# Investigation of Lade-Kim Plastic Potential Applicability under Various Stress Paths for Rockfill Materials

Authors: Liping Chen; Shihai Bian; Xiaokai Niu; Yongbo Zhao
Journal: Advances in Materials Science and Engineering
 (2021)

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## Abstract

The dilatancy behavior of rockfill materials shows obvious stress path dependence. Lade-Kim plastic potential equation has been proposed for a long time to model the mechanical behavior of sand and concrete materials. However, it lacks the verification of rockfill materials, especially under various stress paths. In this paper, the dilatancy performance of coarse-grained materials under various stress paths is investigated, and then the dilatancy equation description and verification method based on Lade-Kim plastic potential are given. The applicability of Lade-Kim plastic potential for different stress path tests, such as conventional triaxial tests, constant P tests, and constant stress (increment) ratio tests, are verified and evaluated. It is found that Lade-Kim plastic potential is difficult to consider the influence of stress path. Finally, the Lade-Kim plastic potential, together with nonlinear dilatancy equation, is evaluated by changing the dilatancy equation in the framework of generalized plasticity. Lade-Kim plastic potential is suitable for constant stress increment ratio loading experiments and special care should be taken when applied to other stress paths. These works are helpful to understand stress path dependence of dilatancy behavior for rockfill materials and is beneficial for the establishment of stress path constitutive model.


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## Body

## 1. Introduction

Rockfill is an important building material for high earth rockfill dam, and its deformation characteristics will significantly affect the safety of the dam. In the process of filling and impoundment, the actual stress path is a complex loading process with variable confining pressure. If the deformation characteristics of rockfill materials under complex loading paths can be described more accurately, it is beneficial to correctly evaluate the safety of high earth rockfill dam and its impervious body. The dilatancy equation is an important part of the elastoplastic constitutive model, and it is an important foundation to establish the constitutive model of rockfill materials which reflects the complex deformation characteristics. Investigating dilatancy characteristics of rockfill materials under various stress paths is beneficial to the establishment of stress path constitutive model.Since Rowe [1] proposed the famous dilatancy equation, many domestic and foreign scholars have done a lot of research on the dilatancy equation. Nova and Wood [2] proposed a nonlinear dilatancy equation, and then Jefferies [3] analyzed Nova dilatancy equation from the perspective of energy. Some domestic scholars, such as Zhang et al. [4], Wang et al. [5], and Jia et al. [6], have also put forward modified dilatancy equations for rockfill materials. The experimental study of Cheng et al. [7] shows that Rowe dilatancy equation can better reflect the dilatancy characteristics of coarse-grained soil. Liu et al. [8] studied the dilatancy law of rockfill materials with different initial void ratios. By incorporating grain-crushing into critical state line and dilatancy equation, Hu et al. [9] establish constitutive models for rockfill materials considering particle breakage. Jia et al. [10] discussed particle breakage of rockfill material during triaxial tests under complex stress paths. Guo and Zhu [11] considered particle breakage effect in energy consumption and deduce dilatancy law of rockfill materials. Liu et al. [12], Liu et al. [13], Zhang et al. [14], Gu et al. [15], Yang et al. [16], and Chen et al. [17] studied the stress deformation characteristics of rockfill materials under different stress paths. However, the work of dilatancy characteristics for rockfill materials mainly focuses on constant confining pressure σ3 stress path. Less attention was paid to dilatancy law of rockfill materials along other loading stress paths.Lade and Kim [18–21] put forward the classic Lade-Kim model for a long time. Its plastic potential is relatively simple and has good application in sand and concrete tests. It has good adaptability to some stress paths. However, there is a lack of Lade-Kim plastic potential to verify the triaxial experiments of rockfill materials. At the same time, the applicability of Lade-Kim plastic potential for different stress path tests, such as conventional triaxial tests, constant P tests, and constant stress (increment) ratio tests, should be verified and evaluated.The objectives of this paper are (1) to analyze the dilatancy characteristics of rockfill materials under different stress paths; (2) to study the adaptability of the dilatancy equation derived from Lade-Kim plastic potential surface for different stress paths and compare Lade-Kim plastic potential parameters for multiple stress path tests; and (3) to apply Lade-Kim plastic potential in generalized plasticity model to predict rockfill materials triaxial test and evaluate its performance by making comparison with nonlinear dilatancy equation.


## 2. Dilatancy Equation for Different Loading Stress Paths

In order to explore the dilatancy equation of rockfill under different stress paths, the data of conventional triaxial, constant P tests and constant stress ratio tests were summed to analyze and compare the dilatancy equation of rockfill materials under various stress path experiments.


### 2.1. Analysis of Dilatancy Equation under Different Stress Paths

For conventional triaxial tests of rockfill materials, various dilatancy equations were proposed, such as linear dilatancy equation [22], Rowe dilatancy equation [1], modified Cambridge dilatancy equation [23], and Fu’s dilatancy equation [24]. Few people pay attention to dilatancy equation under other stress paths, and the dilatancy equation derived from conventional triaxial tests is often applied into such paths in model verification. It is necessary to study the dilatancy equation of rockfill materials under various stress paths.The test results of sand by Sik-Cheung Robert Lo1 and Ian Kenneth Lee [25] are shown in Figure 1. Stress path directions are defined in the literature, which is R=dσ1/dσ3. R=∞ means constant σ3 test which is also called conventional triaxial test; R=−2 means constant P test; R=0 means constant σ1 test; R=4 means constant stress increment ratio test; σ1/σ3 = constant means constant Kc test, which is also called constant stress ratio test. It can be seen from Figure 1 that the dilatancy points of sand under different stress paths are highly discrete, and it is difficult to describe them by unified dilatancy equation. For a single loading stress path, there is a good linear relationship between the dilatancy ratio dg and the stress ratio η, but the different stress paths cannot be described by a unified curve. It can also be found that the dilatancy equation lines translate and rotate in one direction with different loading directions.Figure 1
Stress-dilatancy of sand under different loading directions.Xu et al. [26] carried out the triaxial loading test of the same rockfill with different stress paths, and the test results are rich. Based on the test data, the dilatancy equations of conventional triaxial, constant P loading and constant σ1 loading tests are sorted out in Figure 2. Figure 3 summarizes the development law of dilatancy ratio with stress ratio in four loading directions [27] under confining pressure of 900 kPa. It can be seen that with the continuous change of stress loading direction, the dilatancy characteristics of rockfill materials under different loading modes show great differences. The dilatancy behavior of rockfill materials show obvious stress path dependence in the space of dg−η, in which dg is dilatancy ratio and η is stress ratio. It can be seen from Figures 1–3 that both rockfill materials and sand have similar stress path effect, and it is hard to describe the dilatancy characteristics of rockfill under various stress paths.Figure 2
Stress-dilatancy of gravel along different loading paths.Figure 3
Stress-dilatancy of Yixing rockfill materials along different loading paths.


### 2.2. Lade-Kim Method for Describing Dilatancy Equation

Due to the difference of dilatancy equations under various stress paths and difficulty to give a unified form, a new method which can effectively describe the dilatancy equations under different stress paths is needed.The single hardening model introduced by Lade and Kim [18–21] has predicted the behavior of sand, clay, and concrete under a variety of loading conditions. The true triaxial condition was taken into account in the establishment of the model, which has a strong applicability. Lade-Kim plastic potential may be a good choice to construct dilatancy equation incorporating stress path effect.Lade and Kim [20] proposed a three-dimensional expression of plastic potential surface in principal stress space:(1)Gp=ψ1I13I3−I12I2+ψ2I1paμ.In equation (1), I1, I2, and I3 are the first, second, and third stress invariants; ψ1, ψ2, and μ are the model parameters; pa is the atmospheric pressure. The material parameters ψ2 and μ are dimensionless constants that may be determined from triaxial compression tests. The parameter ψ1 is related to the curvature parameter of the failure criterion as follows:(2)ψ1=0.00155m−1.27,where m is the strength parameter for geomaterials.Two parametersη1 and m describing failure surface can be obtained by data fitting at failure point of conventional triaxial test. The failure surface form under normal triaxial state is as follows:(3)σ1+2σ33σ1σ32−27σ1+2σ3pam=η1.By assuming that(4)X=σ1+2σ3pa,Y=σ1+2σ33σ1σ32−27.Accordingly,(5)Y=η1X−m.Two strength parametersη1 and m are obtained by data collection and fitting on the X-Y plane.The invariant of the stress tensor in the formula can be expressed as follows:(6)I1=σx+σy+σz,(7)I2=τxy2+τyz2+τzx2−σxσy+σyσz+σzσx,(8)I3=σxσyσz+2τxyτyzτzx−σxτxy2+σyτyz2+σzτzx2.In equations (6)–(8), σx, σy, σz, τxy, τyz, and τzx are stress variables for material element.According to the theory of elastoplastic, the increment of plastic strain can be determined by the following formula:(9)dεijp=dλ∂Gp∂σij,where dλ is the plastic multiplier and σij is the stress tensor.Combining equation (1) with equations (6)–(8), it can be obtained that(10)∂G∂σx=I1paμG1−ψ1I13I32σyσz−τyz2−I12I22σz+σy,∂G∂σy=I1paμG1−ψ1I13I32σxσz−τzx2−I12I22σx+σz,∂G∂σz=I1paμG1−ψ1I13I32σxσy−τxy2−I12I22σx+σy,∂G∂τxy=I1paμ2τxyI12I22−2ψ1I13I32τyzτzx−σzτxy,∂G∂τyz=I1paμ2τyzI12I22−2ψ1I13I32τxyτzx−σxτyz,∂G∂τzx=I1paμ2τzxI12I22−2ψ1I13I32τxyτyz−σyτzx,where(11)G1=ψ1μ+3I12I3−μ+2I1I2+μI1ψ2.So, the dilatancy ratiodg of rockfill materials can be obtained as follows:(12)dg=dεvpdεsp=∂Gp/∂σmm2/3∂Gp/∂σij−∂Gp/∂σmmδij/3∂Gp/∂σij−∂Gp/∂σmmδij/3,where εvp and εsp are the plastic volumetric strain and plastic deviatoric strain and δij is the Kronecker delta.It is worth noting that, for coarse-grained soil, particle breakage is an important factor. However, the measurement of particle breakage needs the grading curve before and after experiment, and the soil particle screening is a very time-consuming and laborious work. Particle breakage data for the experiments involved in this paper are very limited. This paper mainly discussed Lade-Kim plastic potential applicability under various stress paths for rockfill materials, and the particle breakage effect is indirectly reflected in parameters of dilatancy equation. Thus, particle breakage was not considered in dilatancy equation.Under the condition of triaxial compressionσ2=σ3,ε2=ε3,(13)∂G∂σ1=I1paμG1−ψ1I13I32σ32−2I12I22σ3,(14)∂G∂σ2=∂G∂σ3=I1paμG1−ψ1I13I32σ1σ3−I12I22σ1+σ3,(15)χ=dε1pdε3p=∂G/∂σ1∂G/∂σ3,where ε1p and ε3p are the plastic axial strain and plastic radial strain and χ is the ratio of plastic axial strain increment to plastic radial strain increment.The dilatancy stress ratiodg can be rewritten as(16)dg=dεvpdεsp=dε1p+2dε3p2/3dε1p−dε3p=3χ+62χ−1.


### 2.3. Verification Method of Lade-Kim Dilatancy Equation

Lade-Kim’s plastic potential dilatancy equation can be verified by triaxial test data, and the specific process method is as follows:The incremental plastic strain ratio is first defined as(17)vp=−dε3pdε1p.Under the condition of triaxial compression, we can obtain equation (23) from equations (1), (9), and (13)–(14), that is,(18)ξy=1μξx−ψ2,where(19)ξx=11+vpI13I22σ1+σ3+2vpσ3+ψ1I14I32σ1σ3++2vpσ32−3ψ1I13I3+2I12I2,and(20)ξy=ψ1I13I3−I12I2.Equation (18) reveals the linear relationship between ξx and ξy. Lade-Kim plastic potential applicability can be investigated by sorting out triaxial experimental data of ξx and ξy for rockfill materials. Correspondingly, Lade-Kim plastic potential surface can be used to characterize the dilatancy law of coarse-grained soil with different stress paths.


## 2.1. Analysis of Dilatancy Equation under Different Stress Paths

For conventional triaxial tests of rockfill materials, various dilatancy equations were proposed, such as linear dilatancy equation [22], Rowe dilatancy equation [1], modified Cambridge dilatancy equation [23], and Fu’s dilatancy equation [24]. Few people pay attention to dilatancy equation under other stress paths, and the dilatancy equation derived from conventional triaxial tests is often applied into such paths in model verification. It is necessary to study the dilatancy equation of rockfill materials under various stress paths.The test results of sand by Sik-Cheung Robert Lo1 and Ian Kenneth Lee [25] are shown in Figure 1. Stress path directions are defined in the literature, which is R=dσ1/dσ3. R=∞ means constant σ3 test which is also called conventional triaxial test; R=−2 means constant P test; R=0 means constant σ1 test; R=4 means constant stress increment ratio test; σ1/σ3 = constant means constant Kc test, which is also called constant stress ratio test. It can be seen from Figure 1 that the dilatancy points of sand under different stress paths are highly discrete, and it is difficult to describe them by unified dilatancy equation. For a single loading stress path, there is a good linear relationship between the dilatancy ratio dg and the stress ratio η, but the different stress paths cannot be described by a unified curve. It can also be found that the dilatancy equation lines translate and rotate in one direction with different loading directions.Figure 1
Stress-dilatancy of sand under different loading directions.Xu et al. [26] carried out the triaxial loading test of the same rockfill with different stress paths, and the test results are rich. Based on the test data, the dilatancy equations of conventional triaxial, constant P loading and constant σ1 loading tests are sorted out in Figure 2. Figure 3 summarizes the development law of dilatancy ratio with stress ratio in four loading directions [27] under confining pressure of 900 kPa. It can be seen that with the continuous change of stress loading direction, the dilatancy characteristics of rockfill materials under different loading modes show great differences. The dilatancy behavior of rockfill materials show obvious stress path dependence in the space of dg−η, in which dg is dilatancy ratio and η is stress ratio. It can be seen from Figures 1–3 that both rockfill materials and sand have similar stress path effect, and it is hard to describe the dilatancy characteristics of rockfill under various stress paths.Figure 2
Stress-dilatancy of gravel along different loading paths.Figure 3
Stress-dilatancy of Yixing rockfill materials along different loading paths.


## 2.2. Lade-Kim Method for Describing Dilatancy Equation

Due to the difference of dilatancy equations under various stress paths and difficulty to give a unified form, a new method which can effectively describe the dilatancy equations under different stress paths is needed.The single hardening model introduced by Lade and Kim [18–21] has predicted the behavior of sand, clay, and concrete under a variety of loading conditions. The true triaxial condition was taken into account in the establishment of the model, which has a strong applicability. Lade-Kim plastic potential may be a good choice to construct dilatancy equation incorporating stress path effect.Lade and Kim [20] proposed a three-dimensional expression of plastic potential surface in principal stress space:(1)Gp=ψ1I13I3−I12I2+ψ2I1paμ.In equation (1), I1, I2, and I3 are the first, second, and third stress invariants; ψ1, ψ2, and μ are the model parameters; pa is the atmospheric pressure. The material parameters ψ2 and μ are dimensionless constants that may be determined from triaxial compression tests. The parameter ψ1 is related to the curvature parameter of the failure criterion as follows:(2)ψ1=0.00155m−1.27,where m is the strength parameter for geomaterials.Two parametersη1 and m describing failure surface can be obtained by data fitting at failure point of conventional triaxial test. The failure surface form under normal triaxial state is as follows:(3)σ1+2σ33σ1σ32−27σ1+2σ3pam=η1.By assuming that(4)X=σ1+2σ3pa,Y=σ1+2σ33σ1σ32−27.Accordingly,(5)Y=η1X−m.Two strength parametersη1 and m are obtained by data collection and fitting on the X-Y plane.The invariant of the stress tensor in the formula can be expressed as follows:(6)I1=σx+σy+σz,(7)I2=τxy2+τyz2+τzx2−σxσy+σyσz+σzσx,(8)I3=σxσyσz+2τxyτyzτzx−σxτxy2+σyτyz2+σzτzx2.In equations (6)–(8), σx, σy, σz, τxy, τyz, and τzx are stress variables for material element.According to the theory of elastoplastic, the increment of plastic strain can be determined by the following formula:(9)dεijp=dλ∂Gp∂σij,where dλ is the plastic multiplier and σij is the stress tensor.Combining equation (1) with equations (6)–(8), it can be obtained that(10)∂G∂σx=I1paμG1−ψ1I13I32σyσz−τyz2−I12I22σz+σy,∂G∂σy=I1paμG1−ψ1I13I32σxσz−τzx2−I12I22σx+σz,∂G∂σz=I1paμG1−ψ1I13I32σxσy−τxy2−I12I22σx+σy,∂G∂τxy=I1paμ2τxyI12I22−2ψ1I13I32τyzτzx−σzτxy,∂G∂τyz=I1paμ2τyzI12I22−2ψ1I13I32τxyτzx−σxτyz,∂G∂τzx=I1paμ2τzxI12I22−2ψ1I13I32τxyτyz−σyτzx,where(11)G1=ψ1μ+3I12I3−μ+2I1I2+μI1ψ2.So, the dilatancy ratiodg of rockfill materials can be obtained as follows:(12)dg=dεvpdεsp=∂Gp/∂σmm2/3∂Gp/∂σij−∂Gp/∂σmmδij/3∂Gp/∂σij−∂Gp/∂σmmδij/3,where εvp and εsp are the plastic volumetric strain and plastic deviatoric strain and δij is the Kronecker delta.It is worth noting that, for coarse-grained soil, particle breakage is an important factor. However, the measurement of particle breakage needs the grading curve before and after experiment, and the soil particle screening is a very time-consuming and laborious work. Particle breakage data for the experiments involved in this paper are very limited. This paper mainly discussed Lade-Kim plastic potential applicability under various stress paths for rockfill materials, and the particle breakage effect is indirectly reflected in parameters of dilatancy equation. Thus, particle breakage was not considered in dilatancy equation.Under the condition of triaxial compressionσ2=σ3,ε2=ε3,(13)∂G∂σ1=I1paμG1−ψ1I13I32σ32−2I12I22σ3,(14)∂G∂σ2=∂G∂σ3=I1paμG1−ψ1I13I32σ1σ3−I12I22σ1+σ3,(15)χ=dε1pdε3p=∂G/∂σ1∂G/∂σ3,where ε1p and ε3p are the plastic axial strain and plastic radial strain and χ is the ratio of plastic axial strain increment to plastic radial strain increment.The dilatancy stress ratiodg can be rewritten as(16)dg=dεvpdεsp=dε1p+2dε3p2/3dε1p−dε3p=3χ+62χ−1.


## 2.3. Verification Method of Lade-Kim Dilatancy Equation

Lade-Kim’s plastic potential dilatancy equation can be verified by triaxial test data, and the specific process method is as follows:The incremental plastic strain ratio is first defined as(17)vp=−dε3pdε1p.Under the condition of triaxial compression, we can obtain equation (23) from equations (1), (9), and (13)–(14), that is,(18)ξy=1μξx−ψ2,where(19)ξx=11+vpI13I22σ1+σ3+2vpσ3+ψ1I14I32σ1σ3++2vpσ32−3ψ1I13I3+2I12I2,and(20)ξy=ψ1I13I3−I12I2.Equation (18) reveals the linear relationship between ξx and ξy. Lade-Kim plastic potential applicability can be investigated by sorting out triaxial experimental data of ξx and ξy for rockfill materials. Correspondingly, Lade-Kim plastic potential surface can be used to characterize the dilatancy law of coarse-grained soil with different stress paths.


## 3. Applicability Verification of Lade-Kim Dilatancy Equation under Different Stress Path Tests

### 3.1. Conventional Triaxial Test

Lade-Kim’s plastic potential surface is proposed based on a large number of sand and concrete tests. Since the plastic potential is established in three-dimensional stress space, it has certain adaptability to different stress paths, but it has not been verified for rockfill material tests.The conventional triaxial tests for three kinds of rockfill materials under different confining pressures are used to verify the applicability of Lade-Kim plastic potential, as shown in Figures4–6 . For the three types of rockfill tests, ξx and ξy show a good linear relationship, indicating that the plastic potential can better describe the dilatancy characteristics of conventional triaxial loading.Figure 4
CT tests of Shuangjiangkou rockfill materials [11].Figure 5
CT tests of limestone rockfill materials [26].Figure 6
Conventional triaxial test of granite rockfills [16].


### 3.2. Constant P Test

Constant P triaxial compression test, e.g., increases the axial stressσ1 and decreases the confining pressure σ3 during loading, so as to keep the mean stress P unchanged. In three-dimensional space, the yield surface is still expanding, but the yield surface expansion is not obvious. By drawing the plastic potential curve, it can be seen that the ξx−ξy curve still conforms to the linear relationship in Figures 7–9.Figure 7
CP tests of granite rockfills [16].Figure 8
CP tests of Yixing rockfill materials [27].Figure 9
CT tests of limestone rockfill materials [27].


### 3.3. Constant Stress Increment Ratio Test

Constant stress increment ratio test, e.g., increases the axial stressσ1 and small principal stress σ3 during loading, so as to keep dσ1/dσ3 constant. In three-dimensional space, the yield surface expansion is obvious.Figures10 and 11 show the results of data arrangement at the same stress increment ratio. When the stress increment ratio is constant, the ξx−ξy curve corresponding to each group of confining pressures is linear, but the slope and intercept of the curves are different. When the stress increment ratio is small, the difference of curve shape under different confining pressures is larger. It can be seen that both confining pressure and stress increment ratio have influence on the dilatancy characteristics of rockfill materials for constant stress increment ratio test, and a set of parameters cannot be used to describe the Lade-Kim dilatancy law.Figure 10
CIR test of limestone rockfill materials [27] (dσ1/dσ3 = 8).Figure 11
CIR test of limestone rockfill materials [27] (dσ1/dσ3 = 3).


### 3.4. Constant Stress Ratio Test

Constant stress ratio test, e.g., increases the axial stressσ1 and small principal stress σ3 during loading, so as to keep R = σ1/σ3 constant. The value of initial confining pressure of this kind of tests is equal to 0, and the stress ratio remains constant in the test. In three-dimensional space, the yield surface expansion is obvious. By drawing ξy-ξx curve, it can be seen from Figure 12 that a certain kind of experiments (such as R = 3.5) has only one data point, and multiple groups of constant stress ratio loading test points still conform to the linear relationship.Figure 12
CR test of granite rockfill materials [16] (σ1/σ3 = 1.5, 2.0, 2.5, and 3.5).


### 3.5. Comparison of Multiple Stress Path Tests

#### 3.5.1. Weakening Weathered Granite Rockfill Experiments [16] under Multiple Stress Paths

Theξy-ξx curve of various stress path tests of granite rockfill materials is exhibited in Figure 13. It is found that CT test and CP test have good consistency by sorting out ξy-ξx curves involved with Lade-Kim plastic potential, while data of three types of experiments (i.e., CT, CP, and CR tests) do not show linear relationship. Meanwhile, CR test under different stress ratios also has a good linear relationship for ξy-ξx curves.Figure 13
Various stress path tests of granite rockfill materials [16].


#### 3.5.2. Liu’s Tests

Yixing rockfill materials are from Yixing Pumped Storage Hydropower Station, located in Yixing, Jiangsu, China. For a variety of stress path tests of Yixing rockfill materials, by sorting out the Lade-Kim plastic potential relatedξy-ξx curves, it can be seen from Figure 14 that the data points of various tests are approximately linear. However, due to the concentration of CIR = 3 test data points, the curve obtained after reducing the scale range of coordinate axis is shown in Figure 11. It can be seen from Figure 11 that the ξy-ξx curves of CIR = 3 test have a large deviation from other tests.Figure 14
Various stress path tests of Yixing rockfill materials [27].The stress loading path direction is defined as follows:(21)β=tanhdpdq,where p is the mean stress, which equals (σ1 + σ2 + σ3)/3, and q is the deviatoric stress that equals (σ1–σ3).By analyzing the triaxial tests of granite rockfill and Yixing rockfill under various stress paths, it is found that Lade-Kim plastic potential data pointsξy and ξx have a certain linear relationship under the condition of β less than 0.3215 or nearby, and the data points for ξy and ξx greater than 0.3215 have greater dispersion. That is to say, Lade-Kim plastic potential is suitable for the case where the loading parameter β is less than 0.3215 or near the value of 0.3215, and it has consistency. When β is greater than 0.3215, the selection of plastic potential parameters should be particularly careful.


## 3.1. Conventional Triaxial Test

Lade-Kim’s plastic potential surface is proposed based on a large number of sand and concrete tests. Since the plastic potential is established in three-dimensional stress space, it has certain adaptability to different stress paths, but it has not been verified for rockfill material tests.The conventional triaxial tests for three kinds of rockfill materials under different confining pressures are used to verify the applicability of Lade-Kim plastic potential, as shown in Figures4–6 . For the three types of rockfill tests, ξx and ξy show a good linear relationship, indicating that the plastic potential can better describe the dilatancy characteristics of conventional triaxial loading.Figure 4
CT tests of Shuangjiangkou rockfill materials [11].Figure 5
CT tests of limestone rockfill materials [26].Figure 6
Conventional triaxial test of granite rockfills [16].


## 3.2. Constant P Test

Constant P triaxial compression test, e.g., increases the axial stressσ1 and decreases the confining pressure σ3 during loading, so as to keep the mean stress P unchanged. In three-dimensional space, the yield surface is still expanding, but the yield surface expansion is not obvious. By drawing the plastic potential curve, it can be seen that the ξx−ξy curve still conforms to the linear relationship in Figures 7–9.Figure 7
CP tests of granite rockfills [16].Figure 8
CP tests of Yixing rockfill materials [27].Figure 9
CT tests of limestone rockfill materials [27].


## 3.3. Constant Stress Increment Ratio Test

Constant stress increment ratio test, e.g., increases the axial stressσ1 and small principal stress σ3 during loading, so as to keep dσ1/dσ3 constant. In three-dimensional space, the yield surface expansion is obvious.Figures10 and 11 show the results of data arrangement at the same stress increment ratio. When the stress increment ratio is constant, the ξx−ξy curve corresponding to each group of confining pressures is linear, but the slope and intercept of the curves are different. When the stress increment ratio is small, the difference of curve shape under different confining pressures is larger. It can be seen that both confining pressure and stress increment ratio have influence on the dilatancy characteristics of rockfill materials for constant stress increment ratio test, and a set of parameters cannot be used to describe the Lade-Kim dilatancy law.Figure 10
CIR test of limestone rockfill materials [27] (dσ1/dσ3 = 8).Figure 11
CIR test of limestone rockfill materials [27] (dσ1/dσ3 = 3).


## 3.4. Constant Stress Ratio Test

Constant stress ratio test, e.g., increases the axial stressσ1 and small principal stress σ3 during loading, so as to keep R = σ1/σ3 constant. The value of initial confining pressure of this kind of tests is equal to 0, and the stress ratio remains constant in the test. In three-dimensional space, the yield surface expansion is obvious. By drawing ξy-ξx curve, it can be seen from Figure 12 that a certain kind of experiments (such as R = 3.5) has only one data point, and multiple groups of constant stress ratio loading test points still conform to the linear relationship.Figure 12
CR test of granite rockfill materials [16] (σ1/σ3 = 1.5, 2.0, 2.5, and 3.5).


## 3.5. Comparison of Multiple Stress Path Tests

### 3.5.1. Weakening Weathered Granite Rockfill Experiments [16] under Multiple Stress Paths

Theξy-ξx curve of various stress path tests of granite rockfill materials is exhibited in Figure 13. It is found that CT test and CP test have good consistency by sorting out ξy-ξx curves involved with Lade-Kim plastic potential, while data of three types of experiments (i.e., CT, CP, and CR tests) do not show linear relationship. Meanwhile, CR test under different stress ratios also has a good linear relationship for ξy-ξx curves.Figure 13
Various stress path tests of granite rockfill materials [16].


### 3.5.2. Liu’s Tests

Yixing rockfill materials are from Yixing Pumped Storage Hydropower Station, located in Yixing, Jiangsu, China. For a variety of stress path tests of Yixing rockfill materials, by sorting out the Lade-Kim plastic potential relatedξy-ξx curves, it can be seen from Figure 14 that the data points of various tests are approximately linear. However, due to the concentration of CIR = 3 test data points, the curve obtained after reducing the scale range of coordinate axis is shown in Figure 11. It can be seen from Figure 11 that the ξy-ξx curves of CIR = 3 test have a large deviation from other tests.Figure 14
Various stress path tests of Yixing rockfill materials [27].The stress loading path direction is defined as follows:(21)β=tanhdpdq,where p is the mean stress, which equals (σ1 + σ2 + σ3)/3, and q is the deviatoric stress that equals (σ1–σ3).By analyzing the triaxial tests of granite rockfill and Yixing rockfill under various stress paths, it is found that Lade-Kim plastic potential data pointsξy and ξx have a certain linear relationship under the condition of β less than 0.3215 or nearby, and the data points for ξy and ξx greater than 0.3215 have greater dispersion. That is to say, Lade-Kim plastic potential is suitable for the case where the loading parameter β is less than 0.3215 or near the value of 0.3215, and it has consistency. When β is greater than 0.3215, the selection of plastic potential parameters should be particularly careful.


## 3.5.1. Weakening Weathered Granite Rockfill Experiments [16] under Multiple Stress Paths

Theξy-ξx curve of various stress path tests of granite rockfill materials is exhibited in Figure 13. It is found that CT test and CP test have good consistency by sorting out ξy-ξx curves involved with Lade-Kim plastic potential, while data of three types of experiments (i.e., CT, CP, and CR tests) do not show linear relationship. Meanwhile, CR test under different stress ratios also has a good linear relationship for ξy-ξx curves.Figure 13
Various stress path tests of granite rockfill materials [16].


## 3.5.2. Liu’s Tests

Yixing rockfill materials are from Yixing Pumped Storage Hydropower Station, located in Yixing, Jiangsu, China. For a variety of stress path tests of Yixing rockfill materials, by sorting out the Lade-Kim plastic potential relatedξy-ξx curves, it can be seen from Figure 14 that the data points of various tests are approximately linear. However, due to the concentration of CIR = 3 test data points, the curve obtained after reducing the scale range of coordinate axis is shown in Figure 11. It can be seen from Figure 11 that the ξy-ξx curves of CIR = 3 test have a large deviation from other tests.Figure 14
Various stress path tests of Yixing rockfill materials [27].The stress loading path direction is defined as follows:(21)β=tanhdpdq,where p is the mean stress, which equals (σ1 + σ2 + σ3)/3, and q is the deviatoric stress that equals (σ1–σ3).By analyzing the triaxial tests of granite rockfill and Yixing rockfill under various stress paths, it is found that Lade-Kim plastic potential data pointsξy and ξx have a certain linear relationship under the condition of β less than 0.3215 or nearby, and the data points for ξy and ξx greater than 0.3215 have greater dispersion. That is to say, Lade-Kim plastic potential is suitable for the case where the loading parameter β is less than 0.3215 or near the value of 0.3215, and it has consistency. When β is greater than 0.3215, the selection of plastic potential parameters should be particularly careful.


## 4. Application of Lade-Kim Plastic Potential in Elastoplastic Model

The generalized plasticity model was first proposed by Pastor et al. [28] to establish a unified static and dynamic constitutive model. In the generalized plasticity model, flow direction, loading direction, and plastic modulus are explicitly defined. The model is simple and convenient to be embedded in program. Since the Lade-Kim dilatancy equation can be deduced from the Lade-Kim plastic potential, the Lade-Kim plastic potential can be evaluated by changing the dilatancy equation in the framework of generalized plasticity. The Lade-Kim dilatancy equation and nonlinear dilatancy equation are two dilatancy equations to be evaluated.


### 4.1. Elastoplastic Stiffness Matrix

In the elastoplastic model, the increment of stress and strain has the following relationship:(22)dσ=Depdε.The elastoplastic stiffness matrixDep in the generalized plasticity model is defined as follows:(23)Dep=De−De:ng⊗nf:DeH+nf:De:ng,where ng is the plastic potential flow direction; nf is the loading direction; H is the loading and unloading plastic modulus; De is the elastic matrix; and Dep is the elastoplastic matrix. The flow direction, loading direction, and plastic modulus are important components of generalized plasticity model.


### 4.2. Flow Direction and Loading Direction

#### 4.2.1. Plastic Flow Direction

The explicit expression of the plastic flow direction can be expressed as follows:(24)ng=1/3dgδij+3sij/2q1/3dg2+3/2,in which(25)dg=dεvpdεsp,q=32sijsij.The Lade-Kim dilatancy equation is equation (12) and the nonlinear dilatancy equation, similar to the one proposed by Fu et al. [24], is listed as follows:(26)dg=dg01−ηMcγ,where dg0 and γ are the dilatancy parameters; η is the stress ratio; and Mc is the dilatancy stress ratio.


#### 4.2.2. Loading Direction

The loading direction tensor is similar to the plastic direction tensor, that is,(27)nf=1/3dfδij+3sij/2q1/3df2+3/2,df=Mf−η.A nonlinear expression [24] is used to describe the peak stress ratio:(28)Mf=Mf0ppa−n0,where Mf0 and n0 are the strength parameters and the peak stress ratio is the function of p.


### 4.3. Plastic Modulus and Elastic Modulus

Fu et al. [29] studied the mechanical properties of rockfill under cyclic loading and deduced an explicit plastic modulus. According to the plastic strain increment relationship,(29)dεp=nf:dσHng.Further considering the elastic deformation, for the triaxial compression test under different stress paths, it can be obtained that(30)dε1dε2dε3=1Ee1−υ−υ−υ1−υ−υ−υ1dσ1dσ2dσ3+1H13ngdg+3dg−32dg−3213nfdf+3df−32df−32Tdσ1dσ2dσ3,where Ee and υ are the elastic modulus and Poisson’s ratio.The norm of the plastic flow direction tensor and that of the loading direction tensor are given as follows:(31)ng=13dg2+32,nf=13df2+32.For triaxial compression stateσ1>σ2=σ3, the tangent modulus Et∗ and conventional tangent modulus Et are defined as follows:(32)Et∗=dqdε1=dσ1−σ3dε1,Et=dσ1dε1,Et∗=1−ηMfα⋅kt⋅pa⋅σcpan,where kt, n, and α are the tangent modulus parameters.Under triaxial loading conditiondσ1>0,RΔ=dσ3/dσ1RΔ<1,(33)Et=Et∗1−RΔ.The plastic modulus can be derived, that is,(34)H=dg+31+2RΔdf+3−3RΔ91/Et−1/Ee1−2υRΔ1/3dg2+3/21/3df2+3/2.It is assumed that the elastic modulus is consistent under various stress paths:(35)Ee=kaupappan,Ge=Ee21+υ,Ke=Ee31−2υ,where kau is the elastic modulus parameter and Ge and Ke are the shear modulus and bulk modulus.


### 4.4. Model Validation and Analysis

In this paper, the applicability of Lade-Kim plastic potential for triaxial tests of rockfill materials is evaluated by using two kinds of dilatancy equations to predict CT, CP, and constant stress increment ratio tests in the framework of generalized plasticity.The three typical tests verified here include conventional triaxial test under 1000 kPa confining pressure, constant P test under 1000 kPa confining pressure, and constant stress increment ratio test under 900 kPa confining pressure (dσ1/dσ3 = 3). The Lade-Kim plastic potential parameters are obtained by sorting out ξy-ξx curves. The parameters of the nonlinear dilatancy equation are obtained by arranging dg-η curves. The calibration methods of other model parameters are shown in the literature [24]. The parameters of generalized plasticity models using two kinds of dilatancy equations are listed in Table 1.Table 1
Model parameters.
Model parameters (excluding dilatancy)Lade-Kim dilatancyNonlinear dilatancyExp.kaunvktαMf0n0ψ1ψ2μdg0McγCT14400.50.22000.82.020.10.0059−2.622.041.21.504.0CP14400.50.23700.82.020.10.0059−2.722.180.841.464.0CR13800.340.35002.32.340.1320.0091−1.510.571.90.843.0It can be seen from the three types of test prediction curves in Figures15 and 16 that the nonlinear dilatancy equation is suitable for describing the dilatancy characteristics of rockfill materials under different stress paths. When Lade-Kim plastic potential is applied to conventional triaxial and constant P tests, it seems that there are some deviations. The reason for this deviation is that νp obtained from the experimental data is quite different from the νp calculated by the model. It can be seen from Figures 17 and 18 that, for Lade-Kim plastic potential, the value of νp is equal to 1 at the beginning of loading and then gradually transits to a constant value. But in the actual test, the initial value of νp is not equal to 1, even a positive value, and then transition to a fixed value. In order to make Lade-Kim’s plastic potential suitable for conventional triaxial and constant P tests, it is necessary to ensure that the initial νp value of the test approaches to 1. As can be seen from Figure 19, the Lade-Kim plastic potential and the nonlinear dilatancy equation show the similar prediction results for the constant stress increment ratio test. Figure 20 presents the evolution of νp for constant stress increment ratio experiments. The main reason for similar prediction results is that the νp obtained from the test data is close to the value calculated by the model, although there are some deviations in the initial stage of loading. In conclusion, for the triaxial test of rockfill materials, Lade-Kim plastic potential is more suitable for constant stress increment ratio loading, and there is a certain deviation for other stress paths, such as conventional triaxial and constant P stress paths. We need to be especially careful when Lade-Kim plastic potential is applied into conventional triaxial and constant P stress paths.Figure 15
Model prediction results for CT experiments.Figure 16
Model prediction results for CP experiments.Figure 17
The evolution ofvp for CT experiments.Figure 18
The evolution ofvp for CP experiments.Figure 19
Model prediction results for constant stress increment ratio experiments.Figure 20
The evolution ofvp for constant stress increment ratio experiments.


## 4.1. Elastoplastic Stiffness Matrix

In the elastoplastic model, the increment of stress and strain has the following relationship:(22)dσ=Depdε.The elastoplastic stiffness matrixDep in the generalized plasticity model is defined as follows:(23)Dep=De−De:ng⊗nf:DeH+nf:De:ng,where ng is the plastic potential flow direction; nf is the loading direction; H is the loading and unloading plastic modulus; De is the elastic matrix; and Dep is the elastoplastic matrix. The flow direction, loading direction, and plastic modulus are important components of generalized plasticity model.


## 4.2. Flow Direction and Loading Direction

### 4.2.1. Plastic Flow Direction

The explicit expression of the plastic flow direction can be expressed as follows:(24)ng=1/3dgδij+3sij/2q1/3dg2+3/2,in which(25)dg=dεvpdεsp,q=32sijsij.The Lade-Kim dilatancy equation is equation (12) and the nonlinear dilatancy equation, similar to the one proposed by Fu et al. [24], is listed as follows:(26)dg=dg01−ηMcγ,where dg0 and γ are the dilatancy parameters; η is the stress ratio; and Mc is the dilatancy stress ratio.


### 4.2.2. Loading Direction

The loading direction tensor is similar to the plastic direction tensor, that is,(27)nf=1/3dfδij+3sij/2q1/3df2+3/2,df=Mf−η.A nonlinear expression [24] is used to describe the peak stress ratio:(28)Mf=Mf0ppa−n0,where Mf0 and n0 are the strength parameters and the peak stress ratio is the function of p.


## 4.2.1. Plastic Flow Direction

The explicit expression of the plastic flow direction can be expressed as follows:(24)ng=1/3dgδij+3sij/2q1/3dg2+3/2,in which(25)dg=dεvpdεsp,q=32sijsij.The Lade-Kim dilatancy equation is equation (12) and the nonlinear dilatancy equation, similar to the one proposed by Fu et al. [24], is listed as follows:(26)dg=dg01−ηMcγ,where dg0 and γ are the dilatancy parameters; η is the stress ratio; and Mc is the dilatancy stress ratio.


## 4.2.2. Loading Direction

The loading direction tensor is similar to the plastic direction tensor, that is,(27)nf=1/3dfδij+3sij/2q1/3df2+3/2,df=Mf−η.A nonlinear expression [24] is used to describe the peak stress ratio:(28)Mf=Mf0ppa−n0,where Mf0 and n0 are the strength parameters and the peak stress ratio is the function of p.


## 4.3. Plastic Modulus and Elastic Modulus

Fu et al. [29] studied the mechanical properties of rockfill under cyclic loading and deduced an explicit plastic modulus. According to the plastic strain increment relationship,(29)dεp=nf:dσHng.Further considering the elastic deformation, for the triaxial compression test under different stress paths, it can be obtained that(30)dε1dε2dε3=1Ee1−υ−υ−υ1−υ−υ−υ1dσ1dσ2dσ3+1H13ngdg+3dg−32dg−3213nfdf+3df−32df−32Tdσ1dσ2dσ3,where Ee and υ are the elastic modulus and Poisson’s ratio.The norm of the plastic flow direction tensor and that of the loading direction tensor are given as follows:(31)ng=13dg2+32,nf=13df2+32.For triaxial compression stateσ1>σ2=σ3, the tangent modulus Et∗ and conventional tangent modulus Et are defined as follows:(32)Et∗=dqdε1=dσ1−σ3dε1,Et=dσ1dε1,Et∗=1−ηMfα⋅kt⋅pa⋅σcpan,where kt, n, and α are the tangent modulus parameters.Under triaxial loading conditiondσ1>0,RΔ=dσ3/dσ1RΔ<1,(33)Et=Et∗1−RΔ.The plastic modulus can be derived, that is,(34)H=dg+31+2RΔdf+3−3RΔ91/Et−1/Ee1−2υRΔ1/3dg2+3/21/3df2+3/2.It is assumed that the elastic modulus is consistent under various stress paths:(35)Ee=kaupappan,Ge=Ee21+υ,Ke=Ee31−2υ,where kau is the elastic modulus parameter and Ge and Ke are the shear modulus and bulk modulus.


## 4.4. Model Validation and Analysis

In this paper, the applicability of Lade-Kim plastic potential for triaxial tests of rockfill materials is evaluated by using two kinds of dilatancy equations to predict CT, CP, and constant stress increment ratio tests in the framework of generalized plasticity.The three typical tests verified here include conventional triaxial test under 1000 kPa confining pressure, constant P test under 1000 kPa confining pressure, and constant stress increment ratio test under 900 kPa confining pressure (dσ1/dσ3 = 3). The Lade-Kim plastic potential parameters are obtained by sorting out ξy-ξx curves. The parameters of the nonlinear dilatancy equation are obtained by arranging dg-η curves. The calibration methods of other model parameters are shown in the literature [24]. The parameters of generalized plasticity models using two kinds of dilatancy equations are listed in Table 1.Table 1
Model parameters.
Model parameters (excluding dilatancy)Lade-Kim dilatancyNonlinear dilatancyExp.kaunvktαMf0n0ψ1ψ2μdg0McγCT14400.50.22000.82.020.10.0059−2.622.041.21.504.0CP14400.50.23700.82.020.10.0059−2.722.180.841.464.0CR13800.340.35002.32.340.1320.0091−1.510.571.90.843.0It can be seen from the three types of test prediction curves in Figures15 and 16 that the nonlinear dilatancy equation is suitable for describing the dilatancy characteristics of rockfill materials under different stress paths. When Lade-Kim plastic potential is applied to conventional triaxial and constant P tests, it seems that there are some deviations. The reason for this deviation is that νp obtained from the experimental data is quite different from the νp calculated by the model. It can be seen from Figures 17 and 18 that, for Lade-Kim plastic potential, the value of νp is equal to 1 at the beginning of loading and then gradually transits to a constant value. But in the actual test, the initial value of νp is not equal to 1, even a positive value, and then transition to a fixed value. In order to make Lade-Kim’s plastic potential suitable for conventional triaxial and constant P tests, it is necessary to ensure that the initial νp value of the test approaches to 1. As can be seen from Figure 19, the Lade-Kim plastic potential and the nonlinear dilatancy equation show the similar prediction results for the constant stress increment ratio test. Figure 20 presents the evolution of νp for constant stress increment ratio experiments. The main reason for similar prediction results is that the νp obtained from the test data is close to the value calculated by the model, although there are some deviations in the initial stage of loading. In conclusion, for the triaxial test of rockfill materials, Lade-Kim plastic potential is more suitable for constant stress increment ratio loading, and there is a certain deviation for other stress paths, such as conventional triaxial and constant P stress paths. We need to be especially careful when Lade-Kim plastic potential is applied into conventional triaxial and constant P stress paths.Figure 15
Model prediction results for CT experiments.Figure 16
Model prediction results for CP experiments.Figure 17
The evolution ofvp for CT experiments.Figure 18
The evolution ofvp for CP experiments.Figure 19
Model prediction results for constant stress increment ratio experiments.Figure 20
The evolution ofvp for constant stress increment ratio experiments.


## 5. Conclusion

In this paper, based on a large number of coarse-grained material stress path tests, the adaptability of Lade-Kim plastic potential equation for different stress paths is evaluated and analyzed. The Lade-Kim plastic potential, together with nonlinear dilatancy equation, is applied in elastoplastic model to model the behavior of rockfill materials under various stress paths. The main conclusions are as follows:(1)
The dilatancy characteristics of coarse-grained materials are different under different loading stress paths. The parameters for Lade-Kim plastic potential equation were calibrated only by a single soil triaxial compression test which may be not enough. It is necessary to calibrate Lade-Kim plastic potential parameters by multiple stress path tests.(2)
Lade-Kim plastic potential has good applicability to single stress path, such as conventional triaxial test, constant P test, constant stress ratio test, and constant stress increment ratio test, but it is difficult to consider the influence of stress paths on original Lade-Kim plastic potential for the same geotechnical materials. The plastic potential parametersψ1 and ψ2 are not obviously affected by the stress path, but the parameter µ is significantly affected by the loading stress path.(3)
The Lade-Kim plastic potential and nonlinear dilatancy equation present different prediction results, when applied to the generalized plasticity model. Lade-Kim plastic potential is more suitable for constant stress increment ratio loading experiments and special care should be taken when applied to other stress paths. When the Lade-Kim plastic potential is applied to the triaxial test of rockfill materials, it needs to be further modified so that the stress path dependency can be considered.



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*Source: 5561427-2021-09-15.xml*