Numerical study of AE and DRA methods in sandstone and granite in orthogonal loading directions

Xu-hua REN, Hai-jun WANG*, Ji-xun ZHANG

College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China

Numerical study of AE and DRA methods in sandstone and granite in orthogonal loading directions

Xu-hua REN, Hai-jun WANG*, Ji-xun ZHANG

College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China

The directional dependency of the acoustic emission (AE) and deformation rate analysis (DRA) methods was analyzed, based on the contact bond model in the two-dimensional particle flow code (PFC2D) in two types of rocks, the coarse-grained sandstone and Aue granite. Each type of rocks had two shapes, the Brazilian disk and a square shape. The mechanical behaviors of the numerical model had already been verified to be in agreement with those of the physical specimens in previous research. Three loading protocols with different loading cycles in two orthogonal directions were specially designed in the numerical tests. The results show that no memory effect is observed in the second loading in the orthogonal direction. However, both the cumulative crack number of the second loading and the differential strain value at the inflection point are influenced by the first loading in the orthogonal direction.

Kaiser effect; deformation rate analysis; PFC2D; orthogonal loading directions; in situ stress measurement

1 Introduction

The importance of reliable evaluation ofin situstress has been recognized not only in mining and waste storage caverns (Filimonov et al. 2001; Tuncay and Ulusay 2008), but also in the stability of hydraulic tunnels. Numerous methods have been developed to determine thein situstress. The measurement methods can be divided into two groups. One is on-site measuring techniques such as borehole relief methods and hydraulic fracturing methods. The other group is the laboratory measurement method done by special loading on samples extracted from cores, such as the acoustic emission (AE) method based on the Kaiser effect (KE) and the deformation rate analysis (DRA) method.

For the first group of techniques, the main deficiency is its time consumption and high cost. In addition, it is very difficult for these methods to measure thein situstress in bedrock at great depth or in remote regions where no access is available from boreholes or mine workings (Filimonov et al. 2001; Villaescusa et al. 2002). However, compared with the firstgroup, the AE method and DRA method are considerably economical and permit large numbers of measurements. They are two of the most promising potential methods for determining thein situstress. Both methods are based on the memory effect of the rock. Besides the Kaiser effect and DRA phenomena, the rock memory effect also includes the ultrasonic memory effect, the electric memory effect (Fujii and Hamano 1977), and the ion emission memory effect (Reed and McDowell 1994).

The AE method is based on the Kaiser effect, which was first discovered by Kaiser (1953). The Kaiser effect takes place in rocks and materials under repetitive uniaxial compressions. When the previous peak stress level is attained in the subsequent loading, the AE activities increase dramatically. The first attempt to determine thein situstress using the AE method was made by Kanagawa and Nakasa (1978). The method is based on the assumption that the Kaiser effect detected by uniaxial compression can be used to detect the normal stress of the previous peak stress (or thein situstress) in the loading direction (Holcomb 1993). Therefore, the steps for stress state determination by the AE method are as follows: (1) six sub-samples with different orientations are extracted from the same core, (2) repetitive uniaxial compressions on the sub-samples are performed and the normal stress component in the loading direction is determined for each sample by the AE method, and, (3) through six independent normal stress components and their orientations in the core, the complete stress tensor of the core can be computed. The detailed methodology is described in Villaescusa et al. (2002).

The DRA method was originally proposed and demonstrated by Yamamoto as a valid and reliable technique forin situstress measurement (Yamamoto et al. 1990; Yamamoto 2009). It is based on the concept that there is a change in the slope of the stress-strain curve of the specimen when the previous peak stress (or thein situstress) is encountered. To detect the change, the strain difference function Δεij(σ)is proposed. The Δεij(σ)is defined for a pair of theith and thejth loading cycle by

whereεi(σ) andεj(σ) are the axial strain of a specimen at an applied stressσin theith andjth loading stages, respectively. Not only the linear and nonlinear components of the elastic strain but also the invariable components of the inelastic strain are removed from the equation above. Thus, the strain difference function represents the difference of the inelastic strain between the two cycles. Through this function, the normal component of the previous peak stress (or thein situstress) in the loading direction can be determined at the inflection point in the Δεij(σ)curve. Therefore, as in the AE method, the stress state of the rock can be identified by conducting the DRA tests on at least six sub-core samples, all in independent axial directions. Because there are already many detailed introductions to the DRA method (Yamamoto et al. 1990; Yamamoto 2009; Villaescusa et al. 2002; Wu and Jan 2010), adescription is not repeated here.

There are a considerable amount of further studies on the two methods, especially the former. A comprehensive review of the fundamental questions of the AE method has been made by Lavrov (2003). However, many fundamental questions need to be answered towards the practical application of the two methods to determination of thein situstress. One important question is the directional dependency of the two methods. For the AE method, there are only a few studies. Michihiro et al. (1991/1992), Stuart et al. (1993, 1994), and Chen et al. (2007) performed experiments on cubic or square rock specimens, while the direction of the second loading was orthogonal to that of the first cycle loading. Their results revealed that no Kaiser effect was observed in the second cycle loading. Chen et al. (2007) employed the two-dimensional rock failure process analysis (RFPA2D) code, based on the linear finite element method and continuum mechanics in the research. Holcomb and Costin (1986) performed a cycle loading on a large granite block in one direction, and then sub-cores from the large block in different directions were tested in the uniaxial compression. According to their conclusions, the Kaiser effect was observed at an intersection angle less than 10°. A similar conclusion was made by Lavrov et al. (2002) through Brazilian disk tests on Belgian blue limestone. In the study of Lavrov et al. (2002), the discontinuity interaction and growth simulation (DIGS) code, based on the two-dimensional boundary element method, was employed in the numerical simulation, and the influence of the first loading on the subsequent loading in the orthogonal direction was examined. As for the DRA method, there has not yet been any study on the directional dependency.

The objective of this study is to explore the directional dependency of the AE and DRA methods based on the contact bond model in the two-dimensional particle flow code (PFC2D). Numerical models of two types of rocks, the coarse-grained sandstone and Aue granite were developed. Each type of rocks had two shapes, the Brazilian disk and a square shape. Three loading protocols were specially designed and performed on the numerical models. The results are compared with those of previous studies.

2 Numerical study of KE and DRA

2.1 Contact bond model in PFC2D

PFC2Dallows the user to explicitly and directly model rock mass using a discrete-element modeling algorithm, and thus has an advantage over a continuum modeling approach in the simulation of the internal damage (Hunt et al. 2003). The contact bond model in PFC2Dhas been widely used to simulate rock behavior and to solve many rock engineering and geo-mechanics problems (Hazzard and Young 2000; Chang et al. 2002).

As for the contact bond model in PFC2D(Itasca Consulting Group 2002), the interaction between two particles is determined by the contact bond with constant normal and shear stiffness acting at the contact point. The shearstrengthFscand normal strengthFncareassigned to the contact bond. If the magnitude of the tensile normal contact forceFnexceedsFnc, the bond breaks, and bothFnand shear contact forceFsare set to zero. IfFsexceedsFsc, the bond breaks but the contact forces remain the same, provided that the shear force does not exceed the friction limitFsmaxand the normal force is compressive. If the shear force exceedsFsmax, the slip occurs. The contact behavior is shown in Fig. 1.

Fig. 1 Constitutive behavior for contact occurring at a point (Itasca Consulting Group 2002) (Unis the normal displacement,Usis the shear d isplacement,Knis the normal stiffness, andKsis the shear stiffness)

It should be noted that each broken bond is analogous to the formation of a crack. Each bond-break event will be recorded into the number of cumulative cracks, which is also used as the cumulative crack number of the AE in this study.

2.2 Numerical models for rocks

Two sets of micro-parameters for the numerical models were chosen. One set was for the numerical model of the coarse-grained sandstone, which had been verified in previous work (Hunt et al. 2003). In the study of Hunt et al. (2003), the results of simulation were in good agreement with those of physical experiments. This verified that the micro-parameters of the contact bond model were available for simulation of the mechanics of the coarse-grained sandstone. The second set was for the model of Aue granite, which was used by Yoon (2007), where a new approach that calculated micro-parameters was developed. Based on the research of Yoon (2007), the quantitative and qualitative comparisons between the results of the contact bond model and the results of laboratory tests revealed that the set of micro-parameters provided reliable results. Two sets of the micro-parameters are listed in Table 1.

Two types of numerical models were designed for the coarse-grained sandstone and Aue granite: the square shape and the Brazilian disk. For the coarse-grained sandstone, the dimension of the square specimen was 100 mm × 100 mm, and the diameter of the Brazilian disk was 100 mm. The sizes of the round particles ranged from 0.8 to 1.2 mm. For the Aue granite, the dimension of the square specimen was 50 mm × 50 mm, and the diameter of the Brazilian disk was 50 mm. The sizes of the round particles ranged from 0.25 to 0.415 mm. An illustration of the numerical models is provided in Fig. 2.

Table 1 Micro-parameters of numerical models for coarse-grained sandstone and Aue granite

Fig. 2 Numerical models

2.3 Numerical experiment protocols

To study the influence of orthogonal loading on KE and DRA in rocks, three loading protocols were performed on the numerical models of two types of rocks. In all the numerical experiments, the loading was controlled by platen velocity, which was equal to 0.1 m/s. The displacement was increased or decreased at equal velocity in the experiment. The loading protocols were as follows:

Loading protocol 1: cyclic loadings in theYdirection on the Brazilian disks and square models of the two types of rocks.

Loading protocol 2: loading in theYdirection after preloading in theXdirection on the Brazilian disks and square models of the two types of rocks.

Loading protocol 3: cyclic loadings in theYdirection after preloading in theXdirection on the square models.

3 Results and discussion

3.1 Loading protocol 1

This loading protocol focused on the capability of the contact bond model to reproduce the KE and DRA phenomena. In the Brazilian disk tests, the peak values of the strain were0.004, 0.005, 0.006, and 0.007 for the sandstone, and 0.0012, 0.0015, 0.0018, and 0.0021 for the granite. The cyclic loadings in terms of strain are shown in Fig. 3. The corresponding results are shown in Fig. 4. In the uniaxial compression tests on the square models, the peak stresses were 30 MPa, 40 MPa, 40 MPa, 50 MPa, and 50 MPa for the sandstone, and 70 MPa, 90 MPa, 90 MPa, 110 MPa, and 110 MPa for the granite. The cyclic loadings in terms of stress on the square models are shown in Fig. 5. The corresponding results are shown in Figs. 6 and 7.

Fig. 3 Cyclic loadings on Brazilian disks

Fig. 4 Cumulative crack number vs. strain in Brazilian tests

Fig. 5 Cyclic loadings on square models

Fig. 4 and Fig. 6 show that there are no cracks generated in the subsequent cyclic loading until the peak strain (stress) is exceeded. The curve of cumulative crack number vs. strain (stress) have clear inflections at the peak strain (stress) attained in the previous loading.Lavrov et al. (2002) performed both the laboratory experiment and numerical simulation on Brazilian disks without direction rotation, and the results show a clear Kaiser effect. The results of the AE method based on the contact bond model described in this study show good agreement with the results obtained by Lavrov et al. (2002).

Fig. 6 Cumulative crack number vs. stress in square models

Fig. 7 Differential strain vs. stress in square models

Fig. 7 shows the Δε2,3(σ)curve and Δε4,5(σ)curve of the square models. The Δε2,3(σ)and Δε4,5(σ)curves had a clear inflection at the peak stress attained in the previous loading. The Δε2,3(σ)curve recollected the peak stress in the first cycle loading. However, the Δε4,5(σ)recollected the peak stress value in the third cycle loading. These results, combined with the results of the AE method, indicate that the contact bond model of the rock has the capability of recollecting the largest stress value reached in the loading history, which was also discussed by Lavrov (2003). The results verify that the numerical models can reproduce the KE and DRA phenomena.

3.2 Loading protocol 2

This loading protocol mainly aimed to study the directional dependency of the AE method and the influence of the preloading on the AE activities in the subsequent loading in the orthogonal direction. In the tests, the number of cumulative cracks was selected as the indicator of the variation in the subsequent loading.

Four cases were analyzed for the Brazilian disks and the square models. Table 2 showsthe four cases of the Brazilian disks and square models. In case 1, there was no preloading in theXdirection. For the other three cases, different levels of the strain for Brazilian disks (stress for square models) were preloaded in theXdirection. Then, the second loading in theYdirection was performed on the same model.

Table 2 Loadings on Brazilian disks and square models in four cases

Fig. 8 shows that the curves of the cumulative crack number vs. strain in cases 2, 3, and 4 were similar to that in case 1. These results demonstrate that no Kaiser effect phenomenon can be observed in the second loading on the Brazilian disks in theYdirection after preloading in theXdirection. Similarly, Fig. 9 shows that there is no specific inflexion in the curve of cumulative crack number vs. stress at the peak stress attained in the preloading. This result is supported by the previous physical experiments. When a sample was subjected to sequential loadings in several orthogonal directions, the Kaiser effect in one direction was unaffected by the loading in the orthogonal direction (Michihiro et al. 1991/1992). There is another kind of experiment, where only two sequential loadings were performed on the samples in perpendicular directions. Holcomb and Costin (1986) performed a uniaxial loading cycle on a large block of rock. Then, sub-samples with different direction deviations with the loading axis were extracted from it. The results indicated that the Kaiser effect could not be observed when the deviation was larger than 10°. This conclusion was confirmed by the experiments on square plate samples (Chen et al. 2007) and cyclic Brazilian tests (Lavrov et al. 2002).

Fig. 8 Cumulative crack number vs. strain for loadings on Brazilian disks inYdirection

Fig. 9 Cumulative crack number vs. stress for loadings on square models inYdirection

However, as shown in Figs. 8 and 9, the preloading in theXdirection has an influence on the AE activities of loading in theYdirection. The number of the cumulative cracks decreased along with the increase of the peak stress (strain) value attained in the first loading. Lavrov et al. (2002) performed similar numerical simulation based on the DIGS. In their study, the previous loading in one direction resulted in fewer cracks in the subsequent loading in the orthogonal direction. The results based on the contact bond model are the same as those based on the DIGS.

3.3 Loading protocol 3

This loading protocol was designed to further study the influence of the preloading in theXdirection on the DRA in the second loading in theYdirection. Six cases were performed on each model, shown in Table 3.

Table 3 Cyclic loadings on square models

For the six cases of the granite and the sandstone, there is no inflection in the Δε2,3(σ) curve. In other words, there is no memory effect in the orthogonal direction after preloading in one direction for the granite and sandstone. It agrees with the result of the AE method for the granite and sandstone.

However, for the fourth and fifth loadings, the results are different. There is a clearinflexion in every Δε4,5(σ) curve at the peak stress value attained in the third loading. The results are listed in Table 4. From the results of the two types of the rock, it can be concluded that the preloading in theXdirection did not affect the ability of the DRA method to recollect the peak stress value attained in the third loading in theYdirection. The felicity ratio ranged from 1.013 to 1.025 for the sandstone and from 1.017 to 1.028 for the granite.

Table 4 Results of DRA method between the fourth and fifth loadings

As shown in Fig. 10, the value of Δε4,5(σ)at the inflection point is influenced by the preloading in theXdirection. Generally speaking, the value of Δε4,5(σ)at the inflection point decreases with the increase of the stress value preloaded in theXdirection in the first loading. However, the influence of the preloading on the subsequent loading is different for different types of the rock. For the granite, the value of Δε4,5(σ)increases when the peak stress is less than 70 MPa and decreases when the peak stress is not less than 70 MPa, which is different from the case for sandstone.

Fig. 10 Value of Δε4,5(σ)at inflection point vs. peak stress preloaded in first loading inXdirection

4 Conclusions

To study the AE and DRA methods in orthogonal loading directions, Brazilian disk tests and uniaxial compression tests were performed on the numerical models of two types of rock, sandstone and granite. The contact bond model, based on PFC2D, was introduced to perform numerical study on the directional dependency of the two methods. The directional dependency of the DRA method was first examined. The specific influence of the orthogonal loading on the memory effect was also examined. The main conclusions are as follows:

(1) No Kaiser effect is observed in the subsequent loading in the orthogonal direction after the preloading in one direction by the AE method. After the preloading in one direction, no memory effect occurs in the Δεij(σ)curve between the subsequent two successive loadings in the orthogonal direction.

(2) The subsequent loadings in one direction are influenced by the first loading in the orthogonal direction. For the AE method, the cumulative crack number of the second loading is influenced by the first loading in the orthogonal direction. For the DRA method, generally speaking, the deformation strain value between the fourth loading and the fifth loading decreases along with the increase of the peak stress attained in the first loading in the orthogonal loading. However, the peak stress value attained in one direction is still recollected by the DRA method in the subsequent loading in the same direction after the first loading in the orthogonal direction.

In this study, only sandstone and granite were examined. For further study, we recommend that more types and different sizes of rocks should be selected. In this study, only the case in which the rotation of loading direction is 90? was examined in the Brazilian disk tests. Further study should expand the rotation angle between cyclic loadings from 0? to any degree to investigate the directional dependency of the AE and DRA methods based on the contact bond model.

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This work was supported by the National Natural Science Foundation of China (Grant No. 50978083), the Fundamental Research Funds for the Central Universities (Grants No. 2009B07714 and 2010B13914) in China, and the Innovation Project for Graduate Students of Jiangsu Province (Grant No. CX10B_215Z).

*Corresponding author (e-mail:hj.wanghhu@gmail.com)

Received May 3, 2011; accepted Sep. 1, 2011