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In order to explore the mechanical response mechanism of rock materials under cyclic loading, uniaxial constant amplitude cyclic loading tests for sandstone specimens were carried out. The images of specimen deformation during the tests were captured by charge-coupled device (CCD) cameras. Based on the digital image correlation method (DICM), the evolution laws of nonuniform deformation and displacements around localization bands during cyclic loading were investigated. The experimental results show that, during the cyclic loading process, the nonuniform deformation continually escalates with the number of cycles increasing and fluctuates with the cyclic loading stress condition; the nonuniform deformation lags behind the variation of loading stress; and the whole nonuniform deformation experiences a slow evolution stage and a fast evolution stage. At the loading stage or unloading stage, the nonuniform deformation of rock deteriorates with the number of cycles increasing under the same stress condition. In each loading cycle, the nonuniform deformation at the unloading stage is more than that at the loading stage under the same stress condition. The time of dislocation displacements and tension displacements meets hysteresis, compared with the time of stress change. In addition, the dislocation displacements and tension displacements around localization bands in general increase with the number of cycles increasing. The displacement evolution around localization bands has the same hysteresis and accumulation laws as that of nonuniform deformation.

Rock is a complex and common heterogeneous geological material. The deformation and mechanical properties in its failure process are the foundation of rock mechanics. At the stage of geotechnical engineering construction and engineering operation, the rock mass is often subjected to cyclic loading. Therefore, finding out the deformation evolution laws of rock materials under cyclic loading is helpful to understand the failure mechanism of rock under cyclic loading, and further to evaluate the long-term stability of engineering rock mass, which benefits the design, construction, and protection of geotechnical engineering, both in theory and in practice.

Researchers have devoted great efforts to studying the deformation failure and mechanical properties of rocks under cyclic loading. Costin and Holcomb [

Conducting uniaxial constant amplitude cyclic loading tests on the sandstone samples aims to reveal the relation between the nonuniform deformation laws and the number of cycles and stress condition, as well as the displacement evolution laws around localization bands with cyclic loading effect based on the digital image correlation method (DICM).

DICM also named digital speckle correlation method (DSCM) [

Schematic diagram of correlation match.

In Figure _{s} and _{t} are the two speckle images before and after deformation, respectively, in which _{s} is the source image, representing the starting state, and _{t} is the target image, representing the state after deformation. If the motion of point _{s} needs to be tracked, the matching point of point _{t} needs to be solved. Therefore, the characteristic speckle pattern

DICM is simple to operate and has the advantages of wide range of applicable test objects, no special requirements for measurement environment, high sensitivity, full field, and noncontacting [

The tested rock specimens are sandstones. Specimens are hexahedrons with a size of

The electric-hydraulic servocontrol test machine was used for uniaxial constant amplitude cyclic loading tests, and CCD cameras were chosen to capture the speckle deformation images of the samples. Meanwhile, computers were used to analyze the deformation and control the test condition. The schematic diagram of the test system is shown in Figure

Schematic diagram of test system.

Physical diagram of test system.

At the beginning of the test, the test machine and the CCD cameras started at the same time. The test machine automatically recorded data such as load, time, and displacements. The CCD cameras continuously collected the speckle deformation images on the surface of specimens in the whole cyclic loading process.

Before the cyclic loading tests, the uniaxial compressive strength tests were done on three samples cut from the same sandstone block, taking the equipment storage and calculation complexity into account, the cyclic tests were carried out by load control, and the loading rate was 0.2 kN/s. The load amplitudes were kept constant in each cycle, and the load amplitudes ranged from 0.3 to 0.8 of the uniaxial compressive strength. According to the measured uniaxial compressive strength, the cyclic load amplitude varied from 28 kN to 70 kN or the cyclic loading stress changed from 11.2 MPa to 28 MPa. The loading frequency is about 2.3467 × 10^{−3} Hz. The relation curve between load and time in the whole cyclic process, recorded automatically by the computer, is shown in Figure

Curve of load and time.

From Figure

Failure pattern of specimen.

Digital image correlation method was used to analyze the evolution laws of nonuniform deformation and displacements around localization bands at the cracks by calculating all the deformation images in the whole cyclic loading process, the samples were proved eventually being destroyed in the tension mode and the exfoliation occurred at the moment of rock failure, and the flow chart of the whole research process is shown in Figure

Flow chart of whole research process.

In the process of cyclic loading, the rock will undergo nonuniform deformation, i.e., the deformation localization of rock. Compared with the uniform deformation field before rock failure, the main differences of the localized deformation field are as follows [_{w}, which can simultaneously describe the “numerical characteristics” and “spatial characteristics” of deformation localization, is introduced to analyze the evolution of nonuniform deformation field of rock. The formula about the statistical index is expressed as follows [_{k} is the deformation amount of each point in the deformation field; _{k};

From the above formulas, under normal circumstances, the variance _{w} obtained by multiplying

Through calculating the deformation field of the specimen in the constant amplitude cyclic loading process, and normalizing the statistical index _{w}, the contrast curve of the statistical index _{w} and stress was obtained, as shown in Figure

Contrast curve of _{w} and stress.

From Figure _{w} of nonuniform deformation fluctuates with the loading and unloading stress. It basically shows that the value of _{w} increases gradually during the loading stage of each cycle and decreases gradually at the unloading stage of each cycle. But careful comparison of the _{w} curve with the stress curve indicates that, in each cyclic loading and unloading process of rock sample, the value of _{w} does not vary with the loading stress synchronously, and the nonuniform deformation always lags behind the variation of loading stress. The occurrence of this hysteresis is caused by the accumulative or fatigue damage in the process of cyclic loading of rock sample [

To explore the change laws of the nonuniform deformation of rock with the number of cycles in the constant amplitude cyclic loading process (the cyclic load amplitude varied from 28 kN to 70 kN or the cyclic loading stress changed from 11.2 MPa to 28 MPa), the value of _{w} when loading and unloading to 15 MPa, 20 MPa, 25 MPa, the peak point of loading stress (PPL), and the bottom point of unloading stress (BPU) during each cycle was selected, and the evolution curves of _{w} which varied with the number of cycles were drawn under the same stress level, as shown in Figure

Evolution curves of _{w} under the same stress: (a) loading or unloading to 15 MPa; (b) loading or unloading to 20 MPa; (c) loading or unloading to 25 MPa; (d) PPL or BPU.

Figure _{w} curves that vary with the number of cycles at 15 MPa. With the number of cycles increasing, the value of _{w} increases too. It shows that, before the 10^{th} cycle, the value of _{w} increases slowly with the number of cycles increasing. After the 10^{th} loading cycle, the value increases rapidly compared with the earlier loading cycles, and the slope of _{w} curves increases with the number of cycles increasing. This indicates that the evolution rate of nonuniform deformation increases or the accumulative damage evolution rate increases. The analysis shows that there are two obvious stages in the process of nonuniform deformation evolution, i.e., the slow nonuniform deformation evolution stage and the fast nonuniform deformation evolution stage; following the fast nonuniform deformation evolution, the failure of rock sample occurs. By analyzing the _{w} curves when loading and unloading to 15 MPa, it can be found that the value of _{w} at the unloading stage is bigger than that at the loading stage under the same stress level, i.e., due to the accumulative effect of cyclic loading, when loading and unloading to the same stress in the same cycle, the nonuniform deformation at the unloading stage is bigger than that at the loading stage. This validates that cyclic loading can cause damage accumulation in rock materials. Likewise, Figures

As to Figure _{w} at the peak point of loading stress is bigger than that at the bottom point of unloading stress before the 15^{th} loading cycle. After the 15^{th} loading cycle, the value of _{w} at the bottom point of unloading stress turns bigger than that at the peak point of loading stress. And when entering the 17^{th} loading cycle, the value of _{w} at the peak point of loading stress and the bottom point of unloading stress both significantly increase. The speckle images at the peak point of loading stress and the bottom point of unloading stress during the 15^{th}, 16^{th}, and 17^{th} cycle were, respectively, selected to analyze this phenomenon.

The speckle image at the initial time was used as the reference image, and based on the DICM [

Nephogram of deformation field: (a) PPL at the 15^{th} cycle; (b) BPU at the 15^{th} cycle; (c) PPL at the 16^{th} cycle; (d) BPU at the 16^{th} cycle; (e) PPL at the 17^{th} cycle; (f) BPU at the 17^{th} cycle.

Characteristic parameters of localization band at each corresponding time.

Corresponding time | Length (mm) | Width (mm) | Deformation value |
---|---|---|---|

15^{th} PPL |
30 | 9 | 3.5 |

15^{th} BPU |
21 | 7 | 2.9 |

16^{th} PPL |
37 | 10 | 4.3 |

16^{th} BPU |
41 | 12 | 4.7 |

17^{th} PPL |
54 | 13 | 5.8 |

17^{th} BPU |
73 | 15 | 6.8 |

As shown in Table ^{th} cycle to the 17^{th} cycle, the length, width, and deformation value of localization band at the peak point of loading stress and the bottom point of unloading stress all keep increasing. The value decreases from the peak point of loading stress to the bottom point of unloading stress at the 15^{th} cycle, and there is still a deformation recovery. However, since the 16^{th} cycle, the value begins to increase from the peak point of loading stress to the bottom point of unloading stress, and at the 17^{th} cycle, the length and deformation value increase greatly.

The above analysis of the nephogram of deformation field indicates that, in the process of constant amplitude cyclic loading, the statistical index _{w} can well reflect the nonuniform deformation of the specimen. In addition, with the increase of the number of cycles, the nonuniform deformation increases gradually, and the fatigue damage accumulates gradually. When the number of cycles increases to a certain one, the final nonuniform deformation of rock still increases, no matter at the loading stage or at the unloading stage, and the parameters of localization band still expand. When the damage accumulates to a certain extent, the final failure of rock will occur along the localization band.

The deformation field evolution in the process of constant amplitude cyclic loading has been analyzed in detail, in which we can see that the nonuniform deformation of rock increases gradually with the number of cycles, and the localization phenomenon is more and more obvious. In the end, the failure occurs along the deformation localization band. To make further research on the relationship between the deformation evolution of rock and stress condition in the cyclic loading process, the laws of displacement evolution around localization band are analyzed as follows.

The position of localization band was determined according to the final failure mode (Figure

Nephogram of deformation field before failure.

The analytical method of displacement evolution around deformation localization band [

Analytical method of displacement evolution around deformation localization band.

The displacement field of each deformation image which was processed by the digital image correlation method was taken in the first step. Then, determine the displacement calculation region. Five groups of pixels (the corresponding region with the point _{1} and the point _{2} as the center points in Figure

The displacement field of the specimen in the cyclic process was processed through the above analytical method, and the displacement evolution curves of localization band A and localization band B were obtained, as shown in Figure

Displacement evolution curves around localization bands: (a) dislocation displacement of localization band A; (b) tension displacement of localization band A; (c) dislocation displacement of localization band B; (d) tension displacement of localization band B.

As shown in Figures

To explore the displacement evolution laws around deformation localization bands in relation to the increase of the number of cycles in the constant amplitude cyclic process, the dislocation displacements and the tension displacements at the peak point of loading stress and the bottom point of unloading stress were chosen to draw the displacement evolution curves with the number of cycles, as shown in Figure

Displacement evolution curves at the peak point of loading stress and the bottom point of unloading stress: (a) dislocation displacement of localization band A; (b) tension displacement of localization band A; (c) dislocation displacement of localization band B; (d) tension displacement of localization band B.

Figure ^{th} cycle, the value of displacements increases slowly. After the 10^{th} loading cycle, the value of displacements increases dramatically compared with the earlier loading cycle, and the slope of displacement curves overall increases with the number of cycles increasing. The displacements have the same evolution laws as that for the nonuniform deformation of rock. And in the cyclic loading process, two stages about displacement evolution exist: slow and fast evolution stage.

So far, in the process of uniaxial constant amplitude cyclic loading of rock, it is rarely seen to choose a scientific index that can reflect the main characteristics of nonuniform deformation field of rock specimens to analyze the evolution of nonuniform deformation and explore the displacement evolution of deformation localization bands based on DICM. The research results are of positive significance for understanding the deformation and failure of rocks during the cyclic process, but there are some insufficiencies. In the latter study, it is necessary to carry out a large number of tests for different types of rocks under different loading methods and different loading rates, as well as combining the method of CT scanning and scanning electron microscopy, to explore the relationships between the nonuniform deformation of rock and the microstructure within the rock, as well as the relationships between the nonuniform deformation of rock and the failure mode of rock.

Uniaxial constant amplitude cyclic loading tests of sandstones were carried out, and the deformation images of specimen surfaces during the tests were captured by CCD cameras. Based on the digital image correlation method, the nonuniform deformation and displacement evolution were analyzed, and the following conclusions can be drawn.

By introducing the statistical index, the nonuniform deformation degree was quantified, the nonuniform deformation of rock gradually increases with the cyclic loading number increasing and fluctuates with the loading stress condition on the whole; compared with the time of stress change, nonuniform deformation experiences hysteresis, no matter at the loading stage or at unloading stage. In general, the nonuniform deformation has two evolution stages with the number of cycles, i.e., slow and fast nonuniform deformation evolution stage.

At the loading stage or unloading stage of the entire cyclic process, the nonuniform deformation of rock increases with the number of cycles increasing under the same stress condition. In each loading cycle, the nonuniform deformation at the unloading stage is bigger than that at the loading stage under the same stress condition. This is caused by the effect of cyclic loading, and cyclic loading can cause damage accumulation in rock materials.

As to the displacement evolution laws around localization bands, the curves of displacements fluctuate with loading stress condition during the cyclic process. Basically, in the loading and unloading stages, the displacements produce dislocation along the opposite direction parallel to the deformation localization bands and cause tension or extrusion along the opposite direction vertical to the deformation localization bands. The dislocation displacements and the tension displacements lag behind the loading and unloading stress, having the same hysteresis laws as nonuniform deformation. Overall, the values of the dislocation displacements and the tension displacements around localization bands increase with the increase of the number of cycles, and there exist slow displacement evolution stage and fast displacement evolution stage, just as the evolution stages of nonuniform deformation.

The raw data will be provided upon request to the corresponding author.

The authors declare that they have no conflicts of interest.

The authors are grateful to the National Natural Science Foundation of China (Grant nos. 50904071 and 51274207) for the financial support to this work.