# Rheological Behavior and Modeling of a Crushed Sandstone-Mudstone Particle Mixture

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## Abstract

**:**

## 1. Introduction

## 2. Test Materials and Testing Methods

#### 2.1. Tested Materials

^{3}and 8%, respectively. The particle grading curve is shown in Figure 1; the maximum particle size was 5 mm. The mean particle size of the particle grading curve, D50, was 0.83 mm, and the non-uniformity and curvature coefficients were 25.56 and 1.16, respectively. The particle contents are listed in Table 1, in terms of the various particles size fractions.

#### 2.2. Confined Uniaxial Compression Tests

#### 2.3. Testing Method

## 3. Test Results

#### 3.1. Results of the Rheological Tests

#### 3.2. Four-Phase Rheological Process

## 4. Segmented Rheological Model for SMPM

#### 4.1. Segmented Rheological Model

_{1}is the axial rheological strain; a and b are the model parameters; ε

_{1f}is the rheological final strain; t is the time of rheological test.

_{1i}, was calculated using the three coarse-grained soil rheological models (Equations (1) to (3)), if it had not yet reached the initial critical value, ε

_{1c}, of the secondary rheology (Figure 4). Otherwise, the rheological time and rheological strain, ε

_{2i}, were recalculated, and the rheological parameters were re-determined. The total rheological strain of the segmented rheological model can be obtained as follows:

_{1}is the total axial rheological strain and ε

_{1i}and ε

_{2i}are the attenuation rheology and secondary attenuation rheology, respectively; they can be calculated using the coarse-grained soil rheological models listed in Table 2. ε

_{1c}is the critical rheological strain of the secondary attenuation, which can be calculated using the attenuation rheological factor and final rheology, as follows:

_{f}is the rheological limit strain and $\alpha $ is the rheological attenuation factor.

#### 4.2. Rheological Limit Strain ε_{f}

_{f}has been previously investigated by several researchers. Zhang et al. (2010) suggested that ε

_{f}of rock-fill materials was associated with the confining pressure and concluded that the relationship between the vertical stress and the limit strain could be calculated using a hyperbola. Cao [29] argued that ε

_{f}(total strain within a test period) was exponentially related to the vertical stress. In the present study, the relationship between the vertical limit strain (total strain within a test period) and axial pressure is shown in Figure 5; ε

_{f}increased with increasing vertical stress, which could be fitted by the following linear relationship:

_{1}and p

_{a}are the axial and atmospheric pressures (kPa), respectively; and m and n are the fitting parameters associated with the material properties. The fitted values of parameters m and n are listed in Table 2.

_{0}and n

_{0}are fitting parameters whose values are 0.0024 and 0.0406, respectively.

_{f}is the rheological limit strain, M is the mudstone particle content (%) and σ

_{1}and p

_{a}are the vertical and atmospheric pressures (kPa), respectively. These three fitting parameters for predicting the rheological limit strain, m, m

_{0}and n

_{0}, are fitting parameters whose values are 0.0163, 0.0024 and 0.0406, respectively. σ

_{1}is the vertical stress in the present study. The vertical stress is the sum of pore pressure and the horizontal effective stress. The pore pressure in the coarse-soil is a small force as compared with the effective stress. Investigated by Qiu [4], the pore pressure can be ignored due to the permeability coefficient of the SMPM being greater than 0.01 cm/s. In addition to minimizing the pore pressure of the sample, the pore stone with its permeability coefficient greater than 0.05 cm/s was used in the test. The drainage of the sample was free during the entire testing process.

#### 4.3. Rheological Attenuation Factor

_{1c}is the critical strain value of the attenuation rheology, which can be obtained using the rheological curve, and ε

_{f}is the strain value of the final rheology.

_{1}and p

_{a}are the axial and atmospheric pressures (kPa), respectively. a

_{α}and b

_{α}are fitting parameters, and their relationship with the mudstone particle content is shown in Figure 8.

_{α}increased with increasing mudstone particle content, whereas b

_{α}decreased. The relationship between a

_{α}, b

_{α}and the mudstone particle content could be fitted with a linearly relationship, as follows:

_{1}and d

_{1}are the fitting parameters of the linear fitting relationship between a

_{α}and the mudstone particle content with values of 0.0003 and 0.03, respectively. c

_{2}and d

_{2}are the fitting parameters of the linear fitting relationship between b

_{α}and the mudstone particle content with values of −0.002 and 0.78, respectively.

_{1}and p

_{a}are the axial and atmospheric pressures (kPa), respectively. There are four parameters for predicting the rheological attenuation factor. c

_{1}, d

_{1}, c

_{2}and d

_{2}are fitting parameters whose values are 0.0003, 0.03, −0.002 and 0.78, respectively.

## 5. Discussions

_{1}, b

_{1}, a

_{2}and b

_{2}are model parameters, c

_{1}, d

_{1}, c

_{2}, d

_{2}, m, m

_{0}and n

_{0}are seven fitting parameters, and these parameters can be obtained by fitting with the experimental data. There are 11 parameters for the suggested model, which can be used to calculate the rheological deformation filled with SMPM of different mudstone content. There are too many parameters for the suggested model. In further study, research on a model for the rheological deformation of SMPM within fewer parameters is very meaningful. The strength of the mudstone particles is lower than that of the sandstone particles, which suggests that they have different critical crushing stresses. The particle-crushing critical stress of the mudstone particles is lower than that of the sandstone particles. In the rheological compression tests, the deformation initially resulted from the mudstone particles being crushed, with further deformation occurring due to the crushing of sandstone particles at a later time. Mudstone particles are usually disintegrative, and mudstone particle surfaces are often covered with fractures. Therefore, the mudstone particle breakage appears to be more serious than that of the sandstone particles. Compared with other materials such as rock-fill materials and coarse-grained soil, SMPM features a unique rheological mechanism because of the mudstone particles, which are associated with various factors, including the mudstone particle content and the particle strength. The suggested rheological model was proven by the validation exercise. It is not an intuitive method to prove the accuracy of the suggested model. This model can be used to calculating the rheological deformation of a dam or foundations filled with SMPM under different mudstone content. However, it cannot be applied to real engineering works now. In future research, the finite element approach may be used to analyze the rheological behavior of SMPM. When the FEM procedures for the suggested constitutive relationships are worked out, it can be applied to the analysis of a real problem, and it could be generalized while its precision is proven.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Compressive rheological curves of sandstone and mudstone particle mixture (SMPM) samples; M = mudstone particle content.

Particle size (mm) | 5–2 | 2–1 | 1–0.5 | 0.5–0.25 | 0.25–0.075 | <0.075 |

Content (%) | 27 | 18 | 15 | 11 | 14 | 15 |

Mudstone Particles Content (%) | m | n | R^{2} |
---|---|---|---|

20 | 0.0163 | 0.0164 | 0.96 |

40 | 0.0165 | 0.0571 | 0.96 |

60 | 0.0168 | 0.0798 | 0.96 |

80 | 0.0172 | 0.1711 | 0.99 |

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**MDPI and ACS Style**

Qiu, Z.; Cao, T.; Li, Y.; Wang, J.; Chen, Y.
Rheological Behavior and Modeling of a Crushed Sandstone-Mudstone Particle Mixture. *Processes* **2018**, *6*, 192.
https://doi.org/10.3390/pr6100192

**AMA Style**

Qiu Z, Cao T, Li Y, Wang J, Chen Y.
Rheological Behavior and Modeling of a Crushed Sandstone-Mudstone Particle Mixture. *Processes*. 2018; 6(10):192.
https://doi.org/10.3390/pr6100192

**Chicago/Turabian Style**

Qiu, Zhenfeng, Ting Cao, Yongsuo Li, Junjie Wang, and Yulong Chen.
2018. "Rheological Behavior and Modeling of a Crushed Sandstone-Mudstone Particle Mixture" *Processes* 6, no. 10: 192.
https://doi.org/10.3390/pr6100192