The Effect of a Moving Boundary on the Shear Strength of Granular Materials in a Direct Shear Test

Abstract

The boundary state significantly influences the soil shear strength. Therefore, it is necessary to overcome the limitations of existing indoor test instruments and determine the differences in the shear properties of granular materials to ensure the economic feasibility and mechanical integrity of engineering structures. In this study, the core formula for the direct shear test was derived from the static balancing analysis of the shear box, the external force on the specimen, and the internal force on the shear surface. Three loading methods were then developed by the staggered state of the upper and lower boxes: the upper box moving shear loading method (UM), the lower box moving shear loading method (LM), and the bidirectional moving shear loading method (BM). Finally, by manipulating the motion boundary, the discrete element method (DEM) was employed to simulate the shear test of granular materials. Among the three loading methods, the order of the peak shear stresses was as follows: UM > BM > LM. Moreover, the order of the sample post-peak stress uniformities was as follows: LM > BM > UM. A shear strength conversion formula was then proposed. The findings of this study promote the advancement of the shear mechanics theory of granular materials in direct shear testing and can serve as a scientific basis for the design and manufacture of shear equipment.

1. Introduction

The mechanical properties of granular materials under external loads have always been the research focus in the field of particle mechanics [1,2,3,4]. As engineering materials, granular materials are widely used in road, airport, and port construction, among other engineering structures. Soil is one of the granular materials used. The failure in soil mass is generally due to shearing, and strain localization can be witnessed along the shearing planes. When soil is subject to an external load, shear stress and shear deformation occur. If the shear stress at a point reaches the shear strength, relative sliding occurs along the shear stress direction, thus causing shear failure. As the load continues to increase, the failure area gradually expands, overall shear failure occurs, and stability is lost. The stability of the soil slope, the bearing capacity of the building foundation, and the calculated earth pressure of the underground structure are closely related to the shear strength of the soil. In soil mechanics, the instability of engineering soil is generally considered as shear failure. Shear strength is one of the most critical mechanical indicators of soil. Among multiple types of shear tests, the direct shear test is one of the most convenient and commonly used methods to determine the shear strength of soil in indoor geotechnical environments [5].

It has been discovered through multiple direct shear testing that boundary circumstances often affect direct shear tests. Tejchman, J. and Bauer, E. [6] compared the difference between the two boundary states of direct shear apparatus and simple shear apparatus with respect to the formation of granular material shear bands. Shang, J. et al. [7] studied the discontinuous failure mechanism of rock in the direct shear test under two boundary conditions of constant normal load and constant normal stiffness. Bahaddini, M. [8] reported that the size of the gap between the upper and lower boxes of the shear box has a significant influence on the direct shear test results. Kostkanová, V. and Herle, I. [9] reported that the upper rigid part of the shear box is subject to lateral restraint during the shear process. The load only requires timeous adjustments during vertical movement according to the soil property during the shear process. Seyyedan, Seyyed Mahdi, et al. [10] found that when the upper box of the shear box moves, only one-third of the particles above the shear box move horizontally and synchronize with the displacement of the sidewalls, thus reflecting the movement relationship between the boundary and particles.

Several scholars found that the direct shear test is subject to uneven stress and inconsistent deformation. Wang, J. et al. [5] studied the development of the shear band of granular soil in the direct shear test as developed from the side boundary to the center, thus forming the primary and secondary shear bands. Zong-Ze, Yin, et al. [11] carried out large-scale direct shear tests on the interface between soil and concrete. The relative slip displacement measured along the interface revealed that the relative displacement distribution is uneven. Nitka, M. and Grabowski, A. [12] found that the direct shear test is not initially uniform. In particular, the material started to shear in the middle part, and the entire sample was mobilized. Zhang, L. and Thornton, C. [13] used the discrete element method (DEM) to simulate a two-dimensional (2D) direct shear test, and the results revealed that the stress and strain of the sample were non-uniformly distributed. Garcia, Fernando E. and Bray, Jonathan D. [14] studied the direct shear test of irregularly shaped particles and found that the shear zone area developed along the middle plane of the sample was not completely horizontal and that the edge of the shear box induced two inclined shear zones. Jewell, R.A. [15] obtained the uneven distribution of stress in the central plane of the free loading cap by studying the application of the loading cap force in the direct shear test. When the loading cap was fixed, the stress distribution on the central plane was more uniform.

At the same time, with the development of numerical simulation technology, the DEM in the form of particle flow is a common method used in the field of soil mechanics to simulate the mesoscopic movement and failure characteristics of particles under shear. It is generally used to establish rigid walls and particles, with rigid walls as the boundary, and to simulate the application of shear force by applying velocity to the rigid walls [16]. However, in the specific implementation process, several scholars fixed the upper box and induced the lower box movement [5,17,18,19], and others fixed the lower box and induced the upper box movement [20,21,22]. As there is no clear definition of the loading method of the moving shear of the upper or lower boxes of the shear box in the direct shear test, the influence of the moving boundary on the test results was rarely considered in previous direct shear test research, especially the calculation of the stress from the macro-force and boundary control methods. At present, systematic research on the variation of the shear strength of granular materials due to the dislocation of the upper and lower boxes under different loading methods is relatively sparse, especially research on the shear loading method of the opposite movement. Moreover, owing to the limitations of indoor test instruments, it is difficult to compare the loading methods of the upper and lower boxes with each other pertinently and to reveal the differences in the shear properties of granular materials. Therefore, based on the basic principle of the direct shear test, in this study, static balance analysis was conducted, and the core equation of the direct shear test was derived. Moreover, Particle Flow Code (PFC) 2D software was employed, in addition to the DEM, and based on the moving boundary, the relationship between the contact force and shear displacement, contact force distribution uniformity analysis, the shear stress–shear displacement relationship, and shear strength conversion in the direct shear test were systematically studied.

2. Basic Principle and Core Formula in a Direct Shear Test

2.1. Structural Principle of the Direct Shear Apparatus

The direct shear apparatus is an instrument used in the direct shear test. The direct shear apparatus is classified into two types: the strain control type and the stress control type. The former causes the sample to undergo shear displacement at an equal strain rate until shear failure occurs, and the latter applies horizontal shear stress and measures the corresponding shear displacement in stages. At present, the strain-controlled direct shear apparatus is widely used.

Figure 1 presents a schematic diagram of a standard strain-controlled direct shear apparatus. The core component of the direct shear instrument is the shear box, which is the mold and loading device of the sample. The shear box is composed of upper and lower boxes. The upper part of the shear box applies a vertical load to the sample via the rigid loading cap. The vertical load generally remains unchanged during the test. In the horizontal direction, the upper box is generally fixed, and the lower box is pushed to cause the sample to shear along the horizontal plane between the upper box and the lower box. The normal stress and shear stress on the shear surface are calculated by measuring the vertical deformation of the test sample, horizontal shear load, and horizontal displacement, to determine the shear strength of the sample.

2.2. Static Balance Analysis

To clarify the basic principle of the direct shear test, it is mainly considered from three aspects: the external force of the shear box, the external force of the sample, and the internal force on the shear surface. The loading force of the shear box and the internal sample were analyzed in this study. The center of the connecting line at the interface of the upper and lower boxes is the coordinate origin, with the horizontal right-side region as the positive x direction and the vertical upward region as the positive y direction.

2.2.1. Taking the Shear Box as the Research Object

Figure 2 presents the force diagram of the shear box used in the direct shear test.

According to the static equilibrium conditions, the static equilibrium equation of the shear box is expressed as follows:

$${\displaystyle \sum F_x}=T_l+T_u=0$$

(1)

$${\displaystyle \sum F_y}=F_b+F_t+G=0$$

(2)

where ${\displaystyle \sum F_x}$ is the resultant force of the shear box in the x direction; T_l is the thrust of the lower box; T_u is the reaction force generated by the upper box and the lower box movement; ${\displaystyle \sum F_y}$ is the resultant force of the shear box in the y direction; F_b is the support force of the lower box in the y direction; F_t is the force acting on the upper cover of the upper box in the y direction; and G is the sum of the gravitational force of the shear box and the specimen in the y direction.

2.2.2. Considering the Sample as the Research Object

Figure 3 presents the force diagram of the sample.

The external force on the outside of the sample satisfies the static equilibrium equation:

$${\displaystyle \sum F_{sx}}=F_1+F_2+F_3+F_4=0$$

(3)

$${\displaystyle \sum F_{sy}}=F_{st}+F_{sb}+G_{sg}=0$$

(4)

where ${\displaystyle \sum F_{sx}}$ is the resultant force of the specimen in the x direction; F₁ is the force of the left wall of the upper box on the sample; F₂ is the force of the right wall of the upper box on the sample; F₃ is the force of the left wall of the lower box on the sample; F₄ is the force of the right wall of the lower box on the sample; ${\displaystyle \sum F_{sy}}$ is the resultant force of the specimen in the y direction; F_st is the force of the upper cover on the specimen in the y direction; F_sb is the force of the lower box on the specimen in the y direction; and G_sg is the gravitational force of the sample in the y direction. It should be noted that the gravity of the specimen in the shear box during shearing is considered to remain unchanged in this study.

The formula can be obtained by the following simultaneous Formulas (1) and (3):

F₁ + F₂ = −(F₃ + F₄)

(5)

2.2.3. Considering the Shear Surface as the Research Object

Figure 4 presents the stress diagram with respect to the shear surface of the sample in the shear box. In the process of shearing, the upper and lower boxes move simultaneously, thus resulting in shearing. The sample is divided into two parts: one half in the upper box and the other half in the lower box.

The balance equation of the sample in the loading cap in the y direction satisfies the following:

$${\displaystyle \sum F_{uy}}=F_{st}+\frac{G_{sg}}{2}+{\displaystyle \int \sigma }\mathrm{d}A=0$$

(6)

where ${\displaystyle \sum F_{uy}}$ is the resultant force of the upper box sample in the y direction; $\sigma$ is the normal stress of the shear surface; and A is the area of the shear surface.

Neglecting the gravitational force G_sg of the sample, Formula (6) is transformed into the following:

$${\displaystyle \sum F_{uy}}=F_{st}+{\displaystyle \int \sigma }\mathrm{d}A=0$$

(7)

Assuming that the normal stress σ on the shear surface is equal at all points, the following is obtained:

$$\sigma =\frac{F_{st}}{A}$$

(8)

The balance equation of the sample in the lower box in the x-direction satisfies the following equation:

$${\displaystyle \sum F_{lx}}=F_4+F_3+{\displaystyle \int \tau \mathrm{d}A}=0$$

(9)

where ${\displaystyle \sum F_{lx}}$ is the resultant force of the lower box sample in the x direction and τ is the shear stress on the shear surface.

Assuming that F₄ is the loading force and F₃ is supported, when F₃ = 0, Formula (9) can be expressed as follows:

$${\displaystyle \sum F_{lx}}=F_4+{\displaystyle \int \tau \mathrm{d}A}=0$$

(10)

Assuming that the shear stress τ on the shear surface is equal at all points, the following is obtained:

$$\tau =\frac{F_4}{A}$$

(11)

3. Numerical Test Scheme

To study the influence of the loading method of the upper and lower boxes on the shear strength of the granular materials in the direct shear test, the loading scheme was formulated as shown in

Table 1. Numerical loading scheme.

Type	Upper Box	Lower Box
UM	Move 4 mm horizontally to the right	Fixed
LM	Fixed	Move 4 mm horizontally to the left
BM	Move 2 mm horizontally to the right	Move 2 mm horizontally to the left
Loading time step	4 × 10−8 m/time step

, where “UM” indicates that the lower box is fixed and the upper box moves horizontally, and “LM” indicates that the upper box is fixed and the lower box moves horizontally. Moreover, “BM” indicates the simultaneous horizontal movements of both the upper and lower boxes in opposite directions. Here, these three loading methods are defined as the upper box moving shear loading method (UM), the lower box moving shear loading method (LM), and the bidirectional moving shear loading method (BM).

Figure 5 presents schematic diagrams of the three loading methods.

Figure 6 presents the numerical model of the direct shear test in PFC 2D, which was mainly composed of a shear box and particles. Six rigid walls were used to simulate the shear box of the physical test. Walls 3, 1, and 4 constituted the upper box of the direct shear test, and Walls 5, 2, and 6 constituted the lower box of the direct shear test. The dimensions of the upper and lower boxes were 61.8 mm × 10 mm. To prevent particles from overflowing during shearing, two horizontal rigid partitions, namely Walls 7 and 8, were generated in the middle of the upper and lower boxes.

To obtain the relationship between the contact force and displacement of the sample, the Walls 3–6 were selected. In addition, at the midpoints of Walls 3–6, four monitoring points were set to monitor the total contact force and displacement value of the corresponding walls, as indicated by the diamond symbol in Figure 6. The measurement point on Wall 3 was denoted as “F₁”, the measurement point on Wall 4 was denoted as “F₂”, the measurement point on Wall 5 was denoted as “F₃”, and the measurement point on Wall 6 was denoted as “F₄”. The contact force on the four sidewalls of the shear box Walls 3, 4, 5, and 6 were recorded, in addition to the horizontal relative displacement of Walls 4 and 6.

The DEM simulation of the direct shear test was divided into the following three stages:

(1)

The test sample was prepared. First, regular disc-shaped particles with a radius of 0.2 mm were generated in the shear box; there were a total of 8360 particles. There is no need to take boundary effects into account because the minimum size of the shear box is 20 mm, which is 50 times the diameter of 0.4 mm spherical particles [12]. The accumulated particles were then dispersed in the shear box by the expansion method. The gravity deposition method was used to apply gravity to particles, such that the particles fell and were deposited naturally under the action of gravity.

In this study, cohesionless granular particles were simulated. For the microscopic contact model between particles, the in-built PFC 2D linear model [23] was used to carry out the simulation study. The model parameter values were set with reference to existing research [24] and were adjusted appropriately. See

Table 2. The PFC simulation parameters.

Particle Radius mm	Particle Density (kg·m−3)	Initial Porosity	Gravity Acceleration (m·s−2)	Effective Modulus (N·m−2)	Normal/Shear Stiffness Ratio	Ball–Ball Friction Coefficient	Ball–Wall Friction Coefficient
0.20	2650	0.15	9.81	1.0 × 108	1.00	0.75	0

for the specific values.

(2)

Normal loading. After sample preparation, the positions of Walls 2–6 were fixed. Thereafter, a normal stress of 100 kPa was applied to move Wall 1 vertically to compress the sample and achieve stability.

(3)

Shear loading. After the sample was compressed and stabilized, the normal stress of Wall 1 was maintained. The lateral wall was controlled to move horizontally at 4 × 10⁻⁸ m/time step to simulate the shear loading process in the direct shear test. When the relative displacement of the upper and lower boxes reached 4 mm, the shear was considered complete.

It is important to note that the loading rate in this case is different from laboratory testing. This is because it reflects the amount of time spent in virtual computation, which does not coincide with the lab testing. In the lab, the direct shear test can be completed within a short amount of time, whereas if the physical loading rate is input into the virtual calculation, it may take days or even weeks to complete. The literature [20,25,26] can prove that the sample is in a quasi-static state when loading at the rate of 4 × 10⁻⁸ m/time step, which is slow enough. The loading rate in this study is appropriate when taking time cost control into account. For details on how loading rate affects materials, please refer to the relevant literature [27,28].

4. Mechanical Property Analysis

4.1. Relationship between Contact Force and Shear Displacement

Figure 7(a₁,b₁,c₁) present the relationship between the contact force and shear displacement of the shear box sidewalls under the three loading methods. Figure 7(a₂,b₂,c₂) present the force chain diagrams corresponding to the maximum contact force peak point. It should be noted that a scalar representation is employed to make it easier to compare the magnitude of the four contact forces.

Figure 7(a₁) reveals that the curve of F₁ initially shifted to the first peak of 7400 N, declined, and then continued to rise and fall with an approximately cyclical fluctuation. The contact force at F₂ was maintained at 0 N. The contact force at F₃ increased gradually to 1370 N and then decreased. The contact force at F₄ first increased to a peak value of 2880 N and then decreased.

Figure 7(b₁) reveals that when the displacement of F₁ was 0.78 mm, the contact force reached a peak value of 3800 N, and then exhibited an approximately linear decrease. When the displacement of F₂ was 2.5 mm, the curve started to shift upward. The F₃ curve remained horizontal, and its value was 0 N. When the displacement of F₄ was 0.67 mm, the contact force reached a peak value of 4120 N. Thereafter, the curve was approximately in the form of a negative exponential function. The peak contact force values of F₁ and F₄ were relatively close (both approximately 4000 N).

Figure 7(c₁) reveals that F₁ reached a peak value of 6830 N when the displacement was 0.97 mm, and then generally exhibited a downward trend. The F₂ curve was maintained at 0 N. Although there were several data fluctuations in the F₃ curve, compared with the data of the F₁ and F₄ curves, the data of the F₃ curve were significantly small. When the displacement of F₄ was 1.12 mm, it reached a peak value of 5550 N and then decreased.

Compared with Figure 7(a₁,b₁,c₁), the UM data were relatively discrete and can be roughly divided into four levels. The LM and BM data were relatively concentrated. Under the three loading methods, the characteristics of the curves before the peak were relatively consistent, which can be regarded as linear elasticity and elastic plasticity. Moreover, the laws of the curves after the peak were significantly different.

As shown in Figure 7(a₂,b₂,c₂), with an increase in the color darkness of the force chain, the contact force between the particles increased. The force chain was influenced by the sidewalls, such that the applied shear force was converted into the contact force between the particles and extended to the inside of the sample. The force chain in the upper left corner and lower right corner of the shear box passed through the middle of the shear box and was distributed diagonally in the box; whereas the contact force between the particles in the lower left corner and the upper right corner were significantly small, and the force chain was considerably weak. The force chain shape, as shown in Figure 7(a₂), was consistent with the literature [29], and the force chain shape, as shown in Figure 7(b₂), was consistent with the literature [18]. This verifies the numerical model and simulation results. The shapes of the force chains formed by the three different loading methods were different, which indicates that the force chain network structure of the particle system in the shear box changed with respect to the different loading methods, further indicating that the capacity of the particles corresponding to the three loading methods to bear the shear effect was not the same.

4.2. Analysis of Contact Force Distribution Uniformity

According to Figure 7, the peak contact forces of the curve data of F₂ and F₃ were smaller than F₁ and F₄, and the force chain was sparse when compared with F₁ and F₄. To facilitate the statistical analysis of data, the following explanation is provided. During the shearing process, F₂ in UM, F₂ and F₃ in LM, and F₂ and F₃ in BM were 0 N. The details of the peak contact force of the sidewalls are summarized in

Table 3. Peak contact force of sidewalls (unit: N).

Loading Methods	F1peak	F2peak	F3peak	F4peak	F1peak − F4peak	F1peak/F4peak
UM	7400	0	1370	2880	4520	2.57
LM	3800	0	0	4120	−320	0.92
BM	6830	0	0	5550	1330	1.24

. In Table 3, F_ipeak is the peak contact force of F_i at (i = 1, 2, 3 and 4).

Table 3 reveals that F_2peak and F_3peak in LM and BM were 0 N, verifying the following equation for F_2peak and F_3peak in LM and BM:

F_2peak = F_3peak = 0

(12)

Table 3 reveals that the contact force was mainly concentrated in F₁ and F₄ under three loading methods. The movement of the wall promoted contact between the particles and the wall. At the same time, the displacement of the wall caused mutual extrusion between the particles in the shear box, forming a force chain that transferred the stress. Granular materials transfer stress through microscopic contact and the arrangement of particles, resulting in a large contact force between F₁ and F₄, which is consistent with the results for the force chain network characteristics shown in Figure 7.

According to Figure 7 and Table 3, the difference between F₁ and F₄ was different under the three loading methods. In continuum mechanics, when forces applied on opposing sides of a member section are equal in size, directed in the opposite direction, and parallel to one another, shear occurs, causing the two parts of the member to deform along the shear plane. Two forces acting in opposing directions demonstrate load symmetry throughout this shearing process. The shear force produced by a symmetrical load applied to a symmetrical continuum structure should be equal; however, in non-continuum mechanics, there are discrepancies between the two, demonstrating asymmetry because of the discontinuity, heterogeneity, and discreteness of the medium. The difference between F_1peak and F_4peak under the three loading methods was in the following order: UM > BM > LM. The ratio of F_1peak to F_4peak in LM was 0.92. The difference between 0.92 and 1 is only 8%, and the error range is less than 10%. Hence, F_1peak in LM can be considered as equivalent to F_4peak:

F_1peak = F_4peak

(13)

The ratio of F_1peak to F_4peak under the UM and BM loading methods was 2.57 and 1.24, respectively, which does not satisfy Formula (13). In contrast, the difference between F_1peak and F_4peak in LM was 8% and F_2peak and F_3peak were equal, indicating that the peak contact force distribution in the sample under the LM loading method was relatively uniform.

4.3. Shear Stress–Shear Displacement Relationship

The curve of the maximum peak stress under the three loading methods was selected for critical analysis, and the shear stress–shear displacement curve shown in Figure 8 was obtained.

It can be seen in Figure 8 that the shear stress–shear displacement curve can be roughly divided into two curve segments: the rising curve segment and the falling curve segment, in accordance with an increase in the shear displacement under the three loading methods. In the rising curve segment, the shear stress at the initial shear was approximately linear with respect to the shear displacement. As the displacement increased, the shear stress increased gradually and finally reached the peak value. After the peak value, the curve declined with a continuous increase in the shear displacement and then transitioned into the descending curve section. When the shear displacement reaches 4 mm, there is still shear strength. It can be seen that the shear stress–shear displacement curve can accurately reflect the shear test process of granular materials.

Figure 8 presents a comparison of the three peak shear stresses of the three curves. As can be seen in the Figure 8, the peak shear stresses of UM and BM were relatively similar, with a difference of 9.22 kPa. It can be concluded that the peak shear stress of UM was the largest, the peak shear stress of LM was the smallest, and the peak shear stress of BM was in the middle. The order of the peak shear stresses under the three loading methods was as follows: UM > BM > LM. Considering the peak value of the shear stress–shear displacement curve as the boundary, the left curve exhibited an upward trend with a large slope, and the right curve exhibited a downward trend with a small slope. There were slight differences between the slopes of the rising curve segments of the three shear stress–shear displacement curves at the initial stage of shearing. However, there were significant differences in the descending curve after the peak point. In the falling curve segment, UM and BM fluctuated significantly, and there were large fluctuations. The decline of LM was relatively gradual and the curve fluctuation was small, which indicate that the stress of the LM sample was more uniform than UM and BM after the peak. According to Figure 7, Table 3, and Figure 8, the order of the uniformities of the post-peak stresses of the sample was as follows: LM > BM > UM.

4.4. Shear Strength Conversion

Shear strength is the ultimate resistance of soil to shear failure, and it generally refers to the shear stress when the sample is damaged. It can be seen in Figure 8 that there was a peak value of shear stress in the shear stress–shear displacement relationship curve. Considering the peak shear stress as the shear strength, the shear strength of the UM curve was 119.74 kPa, the shear strength of the BM curve was 110.52 kPa, and the shear strength of the LM curve was 67.31 kPa. The shear strengths obtained by the three loading methods were not equal. Therefore, during shear test loading, the motion state of the upper and lower boxes should be considered.

To investigate the numerical conversion relationship of the shear strength under the three loading methods, a mathematical operation was implemented, and the results revealed that the shear strength of UM was greater than LM by a factor of 1.78, the shear strength of BM was greater than LM by a factor of 1.64, and the shear strength of UM was greater than BM by a factor of 1.08, indicating a linear correlation between the shear strengths under all three loading methods. The mathematical expression for conversion between the three shear strengths can be expressed as follows:

$${\tau }_{UM}=a{\tau }_{LM}$$

(14)

$${\tau }_{BM}=b{\tau }_{LM}$$

(15)

$$\frac{{\tau }_{UM}}{{\tau }_{BM}}=\frac{a}{b}=\gamma$$

(16)

where ${\tau }_{UM}$ is the shear strength of UM; ${\tau }_{LM}$ is the shear strength of LM; ${\tau }_{BM}$ is the shear strength of BM; a is the conversion coefficient between ${\tau }_{UM}$ and ${\tau }_{LM}$; b is the conversion coefficient between ${\tau }_{BM}$ and ${\tau }_{LM}$; and γ is the ratio of a to b.

Therefore, in the direct shear test, the shear strengths under the three loading methods can be readily obtained by mathematical conversion. The shear strength measured in LM was the smallest among the three; thus, this value is relatively conservative and safer to use in engineering. Moreover, because this value is relatively conservative, it may be associated with large costs and is, therefore, not economical. For UM and BM, the shear strengths obtained were relatively large, which may overestimate the actual shear strength. In engineering applications, it may be necessary to increase the safety factor, thus increasing the safety reserve, to ensure safety.

By studying the effect of three loading methods on shear strength, it was found that the shear strength of granular materials with different loading methods (LM, UM, and BM) is different. It can be seen that in the direct shear test, ignoring the loading method has a significant impact on the engineering cost and design results, which may lead to improper use of resources, in addition to the occurrence of engineering accidents. The selection of the loading method should be combined with the corresponding engineering practice, and analysis should be conducted to ensure economic feasibility and safety. It is, therefore, necessary to study the shear loading method of granular materials, which demonstrates a significant scientific research potential and engineering application value.

5. Conclusions

Based on the static equilibrium equation, the core formula of the direct shear test was derived in this study. With respect to the contact force, shear stress, and shear strength conversion, the influences of different moving boundaries on the shear strength of granular materials were comprehensively analyzed. The main conclusions are as follows:

(1)

The force chain is mainly concentrated near F₁ and F₄, and is diagonally distributed between F₁ and F₄. The force chain near F₂ and F₃ is relatively sparse, and the contact force is small. In the relationship between the contact force and shear displacement, the contact forces of the four sidewalls under three loading methods are significantly different. Although LM and BM data are relatively concentrated, UM data are significantly discrete with a low contact force homogeneity.

(2)

Under the three loading methods, the contact force is mainly concentrated in F₁ and F₄, and the order of the difference between F_1peak and F_4peak is as follows: UM > BM > LM. The F_1peak and F_4peak of UM and LM are significantly different; thus, the position of the dynamometer should be considered when measuring the shear force. Only the F_1peak and F_4peak of the LM on the diagonal is similar, and the peak contact force distribution inside the sample is relatively uniform.

(3)

The shear stress–shear displacement curve is divided into an rising curve segment and a falling curve segment. The slopes of the rising parts of the three shear stress–shear displacement curves at the initial shear time are similar; however, the falling parts after the peak points are significantly different. The order of peak shear stresses under three loading methods is UM > BM > LM, and the uniformity of the post-peak stress of the sample is as follows: LM > BM > UM.

(4)

The shear strength can be mathematically calculated, and the linear correlation between the three loading methods is as follows: UM is greater than LM by a factor of 1.78, BM is greater than LM by a factor of 1.64, and UM is greater than BM by a factor of 1.08. Based on the shear strength conversion formula, the shear strength of the three can be converted numerically.

6. Discussion

According to most scholars, the results for UM and LM in the direct shear test are consistent, and there are no significant differences. The results of this study revealed that UM and LM are two different loading methods, and the shear test results obtained by the simulation were significantly different, among which the measured shear strength was inconsistent, thus increasing the safety risks of the engineering application. However, these differences and hidden risks are not known by researchers. Therefore, it is not appropriate to directly use the shear strength measured under these two loading methods in engineering applications. Accordingly, this paper proposes a third loading method based on the first two loading methods (UM and LM), the bidirectional moving shear loading method (BM).

In this study, the PFC 2D particle flow software and the DEM were used. The shear strength of granular materials under different moving boundary conditions caused by the dislocation of the upper and lower boxes of the shear box under different loading methods (UM, LM, and BM) was investigated based on a numerical simulation. The shear strength characteristics under the three loading methods were compared and analyzed, and it was found that the three can be cross-converted in terms of numerical value by establishing a functional relationship. In practical engineering applications, such as slope engineering, the sliding layer is generally located above the non-sliding layer, which is similar to the state of upper box sliding. Moreover, UM may be more accurate and appropriate for the analysis of landslide sliding. For tunnel excavation works, the disturbance area is generally located below the stable layer, which is similar to the lower box sliding problem. Additionally, LM may be more accurate and appropriate for the analysis of tunnel excavation, and BM is more applicable to remolded soil samples displaced by shear. The selection of the loading method should be comprehensively considered in conjunction with the site conditions. For engineering design, the safety factor should be appropriately increased if necessary.

In practical engineering applications, the shear strength of soil is influenced by multiple factors such as the mineral composition of particles, particle size, particle shape, and water environment. The focus of this study was to consider the influence of the loading method in the moving boundary on the shear strength. A single-diameter disc-shaped particle was used for the simulation, which may be different from well-graded particles with different shapes in terms of mechanical properties. Given that this study was exploratory, a further in-depth study and analysis of the shear characteristics of the three loading methods should be conducted. In future research, a laboratory test will be conducted on actual granular materials with multiple particle groups under different saturation levels and irregular particle shapes while considering the mineral composition of the particles and particle size, among other factors. Moreover, an accurate and comprehensive investigation of the mechanical properties of granular materials under the three loading methods will be conducted to provide a mechanical theory and experimental basis for the shear test.