Simulation of Triaxial Tests for Unsaturated Soils under a Tension–Shear State by the Discrete Element Method

Abstract

In this paper, the discrete element method is used to simulate triaxial tests of unsaturated soil under a tension–shear state. A relationship between water content and uniaxial tensile strength with different void ratios is obtained, which is applied to uniaxial tensile discrete element simulations to establish a relationship between grain-scale parameters and water content from back analysis. A group of triaxial simulations for unsaturated soil under a tension–shear state is then conducted. The discrete element method is used to obtain the relationship between deviatoric stress and axial displacement with different water contents, and also to reveal the effects of water content on peak strength and dilatancy phenomena with different confining pressures. The displacement fields of numerical specimen are analyzed qualitatively, and the mechanism and process of failure are discussed from the prospective of energy and dissipation.

1. Introduction

Soil is a collection of various mineral particles. Generally speaking, the tensile strength of soil is weak, but unsaturated soil has the ability to bear a certain tensile load. The cracks in the core wall of earth dams and the uneven settlement of foundations are related to the tensile failure of soil. Therefore, it is urgent to study the mechanical property of unsaturated soil under a tension state for these problems. In the past, many scholars have studied the tensile strength of soil [1,2,3,4,5,6,7,8,9,10,11,12]. In practical engineering, the failure of soil is not only caused by tensile load, but also by the interaction of tension and shear. From the microscopic point of view, the tensile strength of unsaturated soil is affected by the indirect contact angle of particles, particle spacing, particle size, and so on. Therefore, it is of great significance to study the tensile–shear failure of unsaturated soil to improve the basic theory of unsaturated soil.

Compared with saturated soil, the existence of a gas phase leads to the complexity of mechanical properties for unsaturated soil. Alonso et al. [13] established the first constitutive model of unsaturated soil, i.e., the Barcelona basic model (BBM). Subsequently, in order to accurately describe the coupling relationship between mechanics and water retention behaviors of unsaturated soil, many scholars (e.g., Wheeler et al. [14], Gallipoli et al. [15], Li [16], Sun et al. [17]) have proposed a series of elastoplastic coupled constitutive models. With the development of computer technology, it has become a trend to study geotechnical engineering by numerical methods. Many scholars have developed finite element programs to solve unsaturated soil problems, such as LAGAMINE [18], CODE-BRIGHT [19], THYMER3D [20], and U-DYSAC2 [21]. These studies pay more attention to the macro-mechanical behavior of unsaturated soil.

The influence of microstructure on the properties of geotechnical materials is significant. The discrete element numerical simulation technology was developed by Cundall [22] and Cundall and Strack [23] for dry granular materials. For the discrete element simulation of geotechnical materials, the determinations of the contact constitutive model and relevant micro parameters are significant [24,25,26,27,28,29,30,31,32,33,34,35,36]. The discrete element method can be used to study the rotation of particles, the arrangement of particles, the tangential contact force between particles and particle breakage, and the mechanical behavior of geotechnical materials [37,38].

In the field of unsaturated soil, the formulas of effective stress and shear strength have been studied [39]. The multiphase particle system of unsaturated soil leads to its complicated failure characteristics. Laboratory experiments and numerical simulations are usually used to study the mechanical properties of geotechnical materials. In the aspect of laboratory experiments, most of the existing uniaxial tensile experiments adopt the stress-controlled loading mode; therefore, it is impossible to determine the softening stage of the stress–strain curve. The triaxial tensile test instrument is usually refitted on the existing conventional triaxial apparatus, whose axial connection stiffness is small. Therefore, it is difficult to measure the whole tensile stress–strain curve. As a result, it is difficult to carry out triaxial tension–shear tests of unsaturated soil by means of laboratory experiment.

According to the existing research results, when clay is dried, its macroscopic performance is volume shrinkage, and tensile stress is generated inside the soil. When the tensile stress exceeds the tensile strength, the soil is damaged and displays macroscopic cracks. Moreover, during the drying process, the suction increases with the decreasing of water content, and the mechanical properties of soil also change [40]. Therefore, considering the tensile strength of unsaturated soil is very important to study the cracking behavior of soil.

The discrete element method can not only generate the specified sample according to relevant conditions to ensure the reproducibility of the sample, but can also change the boundary conditions and the shape of the sample, so as to realize the numerical test with different loading modes. A recent study by Konrad et al. [41] showed that the discrete element method has some advantages in simulating the large deformation and failure of materials. Therefore, some scholars began to use the particle discrete element method to simulate the soil shrinkage cracking. El Youssoufi et al. [42] established a discrete element model of expanding and contracting particles, and simulated the phenomenon of cracking for cemented granular materials. Peron et al. [43] used a two-dimensional discrete element model to simulate the vertical cracking of long strip fine-grained soil, and analyzed the influences of boundary conditions and water content distribution gradient on the number and spacing of cracks. Based on the aggregate structure, Sima et al. [44] preliminarily established a three-dimensional discrete element model for the shrinkage of clay, considering the change of soil properties with water content, and simulated the expansion process of surface cracks.

Therefore, based on the basic principle of the discrete element method, this paper calibrates micro parameters, establishes a triaxial tension–shear model, carries out a triaxial tensile–shear simulation of unsaturated soil with different confining pressures and water contents, and analyzes the tensile–shear failure characteristics of unsaturated soil from the microscopic point of view.

2. Selection of the Grain-Scale Contact Model

During numerical simulations by the discrete element method (DEM), a linear elastic contact model is adopted. In other words, a specimen is composed of a series of elastic elements (i.e., spheres) obeying Newton’s second law. Two elements are connected with each other by a breakable spring, and forces appear at the contact point between adjacent elements. The relationships between forces and displacements of the contact model in the normal and tangential directions are shown in Figure 1.

The normal force between elements, i.e., $F_n$, is defined by

$$F_n=K_n\cdot X_n$$

(1)

where K_n is the spring normal stiffness and X_n is the normal relative displacement. The tensile force exists when elements are connected to each other. When X_n exceeds the fracture displacement X_b, the connection is broken. Therefore, the maximum normal force between elements F_nmax is

$$F_{nmax}=K_n\cdot X_b$$

(2)

When the connection is broken, the tensile normal force between elements no longer exists. When the two elements return to the compressive contact state, the compressive normal force is rebuilt:

$$F_n=K_n\cdot X_n,X_n<0$$

(3)

Additionally, the shear force F_s is also considered in the linear elastic contact model. When two elements contact and slide against each other, the sliding friction force opposite to the sliding direction is generated. The two elements are connected by a breakable spring in the tangent direction, and the tangential spring force F_s is defined by

$$F_s=K_s\cdot X_s$$

(4)

where K_s is the shear stiffness and X_s is the tangential relative displacement. For a complete element connection, the maximum shear force F_smax is determined by the Coulomb criterion:

$$F_{smax}=F_{s0}-{\mu }_p\cdot F_n$$

(5)

where F_s0 is the shear resistance in the tangential direction between elements, ${\mu }_p$ is the friction coefficient between elements, and F_n is the normal force (compressive force is negative). When an external force exceeds the maximum shear force F_smax, the tangent connection between elements is broken, and the shear resistance F_s0 disappears. In this case, the tangential force F_s is less than or equal to the maximum shear force F_smax, which is defined by

$$F_{smax}=-{\mu }_p\cdot F_n,\mathrm{when}\mathrm{the}\mathrm{connection}\mathrm{is}\mathrm{broken}$$

(6)

In the case of connection fracture, elements slide when the external force exceeds the maximum shear force F_smax. When the two elements are separated from each other ($X_n>0$), the normal force and the tangential force between elements are zero.

3. Back Analysis of Grain-Scale Parameters

3.1. Determining Grain-Scale Parameters by Complex Uniaxial Tensile Test Simulation

MatDEM is a general discrete element software for geotechnical materials developed by Nanjing University. Based on the MATLAB calculation method, it adopts the innovative GPU matrix algorithm to realize the discrete element simulation of millions of particles. Its calculating number of units and efficiency are more than dozens of times that of other commercial software. On the basis of reaching the standard of hardware, it can complete the large-scale three-dimensional discrete element numerical simulation in a few hours. The software can realize the automatic modeling of discrete element materials and the calculation of energy conservation for discrete element systems. The software integrates pre-processing, calculation, post-processing, and powerful secondary development, provides a perfect function interface and efficient calculation engine, and completes complex multi-field coupling simulation through secondary development. MatDEM 1.32 version is adopted for this research.

For the convenience of discussion, hereafter the laboratory uniaxial tensile test is called a complex uniaxial tensile test. There are two groups of parameters in MatDEM, i.e., material parameters and grain-scale contact parameters. The material parameters include Young’s modulus ($E$)/GPa, Poisson’s ratio ($\nu$), tensile strength ($T_u$)/MPa, compressive strength ($C_u$)/MPa, and internal friction coefficient (${\mu }_i$). The tensile strength $T_u$ refers to the tensile force when connections are broken, which is different from the complex uniaxial tensile strength. The grain-scale contact parameters include normal stiffness ($K_n$)/(MN/m), tangential stiffness ($K_s$)/(MN/m), failure displacement ($X_b$), shear strength ($F_{s0}$)/N, and the friction coefficient (${\mu }_p$). There is a clear relationship between the two groups of parameters in MatDEM.

The water content is a key influencing factor for the tensile strength of unsaturated soil. During a drying path, both the tensile stress and the tensile strength increase. When the tensile stress exceeds the tensile strength, the soil is damaged and the cracks occur. A relationship between tensile strength and water content could be obtained by comprehensive comparison between the results of complex uniaxial tensile tests and numerical simulation tests. Capillary cementation is a kind of apparent cohesion, which is the same as the cohesion contributing to the tensile strength $T_u$. Therefore, a relationship between water content and microscopic tensile strength should be established according to simulations of triaxial tests for unsaturated soils.

3.2. Impact Analysis of Material Parameters on the Discrete Element Simulation

3.2.1. Creating a MatDEM Model for the Complex Uniaxial Tensile Test

A numerical model was generated according to the complex uniaxial tensile test of clay, and particles were also generated according to the particle size distribution and the maximum dry density of clay used in the laboratory test. It is difficult to generate a sample with a given void ratio due to particle equilibrium iterative cycles and compaction processes by the discrete element software. Only the average radius of particles (BallR) and the radius ratio of the maximum particle to the minimum particle (the distriRate parameter in the software represents the particle diameter dispersion coefficient, and the ratio of the maximum radius to the minimum radius is (1 + rate)²) are controlled. Considering that particle grading and void ratio affect the mechanical behavior of soil, the discrete element particle samples with specified particle grading and void ratio were generated by the Monte Carlo method.

The laboratory complex uniaxial tensile experimental apparatus consists of two symmetrical wedges (80 mm long and 10 mm thick). In order to make the specimen fail in the gap between the wedge-shaped molds, the neck width reduces from 40 to 20 mm in the middle part. One wedge is clamped to the press plate of the testing machine, and the other wedge is connected with the end crossbeam through the load sensor. The numerical wedges are generated by a filter and residual strength function, and the sample is generated inside the wedges. The specific wedges and numerical model are shown in Figure 2.

3.2.2. Loading Mode of the Numerical Model and the Calculation Rule of Tensile Strength

For a numerical tensile test, the loading plate moves outward with a constant displacement rate of 0.5 mm/min. During the test, the axial load and displacement are obtained continuously. The tensile stress ${\sigma }_t$ is

$${\sigma }_t=T/S$$

(7)

where T is the axial tensile load and S is the cross-sectional area (20 × 10 mm²). The peak tensile strength is the maximum tensile stress. The two wedges move in the opposite direction until a continuous failure surface is formed. The average value of contact stress along the tensile direction of two wedges at the failure state is taken as the tensile strength of the specimen. The tensile failure surface is shown in Figure 3.

3.2.3. Influences of Material Parameters on the Complex Uniaxial Tensile Strength

Material parameters affect the properties of geotechnical materials by the discrete element method. Back analysis and a calibration method were used to determine the micro material parameters by complex uniaxial tensile strength.

• Impact analysis of the tensile strength on the complex uniaxial tensile strength

If a tetrahedral element is used to study the mechanical properties of the model, the element is composed of four identical particles. As the force and displacement are very small, the analytical solution of tetrahedron deformation can be obtained. With the increasing of the tensile force (F_z) acting on the element, the relative displacement X_n1 of the top particles increases. When X_n1 > X_b, the connection between particles breaks. If X_n1 = X_b, the element tensile strength (T_u) is obtained:

$$T_u=\frac{6\sqrt{2}{K}_n\left(K_n+K_s\right)}{3K_n+K_s}\cdot \frac{X_b}{d^2}$$

(8)

According to the tensile strength expression of the tetrahedral element composed of four elements, the tensile strength T_u is related to the spring normal stiffness $K_n$, tangential stiffness $K_s$, and the critical fracture displacement X_b. The values of the material parameters are shown in

Table 1. Material parameters.

Average Particle Radius rave/m	Particle Diameter Dispersion Coefficient Rate	Specific Gravity Gs	Young’s Modulus E/MPa	Poisson’s Ratio ν	Compressive Strength Cu/kPa	Internal Friction Coefficient of Material μi
0.002	0.6	2.73	20	0.3	20	0.4

.

Through a large number of numerical tests with different tensile strengths T_u, it was found that the critical fracture displacements X_b obtained by automatic material training are different, and they also have a great influence on the tensile failure displacement of the complex uniaxial tensile simulation. As a result, the complex uniaxial tensile strength of the specimen increases with the increasing of tensile strength T_u. The simulation results are shown in

Table 2. Complex uniaxial tensile strength with different tensile strengths T_u.

Variable	Value
Tensile strength Tu/kPa	0.1	0.2	0.5	1	2	4	6	8
Tensile failure displacement /mm	0.230	0.306	0.349	0.470	0.449	0.570	0.669	0.749
Complex uniaxial tensile strength /kPa	8.081	13.872	18.182	26.263	33.401	48.754	62.896	77.979

and Figure 4.

The mechanical properties of the model were also studied by the tetrahedral elements. The vertical compression and lateral expansion of an element are generated by tensile force F_z. When the relative displacement of the bottom particles exceeds the limit deformation (X_n2 > X_b), the horizontal connection breaks. C_u is the stress value in the vertical direction when the connection is broken horizontally, which is obtained by

$$C_u=\frac{6\sqrt{2}{K}_n\left(K_n+K_s\right)}{K_n-K_s}\cdot \frac{X_b}{d^2}$$

(9)

By fixing the tensile strength T_u, it is found that C_u affects the simulation results, and affects the spring normal and tangential stiffnesses.

• Impact analysis of Young’s modulus on the complex uniaxial tensile strength

The numerical simulation results by Boutt and Mcpherson [45] and Ergenzinger et al. [46] showed that the Young’s modulus increases with the increasing of particle stiffness. The complex uniaxial tensile numerical simulation was carried out by changing only the Young’s modulus (i.e., 1.1 × 10⁷, 2 × 10⁷, 3 × 10⁷, 4 × 10⁷, 5 × 10⁷, and 6 × 10⁷ Pa). Figure 5 shows the tensile stress–displacement curve with different Young’s moduli. It is shown that with the increasing of the Young’s modulus, the tensile displacement decreases.

• Impact analysis of Poisson’s ratio on the complex uniaxial tensile strength

Poisson’s ratio ν increases with the decreasing of the ratio of tangential stiffness to normal stiffness. In the linear elastic model, Poisson’s ratio ν can be calculated by

$$\nu =-\frac{{\epsilon }_{xx}}{{\epsilon }_{zz}}=\frac{K_n-K_s}{5K_n+K_s}=\frac{1-\gamma }{5+\gamma }$$

(10)

where $\gamma$ is the ratio of shear stiffness to normal stiffness. Equation (10) shows that ν decreases with the increasing of $\gamma$. When $\gamma$ > 1, the negative Poisson’s ratio material can be obtained. The complex uniaxial tensile numerical simulation was carried out by changing only the Poisson’s ratio (i.e., 0.35, 0.2, and 0.15). Figure 6 shows the tensile stress–displacement curve with different Poisson’s ratios. The effect of Poisson’s ratio on the grain-scale parameters of particles is not significant.

3.3. Relationship between Water Content and the Complex Uniaxial Tensile Strength of Unsaturated Clay

Thirty-two tensile strength tests by Tang et al. [10] were simulated. The clay, which was medium plastic clay, was from Nanjing, and the physical properties are shown in

Table 3. Physical properties of clay.

Soil Properties	Specific Gravity	Consistency Limit			USCS Classification	Compaction Characteristics		Particle Size Analysis
		Liquid Limit (%)	Plastic Limit (%)	Plasticity Index (%)		Optimum Water Content (%)	Maximum Dry Density (g/cm3)	Sand (%)	Silt (%)	Clay (%)
Value	2.73	37	20	17	CL	16.5	1.7	2	76	22

. The specific gravity was 2.73, the proportion of sand, silt, and clay was 2:76:22, and the elastic modulus was between 4 and 18 MPa. The initial dry densities were 1.5, 1.6, and 1.7 g/cm³, respectively, and the corresponding initial void ratios were 0.820, 0.706, and 0.606, respectively.

The complex uniaxial tensile test results are shown in Figure 7. For a given dry density, the uniaxial tensile strength σ_t significantly depends on the water content w. With the increasing of w, σ_t increases rapidly, and reaches the maximum value at the critical water content w_c, corresponding to the peak value of uniaxial tensile strength. When the water content exceeds a certain value, the change of σ_t is very small with the further increasing of water content.

The experimental results with different initial void ratios were fitted by a second-order function.

(1)

Dry side (0 < w < 11.5%)

$${\sigma }_t=A_d+B_d\cdot w+C_d\cdot w^2$$

(11)

where ${\sigma }_t$ is the uniaxial tensile strength, w is the water content, and A_d, B_d, and C_d are coefficients related to the initial void ratio under dry side. The values of A_d, B_d, and C_d for specimens with different initial void ratios are shown in

Table 4. Fitting parameters for dry side.

Initial Void Ratio e	Ad	Bd	Cd	Determination Factor
0.820	−3.14821	3.88897	−0.08224	0.99546
0.706	−12.37397	8.52281	−0.22218	0.98924
0.606	−29.40861	18.88517	−0.81092	0.99389

.

(2)

Wet side (11.5% < w < 35%)

$${\sigma }_t=A_w+B_w\cdot w+C_w\cdot w^2$$

(12)

where A_w, B_w, and C_w are coefficients under wet side. The values of A_w, B_w, and C_w for specimens with different initial void ratios are shown in

Table 5. Fitting parameters for wet side.

Initial Void Ratio e	Aw	Bw	Cw	Determination Factor
0.820	55.8452	−2.40208	0.03514	0.9753
0.706	132.56575	−8.20754	0.15516	0.9791
0.606	184.43806	−11.7487	0.24232	0.99729

.

3.4. Relation between the Tensile Strength T_u and Complex Uniaxial Tensile Strength σ_t

Based on the analysis results of the complex uniaxial tensile strength, it can be concluded that the influence of tensile strength T_u on the complex tensile strength σ_t is significant. Therefore, a series of numerical tests with different tensile strengths T_u was carried out. The numerical simulation of complex uniaxial tensile strength can be divided into three steps: (1) creating samples with the same particle size distribution as the laboratory experiment; (2) cutting the model, giving the material parameters, and balancing the model; and (3) applying the strain-controlled load by the user-defined function, and outputting the calculation results.

Based on the Monte Carlo method and the secondary development function of MatDEM, three groups of discrete element particle samples were generated with initial void ratios of 0.820, 0.706, and 0.606, respectively. The percentages of sand, silt, and clay for all of three groups were 2%, 76%, and 22%, respectively. If numerical particle sizes are the same as the real ones, millions of particles will be generated in this model. Therefore, considering the hardware limit, the radiuses of particles were uniformly reduced by a certain multiple. The average radius of the sample particles (BallR) was 5 × 10⁻⁴ m, the particle diameter dispersion coefficient (rate) was between 0 and 0.8, the minimum particle radius was 1.25 × 10⁻⁴ m, the maximum particle radius was 7.31 × 10⁻⁴ m, and the total number of particles was 65,139. The information of discrete element samples with specific initial void ratios is shown in

Table 6. Numerical samples with specific initial void ratios.

Initial Dry Density g/cm3		Particle Size Analysis %		Initial Void Ratio e	Total Number of Particles
1.5	Laboratory test	Sand/Silt/Clay	2/76/22	0.820	65,139
	Numerical test	Sand/Silt/Clay	1.3/74/24.7	0.872
1.6	Laboratory test	Sand/Silt/Clay	2/76/22	0.706	65,338
	Numerical test	Sand/Silt/Clay	1.8/78/20.2	0.816
1.7	Laboratory test	Sand/Silt/Clay	2/76/22	0.606	66,327
	Numerical test	Sand/Silt/Clay	2.2/76/21.8	0.762

.

Before numerical simulation, the particles of the model are piled up by gravity twice, so that all the particles are stable, and a pre-equilibrium model is obtained. Then, the model is cut and assigned material parameters. The values of the material parameters except for T_u are listed in Table 1.

After the model is established, the specimen is loaded by uniaxial tension. In order to obtain the completed stress–strain curve of the specimen, the loading speed is as slow as possible, and the change of stress is recorded after each displacement loading. It takes about 400 min for each specimen to complete the simulation. During loading, the sample is locked in the Y and Z directions. The model function d.mo.nfnx is used to monitor the change of contact force in the X direction. The complex uniaxial tensile simulation results of the samples with different tensile strengths T_u are shown in

Table 7. Complex uniaxial tensile simulation results.

Initial Void Ratio for Laboratory Test e0	Initial Void Ratio for Numerical Test e0	Total Number of Particles	Tensile Strength Tu/kPa	Complex Uniaxial Tensile Strength /kPa	Complex Uniaxial Tensile Failure Displacement /mm
0.820	0.872	65,139	0.5	10.12	0.12
			1	12.23	0.15
			2	13.45	0.25
			3	14.11	0.34
			4	15.45	0.38
			5	16.89	0.40
			6	17.98	0.42
			7	19.56	0.45
			8	22.46	0.48
			9	24.56	0.51
			10	25.79	0.58
			11	26.89	0.67
			12	27.36	0.75
			13	28.45	0.80
			14	29.35	0.86
			15	30.15	0.89
			16	32.12	0.95

.

For the sample with an initial void ratio of 0.872, the numerical simulation results are fitted by a polynomial function, and the fitting relationship between the tensile strength T_u and complex uniaxial tensile strength σ_t is obtained by

$${\sigma }_t=9.48315+1.72723T_u-0.02009T_u^2$$

(13)

The value range of T_u is 0 to 20 kPa, and the correction determination coefficient is 0.98722. Figure 8 is the simulation result with a tensile strength T_u of 4 kPa.

3.5. Relationship between MatDEM Material Parameters and the Water Content w of Unsaturated Clay

The complex uniaxial tensile strength expression of compacted clay with different dry densities and water contents is obtained according to the laboratory test results, and the relationship between the material parameter T_u and the complex uniaxial tensile strength with a given initial void ratio is obtained by numerical simulation. In the following, the relationship between the tensile strength T_u and water content w with a given initial void ratio is obtained through the intermediate value of complex uniaxial tensile strength.

According to the fitting results, when the initial void ratio e is 0.820, the fitting relationship between the material parameter tensile strength T_u and the complex uniaxial tensile strength σ_t is

$${\sigma }_t=9.48315+1.72723T_u-0.02009T_u^2(0<T_u<20\mathrm{kPa})$$

(14)

(1)

For the dry side (0 < w < 11.5%),

$${\sigma }_t=-3.14821+3.88897\cdot w-0.08224\cdot w^2$$

(15)

Substituting Equation (15) into Equation (14) obtains

$$T_u=\frac{1.7272-\sqrt{0.0066\cdot w^2-0.3125\cdot w+3.9980}}{0.0402}$$

(16)

(2)

For the wet side (11.5% < w < 35%),

$${\sigma }_t=55.8452-2.40208\cdot w+0.03514\cdot w^2$$

(17)

Substituting Equation (17) into Equation (14) obtains

$$T_u=\frac{1.7272-\sqrt{-0.0028\cdot w^2+0.1930\cdot w-0.7423})}{0.0402}$$

(18)

Then, the material parameter T_u is obtained with different water contents. In order to verify the applicability of the fitting formula, the complex uniaxial tensile numerical simulation was carried out for unsaturated compacted clay with an initial void ratio of 0.706 and 0.606, respectively. According to the fitting results, numerical simulations for samples with different water contents were carried out, and the simulation results of the relationship between tensile strength and water content for the sample with an initial void ratio of 0.706 are shown in Figure 9. Figure 10 is the simulation result of the sample with an initial void ratio of 0.816 and water content of 20%.

For the sample with an initial void ratio of 0.606, the simulation results of the relationship between tensile strength and water content are shown in Figure 11. Figure 12 is the simulation result of the sample with an initial void ratio of 0.762 and water content of 15%. The fitting results show that the relationship between the material parameter tensile strength T_u and the water content w is suitable for simulating the tensile strength of unsaturated clay. According to this relationship, the discrete element numerical simulation of the triaxial tension–shear test for unsaturated soil will be carried out as the follows.

4. Simulation of Triaxial Tests for Unsaturated Soils under a Tension–Shear State

In this part, a group of triaxial test simulations for unsaturated soils under a tension–shear state is carried out. The stress path of triaxial test simulations is shown as follows: the sample is consolidated by confining pressure, then the axial stress gradually decreases. At the beginning of unloading, the specimen is elongated axially until the axial stress σ₃ < 0. Finally, the specimen undergoes tensile failure. According to the results of numerical simulation, the variation of tensile–shear strength for unsaturated soil with confining pressures and water contents is revealed, and the triaxial tensile failure modes of unsaturated soil are analyzed.

4.1. Simulation Steps

The steps of the numerical test are introduced as follows. (1) The Monte Carlo method is used to generate discrete element particle samples close to the specified particle size gradation and void ratio. (2) According to the relationship between the water content w and tensile strength T_u, materials are trained automatically, and the trained materials are given to the model. (3) The stress is applied by the upper and lower pressure plates with the water confining pressure σ₁. (4) The upper and lower pressure plates move to the opposite direction with a constant speed until the sample is damaged or a continuous failure surface is formed.

4.2. Simulation Results

4.2.1. The Relationship between Deviatoric Stress and Axial Displacement

During the triaxial tensile simulation, the confining pressure is the large principal stress σ₁, and the axial stress is the small principal stress σ₃. The deviatoric stress is defined by σ₁–σ₃. Figure 13 shows the discrete element numerical simulation results of samples with an initial void ratio of 0.872 with different confining pressures and water contents. For a given water content, the initial slopes of the deviatoric stress–axial displacement curve with different confining pressures are basically the same; however, the peak and residual strengths with different confining pressures are different. When the confining pressure is small (0 < σ₁ < 200 kPa), the peak strength appears early, and the peak value is larger than the confining pressure σ₁, which means that the axial stress σ₃ reaches a tensile state. The deviatoric stress drops fast after the peak strength. Finally, the sample undergoes tensile failure, and the axial stress goes to 0. When the confining pressure is large (200 < σ₁ < 500 kPa), the peak deviatoric stress increases with the increasing of confining pressure. The peak deviatoric stresses are basically less than or equal to the confining pressure σ₁, which indicates that the axial stress σ₃ is greater than 0. It should be noted that the water content affects the peak deviatoric stress and the hardening/softening characteristic. The strength increases with the decreasing of water content and the increasing of confining pressure. Moreover, the dilatancy phenomena is obvious for the samples with a low confining pressure and water content.

4.2.2. Displacement Field

Figure 14 shows the displacement field of specimens with different confining pressures. The particle color represents the displacement field. A brighter color means a larger displacement. The simulated results illustrate that the confining pressure affects the failure form of a specimen. When the confining pressure is small (σ₁ = 100 kPa), the axial stress σ₃ gradually changes from a compressive state to a tensile state. The fracture surface for tensile failure is basically horizontal.

When the confining pressure is in the middle range (σ₁ = 200, 300 kPa), the failure modes of shear elongation and tensile fracture occur simultaneously. At the initial stage of loading, the four sides of the specimen remain straight. Then, a local inclined shear plane on the surrounding side of specimen appears when a threshold of tensile axial displacement is achieved. However, the shear plane does not develop to the interior of the specimen. Finally, the specimen fractures with the continuous increasing of the tensile axial displacement. According to the displacement field, the middle part of the fracture surface is basically horizontal, and local shear zones generate around the specimen.

When the confining pressure is large (σ₁ = 400 kPa), the axial stress σ₃ of specimen is always in the compressive state under the tensile loading path. With the increasing of tensile displacement, pure shear failure occurs; however, axial tensile stress does not appear. According to the displacement field, a shear band generates inside the specimen.

4.2.3. Heat and Energy Field

The simulation results of the heat field for samples with an 11% water content and different confining pressures are shown in Figure 15. The generated heat increases with the increasing of confining pressure. Based on the law of energy conservation, the total energy of the isolation system is constant. When the sample deforms under external forces, the increment of energy must be equal to the work done by the external force. For the same axial displacement, the mechanical energy of a sample with a high confining pressure is larger than that with a low confining pressure. The mechanical energy required for specimen failure increases with the increasing of confining pressure.

5. Conclusions

In this paper, a relationship between tensile strength and water content was obtained by comprehensive comparison between the results of complex uniaxial tensile tests and numerical simulation tests. Moreover, the discrete element simulation of triaxial tension–shear tests for unsaturated soils was carried out. The simulation results demonstrate the following:

• The water content affects the peak deviatoric stress, dilatancy behavior, and failure mode.
• The strength increases with the decreasing of water content and the increasing of confining pressure.
• The dilatancy phenomena is obvious for the specimens with a low confining pressure range and water content.
• The specimens undergo pure tensile failure under a small confining pressure condition, shear elongation and tensile failure under a middle confining pressure condition, and shear failure under a large confining pressure condition.