# The Creep-Damage Model of Salt Rock Based on Fractional Derivative

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Experiment Results and Discussion

^{−9}/s until 319 days. This stage was considered to be the steady stage of creep. After 319 days, the axial strain rate sharply increased, which means that the creep process entered the accelerated stage (Stage III). The salt rock quickly deformed until failure.

^{4}/h. Meanwhile, the released energy also reached a peak of 3.16 × 10

^{−6}J. The irregular fluctuation of the AE rate in this stage was mainly caused by the inhomogeneous distribution of different components in natural salt rock.

## 4. Creep-Damage Model of Salt Rock Based on a Fractional Derivative

#### 4.1. The Maxwell Model

#### 4.2. The Abel Dashpot: A Fractional Derivative Element

#### 4.3. Maxwell Model Based on a Fractional Derivative

#### 4.4. The Creep-Damage Model Based on a Fractional Derivative

#### 4.5. Parameter Determination by Fitting Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The strain rate and spatial distribution of AE events. The inset images show the spatial distribution characteristics of AE events in the different creep stages.

**Figure 4.**The curves for the AE count rate and released energy: (

**a**) AE count rate vs. strain-time curve; (

**b**) Released energy rate vs. strain-time curve.

**Figure 5.**Creep strain of the Maxwell model based on a fractional derivative given by: σ = 20 MPa and η

^{β}= 8 GPa · h.

Model | E (GPa) | η^{β} (GPa) | β | A | n | Correlation Coefficient |
---|---|---|---|---|---|---|

Creep-damage model | 1.102 | 0.178 | 0.486 | 2.263 × 10^{−13} | 0.998 | 0.989 |

Maxwell model | 4.533 | 13.702 | / | / | / | 0.627 |

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**MDPI and ACS Style**

Zhou, H.; Liu, D.; Lei, G.; Xue, D.; Zhao, Y.
The Creep-Damage Model of Salt Rock Based on Fractional Derivative. *Energies* **2018**, *11*, 2349.
https://doi.org/10.3390/en11092349

**AMA Style**

Zhou H, Liu D, Lei G, Xue D, Zhao Y.
The Creep-Damage Model of Salt Rock Based on Fractional Derivative. *Energies*. 2018; 11(9):2349.
https://doi.org/10.3390/en11092349

**Chicago/Turabian Style**

Zhou, Hongwei, Di Liu, Gang Lei, Dongjie Xue, and Yang Zhao.
2018. "The Creep-Damage Model of Salt Rock Based on Fractional Derivative" *Energies* 11, no. 9: 2349.
https://doi.org/10.3390/en11092349