3.1 Introduction

Cement is grey fine powder that is widely used in civil engineering and mining engineering [1, 2]. In civil engineering, as an important ingredient, cement is always mixed with water, sand and rocks to form concrete, which is the commonly used construction or building material [3, 4]. In mining engineering, cement is usually mixed with water to form the cement grout, which is then poured into drilled boreholes to bond the reinforcing tendon, such as rock bolts and cable bolts, with the surrounding rock masses [5]. Moreover, the cement can be used as the grouting material to be poured into rock mass to improve its mechanical behaviour.

The mechanical properties of the cement grout have a significant effect in deciding the performance of the rock reinforcement systems [6]. Consequently, a vast of research has been conducted to study the mechanical behaviour of the cement grout [7]. Those studies can be basically divided into studying the axial behaviour of the cement grout and shear behaviour of the cement grout.

To study the axial behaviour of the cement grout, researchers mainly used the Uniaxial Compressive Strength (UCS) tests. Feldman and Beaudoin [8] indicated that the strength and elastic modulus of the Portland cement grout were largely influenced by the porosity. Domone and Thurairatnam [9] found that with the w/c ratio decreasing, the strength of the cement grout increased. Hyett et al. [10] evaluated the influence of curing time on the grout strength and found that the grout UCS increased nonlinearly with the hydration time ranging from 1 to 7 days. Boumiz et al. [11] adopted the ultrasonic and calorimetric methods on cement grout. They found that the larger the hydration degree, the higher the elastic modulus of the cement grout. Li et al. [12] studied the influence of nanoparticles on the performance of cement grout. They proved that adding nanoparticles was beneficial for improving the grout strength. Mirza et al. [13] revealed that the grout strength increased within the curing period, independent of the cement type.

The above research played an important role in revealing the mechanical performance of cement grout. However, experimental tests proved that in the fully grouted rock reinforcing system, failure of the system usually occurred at the bolt/grout interface [14]. This was resulted by the low shear strength of the interface [15]. The shear behaviour of the cement grout has a significant role in determining the shear strength of the bolt/grout interface [16]. Therefore, better understanding the shear behaviour of the cement grout was beneficial for preventing failure of the bolt/grout interface.

To study the shear behaviour of cement grout, researchers commonly used tri-axial tests and direct shear tests. For tri-axial tests, Hyett et al. [17] studied the performance of Portland cement grout with the confining pressure ranging from 5 to 15 MPa. The results showed that the larger the confining pressure, the higher the strength of the cement grout. Moreover, the cohesion and the internal friction angle were successfully acquired. Simms and Grabinsky [18] performed tri-axial tests on cement material under the confined condition. The results showed that in the shearing process, the cementitious material had an apparent tendency to dilate. Sakai et al. [19] studied the performance of the cement paste under the tri-axial condition. The results proved that increasing the confining pressure changed the sample failure mode. Specifically, when the confining pressure was 30 MPa, oblique shear failure plane occurred in the sample. However, once the confining pressure was much larger, horizontal failure plane occurred in the sample. Aziz et al. [20] conducted tri-axial tests on two types of cement grout and used Mohr–Coulomb (MC) model to fit the shear strength envelope. A vast of tri-axial tests were conducted by Chen and Xu [21] to study the performance of the cement grout under the high confining pressure condition. It was found that with the confining pressure increasing, the performance of the cement grout changed from brittle to ductile. Moreover, this phenomenon was valid independent of the w/c ratio. Those tri-axial testing research improved the understanding in the shear behaviour of cement grout. However, in tri-axial tests, the shear failure plane occurred dependent on the properties of the grout and the loading status [22]. This was not a true reflection of the shear failure of the bolt/grout interface in the rock reinforcing system, where the shear failure plane was a pre-defined plane [23, 24].

Therefore, the direct shear test was a better approach to study the shear behaviour of the cement grout for the rock reinforcement scenario. The direct shear test is a traditional method to evaluate the shear strength of a material. Normally, a shear box is prepared. Then, the tested sample is cut to the desired geometry with the required dimension. After that, the sample was set in the shear box and installed in the apparatus. Operators can apply normal stress on the sample and then apply shear velocity along the shear direction. This test is continued until the shear strength was obtained. In this way, the shear strength of the material can be acquired. Moosavi and Bawden [25] designed a shear box to conduct direct shear tests on Portland cement grout. The results showed that the direct shear test was likely to provide a smaller shear strength of the cement grout, when compared with tri-axial tests. Lu and Wang [26] performed direct shear tests on cementitious samples with two different w/c ratios: 0.3 and 0.4. They found that when the w/c ratio was 0.3, shear strength of the cement-based sample increased apparently with normal stress. However, when the w/c ratio was 0.4, shear strength of the cement-based sample did not increase much with the normal stress. Assaad et al. [27] proposed using the direct shear test to study the yield stress of cement material. The results revealed that the material shear strength was largely dependent on the normal stress, following the MC rule. Chen et al. [28] compared the shear performance of a Portland cement grout under the constant normal load and constant normal stiffness condition. It was found that the boundary condition might influence the performance of the cement grout. Li et al. [29] adopted the direct shear tests to investigate the influence of particle size on the shear strength of cement grout. They found that through adding superfine cement, the early-age shear strength parameters of the cement grout can be largely enhanced.

That research provided new insights and knowledge in revealing the shear performance of cement grout. However, most research was conducted with experimental tests. Compared with that, much less work has been performed via numerical simulation. Therefore, this study aims at studying the shear behaviour of the cement grout with numerical simulation.

First, a numerical calculating programme was selected to conduct this research. After that, numerical direct shear tests were performed to evaluate the shear behaviour of the cement grout. Then, the numerical calculating results were validated with experimental results. Last, the shear stress distribution on the shear failure plane was investigated.

3.2 Selection of the Software

In this study, the three-dimensional software FLAC3D was used. This programme was developed by the Itasca Consulting Group.

The reason to use this programme is that the Itasca Company developed the interface element in it. This interface element is composed of triangular elements and corresponding nodes. Moreover, the interface element is defined with the shear coupling constitutive model. In this case, once shear displacement occurs in the interface element, shear stress at the interface accumulates and can be calculated. Relied on this function, the shear behaviour of a material can be studied.

The authors of this study previously used the two-dimensional software FLAC to simulate the shear failure process of the bolt/grout interface [30]. However, the drawback was that the problem was simplified to be two dimensional. Compared with FLAC, the advantage of FLAC3D was that it can simulate objects in three dimensions. Therefore, it can better reflect the mechanical behaviour of objects. Then, in this study, the FLAC3D was used to simulate the shearing process of the cement grout.

3.3 Numerical Simulation Process

A vast of direct shear tests were performed by Moosavi and Bawden [25]. Those tests were used in this study and simulated with FLAC3D. In their study, two different w/c ratios were used, namely 0.4 and 0.5. Then, in the following section, numerical tests on those two cement grouts were performed.

3.3.1 Simulation of the Cement Grout with a Higher Strength

3.3.1.1 Numerical UCS Test and Validation

Before the direct shear test was conducted, UCS tests were performed to acquire the basic mechanical properties of the grout, including the UCS, elastic modulus and the Poisson's ratio. Detailed information about the UCS test process and results were reported by Moosavi [24]. Experimental results showed that for the cement grout with a w/c ratio of 0.4, the UCS, elastic modulus and Poisson's ratio were 50.6 MPa, 12.66 GPa and 0.25. More specifically, in the UCS test, the cement grout was filled in PVC tubes with the diameter of 50 mm and the length of 150 mm.

In numerical simulation in FLAC3D, the w/c ratio cannot be directly simulated. As a compromise, the UCS of the numerical mesh was selected as the index to compare with the experimental result. In FLAC3D, a numerical UCS test was created (Fig. 3.1).

Fig. 3.1
A shaded 3-D illustration of numerical simulation U C S test. The plates at the top and bottom with the sample in between are indicated.

Geometry of the UCS test created in numerical simulation

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In the numerical UCS test, a cylindrical mesh was established. At the top and bottom of the cylinder, two square steel plates were established to simulate the loading plates. The whole mesh was composed of 16,032 zones and 16,531 grid points. For the numerical plate, there were 32 zones and 100 grid points. For the numerical sample, there were 16,000 zones and 16,441 grid points.

A strain softening model was used for the cylindrical sample. It was developed based on the MC model. Compared with the MC model, the strain softening model had additional input parameters: the cohesion table and the internal friction angle table. With those two parameters, the cohesion and internal friction angle of the materials can decrease with the increasing plastic shear strain. Therefore, this model can simulate the post-peak behaviour of materials.

The reason to use this strain softening model was that this model can simulate the post-peak behaviour of materials. Therefore, it is better in reflecting the mechanical behaviour of materials.

An elastic model was used for the square plates. The reason to use the elastic model to simulate the square plate was that in the experimental compressive test, the cement-based sample was compressed by the steel plate, since the strength of the steel plate was much higher than the cement-based sample. Failure only occurred in the cement-based sample. The steel plate consistently kept the elastic state. The input parameters for the cylindrical sample and the square plates are tabulated in Table 3.1.

Table 3.1 Input parameters for the sample and the plate
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The velocity of the plates was fixed. Moreover, the initial velocity of the top plate along the z-axis was set as −1e−7 m per step (minus represented that the velocity was along the negative direction of the z-axis). Since the velocity of the plate was fixed during the simulation process, the top plate would decline with a constant velocity. Additionally, no velocity was applied on the bottom plate. Consequently, the bottom plate would be stable in the whole simulation process. With this method, the sample in the middle would be compressed by the top plate to simulate the UCS process.

In the simulation process, the displacement along the z-axis of four grid points in the top plate was monitored (Fig. 3.2).

Fig. 3.2
A shaded 3-D illustration of the simulation process. 4 grid points are indicated on the sample and 4 on the top plate. The plate, sample, and 8 grid points are indicated in different shades.

Grid points monitored in the simulation process

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Those four displacements were averaged as the uniaxial displacement of the sample, as shown in Eq. (3.1).

$$s_{{\text{a}}} = \frac{{s_{{{\text{a}}1}} + s_{{{\text{a}}2}} + s_{{{\text{a}}3}} + s_{{{\text{a}}4}} }}{4}$$
(3.1)

where sa: the axial displacement; and sai: the axial displacement of the grid point i in the top plate.

In UCS test, to calculate the Poisson's ratio, the circumferential strain gauge was usually attached at the middle of the sample, to record the circumferential displacement of the sample. Then, in this simulation, the circumferential displacement of four grid points at the middle of the outside surface of the sample was monitored. Those four displacements were also averaged as the circumferential displacement of the sample, as shown in Eq. (3.2).

$$s_{{\text{c}}} = \frac{{s_{{{\text{c}}1}} + s_{{{\text{c}}2}} + s_{{{\text{c}}3}} + s_{{{\text{c}}4}} }}{4}$$
(3.2)

where sc: the circumferential displacement of the sample; sci: the circumferential displacement of the grid point i at the middle of outside surface of the sample.

Since the bottom plate was stabilised in the whole simulation process, the reaction force along the z-axis of the grid points at the bottom surface of bottom plate was summed. Then, this summed force was divided by the cross-section area of the sample, and the result was regarded as the uniaxial compressive stress of the sample.

Then, the numerical UCS test was conducted. The stress-to-strain relationship of the sample is shown in Fig. 3.3. In the numerical simulation, the UCS of the sample was 55.87 MPa.

Fig. 3.3
A line graph of stress versus strain plots 2 lines. The line for simulation results follows a fluctuating trend. The line for linear regression decreases, cuts in between, and increases.

Stress-to-strain relationship of the cement grout sample

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In the experimental test, the UCS of the cement grout sample was 50.6 MPa, which was consistent with the numerical simulation results.

It also shows that in numerical simulation, after the stress of the sample reached the peak, the circumferential strain decreased. To check the reason, the circumferential displacement of a grid point was taken as an example and plotted. The grid point at the right middle of the sample (with an ID of 1430) was selected. The circumferential displacement of the grid point with the calculating time step is shown in Fig. 3.4. The results show that before the calculating time step reached 6650, the circumferential displacement of the grid point increased. This indicated that the diameter of the sample at the middle increased because of the compression. However, after the calculating time step reached 6650, the circumferential displacement of the grid point started decreasing. This indicated that after the sample reached the peak strength, the diameter of the sample in the middle also reached the peak. After the sample failed, the diameter of the sample in the middle started decreasing. This explained the phenomenon that after the sample reached the peak strength, the circumferential strain started decreasing.

Fig. 3.4
A line graph of displacement versus time step. The line, simulation result rises, peaks, and then drops. The peak position is plotted at (7000, 0.028), approximately.

Circumferential displacement of the grid point at the right middle of the sample

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Based on the numerical simulation, the elastic modulus and the Poisson's ratio can also be acquired. The results showed that the elastic modulus in numerical simulation was 13.47 GPa. The Poisson's ratio can be calculated with Eq. (3.3) [31]:

$$\mu = - \frac{{\varepsilon_{{\text{c}}} }}{{\varepsilon_{{\text{a}}} }}$$
(3.3)

where μ: Poisson's ratio; εc: the circumferential strain of the sample; and εa: the axial strain of the sample.

Then, the Poisson's ratio can be calculated as 0.26. Comparing the elastic modulus and Poisson's ratio acquired from numerical simulation with those acquired from experimental tests, there was a close match between them, as shown in Table 3.2. This further confirmed that the input mechanical parameters were capable for simulating the cement grout.

Table 3.2 Comparison between the mechanical properties of the cement grout acquired from laboratory tests conducted by Moosavi [24] and calculated from numerical simulation
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3.3.1.2 Numerical Direct Shear Test and Validation

The input mechanical parameters used in the UCS simulation were then used in this section to simulate the direct shear process. Moosavi and Bawden [25] conducted experimental direct shear tests on cylindrical samples with a diameter of 100 mm and length of 200 mm. The test process was simulated in this section, and the simulation result was then validated with experimental results.

In FLAC3D, a numerical cylindrical sample whose dimension was same as the experimental sample was established. The sample was composed of two sections, namely the top section and the bottom section. In the whole numerical model, there were 18,000 zones and 19,232 grid points. It should be mentioned that in the direct shear test, the shear failure plane is a pre-defined plane. Consequently, in numerical simulation, an interface was created at the middle of the sample, namely between the top section and the bottom section, to simulate the shear failure plane. The interface was composed of 49,128 elements and 24,725 nodes.

The constitutive model of the created interface was MC model, and the input parameters of the interface are tabulated in Table 3.3.

Table 3.3 Input parameters for the interface
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The top section was fixed along the x-axis. For the bottom section, the external surface was fixed along the x-axis and y-axis. This was to simulate that in the experimental test, the bottom sample was restricted by the shear box and no movement could occur. As for the bottom surface of the sample, it was fully fixed to simulate that in the experiment, the whole sample was supported with a plate. A compressive normal stress of 2 MPa was applied on the top surface of the sample. Then, this numerical model was calculated until the unbalanced force ratio decreased to 1e−5. With this method, the initial normal stress equilibrium condition was acquired. The normal stress distribution in the sample is shown in Fig. 3.5. It is apparent to see that there was basically a uniform normal stress distribution in the sample. Under compression of the normal stress of 2 MPa, the normal stress in the sample was around 2 MPa.

Fig. 3.5
A contour plot depicts the normal stress distribution in a sample. The applied force with arrows pointing downwards is indicated on the top of the sample. The sample is shaded based on a scale of Z Z stress mostly with average values.

Normal stress distribution in the sample

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Before the direct shear test was performed, the displacement along the shearing direction of four grid points at the top surface of the sample was monitored. In the simulation process, the displacements of those four grid points were averaged as the shear displacement of the sample.

It is more important to monitor the shear stress in the direct shear test process. To acquire this, in each step in the direct shear test process, each node on the interface was checked whether that node was still contact. If it was, the shear force of that node was recorded and summed. The accumulated shear force was the whole shear force of the interface and was then divided by the area of the interface. The result was regarded as the shear stress of the interface.

Then, the direct shear test was performed. A shearing velocity of 1e−6 m per step was applied on the top section of the sample. This velocity was kept constant in the whole simulation process, to simulate the servo control. The direct shear test was conducted until the sample failed.

The comparison between the numerical simulation result and the experimental test result is shown in Fig. 3.6. In numerical simulation, when the shear displacement increased to 0.94 mm, the shear stress of the interface increased to the peak of 6.48 MPa. After that, the shear stress decreased gradually. Also, there was a close match between numerical and experimental results. This confirmed that numerical simulation was capable in simulating the shear behaviour of the cement grout. It should be mentioned that this comparison was compared with the final shear displacement being less than 2 mm.

Fig. 3.6
A line and scatter plot of shear stress versus shear displacement. The plots for experimental results follow an increasing trend and then follow a lateral trend. The line for numerical results follows the same trend as the plots.

Comparison between numerical simulation result and the experimental test conducted by Moosavi and Bawden [25]

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In the shearing process, the averaged normal stress of the interface was also monitored. The variation of the normal stress on the interface is shown in Fig. 3.7. Before the shear displacement increased to 0.94 mm, the normal stress of the interface increased slightly to the peak of 2.15 MPa. After that, the normal stress of the interface decreased gradually. However, in the whole process, the minimum normal stress of the interface was 1.99 MPa, which was only a little bit smaller than the maximum normal stress of the interface. This indicated that in the direct shear process, the normal stress of the interface was not constant although the applied normal stress was constant. This was probably due to the separation between the top sample and the bottom sample, and the frictional failure of the interface.

Fig. 3.7
A line graph of normal stress versus shear displacement. The line from (0, 2) slightly increases and then decreases to (2, 2). A slope is plotted at (0.9, 2.2), approximately.

Variation of the normal stress of the shear failure plane

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After the direct shear test, a plot of the normal stress distribution on the interface is shown in Fig. 3.8.

Fig. 3.8
A contour plot depicts the shear failure plane with normal stress distribution. The circular plane is shaded based on a scale of interface normal stress ranging from 0.0000 E + 00 to 3.5626 E + 00 in megapascals.

Normal stress distribution on the shear failure plane

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The results showed that the normal stress at the left side of the interface was zero. This was due to the separation of the top and bottom samples. Consequently, the interface was non-contact around the left side, as shown in Fig. 3.9.

Fig. 3.9
2 circular diagrams. A presents a contacted interface before the shear test. B presents a contacted interface with a non-contacted interface on the left after the shear test.

State of the interface in the direct shear test: a before the direct shear test; b after the direct shear test

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Since it is more important to study the state of the interface that was contact, the stresses on the contacted interface were further studied. At the left side of the contacted interface, the normal stress was apparently much higher, with a maximum normal stress of 3.56 MPa. Moreover, the normal stress on the interface decreased towards the shearing direction. At the right side, the normal stress was much lower, with a minimum normal stress around 0.75 MPa.

As for the shear stresses on the interface, its distribution is shown in Fig. 3.10. Around the left side of the whole interface, the shear stress was also zero, due to the non-contact of the interface.

Fig. 3.10
A contour plot depicts the shear failure plane with shear stress distribution. The circular plane is shaded based on a scale of interface shear stress ranging from 0.0000 E + 00 to 7.3569 E + 00. The plane has maximum values.

Shear stress distribution on the shear failure plane

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As for the contacted interface, the shear stress distribution was apparently not uniform. At the left side of the contacted interface, the shear stress was much higher, with a maximum of 7.36 MPa. Then, the shear stress decreased towards the shearing direction. At the right side, the shear stress was relatively lower, with a minimum around 6.0 MPa. This trend was basically consistent with the distribution of the normal stress on the interface.

However, attention should be paid that Figs. 3.8 and 3.10 depicted the interface stress state after the test. In the middle shearing process, whether the distribution state was consistent with them was uncertain. And this will be studied in the following section.

After the direct shear test, the plastic state of the sample is shown in Fig. 3.11. In this figure, “tension-n” means that the zone was experiencing tensile failure now. “tension-p” means that the zone previously experienced tensile failure. It is apparent to see that at the left side of the bottom sample, tensile failure occurred in the sample. Specifically, in the bottom sample, tensile failure occurred around the left side. In fact, this was consistent with the laboratory experimental test. Moosavi and Bawden [25] indicated that in the experimental test, tension crack occurred in the cement grout sample around the shear failure plane.

Fig. 3.11
A shaded 3-D illustration depicts the plastic zone distribution. 3 zones in the sample indicate the tensile failure as none, tension n tension p, and tension p. The tension zones are on the center-left.

Plastic zone distribution in the sample

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3.3.2 Simulation of the Cement Grout with a Lower Strength

Numerical simulation was also conducted on the same cement grout type while with a relative lower strength. First, the numerical UCS test was performed and compared with the experimental result, to confirm the proper input parameters.

3.3.2.1 Numerical UCS Test and Validation

The UCS test was simulated in FLAC3D. The simulation process was generally consistent with the procedures described above. Therefore, detailed simulation procedures were not described here. Only the simulation results were presented.

After the simulation, the stress-to-strain relationship of the sample was obtained. When the axial strain increased to 0.53%, the axial stress of the sample reached the peak of 46.09 MPa. In experimental test, the measured UCS was 40.3 MPa. Therefore, there was a close match between them.

The elastic modulus and Poisson's ratio were also obtained. In numerical simulation, the elastic modulus and Poisson's ratio of the cement grout sample were 8.33 GPa and 0.26. A comparison between experimental and numerical results is shown in Table 3.4. There was a good correlation between them. This further confirmed that the input parameters were robust for simulating the cement grout and can be used in the following direct shear test.

Table 3.4 Comparison between the experimental result reported by Moosavi [24] and calculated from numerical simulation
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3.3.2.2 Numerical Direct Shear Test and Validation

Then, the numerical direct shear test was performed. And the input parameters of the sample were same as those used in the UCS simulation. A comparison between the experimental and the numerical results is shown in Fig. 3.12. In numerical simulation, when the shear displacement increased to 1.41 mm, the shear stress of the interface reached the peak, which was 6.02 MPa. Following this, the shear stress reduced gradually with the shear displacement. The results also showed that there was a good correlation between experimental and numerical results. This further confirmed the effectiveness of numerical simulation in evaluating the shear behaviour of the cement grout. Attention was paid that in this comparison, the maximum shear displacement was less than 1.8 mm.

Fig. 3.12
A line and scatter plot of shear stress versus shear displacement. The plots for experimental results follow an increasing trend and then follow a lateral trend. The line for numerical results follows the same trend as the plots.

Comparison between the experimental result reported by Moosavi and Bawden [25] and the numerical simulation result

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Also, the variation of the normal stress on the shear failure plane was checked. In the simulation process, the normal stress on the shear failure plane was not constant. It had a slight increasing trend at the beginning and then decreased gradually.

3.4 Analysis on the Shear Stress Distribution

Figures 3.8 and 3.10 showed that after the direct shear test, the normal stress and shear stress distribution on the shear failure plane were not uniform. However, in the middle process of the direct shear test, the normal stress and shear stress distribution state were uncertain. Therefore, in this section, the distribution of the shear stress and normal stress on the shear failure plane was further studied.

On the numerical shear stress versus shear displacement curve for the cement grout with a w/c ratio of 0.4, another three different points when the shear displacement increased to 0.6 mm, 0.94 mm and 1.2 mm, were selected. At those points, the shear stress and normal stress distribution on the shear failure plane was plotted.

When the shear displacement was 0.6 mm, the shear stress and normal stress distribution on the shear failure plane was shown in Fig. 3.13. At this point, the shear stress distribution on the interface was generally uniform. On the contacted interface, the maximum shear stress was 4.97 MPa while the minimum shear stress was 4.63 MPa. The relative difference between the maximum and the minimum shear stresses was only 0.34 MPa and the standard deviation was 0.09. Since this relative difference was too small, it can be neglected.

Fig. 3.13
2 contour plots and a line graph. A and B have circular planes for shear and normal stress. The shear stress is high, while the normal stress fluctuates in the shear plane. Graph C of shear stress versus shear displacement plots a line that rises and stabilizes. A selected point is at (0.6, 4.73).

Distribution of the shear stress and normal stress on the shear failure plane when the shear displacement arrived at 0.6 mm: a shear stress distribution; b normal stress distribution; c point selected on shear stress–shear displacement curve

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As for the normal stress, on the contacted interface, there was a non-uniform distribution. Specifically, the maximum normal stress was around 2.96 MPa while the minimum normal stress was around 1.69 MPa. The relative difference between the maximum and the minimum normal stresses was 1.27 MPa, and the standard deviation was 0.29.

When the shear displacement increased to 0.94 m, the distribution of the shear stress and normal stress is shown in Fig. 3.14. Currently, there was an apparent non-uniform distribution of the shear stress on the contacted interface. At the left side, the shear stress was relatively higher, with a maximum of 7.33 MPa. As for the right side, the minimum was around 6.20 MPa. The relative difference between the maximum shear stress and minimum shear stresses was 1.13 MPa, and the standard deviation was 0.28.

Fig. 3.14
2 contour plots and a line graph. A and B have circular planes for shear and normal stress. The shear stress is high, while the normal stress fluctuates in the shear plane. Graph C of shear stress versus shear displacement plots a line that rises and stabilizes. A selected point is at (0.94, 6.48).

Distribution of the shear stress and normal stress on the shear failure plane when the shear displacement arrived at 0.94 mm: a shear stress distribution; b normal stress distribution; c point selected on shear stress–shear displacement curve

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The results also revealed that the non-uniform distribution of the normal stress became more apparent. On the contacted interface, the maximum normal stress was 3.63 MPa, occurring at the left side while the minimum normal stress was 1.55 MPa, occurring at the right side. The relative difference between the maximum normal stress and minimum normal stress was 2.08 MPa, and the standard deviation was 0.5.

When the shear displacement reached 1.2 mm, the non-uniform shear stress distribution was still maintained, as shown in Fig. 3.15. On the contacted interface, the maximum shear stress was 7.39 MPa while the minimum shear stress was 6.18 MPa. The relative difference between the maximum shear stress and the minimum shear stress was 1.21 MPa, and the standard deviation was 0.29.

Fig. 3.15
2 contour plots and a line graph. A and B have circular planes for shear and normal stress. The shear stress is high, while the normal stress fluctuates in the shear plane. Graph C of shear stress versus shear displacement plots a line that rises and stabilizes. A selected point is at (1.2, 6.45).

Distribution of the shear stress and normal stress on the shear failure plane when the shear displacement arrived at 1.2 mm: a shear stress distribution; b normal stress distribution; c point selected on shear stress–shear displacement curve

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As for the normal stress distribution, it was still non-uniform. At the left side of the contacted interface, the normal stress was much higher, with a maximum of 3.61 MPa. At the right side, the normal stress was smaller, with a minimum of 1.52 MPa. The relative difference between the maximum normal stress and minimum normal stress was 2.09 MPa, and the standard deviation was 0.5.

The above three figures showed that at the beginning of the direct shear test, on the contacted area of the shear failure plane, the relative difference between the maximum shear stress and minimum shear stress was quite small. However, with the shear displacement increasing, the relative difference became larger.

To show the variation of the relative difference of the shear stresses, in the calculating process, the relative difference between the maximum shear stress and the minimum shear stress was monitored, as shown in Fig. 3.16. When the shear displacement was smaller than 0.81 mm, the relative difference of shear stresses was quite small. This indicated that in this stage, the shear stress can be regarded that it uniformly distributed on the shear failure plane. However, after that, the relative difference of the shear stresses increased dramatically. When the shear displacement increased to 0.97 mm, which was around the shear strength point, the relative difference of the shear stresses was maximal. After that, the relative difference became almost constant.

Fig. 3.16
A line graph of relative difference for shear stress versus shear displacement. A solid line increases and then remains stable. 2 horizontal and 2 vertical dashed lines meet the solid line at (0.81, 0.49) and (0.97, 1.21).

Variation of the relative difference of shear stresses in the numerical test

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Meanwhile, the relative difference between the maximum normal stress and minimum normal stress was monitored, as shown in Fig. 3.17. It shows that before the shear displacement increased to 0.97 mm, which was around the shear strength point, the relative difference increased monotonously. After that, the relative difference became also constant.

Fig. 3.17
A line graph of relative difference for normal stress versus shear displacement. A solid line follows an increasing trend and then remains stable. Horizontal and vertical dashed lines meet the solid line at (0.97, 2.1).

Variation of the relative difference of normal stresses in the numerical test

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3.5 Recommendation for Further Work

Following the current study, the authors plan to continue the numerical study on the shear behaviour of cement grouts. Considering that the shear behaviour of cement grouts can also be studied with tri-axial tests, the authors may use the numerical method to evaluate the mechanical response of cement grouts under the tri-axial loading condition. Its purpose is to reveal the shear failure mechanism of cement grout during the tri-axial loading process.

3.6 Conclusions

Shear behaviour of the cement grout plays a significant role in determining the stability of systems and structures in civil and mining engineering. To better understand the shear behaviour of the cement grout, numerical direct shear tests were performed with the FLAC3D software.

Cylindrical cement grout samples with two different strengths were established and numerically simulated. The numerical simulation results were compared with experimental results. There was a good correlation between them, confirming that numerical simulation can be used to evaluate the shear behaviour of the cement grout.

The numerical direct shear test results showed that in the shearing process, along the shear failure plane, the shear stress at each point was not identical. There was a shear stress distribution on the contacted shear failure plane. At the beginning of the direct shear test, the relative difference of the shear stresses was quite small. Then, in this stage, it can be assumed that there was a uniform shear stress distribution on the contacted shear failure plane. However, once the shear displacement increased to around the shear strength point, the relative difference of the shear stresses became much larger. In this stage, the shear stress was non-uniformly distributed on the contacted shear failure plane. After that, the relative difference of the shear stress became stable.

Additionally, although the applied normal stress on the sample was constant, the normal stress on the contacted shear failure plane was variable. Before the shear strength point, the normal stress increased slightly. After that, the normal stress gradually decreased. There was also a distribution of the normal stress on the contacted shear failure plane. This normal stress distribution was not apparent at the beginning of the test. However, when the shear displacement increased to around the shear strength point, it became much obvious. Moreover, after that, this non-uniform normal stress distribution became stable.