# Nonlinear Viscoelastic–Plastic Creep Model Based on Coal Multistage Creep Tests and Experimental Validation

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Multistage Creep Tests

#### 2.1. Specimen Preparation and Experimental Set-Up

- (a)
- The main engine adopts a framework structure with stiffness of more than 5000 kN/mm.
- (b)
- The maximum axial capacity is 500 kN, with measure resolution and precision less than 2.5 N and 1%, respectively. The maximum confining pressure is 50 MPa, with a measure resolution and precision of less than 0.001 MPa and ±1%, respectively.
- (c)
- Axial and radial extensometer deformation measuring ranges are 0–15 mm and 0–7 mm, respectively; and the measuring accuracy is ±0.5%.
- (d)
- Limit control system: When the parameters of axial deformation, radial deformation, or test time reach a preset value, or the specimen split, oil pipe blockage, and oil temperature too high, the device can be automatically stopped for self-protection.

#### 2.2. Experimental Scheme

#### 2.3. Experimental Results

- (1)
- In decelerating creep, creep rate decreases with the increase in axial strain, that is, creep rate $\dot{\epsilon}$ > 0 and creep acceleration $\ddot{\epsilon}$ < 0. The creep curve in this stage presents a convex shape.
- (2)
- In steady state creep, the coal strain tends to be constant along with time, that is, creep rate $\dot{\epsilon}$ > 0 and creep acceleration $\ddot{\epsilon}$ = 0, with a linear change tendency of creep curve.
- (3)
- In accelerating creep, when the coal strain goes beyond the yield point, the axial strain and creep rate increase rapidly, that is, creep rate $\dot{\epsilon}$ > 0 and creep acceleration $\ddot{\epsilon}$ > 0, and the creep curve is presented as a concave shape.

## 3. Analysis of Classical Nishihara Model

_{1}− σ

_{3}) < S

_{h}, then the creep equation of the classical Nishihara model under equal confining pressure conditions is given by the following [24]:

_{1}and σ

_{3}are the first and third principal stresses, respectively; G

_{1}and G

_{2}are the shear modulus of generalized Kelvin and ideal viscoplastic body, respectively; η

_{1}and η

_{2}are the viscosity coefficients; S

_{h}is the deviatoric yield stress of the model; and t is the creep time.

^{2}of the model parameters are above 0.925, which indicates that the classical Nishihara model is well fit to the data of decay creep of coal. The data from Table 2 are substituted into Equation (1) to obtain the creep curves of coal specimens (Figure 8). The fitting curves coincide with the creep test data, indicating that the classical Nishihara model can describe the decelerating and steady state creep behavior satisfactorily. Thus, fitting decay creep characteristic using the classical Nishihara model is appropriate.

## 4. Establishment of Nonlinear Viscoelastic–Plastic Creep Model

#### 4.1. Model Development

#### 4.1.1. Improvement of Non-Newtonian Component

_{0}is the initial strain of the accelerating stage; A is the fitting parameter; and n is the acceleration index, n > 0.

^{2}of the fitting results reaches up to 0.96, demonstrating that the use of the exponential function is a feasible method for describing the accelerating creep stage curve of coal. Hence, this non-Newtonian component with exponential change law can be used instead of the ideal viscous component of the classical Nishihara model, which has a constitutive equation as follows:

_{0}, C = ε

_{0}as follows:

_{0}are the nonlinear and initial viscosity coefficients of non-Newtonian unit, respectively.

#### 4.1.2. Introduction of Negative Elastic Modulus E_{3} (Shear Modulus G_{3})

_{3}(corresponding to the shear modulus G

_{3}under 3D stress condition), can be introduced parallel with the viscoplastic component to construct a new nonlinear viscoelastic–plastic body (Figure 10). This viscoelastic–plastic body characterizes the yield that weakens with the increase in coal strain in the accelerating creep stage. It also indicates the process of releasing elastic potential energy.

_{3}< 0 can the creep rate and creep acceleration be above zero, because the acceleration index n, yield stress σ

_{s}, and nonlinear viscosity coefficient η(t) are the constants that are greater than zero, indicating that the strain of the nonlinear viscoelastic–plastic body will increase over time. Therefore, E

_{3}< 0 not only has mathematical significance, but also has definite physical meaning that describes the process of releasing elastic potential energy in the accelerating creep stage. Therefore, this non-linear viscoelastic–plastic unit can be used to capture the changing regularity of accelerating creep.

#### 4.2. Nonlinear Viscoelastic–Plastic Creep Model

_{s}, then the model is in a viscoplastic state, that is, Part 2, is rigid and only Part 1 deforms in the model. Therefore, the improved model degenerates into a generalized Kelvin body. However, if the constant stress σ is more than σ

_{s}, Part 1 and Part 2 both deform in the model, and the nonlinear viscoelastic–plastic creep model meets the following formulation:

_{ij}can be decomposed as a spherical stress tensor σ

_{m}and deviatoric stress tensor S

_{ij}, which can be represented as follows [26]:

_{ij}is the Kronecker function.

_{m}only changes the volume of an object, and deviatoric stress tensor S

_{ij}only changes the shape. Similarly, strain tensor ε

_{ij}can also be divided into spherical strain tensor ε

_{m}and partial strain tensor e

_{ij}. Thus, ε

_{m}and e

_{ij}are given by the following:

_{ij}. Thus, the 3D creep equation can be described as follows:

_{2}= σ

_{3}. By substituting S

_{ij}= (σ

_{2}− σ

_{3}) into Equation (14), the axial strain in nonlinear viscoelastic–plastic creep model can be deduced as follows:

## 5. Model Validation and Parametric Analysis

#### 5.1. Model Validation

_{s}) in Equation (15). Meanwhile, the fitting results were also compared with the classical Nishihara model. The parameter fitting results of different creep models are listed in Table 4, and the comparison results of the fitting curves and test data are shown in Figure 12.

^{2}of the nonlinear viscoelastic–plastic creep model are above 0.98, which is higher than those of the classical Nishihara model, thereby showing good agreement with the experimental data. Combining the comparison results in Figure 12, the classical Nishihara model can reflect the decelerating creep and steady state creep stages, but the fitting curves still maintain a linear tendency during the accelerating creep, thus deviating from the experimental data and resulting in a large discrepancy. By contrast, the fitting curves obtained by the improved model basically coincide with the experimental data and exhibit an exponential increasing law that can describe the full creep stage of coal, especially the accelerating creep stage, in which the coal strain gradually increased with time. Furthermore, the fitting results demonstrate the high applicability and superiority of the proposed nonlinear viscoelastic–plastic creep model.

#### 5.2. Parametric Sensitivity Analysis

_{3}(corresponding to the shear modulus G

_{3}) and a non-Newtonian component with exponential change laws are introduced in the improved model. The introduction is necessary to describe the non-decay creep characteristics of coal, among which the negative elastic modulus E

_{3}represents the release of elastic potential energy during the accelerating creep stage. In addition, the acceleration creep index n of the non-Newtonian component can embody the accelerated creep rate. To further analyze the sensitive effect of n value on coal creep deformation, the creep acceleration equation can be deduced from Equation (15), as shown in Equation (16). With specimen JC-2 as a sample, the creep curves and creep acceleration curves with different acceleration creep indexes n (n = n

_{1}, n

_{2}, n

_{3}, n

_{4}, n

_{5}) are shown in Figure 13.

## 6. Conclusions

- (1)
- A series of multistage creep tests was performed on coal specimens with different stress levels and found that the concavity–convexity in creep curves will be changed when coal enters accelerating creep stage. That is, a creep inflection point is found on the creep curve. If the occurrence time of the creep inflection point can be accurately predicted, then some effective protective measures can be taken in time to avoid the potential safety hazards caused by coal instability and failure.
- (2)
- The exponential function was used to fit the accelerating creep data of coal specimens and achieved the desired results. Thus, a non-Newtonian component was proposed to replace the ideal Newtonian component. Furthermore, a negative elastic modulus E
_{3}(shear modulus G_{3}) was introduced to establish a nonlinear viscoelastic–plastic body to describe the accelerating creep characteristics of coal. Therefore, a new nonlinear viscoelastic–plastic creep model that consists of a generalized Kelvin and improved viscoelastic–plastic body was proposed, and the creep equation of the improved model under constant confining pressure was also derived. - (3)
- 1 stop analysis software was used to fit the experimental data of the non-decay creep characteristics in the coal multistage creep tests based on the proposed nonlinear viscoelastic–plastic creep model. The results show that the fitting correlation coefficients have R
^{2}values above 0.98, and the fitting curves are highly consistent with the experimental data, which can accurately describe the non-decay creep characteristic of coal and verify the correctness and applicability of the model. - (4)
- The analyses of creep rate and creep acceleration curves using different acceleration creep indexes n show that if n = 0, then the non-Newtonian component will degenerate into the ideal Newtonian body, and only the decelerating and steady-state creeps of coal will occur. If n > 0, then coal specimens will present non-decay creep characteristic. When the acceleration creep index n is large, the level of creep acceleration increases, and the occurrence of coal instability appears earlier.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 9.**Comparison between creep test data and fitting curves of accelerating creep stage of coal specimens.

Parameter | Value | Parameter | Value |
---|---|---|---|

Adsorption constant a | 29.412 m^{3}/t | M_{ad} | 0.44% |

Adsorption constant b | 2.252 MPa^{−1} | A_{d} | 8.12% |

True density | 1.56 g·cm^{−3} | V_{daf} | 10.58% |

Apparent density | 1.49 g·cm^{−3} | FC_{ad} | 80.86% |

Specimen | σ_{1}/MPa | G_{1}/GPa | G_{2}/GPa | η_{1}/(GPa·h) | R_{2} |
---|---|---|---|---|---|

JC-1 | 5 | 15.049 | 39.493 | 5.339 | 0.963 |

15 | 24.120 | 102.695 | 18.413 | 0.969 | |

25 | 23.091 | 97.723 | 16.608 | 0.956 | |

35 | 18.025 | 286.917 | 352.267 | 0.934 | |

JC-2 | 5 | 9.814 | 44.575 | 6.066 | 0.937 |

15 | 17.452 | 128.684 | 9.252 | 0.925 | |

25 | 20.630 | 194.027 | 33.941 | 0.931 | |

35 | 19.599 | 317.160 | 136.092 | 0.953 |

Specimen | ε_{0} | A | n | R^{2} |
---|---|---|---|---|

JC-1 | 0.8306 | 1.658×10^{−3} | 3.1724 | 0.9656 |

JC-2 | 0.7946 | 2.24×10^{−5} | 6.4569 | 0.9765 |

Model | Specimen | G_{1}/GPa | G_{2}/GPa | G_{3}/GPa | η_{1}/(GPa·h) | η_{0}/(GPa·h) | η_{2}/(GPa·h) | n | S_{h}/MPa | R^{2} |
---|---|---|---|---|---|---|---|---|---|---|

Improved model | JC-1 | 20.080 | 86.918 | −164.902 | 37.604 | 451.877 | - | 1.507 | 41.027 | 0.987 |

JC-2 | 21.213 | 144.289 | −11.506 | 34.096 | 8.211 | - | 1.114 | 43.991 | 0.984 | |

Nishihara model | JC-1 | 20.815 | 455.497 | - | 47.673 | - | 0.010 | - | 43.999 | 0.936 |

JC-2 | 21.326 | 305.679 | - | 24.715 | - | 0.361 | - | 43.969 | 0.923 |

_{1}, G

_{2}, and G

_{3}are the shear modulus; η

_{1}and η

_{2}are the viscosity coefficient, respectively; η

_{0}and η(t) are the initial and nonlinear viscosity coefficients, respectively; n is the acceleration index; S

_{h}is the deviatoric yield stress of the model.

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**MDPI and ACS Style**

Zhang, J.; Li, B.; Zhang, C.; Li, P.
Nonlinear Viscoelastic–Plastic Creep Model Based on Coal Multistage Creep Tests and Experimental Validation. *Energies* **2019**, *12*, 3468.
https://doi.org/10.3390/en12183468

**AMA Style**

Zhang J, Li B, Zhang C, Li P.
Nonlinear Viscoelastic–Plastic Creep Model Based on Coal Multistage Creep Tests and Experimental Validation. *Energies*. 2019; 12(18):3468.
https://doi.org/10.3390/en12183468

**Chicago/Turabian Style**

Zhang, Junxiang, Bo Li, Conghui Zhang, and Peng Li.
2019. "Nonlinear Viscoelastic–Plastic Creep Model Based on Coal Multistage Creep Tests and Experimental Validation" *Energies* 12, no. 18: 3468.
https://doi.org/10.3390/en12183468