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Soil and stone mixture is widely distributed in China, and its physical and mechanical properties are complex, which has a significant impact on geotechnical engineering. Usually soil and stone mixture shows anisotropic features along or perpendicular to the direction of settled layers, and the strength will be significantly affected by intermediate principal stress. True triaxial tests were carried on, Paul–Mohr–Coulomb (PMC) failure criterion was used for the strength analysis in soil-stone mixture, and related parameters of PMC model were obtained. A user-defined PMC numerical constitutive model was developed for FLAC3D. Composite failure criteria of shear failure and tension cut-off were applied for numerical analysis, and nonassociated flow rule was proposed based on the Mohr–Coulomb model. Verification modelling was applied as well, and deviation between analytical and numerical solutions in strains of

Soil and stone mixture is widely distributed in China, which has a great impact on geotechnical engineering. Due to the river scouring and accumulation settlement, the gravels or stones in the soil-stone mixture are arranged in a certain regularity, resulting that the physical and mechanical properties are different from the isotropic soil material. Soil-stone mixture has strong anisotropy in the layer strike direction and vertical direction. Meanwhile, its triaxial strength characteristics and failure characteristics are significantly affected by the intermediate principal stress [

At present, a lot of experiments and theoretical research on soil-stone mixture were carried out, and achievements in the aspects of mechanical properties and strength criteria of soil-stone mixture were obtained. In the experimental research, Wu Ming carried out 4 groups of tests of soil-stone mixture and the results showed that the common feature of specimens composed of coarse and fine materials depended on the content of components [

Wang et al. carried out a series of numerical uniaxial compression tests to obtain the deformation and failure characteristics of the soil-rock mixture with different compositions and structures. Obvious bedding phenomena usually appeared in the direction of rock block inclination, and the ultimate shear strain increases with the rock block proportion increase [

In terms of constitutive model research, Vallejo and Lobo-Guerrero [

Lee and Pyo [

Ma et al. [

Xiaochun and Dongjun [

Paul [

In this paper, the PMC failure criterion is carried out to fit the result of the triaxial test and to research the strength and failure rules. Material parameters of soil and stone mixture under the PMC failure criterion were determined, a user-defined FLAC3D numerical constitutive model based on the PMC strength criterion was developed, and verification analysis was involved as well.

The PMC failure criterion formula is as follows:

In equation (_{c} and _{e} are internal friction angles in compression and tension, respectively, and _{0} is the theoretical value of triaxial tensile strength, as shown in Figure

Six-sided pyramidal failure surface of the PMC model.

Zeng et al. [_{c}, _{e}, and _{0} of geotechnical materials in the PMC failure criterion.

The relationship between three-dimensional principal stress and mean stress, shear stress, and Lode angle is as follows:

Based on the new fitting method, in the triaxial compression test (

In the triaxial compression test (_{c} and _{e} are compressive cohesion and tensile cohesion, respectively, as follows:

When _{c} = _{e} and _{c} = _{e}, _{c} in the PMC model is the internal friction angle _{c} is the cohesive force

True triaxial testing of soil-stone mixture materials was applied on the machine of TSW-50 (see Figure

TSW-50 soil true triaxial testing machine.

There were 8 groups of experiments involved, 1∼4 groups were soil-stone mixture experiments with medium principal stress ratio coefficient of 0 and 5∼8 groups were experiments with the coefficient of 1. When the intermediate principal stress coefficient _{3} unchanged, and _{1} gradually increased until the failure occurred. When the intermediate principal stress coefficient _{3} remained unchanged, and gradually increased _{1} to the failure strength. The curve of stress deviation vs. the axial principal strain (_{1}–_{3}∼_{1}) is shown in Figure

_{1}–_{3}∼_{1} curve of the soil and stone mixture triaxial test: (a)

The results show that with the increase in stress level, the triaxial failure strengths are obtained. The failure surface and test failure data are fitted in

Comparison of PMC strength curve and test data in

The correlation coefficient ^{2} of soil-stone mixture test data is 0.92. The true triaxial test results of soil-stone mixture fit the PMC criterion well, and the trend is consistent. The compression friction angle is smaller than the tensile internal friction angle. The difference between _{c} and _{e} (see Table

PMC material parameters of soil-stone mixture.

Geotechnical materials | _{c} (°) | _{e} (°) | _{0} (kPa) | ^{2} |
---|---|---|---|---|

Soil-stone mixture | 38.1 | 42.7 | 6.4 | 0.92 |

Through the applicability analysis of the PMC strength criterion in soil-stone mixture material, the fitted PMC failure surface is consistent with the test data, which are in line with the expected trend of soil-stone mixture material. The PMC strength criterion can be used to describe the strength and failure behavior of soil-stone mixture material.

In general, Hooke’s Law is used to describe the linear elastic behavior of the PMC model as follows:

In the plastic stage, the material constitutive in FLAC3D software follows the incremental forms. The specific calculation steps are as follows:

Strain increments are divided into elastic strains and plastic strains as follows:

The stress-strain constitutive equation for flow rule is as follows:

where

The stress increment can be determined by the following equation:

where _{i} is the linear equation which obeys Hooke’s Law.

The new stress state also satisfies the yield equation as follows:

where

The expression of coefficient

The failure criterion of the PMC model is a composite failure criterion, as shown in Figure

Schematic diagram of the composed strength model.

To neglect hardening of materials, the strain increment can be divided into elastic and plastic strain increment as follows:_{1}, _{2}, _{3}) is the plastic potential function.

The plastic potential functions _{1}, _{2}, _{3}) and _{3}) of the PMC model are chosen for shear failure and tension failure, which can be derived according to the Mohr–Coulomb model of FLAC3D.

PMC criteria can be expressed as follows:_{1}, _{2}, and _{3} are the maximum principal stress, intermediate principal stress, and minimum principal stress, respectively; _{0} is the theoretical triaxial tensile strength.

Let

Then,

And the plastic potential function is as follows:

If ^{t} = 0 is used as the tension strength, then

Then, the plastic potential function is as follows:

The stress of material according to the elastic-to-plastic model defined by its elastic component can be written as follows:

For shear yield correction, _{1} and _{2} are the material parameters defined based on shear modulus

The yield function of PMC _{n}) is the constant term of the linear failure criterion.

In equation (

After the material enters the plastic state, the flow criterion is as follows:

Then,

When the model does not consider the dilatancy, if _{ψ} = 1. The plastic volume strain increment is 0.

For tension failure,

_{n}) is the constant term of the linear failure criterion.

Based on the software of “Fast Lagrangian Analysis of Continua in 3 Dimensions” (FLAC3D, Itasca Consulting Group Inc.), a user-defined PMC model is developed.

The parameters and corresponding keywords of the PMC model defined in this paper are shown in Table

PMC model parameters and keywords corresponding table.

Model keywords | Model parameters |
---|---|

Bulk | Bulk modulus, |

Shear | Shear modulus, |

c_friction | Internal friction angle in compression, _{c} |

e_friction | Internal friction angle in tension, _{e} |

v_para | Theoretical triaxial tensile strength, _{0} |

Tension | Ensile strength, _{t} |

Dilation | Dilation, |

When calling the custom PMC numerical constitutive model for numerical model calculation, it is necessary to load the PMC constitutive model into the FLAC3D. The validity of the numerical constitutive model of PMC was evaluated by modelling in the true triaxial test process.

The PMC failure criterion is derived from the M-C strength criterion. When

Parameter table of the triaxial compression test model.

Working conditions | _{c} (°) | _{e} (°) | _{c} (MPa) | _{e} (MPa) | _{0} (MPa) | |||
---|---|---|---|---|---|---|---|---|

M-C | 10.0 | 0.33 | 28.0 | — | 8.5 | — | 16.0 | 28.29 |

PMC-1 | 10.0 | 0.33 | 28.0 | 28.0 | 8.5 | 8.5 | 16.0 | 28.29 |

PMC-2 | 10.0 | 0.33 | 28.0 | 33.0 | 8.5 | 10.4 | 16.0 | 28.29 |

_{ci} is the uniaxial compressive strength.

A cube element (size: 1 × 1 × 1) is created, and the fixed boundary surface

Model boundary condition diagram.

To ensure the quasi-static loading, the loading speed is set at 10^{−8} per step, and a total of 450,000 steps are calculated. The MC model and PMC-1 model entered the plastic stage in 372,000 steps, and the PMC-2 model entered the plastic stage in 385,000 steps. After 450,000 steps of calculation, the strain in ^{−3}. At this time, the triaxial test blocks under each constitutive model have entered the plastic state. By analyzing the strength peak value, elastic strain, and plastic strain of the test block, the user-defined PMC model can be checked and the failure characteristics of geotechnical materials can be explored.

Figure _{z}|_{A} = 3851.00 × 10^{−6}. The plastic strain is obtained by subtracting the elastic strain from the total strain, and the plastic deformation at point B is _{z}|_{B} = 654.13 × 10^{−6}.

Vertical stress curve of the PMC-2 model.

Therefore, it can be verified that the numerical constitutive model of self-defined PMC can be called normally, and the calculation results in special cases are consistent with those of the M-C model in special case.

Table

Comparison table of numerical simulation results of the true triaxial test.

Working conditions | M-C | PMC-1 | PMC-1/M-C | |
---|---|---|---|---|

Peak strength (MPa) | 42.14 | 42.17 | 1.0007 | |

Elastic strain (×10^{−6}) | 1234.03 | 1234.20 | 1.0001 | |

Plastic strain (×10^{−6}) | 772.77 | 770.99 | 0.9977 | |

Total strain (×10^{−6}) | 2006.80 | 2005.19 | 0.9992 | |

Elastic strain (×10^{−6}) | 555.50 | 556.30 | 1.0014 | |

Plastic strain (×10^{−6}) | 0.00 | 0.00 | ||

Total strain (×10^{−6}) | 555.50 | 556.30 | 1.0014 | |

Elastic strain (×10^{−6}) | −3733.91 | −3734.15 | 1.0001 | |

Plastic strain (×10^{−6}) | −772.76 | −770.98 | 0.9977 | |

Total strain (×10^{−6}) | −4506.67 | −4505.13 | 0.9997 | |

Average value | 0.9998 | |||

Coefficient of variation | 0.0013 |

Table

Comparison table of numerical simulation results of the true triaxial test.

Computation parameter | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Constitutive model | Analytical solution | Numerical simulate solution | Analytical solution/numerical simulate solution | |||||||

_{x} (×10^{−6}) | _{y} (×10^{−6)} | _{z} (×10^{−6}) | _{x} (×10^{−6}) | _{y} (×10^{−6}) | _{z} (×10^{−6}) | _{x} (×10^{−6}) | _{y} (×10^{−6}) | _{z} (×10^{−6}) | ||

PMC-1 | Elastic strain | 1221.61 | 556.61 | −3722.00 | 1234.20 | 556.30 | −3734.15 | 0.9898 | 1.0006 | 0.9967 |

Plastic strain | 783.13 | −783.13 | 770.99 | −770.98 | 1.0157 | 1.0158 | ||||

Total strain | 2004.74 | 556.61 | −4505.13 | 2005.19 | 556.30 | −4505.13 | 0.9998 | 1.0006 | 1 | |

PMC-2 | Elastic strain | 1264.18 | 599.22 | −3851.00 | 1267.07 | 599.22 | −3854.15 | 0.9977 | 1 | 0.9992 |

Plastic strain | 654.13 | −654.13 | 650.98 | −650.98 | 1.0048 | 1.0048 | ||||

Total strain | 1918.31 | 599.22 | −4505.13 | 1918.05 | 599.22 | −4505.13 | 1.0001 | 1 | 1 | |

Average value | 1.0013 | 1.0003 | 1.0028 | |||||||

Coefficient of variation | 0.0078 | 0.0003 | 0.0013 |

It can be seen from the table that the ratio of each value of the theoretical analytical solution and the numerical simulation solution is equal to 1 or approaches to 1, and the coefficient of variation approaches to 0.

So far, it can be verified that the user-defined PMC numerical constitutive model developed based on the PMC strength criterion can be normally used in numerical calculation, and the numerical settlement results are consistent with the theoretical calculation results.

The true triaxial numerical simulation test results show that the developed PMC strength numerical constitutive model can reflect the influence of intermediate principal stress on the triaxial strength of soil-stone mixture and accurately simulate the strength peak value, elastic strain, and plastic strain of soil and stone mixture.

Through the research on the applicability of the PMC strength criterion in soil-stone mixture, the numerical constitutive model based on the PMC failure criterion is developed, and the true triaxial numerical simulation test is carried out. The main conclusions are as follows:

Through the true triaxial test, it is found that the triaxial strength characteristics of soil-stone mixture are significantly affected by medium principal stress. The material parameters of soil-stone mixture under the PMC model are determined.

Combined with the composite failure criterion of shear failure, a user-defined numerical constitutive model of PMC was developed and embedded in the software platform of FLAC3D. The numerical simulation solution is compared with the theoretical analytical solution, and the self-defined PMC elastic-plastic constitutive model is verified.

The self-defined PMC numerical constitutive model can reflect the influence of medium principal stress on the triaxial strength of soil-stone mixture and accurately simulate the strength peak value, elastic strain, and plastic strain of soil-stone mixture, which provides an effective method for numerical simulation analysis of underground engineering of soil-stone mixture stratum.

The data used to support the findings of this study are included within the paper.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The study was supported by the National Key Research and Development Project of China (2018YFE0101100) and General Project of Science and Technology Plan of Beijing Education Committee (KM202011418001).