# A Unified Nonlinear Elastic Model for Rock Material

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## Abstract

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_{ni}and the modulus change rate m. This model covers a variety of existing models, including the simple exponential model, BB model and two-part Hooke’s model, etc. Based on the RMT experimental system, a large number of uniaxial compression tests for dolomite, granite, limestone and sandstone were performed, and their nonlinear deformation stress‒strain curves were obtained and fit with the unified nonlinear elastic model. The results show that the rocks have obvious nonlinear elastic characteristics in their initial compression stage, and that the unified nonlinear elastic model is able to describe these phenomena rather well. In addition, an empirical formula for predicting the uniaxial compressive strength of the rock was constructed, corresponding to the unified nonlinear elastic model.

## 1. Introduction

## 2. Common Nonlinear Elastic Models

#### 2.1. Nonlinear Elastic Models of Joints

_{max}is the maximum allowable closure, σ

_{0}is the initial normal stress, σ

_{n}is the normal stress and d

_{n}is the normal closure.

_{ni}as an important parameter for joints, the classical BB model was established based on the Goodman’s hyperbolic model. The formula of the BB model is as follows:

_{1/2}as a new parameter, which is the stress when the joint closure reached half of the maximum allowable closure, the modified exponential model was proposed. The modified exponential model excludes initial joint stiffness K

_{ni}in its formula, which is an important parameter [10], and its physical meaning is not clear.

#### 2.2. Nonlinear Elastic Models of Joints

_{e}and E

_{t}are the elastic modulus of the hard and soft parts, respectively, and γ

_{e}and γ

_{t}are the proportions of the hard and soft parts of the rock, respectively. This model can be used for composite materials [36], such as material combined with coal and rock [37].

_{t}, the nonlinear elastic deformation of the soft part in the two-part Hooke’s model can be expressed as

## 3. A New Unified Nonlinear Elastic Model

_{0}is the initial length of the material.

_{ni}parameter is introduced, which represents the initial elastic modulus of the material, and we have the following:

_{ni}and m, where E

_{ni}is the initial elastic modulus and m represents its change rate, and both have reasonable physical significance. The unified nonlinear elastic model can include the existing main nonlinear elastic models of the rocks and joints and can also express some new models.

## 4. The Characteristics of the New Model

#### 4.1. Sensitivity Analysis of the Parameters

_{ni}and m. The influence of the E

_{ni}values was studied by fixing the m value to 1.0. The stress‒strain curves of the new model when E

_{ni}is equal to 1 MPa, 5 MPa, 10 MPa, 15 MPa and 20 MPa are listed in Figure 4. All the curves rise linearly when the strain is less than 0.4, and nonlinearity gradually appears when the strain is greater than 0.4. The higher the E

_{ni}values are, the higher the position of the curves, and the distance between adjacent curves is equal when the strain is the same. The slope of the curve for the unified nonlinear elastic model in its initial stage is determined by the E

_{ni}value.

_{ni}value to 10. The stress‒strain curves of the new model when the m value is equal to 0, 0.5, 1.0, 1.5 and 2.0 are listed in Figure 5. All the curves are coincident when the strain is less than 0.4, and differences appear quickly when the strain is greater than 0.4. The curvature of the curves for the unified nonlinear elastic model in its middle–late stages is determined by the m value.

_{ni}value, and its middle–late stages are determined by the m value. The unified nonlinear elastic model uses the above two parameters to jointly adjust and control in order to achieve accurate descriptions of the nonlinear elasticity of rock.

#### 4.2. Approximating to a Single Stress‒Strain Curve

_{ni}and m for the unified nonlinear elastic model are obtained, and the correlation coefficient R

^{2}values were recorded. The horizontal axis in Figure 6 represents the different ranges of the stress‒strain curve, with the E

_{ni}, the m values and the R

^{2}values located on different vertical axes. The red line represents the R

^{2}values, and its fluctuation range is very small. All the R

^{2}values are greater than 0.98, which showed a good relevant fit for the different ranges. The values of E

_{ni}decrease with a larger range, whereas the m value increases with it. These trends become obvious when the strain is greater than 0.7 and very obvious when it is greater than 0.95. There is a negative correlation between the E

_{ni}values and the m values.

_{ni}values and m values, which were both obtained after fitting different curve ranges, are listed in Figure 7. As the fitting curve range increases, the calculated curve gradually approaches the target curve. The red line was calculated using the parameters after fitting the curve from 0 to 1.0, and is the curve of best fit.

## 5. Experimental Validation of the New Model

#### 5.1. Experimental Methods

#### 5.2. Experimental and Fitting Results

_{ni}and m values that were obtained by fitting the curves and the coefficients of correlation R

^{2}values are listed in Table 1. This new model shows a better goodness of fit and a higher accuracy of fit since all the R

^{2}values of the specimens exceed 0.97. The UCS values, the E

_{ni}values and the m values of each rock specimen were different. The E

_{ni}values increased as the UCS value increased for each rock type, whereas the m values changed slightly.

#### 5.3. The m Value Range of the Rock Material

_{ni}values were not fixed. The scatter plots of the coefficients of correlation R

^{2}values and the m values are listed in Figure 11, and the scatter plots of the E

_{ni}and m values are listed in Figure 12. All the curves in Figure 11a–d are parabolic in their shape; that is, the R

^{2}values near the extreme point are high and the R

^{2}values far from the extreme point are low, which conforms to the local optimal feature of the least squares algorithm method. It is clear that the parameters fit to the unified nonlinear elastic model were optimal and unique. The highest points of the different curves in each picture are close, and the two side curves decrease rapidly in Figure 11a–d, indicating that the appropriate m values are in a small range and the R

^{2}values decrease rapidly when they are greater than this range.

^{2}, an R

^{2}greater than 0.95 and an R

^{2}greater than 0.8. The maximum and minimum values of m were used to describe its range in each condition, and the results are listed in Table 2. The ranges of the m values were 0.65~1.05, 0.28~1.26 and 0.00~1.52 in the above three conditions for the R

^{2}values, the differences were small and the upper and lower limits were close. The fitting degree will be poor when the m values are greater than 1.52, and the unified nonlinear elastic model will not be applicable. The nonlinear elastic deformation characteristics of each rock type were different and are easy to ignore since their differences are small. In previous studies, there was no model or method that could accurately describe these behaviors, whereas the unified nonlinear elastic model can distinguish and accurately describe them.

#### 5.4. The Range of E_{ni} Values for the Rock Material

_{ni}and m values are listed in Figure 12. All the curves in Figure 12a–d show a downwards trend, indicating that the E

_{ni}values are negatively correlated with the m values, which follows the same rule of fit for a single curve shown in Figure 6. The E

_{ni}values increase as the serial numbers of the different rocks increases, and the position of the curve is also higher. The linear relationship of all the curves was good when the m values were less than 1.2, and the curves gradually approach each other with decreasing rates when the m values were greater than 1.2, as shown in Figure 12a–d. The scatter points hardly overlap with one another, that is, the E

_{ni}values have good independence and non-repeatability. The E

_{ni}values depend on the basic properties of the rocks, such as the m values, and the basic attributes are natural. It is not feasible to obtain the E

_{ni}values of all rocks by using the same m value, which is why the unified nonlinear elastic model contains both E

_{ni}and m values.

_{ni}values for the different rock types were determined and are listed in Table 3. The range of E

_{ni}values of different rock types is slightly different, and dolomite and granite have larger ranges than limestone and sandstone. In general, the maximum E

_{ni}values for all the rock types were less than 21 MPa.

## 6. Application of the New Model

_{ni}values for the different rocks were counted, and the scatter diagrams are shown in Figure 13a–d. The results of the linear fitting of the data are also shown in these diagrams. The coefficients of correlation R

^{2}values for the four rock types are greater than 0.90, indicating that the UCS values and the E

_{ni}values have a good linear relationship [39]. The slope and intercept of different linear fittings and the R

^{2}values are listed in Table 4. Granite and limestone have the highest R

^{2}values, followed by dolomite and sandstone. The slopes of the linear formula for the different rocks were close, whereas the intercepts were different. When marble was added as another rock type, the scatter plots of the UCS and E

_{ni}values for the five rock types were determined and are listed in Figure 14. The results of the linear fitting of the data are also shown in Figure 14. In order to ascertain whether the experimental values were, in fact, unique to the present model, the 95% prediction band for the present model was determined and is represented by the red band. Almost all the experimental data fall within the 95% prediction band of the present model, which means that the UCS values can be predicted by the E

_{ni}values. The linear fitting formula is listed below:

## 7. Summary and Conclusions

_{ni}and modulus change rate m are introduced in order to establish a new unified nonlinear elastic model for the nonlinear elastic deformation of intact rocks. Based on the RMT experimental system, a large number of uniaxial compression tests for dolomite, granite, limestone and sandstone were performed, and their nonlinear deformation stress‒strain curves were employed to fit the unified nonlinear elastic model. The main outcomes are summarized as follows:

_{ni}representing the initial elastic modulus and the m representing its change rate, were clearly defined. E

_{ni}determines the slope in the initial stage of deformation, and m determines the degree of nonlinearity in the middle and later stages. The unified nonlinear elastic model not only covers the existing nonlinear elastic models, such as the simple exponential model (m = 1) and the BB model (m = 2) of the joint and the two-part Hooke’s model (m = 1) of the rock, but is also able to provide some new models.

_{ni}values have a good linear correlation with the uniaxial compressive strength (UCS) values of the intact rocks. Based on this observation, a new empirical formula for the prediction of the UCS value using the E

_{ni}value was proposed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The stress‒strain curve of a uniaxial compression test adapted from Corkum. Adapted from [21].

**Figure 2.**Linear and nonlinear elastic deformation of rock adapted from Liu. Adapted from [30].

**Figure 11.**The R

^{2}values with different m values for different rock types. (

**a**—Dolomite;

**b**—Granite;

**c**—Limestone;

**d**—Sandstone).

**Figure 12.**The E

_{ni}values with different m values for different rock types. (

**a**—Dolomite;

**b**—Granite;

**c**—Limestone;

**d**—Sandstone).

**Figure 13.**Correlations between UCS and E

_{ni}for different rock types. (

**a**—Dolomite;

**b**—Granite;

**c**—Limestone;

**d**—Sandstone).

Type of Rock | UCS/MPa | E_{ni}/MPa | m | Coefficients of Determination R^{2} |
---|---|---|---|---|

D01 | 62.15 | 5.82 | 0.9117 | 0.9951 |

D02 | 115.06 | 11.01 | 0.8911 | 0.9977 |

D03 | 124.10 | 12.79 | 0.9039 | 0.9792 |

D04 | 150.59 | 13.48 | 0.9626 | 0.9935 |

D05 | 136.74 | 14.38 | 0.8759 | 0.9917 |

D06 | 176.04 | 16.30 | 0.9216 | 0.9907 |

D07 | 168.37 | 17.41 | 0.9389 | 0.9922 |

D08 | 189.53 | 19.17 | 0.8630 | 0.9817 |

D09 | 192.21 | 19.94 | 0.9478 | 0.9976 |

D10 | 185.34 | 20.28 | 0.8880 | 0.9927 |

G01 | 36.24 | 3.14 | 0.6508 | 0.9939 |

G02 | 49.95 | 5.57 | 0.7621 | 0.9840 |

G03 | 69.71 | 6.74 | 0.8872 | 0.9859 |

G04 | 70.76 | 8.28 | 0.7335 | 0.9998 |

G05 | 84.64 | 9.97 | 0.7789 | 0.9934 |

G06 | 108.23 | 11.70 | 0.8899 | 0.9977 |

G07 | 140.88 | 13.81 | 0.9517 | 0.9861 |

G08 | 153.39 | 14.42 | 0.9488 | 0.9758 |

G09 | 171.11 | 17.03 | 0.9455 | 0.9846 |

G10 | 202.02 | 19.76 | 0.9919 | 0.9951 |

L01 | 23.72 | 1.91 | 0.8868 | 0.9907 |

L02 | 48.96 | 4.09 | 0.9927 | 0.9904 |

L03 | 48.28 | 5.12 | 0.9867 | 0.9850 |

L04 | 62.50 | 5.58 | 0.8569 | 0.9850 |

L05 | 75.60 | 6.81 | 1.0282 | 0.9733 |

L06 | 84.66 | 7.58 | 0.8631 | 0.9943 |

L07 | 99.42 | 8.90 | 0.8684 | 0.9897 |

L08 | 107.62 | 10.07 | 0.8406 | 0.9888 |

L09 | 116.59 | 10.23 | 0.8039 | 0.9948 |

L10 | 136.49 | 12.40 | 0.8981 | 0.9870 |

S01 | 32.69 | 2.88 | 1.0102 | 0.9954 |

S02 | 46.07 | 3.93 | 1.0529 | 0.9868 |

S03 | 70.80 | 5.37 | 0.9518 | 0.9947 |

S04 | 44.31 | 5.66 | 0.8245 | 0.9940 |

S05 | 89.10 | 8.69 | 0.9203 | 0.9853 |

S06 | 76.43 | 7.48 | 1.0095 | 0.9762 |

S07 | 104.29 | 9.61 | 1.0199 | 0.9775 |

S08 | 95.49 | 10.69 | 0.9901 | 0.9780 |

S09 | 155.85 | 14.17 | 1.0273 | 0.9916 |

S10 | 132.84 | 14.87 | 1.0529 | 0.9922 |

Type of Rock | Best-Fit Range | Range When R^{2}Values Exceed 0.95 | Range When R^{2}Values Exceed 0.80 |
---|---|---|---|

Dolomite | 0.86~0.96 | 0.58~1.20 | 0.08~1.46 |

Granite | 0.65~0.99 | 0.28~1.23 | 0.00~1.47 |

Limestone | 0.80~1.03 | 0.51~1.22 | 0.04~1.52 |

Sandstone | 0.82~1.05 | 0.49~1.26 | 0.00~1.32 |

Type of Rock | Range of E_{ni}/MPa |
---|---|

Dolomite | 5.82~20.28 |

Granite | 3.14~19.76 |

Limestone | 1.91~12.40 |

Sandstone | 2.88~14.87 |

Type of Rock | Linear Fitting y = ax + b | Coefficients of Determination R ^{2} | |
---|---|---|---|

a | b | ||

Dolomite | 8.92 | 15.74 | 0.950 |

Granite | 10.44 | −6.52 | 0.978 |

Limestone | 10.94 | 0.87 | 0.988 |

Sandstone | 9.29 | 7.38 | 0.915 |

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**MDPI and ACS Style**

Chen, C.; Chen, S.; Zhang, Y.; Lin, H.; Wang, Y. A Unified Nonlinear Elastic Model for Rock Material. *Appl. Sci.* **2022**, *12*, 12725.
https://doi.org/10.3390/app122412725

**AMA Style**

Chen C, Chen S, Zhang Y, Lin H, Wang Y. A Unified Nonlinear Elastic Model for Rock Material. *Applied Sciences*. 2022; 12(24):12725.
https://doi.org/10.3390/app122412725

**Chicago/Turabian Style**

Chen, Chong, Shenghong Chen, Yihu Zhang, Hang Lin, and Yixian Wang. 2022. "A Unified Nonlinear Elastic Model for Rock Material" *Applied Sciences* 12, no. 24: 12725.
https://doi.org/10.3390/app122412725