# Dilatancy Equation Based on the Property-Dependent Plastic Potential Theory for Geomaterials

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Dilatancy Equation Based on the Potential Theory

#### 2.1. Establishment Method

_{1}, F

_{2,}and F

_{3}, respectively, ${\theta}_{\sigma}={\theta}_{\mathrm{d}\epsilon}$, and the stress and strain increments are coaxial. In the triaxial tensile state, ${\sigma}_{1}={\sigma}_{2}{\sigma}_{3}$, the principal stress acts on F

_{2}, F

_{3}, and F

_{1}, respectively, the stress lode angle is equal to the strain lode angle, and the stress and strain increment are coaxial. In other stress states of the deviatoric plane, ${\theta}_{\sigma}\ne {\theta}_{\mathrm{d}\epsilon}$. Accordingly, the description of noncoaxiality under different stress states is associated with the meso-fabric properties of materials, which is different from the noncoaxial coefficients commonly used in previous research, has clearer physical significance, and is consistent with the experimental phenomenon in reference [14].

#### 2.2. Description of Dilatancy under Plane Stress State

_{dss}and ϕ

_{p}are the friction angle and the peak friction angle, respectively. The relation between stress components σ

_{x}, σ

_{y}, σ

_{z}, normal stress s, and shear stress t is shown in Equation (14). Similarly, the relation between the strain component increments dε

_{x}, dε

_{y}, and dε

_{z}, volumetric strain, and shear strain in the strain Mohr’s circle can be written as Equation (15).

^{(K)}and α

^{(K)}are shown in Figure 3. When ${\alpha}^{\left(K\right)}=\pi /4$, ${\mathrm{sin}(\alpha}^{\left(K\right)}{)=\mathrm{cos}(\alpha}^{\left(K\right)})$, ${a}_{2}={a}_{3}$, which is transversely isotropic, and the value of a is shown in Equation (27). In Equations (26) and (27), N is the particle number of the sample, and it represents the number of contact fabrics when F

_{ij}is employed to describe the contact of particles.

_{0}represents the initial average principal stress. It can be seen that the dilatancy equation can reflect the noncoaxiality of stress and plastic strain increments in the plane stress state and also reflect the coaxiality when a = 0. According to Equation (23), the more obvious the anisotropy is, the more significant the noncoaxiality is. The dilatancy and stress ratio under plane stress are shown in Figure 4, which is drawn by setting different model parameters, in which d

_{0}was defined by Li [2] and anisotropic parameter a was under the same stress condition. With the gradual increase in the stress ratio, dilatancy presents as shear contraction followed by dilatancy. With the increase in the fabric parameter a, the difference in dilatancy is more significant with different anisotropy. With the decrease in material coefficient d

_{0}, the volume contraction is lower, and the difference in dilatancy is lower.

## 3. Model Verification

_{y}= 400 kPa. After the loading starts, shear is carried out at a constant shear rate until the peak value is reached. The test can reflect the rotation of principal stress and the noncoaxiality between the stress direction and strain increment direction, which is consistent with the research goal of this paper.

#### 3.1. Noncoaxiality Verification

#### 3.2. Verification of Dilatancy

## 4. Conclusions

- (1)
- For noncoaxial conditions, calculation using stress invariants and strain increment invariants will overestimate the energy dissipated during loading. The energy transformation relation based on the potential theory introduces a new noncoaxial coefficient with values of 0–1, which can reasonably correct the influence of noncoaxiality on energy dissipation. Meanwhile, the influence of material microscopic properties on energy dissipation is introduced, which is closer to the actual condition.
- (2)
- The new noncoaxial coefficient is different from previous research, which is not only related to the stress level and stress direction but also related to the material microscopic fabric characteristics. The potential theory can be used to calculate the newly defined noncoaxial coefficient to provide a dilatancy equation considering noncoaxiality. When the microscopic fabric is isotropic, the noncoaxial coefficient is naturally 1, and the dilatancy equation can be reduced to the form of the critical state theory. When the fabric is anisotropic, the noncoaxial angle is related to the material anisotropy, the geometric relation between the fabric and the stress direction. The dilatancy equations can naturally describe noncoaxial effects, and the physical meaning is clearer.
- (3)
- Under the simple shear stress state, after introducing the noncoaxial coefficient, the dilatancy equation can naturally reflect the influence of noncoaxiality on the dilatancy under the condition of principal stress rotation. At the low-stress ratio, the generation of noncoaxiality depends on the material properties and has a significant effect on dilatancy. When the stress ratio is high, the influence of material properties on stress and strain is not obvious, the stress and strain naturally tend to be coaxial, and the influence on dilatancy is weakened. The experimental results verify the effectiveness of the proposed dilatancy equation.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Stress dilatancy relation in plane stress state: (

**a**) d

_{0}= 0.8; (

**b**) d

_{0}= 0.6; (

**c**) d

_{0}= 0.4.

**Figure 6.**The predicted and tested values of principal stress direction and noncoaxial angle: (

**a**) direction of principal stress; (

**b**) loose sand; (

**c**) medium dense sand; (

**d**) dense sand.

**Figure 7.**Comparison of predicted dilatancy values with test values: (

**a**) loose sand; (

**b**) medium dense sand; (

**c**) dense sand.

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**MDPI and ACS Style**

Li, X.; Zhu, H.; Yuan, Q.
Dilatancy Equation Based on the Property-Dependent Plastic Potential Theory for Geomaterials. *Fractal Fract.* **2023**, *7*, 824.
https://doi.org/10.3390/fractalfract7110824

**AMA Style**

Li X, Zhu H, Yuan Q.
Dilatancy Equation Based on the Property-Dependent Plastic Potential Theory for Geomaterials. *Fractal and Fractional*. 2023; 7(11):824.
https://doi.org/10.3390/fractalfract7110824

**Chicago/Turabian Style**

Li, Xuefeng, Houying Zhu, and Qi Yuan.
2023. "Dilatancy Equation Based on the Property-Dependent Plastic Potential Theory for Geomaterials" *Fractal and Fractional* 7, no. 11: 824.
https://doi.org/10.3390/fractalfract7110824