# A Constitutive Model of Time-Dependent Deformation Behavior for Sandstone

^{1}

^{2}

^{3}

## Abstract

**:**

## 1. Introduction

_{1}–σ

_{3}) over extended periods of time, a phenomenon known as brittle creep (or time-dependent deformation, abbreviated to TDD in the following) [1], especially for sandstone (a rock often seen in the roofs and floors of coal seams) [2,3]. Sandstone taken from deep coal mines and deformed in creep experiments generally exhibits three regimes of strain vs. time: (1) primary creep, or decreasing strain rate stage; (2) secondary creep, or constant strain rate stage; (3) tertiary creep, or increasing strain rate stage ending in failure [4,5]. However, some researchers have divided the creep behavior of rock into three sections: elastic instantaneous deformation, viscoelastic TDD, and viscoplastic TDD [6,7,8,9,10,11]. This categorization—identifying three kinds of deformation—is convenient for building a constitutive model that can be used to reproduce the creep deformation of rock and further understand the failure evolution of rock.

## 2. Numerical Model

#### 2.1. Instantaneous Elastic-Plastic Model

#### 2.1.1. Heterogeneous Damage Model

#### 2.1.2. Instantaneous Elastic-Plastic Model

#### 2.1.3. Model Validation

- (1)
- Analytical solution model

_{h}, and the vertical stress is σ

_{v}. The dashed line A-A′ in Figure 3 extends from the right side of the central circular hole to the right boundary of the stress field. In addition, the tangential (vertical) and radial (horizontal) stress data of the elements on the dashed line are extracted from the numerically calculated result and from the analytical solution for further comparison.

- (2)
- Numerical result validation

#### 2.2. Time-Dependent Viscoelastic Plastic Model

#### 2.2.1. Nishihara Model

#### 2.2.2. Improved Nishihara Model

#### 2.2.3. Analytical Solution Verification

- (1)
- Analytical solution of Nishihara model based on D–P criterion

_{0}is P

_{0}, and the distance from a point in the stress field to the center of the circular hole is r. According to the work of Hou and Niu [53], the plastic stress distribution around the circular hole, as derived using the D–P criterion, is as follows:

_{0}are, respectively, the following:

- (2)
- Verification results

_{0}to the boundary of the borehole is displayed in Figure 10A. Similarly, the stress distribution around the circular hole plotted against the distance to the boundary of the borehole is shown in Figure 10B.

## 3. Comparisons with Laboratory Tests

#### 3.1. Consistent Strain Rate Tests

^{−5}s

^{−1}until macroscopic failure. The stress–strain curves obtained from the numerical simulation and the laboratory tests are compared in Figure 11. From the comparison of the stress–strain curves, it can be found that the numerical damage model can exhibit the full stress–strain process of the sandstone, and can also show compression and dilation behavior of volumetric deformation. This feature is basically consistent with that illustrated by the Vosges sandstone in ref. [13]. Additionally, the compression and dilation behavior of the volumetric strain in the numerical simulation can verify the validity of the non-associated flow rule adopted in the damage model.

#### 3.2. Multi-Step Creep Tests

#### 3.3. Discussion

## 4. Conclusions

- (1)
- The instantaneous elastic-plastic damage model can display the process of volumetric strain for sandstone from compression to dilation. Many small fractures are formed within a very short period of time in the specimen before the main fracture emerges. The damage model captures the phenomenon that small damage of the specimen can lead to the generation of micro-cracks, the accumulation of which causes the specimen to fracture. The final failure mode provided by the numerical simulation is consistent with that extracted from laboratory test results.
- (2)
- The viscoelastic plastic damage model can reproduce the viscoelastic plastic deformation behavior. However, unlike the result of the conventional constant-strain-rate test, cracks of the rock appear through all of the stress loading stages in the multi-step creep tests. The final macroscopic fracture is the result of gradual accumulation of the cracks, a conclusion that is similar to that of the UCS tests.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Heterogeneous model of sandstone in the mesoscale. (

**A**) Optical microscope image at a 100 μm scale [45]. (

**B**) Heterogeneous distribution of strength elements in the model.

**Figure 2.**Elastic-plastic damage constitutive rule for an element under uniaxial compression and tension.

**Figure 4.**Fitted yield and fracture surfaces from experimental data. (

**A**) Axial differential stress vs. axial strain curves under different confining pressures. (

**B**) Effective deviatoric stress vs. mean effective stress for the yield surface and the failure surface.

**Figure 5.**Comparison between numerical simulation results at different degrees of homogeneity and the analytical solutions. (

**A**) Stress curves of radial stress vs. distance to the center of the borehole. (

**B**) Stress curves of circumferential stress vs. distance to the center of the borehole.

**Figure 6.**Cloud map of plastic strain around the borehole calculated from the damage model. (

**A**) Plastic deformation zone around the borehole. (

**B**) Enlarged view.

**Figure 10.**Comparison of displacement and circumferential stress distribution between the analytical solution and the proposed damage model at different homogeneities.

**Figure 11.**Stress–strain curves of Neijiang sandstone obtained from numerical simulation and laboratory tests. Lateral strain, axial strain and volumetric strain are obtained from the numerical simulation.

**Figure 12.**Strength failure snapshots extracted from the simulation result of the damage model. (

**A**) An undamaged model. (

**B**) Some cracks appear in the specimen. (

**C**) Some cracks appear to accumulate and coalesce. (

**D**) Main fracture surface is generated and some new cracks appear around the main fracture.

**Figure 13.**Failure mode of Neijiang sandstone in uniaxial compression strength tests. (

**A**) The shear failure angle is about 25 degrees. There are some small cracks around the main fracture. (

**B**) A main shear fracture on another side opposite to (

**A**).

**Figure 14.**Comparison of axial and transverse strain vs. time curves in numerical simulation and in laboratory tests.

**Figure 16.**Failure mode of Neijiang sandstone in multi-step creep test. (

**A**) The failure mode focuses on the shear failure, and a small crack (highlighted by the yellow circle) occurs near the main fracture similar to that in the failure mode in the UCS tests. (

**B**) A main shear fracture on another side opposite to (

**A**).

**Figure 17.**Relationships between creep strain rate and applied stress in the multi-step creep tests in the laboratory and in numerical simulation.

Mechanical Parameters | Value |
---|---|

Specimen size | 50 mm × 100 mm |

Element size | 0.5 mm |

Uniaxial compressive strength (UCS), σ/MPa | 110 |

Young’s modulus, E/GPa | 10.0 |

Poisson’s ratio, μ | 0.3 |

Homogeneity index, m | 9.0 |

Increment per step, dy/mm | 10^{−5} |

Parameter related to internal friction, A | 1.88 |

Plastic hardening function in the strength criterion, ${\alpha}_{f}$ | 1.37 |

Plastic hardening function in the field criterion, ${\alpha}_{0}$ | 1.0 |

Cohesive coefficient, C_{s} | 0.01 |

Coefficient controlling plastic hardening rate, B | 4 × 10^{−5} |

Physical and Mechanical Parameters | Value |
---|---|

Hole radius R_{0}/m | 0.05 |

In situ stress P_{0}/MPa | 30 |

UCS of surrounding rock σ_{0}/MPa | 20 |

Internal friction angle φ/° | 25 |

Cohesion C/MPa | 6.0 |

Elastic modulus E_{0}/MPa | 6.0 |

Elastic modulus E_{1}/MPa | 11.76 |

Viscosity coefficient η_{0}/GPa∙d | 3.57 |

Viscosity coefficient η_{1}/GPa∙d | 32.0 |

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**MDPI and ACS Style**

Chen, C. A Constitutive Model of Time-Dependent Deformation Behavior for Sandstone. *Materials* **2023**, *16*, 135.
https://doi.org/10.3390/ma16010135

**AMA Style**

Chen C. A Constitutive Model of Time-Dependent Deformation Behavior for Sandstone. *Materials*. 2023; 16(1):135.
https://doi.org/10.3390/ma16010135

**Chicago/Turabian Style**

Chen, Chongfeng. 2023. "A Constitutive Model of Time-Dependent Deformation Behavior for Sandstone" *Materials* 16, no. 1: 135.
https://doi.org/10.3390/ma16010135