# Resilient Modulus—Physical Parameters Relationship of Improved Red Clay by Dynamic Tri-Axial Test

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material Properties and Testing Method

#### 2.1. Materials

_{2}Si

_{2}O

_{5}(OH)

_{4}), goethite (i.e., FeO(OH)), and diaspore (i.e., SiO

_{2}). The amount of kaolinite is 60.87%. Therefore, kaolinite dominates the mineral compositions of the red clay.

#### 2.2. Testing Method

## 3. Analysis of Results

#### 3.1. Influence of Stress State on Resilient Modulus

#### 3.2. Prediction Models of Resilient Modulus

#### 3.2.1. Uzan-Witczak Model

#### 3.2.2. Andrei Model

#### 3.2.3. Ni Model

#### 3.3. Regression Analysis Results

^{2}, defined as the ratio of regression sum of squares (SSR) to total sum of square (SST), that is ${\mathrm{R}}^{2}$ = SSR/SST.

## 4. The Dependence of Resilient Modulus on Material Properties

#### 4.1. The Influence of Moisture Content and Compactness on Resilient Modulus

#### 4.2. The Influence of Additive Content on Resilient Modulus

#### 4.3. New Comprehensive Prediction Model

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Definition of dynamic resilient modulus (Hopkins et al., 2001 [51]).

**Figure 2.**Variation of resilient modulus with deviatoric stress and bulk stress. (

**a**) Lime 3% OMC ${M}_{\mathrm{r}}-{\sigma}_{d}$; (

**b**) Lime 3% OMC ${M}_{\mathrm{r}}-{\sigma}_{\theta}$; (

**c**) Cement 3% OMC ${M}_{\mathrm{r}}-{\sigma}_{d}$; (

**d**) Cement 3% OMC ${M}_{\mathrm{r}}-{\sigma}_{\theta}$.

**Figure 3.**Predicted resilient modulus versus measured resilient modulus by three prediction models (Andrei model, NI model and Uzan-Witczak model). (

**a**) 93% compactness and 6% cement; (

**b**) 93% compactness and 6% lime.

**Figure 4.**${M}_{R}-w$ relation curve at different stress states. (

**a**) $C6\%,\text{}P93\%,\text{}{\sigma}_{d}=30\text{}\mathrm{kpa}$; (

**b**) $C6\%,\text{}P93\%,\text{}{\sigma}_{d}=55\text{}\mathrm{kpa}$; (

**c**) $L6\%,\text{}OMC,\text{}{\sigma}_{d}=30\text{}\mathrm{kpa}$; (

**d**) $L6\%,\text{}P93\%,\text{}{\sigma}_{d}=55\text{}\mathrm{kpa}$.

**Figure 5.**${M}_{R}-P$ curve at different stress states. (

**a**) $C6\%,\text{}OMC,\text{}{\sigma}_{d}=30\text{}\mathrm{kpa}$; (

**b**) $C6\%,\text{}OMC,\text{}{\sigma}_{d}=55\text{}\mathrm{kpa}$; (

**c**) $L6\%,\text{}OMC,\text{}{\sigma}_{d}=30\text{}\mathrm{kpa}$; (

**d**) $L6\%,\text{}OMC,\text{}{\sigma}_{d}=55\text{}\mathrm{kpa}$.

**Figure 6.**Relationship between resilient modulus (M

_{r}) and lime content for various stress states. (

**a**) $OMC,\text{}P93\%,\text{}{\sigma}_{d}=30\text{}\mathrm{kpa}$; (

**b**) $OMC,\text{}P93\%,\text{}{\sigma}_{d}=55\text{}\mathrm{kpa}$.

**Figure 7.**Relationship between resilient modulus (M

_{r}) and cement content for various stress states. (

**a**) $OMC,P93\%,{\sigma}_{d}=30\text{}\mathrm{kpa}$; (

**b**) $OMC,\text{}P93\%,\text{}{\sigma}_{d}=55\text{}\mathrm{kpa}$.

**Figure 8.**Predicted resilient modulus versus measured resilient modulus at different optimum moisture content, compactness and cement by four prediction models (comprehensive model, Andrei model, NI model &Uzan-Witczak model). (

**a**) 93% compactness and 6% cement; (

**b**) optimum moisture content (OMC)and 6% cement; (

**c**) 93% compactness and OMC.

**Figure 9.**Predicted resilient modulus versus measured resilient modulus at different optimum moisture content, compactness and lime by four prediction models (Comprehensive model, Andrei model, NI model and Uzan-Witczak model). (

**a**) 93% compactness and 6% lime; (

**b**) OMC and 6% lime; (

**c**) 93% compactness and OMC.

Samples | Grain Density | Natural Dry Density | Maximum Dry Density | Optimum Moisture Content | Liquid Limit | Plastic Limit | <0.075mm Grain Size |
---|---|---|---|---|---|---|---|

/(g/cm) | /(g/cm) | /(g/cm) | /% | /% | /% | Soil particle content/% | |

Red clay | 2.72 | 1.28 | 1.73 | 27.8 | 52.6 | 30.7 | 90 |

Constituents | SiO_{2} | Al_{2}O_{3} | Fe_{2}O_{3} | K_{2}O | MgO | CaO | Na_{2}O | LOI |
---|---|---|---|---|---|---|---|---|

Content(%) | 40.27 | 32.65 | 18.52 | 2.51 | 0.91 | 0.72 | 0.24 | 4.18 |

Mineral | Mineral Compositions/(%) | ||||
---|---|---|---|---|---|

Al_{2}Si_{2}O_{5}(OH)_{4} | Al(OH)_{3} | FeO(OH) | SiO_{2} | Deviation | |

Content(%) | 60.87 | 11.72 | 13.13 | 11.96 | −2.32 |

Cement Content/% | Optimum Moisture Content/% | Maximum Dry Density/(g/cm^{3}) | Lime Content/% | Optimum Moisture Content/% | Maximum Dry Density/(g/cm^{3}) |
---|---|---|---|---|---|

3 | 25.6 | 1.75 | 3 | 26.2 | 1.74 |

6 | 24.3 | 1.76 | 6 | 25.1 | 1.75 |

9 | 23.2 | 1.78 | 9 | 24.2 | 1.76 |

(a) Improved red clay with 3% cement | ||||||

Source of Variation | Variance Sum | Degree of Freedom | Mean Square Error | F Value | Critical Value a = 0.05 | Significance |

Deviatoric stress | 2397.41 | 3 | 799.136 | 134.39 | 3.86 | Significant |

Confining pressure | 1832.63 | 3 | 610.877 | 102.73 | 3.86 | Significant |

Error | 53.52 | 9 | 5.946 | |||

Sum | 4283.56 | 15 | ||||

(b) Improved red clay with 3% lime | ||||||

Source of Variation | Variance Sum | Degree of Freedom | Mean Square Error | F Value | Critical Value a = 0.05 | Significance |

Deviatoric stress | 1853.83 | 3 | 617.942 | 178.36 | 3.86 | Significant |

Confining pressure | 1870.63 | 3 | 623.544 | 179.98 | 3.86 | Significant |

Error | 31.18 | 9 | 3.465 | |||

Sum | 3755.64 | 15 |

Additive Content | Compactness/% | Moisture Content/% | Uzan-Witczak Model | NI Model | Andrei Model | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

k1 | k2 | k3 | R2 | k1 | k2 | k3 | R2 | k1 | k2 | k3 | R2 | |||

C6% | 93 | OMC minus 3% | 139.6 | 0.2047 | −0.2176 | 0.9260 | 1791.6 | 0.5045 | −0.3637 | 0.9556 | 2033.8 | 0.1974 | −0.9476 | 0.9618 |

OMC | 148.88 | 0.1953 | −0.2225 | 0.9146 | 1733 | 0.5054 | −0.4431 | 0.9421 | 1960.5 | 0.1999 | −1.0855 | 0.9569 | ||

OMC plus 3% | 152.38 | 0.1083 | −0.1391 | 0.9191 | 1585 | 0.2844 | −0.2882 | 0.9669 | 1697.6 | 0.1116 | −0.6782 | 0.9739 | ||

L6% | OMC minus 3% | 123.97 | 0.2168 | −0.2198 | 0.9375 | 1578.1 | 0.5632 | −0.4147 | 0.9718 | 1816.1 | 0.2214 | −1.0717 | 0.9812 | |

OMC | 149.13 | 0.1901 | −0.2585 | 0.934 | 1513 | 0.5253 | −0.5224 | 0.9782 | 1716.3 | 0.2016 | −1.2226 | 0.9721 | ||

OMC minus 3% | 129.48 | 0.1445 | −0.1667 | 0.9182 | 1439.2 | 0.3776 | −0.3333 | 0.9662 | 1579.2 | 0.1489 | −0.8169 | 0.9788 | ||

C6% | 91 | OMC | 155.61 | 0.1811 | −0.2402 | 0.8621 | 1667.8 | 0.4873 | −0.5172 | 0.9342 | 1874.8 | 0.189 | −1.1987 | 0.9434 |

93 | 148.88 | 0.1953 | −0.2225 | 0.9146 | 1733 | 0.5054 | −0.4431 | 0.9421 | 1960.5 | 0.1999 | −1.0855 | 0.9569 | ||

95 | 165.93 | 0.198 | −0.2418 | 0.8084 | 1851.3 | 0.5399 | −0.5119 | 0.9005 | 2112.9 | 0.2086 | −1.2227 | 0.909 | ||

L6% | 91 | 115.10 | 0.2107 | −0.2136 | 0.9266 | 1446.8 | 0.557 | −0.3953 | 0.962 | 1664 | 0.2123 | −1.0225 | 0.9374 | |

93 | 149.13 | 0.1901 | −0.2585 | 0.934 | 1513 | 0.5253 | −0.5224 | 0.9782 | 1716.3 | 0.2016 | −1.2226 | 0.9721 | ||

95 | 144.19 | 0.2147 | −0.244 | 0.8937 | 1708 | 0.5644 | −0.4923 | 0.945 | 1961.7 | 0.2216 | −1.2071 | 0.96 | ||

C3% | 93 | OMC | 94.43 | 0.2335 | −0.1859 | 0.9596 | 1405 | 0.5954 | −0.2959 | 0.9664 | 1637.3 | 0.236 | −0.8898 | 0.9753 |

C6% | 148.88 | 0.1953 | −0.2225 | 0.9146 | 1733 | 0.5054 | −0.4431 | 0.9421 | 1960.5 | 0.1999 | −1.0855 | 0.9569 | ||

C9% | 107.73 | 0.2577 | −0.2073 | 0.9224 | 1655.5 | 0.6738 | −0.3373 | 0.9656 | 1969.8 | 0.2615 | −1.0027 | 0.9514 | ||

L3% | 86.24 | 0.2671 | −0.2462 | 0.9399 | 1241.3 | 0.6923 | −0.4427 | 0.9681 | 1477.3 | 0.2719 | −1.1974 | 0.974 | ||

L6% | 149.13 | 0.1901 | −0.2585 | 0.934 | 1513 | 0.5253 | −0.5224 | 0.9782 | 1716.3 | 0.2016 | −1.2226 | 0.9721 | ||

L9% | 155.61 | 0.1885 | −0.2449 | 0.9156 | 1658.2 | 0.4997 | −0.5133 | 0.9545 | 1868.9 | 0.1928 | −1.1943 | 0.9543 |

Mass Stress θ/kPa | Octahedral Shear Stress/kPa | Measured Data/Mpa | Resilient Modulus | |||||
---|---|---|---|---|---|---|---|---|

Andrei Model | NI Model | Uzan-Witczak Model | ||||||

Predicted Value/Mpa | Error/% | Predicted Value/Mpa | Error/% | Predicted Value/Mpa | Error/% | |||

75 | 14.14 | 160 | 160.3371 | 0.2102 | 165.572 | 3.3653 | 162.302 | 1.4184 |

100 | 25.93 | 150 | 152.6465 | 1.7338 | 153.1579 | 2.0618 | 150.0214 | 0.0143 |

120 | 35.36 | 144 | 146.379 | 1.6252 | 145.1393 | 0.785 | 145.0932 | 0.7534 |

150 | 49.5 | 132 | 137.405 | 3.9336 | 135.3122 | 2.4479 | 140.6242 | 6.1328 |

120 | 14.14 | 180 | 176.1317 | 2.1962 | 176.1559 | 2.1822 | 177.9052 | 1.1775 |

145 | 25.93 | 167.4 | 164.4161 | 1.8148 | 162.9482 | 2.732 | 161.3126 | 3.7736 |

165 | 35.36 | 158 | 156.0003 | 1.2819 | 154.4171 | 2.3203 | 154.4036 | 2.3292 |

195 | 49.5 | 145.9 | 144.8037 | 0.7571 | 143.9618 | 1.3463 | 148.0175 | 1.4306 |

165 | 14.14 | 187 | 187.7086 | 0.3775 | 186.1511 | 0.456 | 189.3211 | 1.226 |

190 | 25.93 | 180 | 173.5441 | 3.7201 | 172.194 | 4.5333 | 170.0567 | 5.847 |

210 | 35.36 | 165.3 | 163.705 | 0.9743 | 163.1788 | 1.2999 | 161.8498 | 2.1317 |

240 | 49.5 | 153.5 | 150.9406 | 1.6956 | 152.1303 | 0.9003 | 154.1433 | 0.4173 |

210 | 14.14 | 190.5 | 196.9794 | 3.2894 | 195.6467 | 2.6306 | 198.4513 | 4.0067 |

235 | 25.93 | 182.7 | 181.077 | 0.8963 | 180.9776 | 0.9517 | 177.2649 | 3.0661 |

255 | 35.36 | 166 | 170.1836 | 2.4583 | 171.5026 | 3.2084 | 168.1048 | 1.2521 |

285 | 49.5 | 155.9 | 156.2159 | 0.2022 | 159.8905 | 2.4958 | 159.4045 | 2.1985 |

Additive | Comprehensive Prediction Model Regression Equation | R^{2} |
---|---|---|

Cement | ${M}_{R}=0.6896{P}_{a}{\left(\frac{P}{W}\right)}^{0.5197}{\left(\frac{C}{W}+1\right)}^{1.2702}{\left(\frac{\theta}{{P}_{a}}\right)}^{0.2042}{\left(\frac{{\tau}_{oct}}{{P}_{a}}+1\right)}^{-0.9383}$ | 0.84 |

Lime | ${M}_{R}=0.8096{P}_{a}{\left(\frac{P}{W}\right)}^{0.4186}{\left(\frac{C}{W}+1\right)}^{0.9594}{\left(\frac{\theta}{{P}_{a}}\right)}^{0.2191}{\left(\frac{{\tau}_{oct}}{{P}_{a}}+1\right)}^{-1.1321}$ | 0.86 |

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## Share and Cite

**MDPI and ACS Style**

Yuan, H.; Li, W.; Wang, Y.; Lin, H.; Liu, Y.
Resilient Modulus—Physical Parameters Relationship of Improved Red Clay by Dynamic Tri-Axial Test. *Appl. Sci.* **2019**, *9*, 1155.
https://doi.org/10.3390/app9061155

**AMA Style**

Yuan H, Li W, Wang Y, Lin H, Liu Y.
Resilient Modulus—Physical Parameters Relationship of Improved Red Clay by Dynamic Tri-Axial Test. *Applied Sciences*. 2019; 9(6):1155.
https://doi.org/10.3390/app9061155

**Chicago/Turabian Style**

Yuan, Haiping, Weiqiang Li, Yixian Wang, Hang Lin, and Yan Liu.
2019. "Resilient Modulus—Physical Parameters Relationship of Improved Red Clay by Dynamic Tri-Axial Test" *Applied Sciences* 9, no. 6: 1155.
https://doi.org/10.3390/app9061155