ratio. Input energy density, elastic strain energy density, dissipation energy density, and storage energy ratio were obtained from stress-strain curves under different confining pressures, as shown in Figure
Energy evolution under six different confining pressures. (a) 1 MPa. (b) 10 MPa. (c) 20 MPa. (d) 30 MPa. (e) 40 MPa. (f) 45 MPa.
As shown in Figure
The quantitative relationship between elastic energy density and relative stress level was represented by an exponential function (Figure
Elastic energy density with relative stress level.
Energy limit density with confining pressure.
The variation in storage energy ratio with strain under six different confining pressures was plotted as shown in Figure
Storage energy ratio with strain.
The stress-strain curve subjected to loading-unloading cycles manifested obvious strain hysteresis, and hysteresis loop area could be used to reflect development of dissipation energy. Hysteresis loop area in the compaction and elasticity stage grew slowly within the strain range of 0–0.2%. The effect of confining pressure on dissipation energy was insignificant (Figure
Variation of hysteresis loop area.
The energy consumption ratio varied in the range of 0.05–0.14 and manifested a U-shaped distribution as shown in Figure
Variation of energy consumption.
Dilatancy is a general inelastic volume deformation phenomenon of rock and is a result of unstable development of cracks. The dilatancy point is the turning point of volume changing from compression to expansion and indicates that rock is about to enter the fracture instability development stage. Dilatancy is inevitably related to dissipation energy, and the dilatancy point can be used as the starting point for the unstable development of internal cracks. Dilatancy stress in this experimental study was about 68%–83% of the peak stress level under different confining pressures shown in Table
Variation in volumetric strain and dissipative energy under 1 MPa confining pressure. (a) Outer envelope of volume strain. (b) Variation in dissipation energy with strain.
Internal structure is constantly adjusted so that the mechanical parameters change significantly under cyclic loading and unloading. The internal structure determines energy evolution, and hence, establishment of a relationship between mechanical parameters and energy evolution is useful to predict rock failure. The correlation between elastic modulus, Poisson’s ratio, loading/unloading response ratio, and energy evolution were analysed. Elastic modulus [
Elastic modulus and Poisson’s ratio are the direct reflection of rock stiffness which is an ability index to resist elastic deformation. The limit of storage energy is mainly decided by rock stiffness. Elasticity modulus with loading/unloading cycles revealed an inverted “U” shape (Figure
Variation of elastic modulus with cycles.
Poisson’s ratio during loading/unloading cycles was within the range of 0.15∼0.33 and increased nonlinearly with the number of cycles as shown in Figure
Variation of Poisson’s ratio with cycles.
The increase in Poisson’s ratio indicated the deterioration of internal structure and the enhancement of transverse deformation. Poisson’s ratio was influenced by confining pressure effect during the whole process of rock deformation and failure, and transverse deformation was restricted with an increase in confining pressure as shown in Table
Poisson’s ratio of whole loading process.
Confining pressure (MPa) | Initial Poisson’s ratio | Failure Poisson’s ratio | Increment | Rate of increase (%) |
---|---|---|---|---|
1 | 0.15 | 0.3 | 0.15 | 100 |
10 | 0.14 | 0.28 | 0.14 | 100 |
20 | 0.15 | 0.29 | 0.14 | 93 |
30 | 0.18 | 0.26 | 0.08 | 44 |
40 | 0.20 | 0.27 | 0.07 | 35 |
45 | 0.20 | 0.26 | 0.06 | 30 |
The loading/unloading response ratio decreased first, then stabilized subsequently, and decreased finally. It was generally greater than one under six different confining pressures, thus indicating that elastic modulus in the unloading stage is always greater than in the loading stage (Figure
Variation of loading/unloading response ratio.
Elastic modulus decreased faster in the unloading stage than in the loading stage. The loading/unloading response ratio decreased rapidly. Therefore, the change in loading/unloading response ratio from the stable stage to the stage of secondary decrease can provide an important precursor information for unsteady fracture development. The accumulated damage caused by internal structural adjustment resulted in the decrease of elastic modulus and the increase of Poisson’s ratio, especially entering into the fracture instability development stage. Test results showed that there was a positive correlation between energy storage limit and elastic modulus but a negative correlation between Poisson’s ratio and energy dissipation.
Confining pressure effects on variation and allocation of energy were revealed based on the triaxial cyclic loading and unloading testing of granite under six different confining pressures. The correlation between mechanical parameters variation and energy characteristics was also discussed. The outer envelope of stress-strain curves under six different confining pressures is similar to conventional loading. The failure modes developed from a well-developed surface fracture to a single through fracture. Allocation of energy was determined by internal structure, and the containing energy level of rock depends on the stress level. Elastic energy density was always greater than dissipation energy density, indicating that accumulated energy was dominant before the peak stress. Storage energy limit improved nonlinearly with the confining pressure, and the energy consumption ratio showed an inverted U shape with strain. The hysteresis loop area had an obvious quadratic relationship with strain. As the confining pressure increased, the elastic energy density increased nonlinearly at the same stress level and the growth gradient also improved gradually at the same time. High confining pressure improved the stiffness of rock and increased the energy storage level. Elastic modulus and energy consumption ratio presented an inverted “U” type and a “U” type, respectively, which confirmed good correlation between mechanical parameters and energy evolution characteristics. Elastic modulus and Poisson’s ratio are characteristics of rock stiffness. Larger elastic modulus and smaller Poisson’s ratio indicate that rock stiffness and storage energy limit are greater. The characteristic of loading/unloading response ratio from the stable stage to secondary decrease could be regarded as the precursor information of unsteady fracture development. There were four distinct stages in the curve of dissipation energy with strain. The dilatancy point was the key point after which dissipation energy was accelerated. The dilatancy point could provide an important information for the unsteady development of fracture.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This paper was supported by Project no. 2016YFC0600801 of the National Key Research and Development Plan and Key Program of National Natural Science Foundation of China (51534002). The financial aids are gratefully acknowledged.