# Experimental Investigation on Crack Behavior and Stress Thresholds of Sandstone Containing a Square Inclusion under Uniaxial Compression

^{*}

## Abstract

**:**

## Featured Application

**The present work is conducive to understand the effect of filling on the deformation and failure of rock and to accurately determine the crack stress thresholds of rock with holes or inclusions.**

## Abstract

## 1. Introduction

## 2. Test Preparation and Procedures

#### 2.1. Specimen Preparation

#### 2.2. Experimental Setup and Testing Method

## 3. Test Results and Discussion

#### 3.1. Influence of Heterogeneity

#### 3.2. Axial Stress–Strain Response

#### 3.3. Strength and Deformation Properties

_{x}and ΔD

_{y}) before the peak stress, as shown in Figure 7. It can be seen that before about 5 s, the deformation of the hole in the horizontal and vertical directions is basically 0. After that, the value of ΔD

_{x}is always positive and increases with time, which indicates that the hole expands horizontally under the external force, resulting in interface debonding or slipping between the rock matrix and inclusion [26]. On the contrary, the value of ΔD

_{y}is always negative and decreases with time, indicating the hole shrinks in the vertical direction under the external force. Obviously, the deformation of the holes under the three filling modes is different. Generally, the deformation of the unfilled hole is the largest, followed by the hole filled with Type I inclusion, and finally with the Type II inclusion. On the other hand, by comparing Figure 7a,b, it can be found that the inclination angle also has a significant effect on the deformation of holes or inclusions. The deformation of the hole with α = 0° is more sensitive to the change of the filling mode, and the resistance effect of filling on hole deformation improves with the time. This indicates that as the load increases, the inclusion will gradually produce a passive supporting effect on the surrounding rock matrix, especially the inclusion with better strength and stiffness, which can effectively improve the stress state of the surrounding rock, thereby improving the strength and deformation behavior of the specimens with inclusions [10,32].

#### 3.4. AE Behavior and Fracture Process

_{cc}, σ

_{ci}, σ

_{cd}and σ

_{c}) [35,36]. Where σ

_{cc}and σ

_{ci}represent the stresses at the closure of initial defects and the initiation of new cracks, respectively. The crack damage stress σ

_{cd}refers to the stress that unstable cracks begin to appear, and the peak stress σ

_{c}is equal to the UCS. In regard to the values of σ

_{ci}and σ

_{cd}for sandstone specimens in this test, they can be qualitatively and quantitatively determined by combining DIC and AE results [37,38]. Firstly, find out the time when AE counts increase significantly or CAEC deviate from the linear segment; then, observe the feature of major principal strain fields at these moments. If the characteristic AE behavior corresponds to crack initiation, propagation or coalescence in the major principal strain fields, the corresponding stress is the crack initiation or damage stress. However, the crack closure stress σ

_{cc}is difficult to be directly identified by this method. It is generally considered that the axial stress at the critical point of the stress–strain curve from nonlinear to linear is the crack closure stress. Based on this, an objective approximate estimation method of crack closure stress was proposed, which is helpful to reduce human subjective error. Taking the specimen Sq-A-0-1 as an example, the calculation steps are as follows: first, select two points on the stress–strain curve where the stresses are 0.2 and 0.3 times of the peak stress (i.e., 10.33 MPa and 16.05 MPa), and mark them as P1 and P2 respectively; second, the stress–strain curve is linearly fitted between the two points (P1 and P2) to obtain the slope of 10.97 GPa, and then a line (σ = 10.97ε) is drawn from the origin Point 0 (see Figure 11a); third, calculate the axial strain difference (Δε = ε

_{x2}− ε

_{x1}) between the straight line and the stress–strain curve for the same axial stress σ

_{x}; finally, plot the curve of axial strain difference versus axial stress, and find out the Point A with the largest axial strain difference. The stress at Point A is the crack closure stress σ

_{cc}, as shown in Figure 11b.

_{cc}), crack initiation stress (σ

_{ci}), crack damage stress (σ

_{cd}) and peak stress (σ

_{c}), respectively. The stress Point E is at the post-peak. It should be noted that the crack initiation stress σ

_{ci}mentioned in this paper is the stress corresponding to the cracking of rock matrix. However, for the filled specimens, since the interface between the filling material and the hole is the weakest plane, it can be clearly seen from Figure 9b,c and Figure 10b,c that, comparing the principal strain fields at Points A and B′, strain localizations first concentrate at the boundary or inside of the filled holes and initiate from it, which indicates that the corresponding stress at Point B′ can be regarded as the crack initiation stress σ′

_{ci}of the inclusions in the filled specimens.

#### 3.5. Crack Behavior and Failure Pattern

#### 3.6. Crack Stress Thresholds

_{cc}, initiation stress σ

_{ci}and damage stress σ

_{cd}of each group of specimens and their corresponding stress levels are shown in Figure 14. It can be seen from Figure 14a that the σ

_{cc}of the pre-holed specimens with α = 0° and 45° are 12.14 MPa and 12.15 MPa, which are 39.30% and 39.25% lower than that of intact specimens, respectively. After filling, with the increase of the mechanical properties of the filling material, the σ

_{cc}of the pre-holed specimens increases, but it is still smaller than that of the intact specimen. Among them, the increase rates of σ

_{cc}of specimens with α = 0° and 45° are 13.10% and 6.01% in filling mode B, and 24.88% and 22.88% in Filling Mode C, respectively. The above results show that the crack closure stress of rock can be significantly reduced by the existence of holes and improved by the filling treatment, but it seems to be less affected by the hole angle. In addition, the crack closure stress levels σ

_{cc}/σ

_{c}of the pre-holed specimens also increase with the change of Filling Mode from A to C, but the increase rate is very small. The stress levels of specimens with α = 0° are 0.23–0.25, and that of specimens with α = 45° are 0.26–0.28. However, the stress levels of specimens with α = 45° are generally higher than that of specimens with α = 0°, especially in Filling Mode C, the stress level of the specimen with α = 45° is even greater than that of intact specimen. This indicates that the effect of hole angle on the crack closure stress levels of the pre-holed specimens is more significant than that of the filling effect.

_{ci}and σ

_{ci}, which are the crack initiation stress of inclusions and rock matrix, respectively, as mentioned in Section 3.4. Due to the weak mechanical properties of Type I inclusion and strong mechanical properties of Type II inclusion, the σ′

_{ci}of the pre-holed specimens first decreases and then increases with the change of filling mode from A to C, but the difference of σ′

_{ci}between 0° and 45° specimens is very small. However, the crack initiation stress levels σ′

_{ci}/σ

_{c}of 0° specimens in Filling Modes B and C are 0.29 and 0.39, while those of 45° specimens are 0.36 and 0.43, respectively. The above results show that the hole angle has little effect on the σ′

_{ci}of inclusions, but has a significant enhancing effect on the corresponding stress levels, and the effect of 45° hole is greater than the 0° hole. On the other hand, the σ

_{ci}and σ

_{ci}/σ

_{c}of rock matrix in the pre-holed specimens increase with the increase of the mechanical properties of the filling material, and the filled specimens with α = 0° is much larger than those with α = 45°. Among them, the σ

_{ci}/σ

_{c}of 0° specimens under Filling Modes A, B and C are 0.37, 0.62 and 0.69, while that of 45° specimens are 0.43, 0.54 and 0.58, respectively. All in all, the hole and inclusion in specimens with α = 0° are easier to crack than that with α = 45°, but the rock matrix in filled specimens with α = 0° is more difficult to crack than that with α = 45°. This indicates that the filling effect can significantly change the stress distribution around the hole and produce similar effect of crack arrest, so as to significantly increase the crack initiation stress of rock matrix, and the degree of increase is related to the hole angle.

_{cd}and corresponding stress levels σ

_{cd}/σ

_{c}of each group of specimens. Among them, the σ

_{cd}of pre-holed specimens with α = 0° and 45° are 40.45 MPa and 36.11 MPa, which are 29.75% and 37.29% lower than that of intact specimens, respectively. After filling, the change of σ

_{cd}with the filling mode is similar to that of UCS, which is basically linearly increasing with the change of Filling Mode from A to C, and the σ

_{cd}of specimens with α = 0° is generally greater than that with α = 45°. However, the σ

_{cd}/σ

_{c}of specimens with α = 45° is slightly higher than that with α = 0°, but there is no significant difference among each group. The damage stress levels of specimens with α = 0° is 0.76 under different filling modes, while that of specimens with α = 45° is 0.76–0.80. The above results show that both the hole angle and the filling mode have an effect on the crack damage stress of the pre-holed specimen, but have little effect on the corresponding stress level.

## 4. Conclusions

- (1)
- Compared with the intact sandstone, the strength and deformation properties of specimens with a square hole are decreased, and the decrease rate is closely related to hole angle. After filling, the mechanical parameters of the pre-holed specimens are strengthened, and the strengthening effect of relatively rigid inclusions is much stronger than that of weak inclusions. Moreover, virtual extensometers were arranged on hole boundaries by DIC to measure hole deformation. It is found that the inclusions can restrain the hole deformation, especially the rigid inclusion can gradually passively support the surrounding rock matrix with the increase of load, thereby effectively improving the mechanical behavior of the pre-holed specimen.
- (2)
- The crack behavior and final failure pattern of the specimens with a square hole, especially the crack initiation position, are affected by the filling mode and the hole angle. The crack pattern around the square hole can be categorized into four types, while the number of cracks around the square inclusion increases and the crack pattern is more complex, which can be mainly divided into five types. Furthermore, the interface debonding tends to occur at the inclusion interface parallel to major principal stress direction, and the interface slipping tends to propagate along the inclined interface. The fracture evolution of the specimen during uniaxial compression proceeds from primary tensile cracks through secondary cracks to shear cracks, and the rock failure is mainly caused by secondary crack propagation and shear crack coalescence. The failure mode can be generally considered as mixed tensile–shear failure, and the difference is dominated by tension or shear. Notably, the inclusion not only has a certain crack arrest effect on rock matrix, but also can effectively inhibit sidewall spalling.
- (3)
- The damage and fracture evolution during uniaxial compression tests on sandstone specimens can be qualitatively and quantitatively characterized by DIC and AE techniques, and their results are in good agreement with the rock mechanical behavior. The AE counts and cumulative AE counts can be roughly divided into five periods or intervals, which correspond to five stages in the stress–strain curve of the specimen, i.e., pore compaction stage, elastic deformation stage, stable crack growth stage, unstable crack growth stage and rock failure stage. By verifying the major principal strain fields obtained by DIC, the critical points between these characteristic periods or intervals can be identified as the crack stress thresholds.
- (4)
- Since the crack closure stress is hard to be directly determined by the strain field, an objective approximate estimation method based on stress–strain curves was proposed. In addition, based on the observation of strain localizations on specimen with inclusions, the crack initiation stresses of inclusion and rock matrix were defined. The crack stress thresholds of sandstone can be significantly reduced by the hole and improved by the inclusion. The improvement effect of inclusion increases with the increase of mechanical properties of filling materials and is affected by the hole angle. The hole angle has little effect on crack closure stress, damage stress levels and initiation stress of inclusion, but has obvious effect on crack closure stress levels, damage stress, initiation stress of rock matrix and corresponding stress levels.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Effect of heterogeneity on axial stress–strain curves of sandstone specimens under uniaxial compression.

**Figure 5.**Typical stress–strain curves of specimens containing a square hole under different filling modes: (

**a**) α = 0°; (

**b**) α = 45°.

**Figure 6.**Effect of filling mode and hole angle on the mechanical parameters of specimens: (

**a**) peak strength σ

_{c}; (

**b**) peak strain ε

_{c}; and (

**c**) elastic modulus E

_{t}.

**Figure 7.**Layout of virtual extensometers around 0° (

**a**) and 45° (

**b**) square hole and the variation in hole apertures with filling mode.

**Figure 9.**AE behavior and strain fields of pre-holed specimens with α = 0° under different filling modes: (

**a**) Sq-0-A-1; (

**b**) Sq-0-B-2; and (

**c**) Sq-0-C-1.

**Figure 10.**AE behavior and strain fields of pre-holed specimens with α = 45° under different filling modes: (

**a**) Sq-45-A-1; (

**b**) Sq-45-B-2; and (

**c**) Sq-45-C-1.

**Figure 11.**Determination of crack closure stress of specimen Sq-0-A-1: (

**a**) schematic diagram of axial strain difference and (

**b**) location of Point A with the maximum axial strain difference.

**Figure 13.**Crack patterns around a square hole under different filling modes and hole inclination angles: (

**a**) Sq-0-A; (

**b**) Sq-0-B; (

**c**) Sq-0-C; (

**d**) Sq-45-A; (

**e**) Sq-45-B; and (

**f**) Sq-45-C.

**Figure 14.**Effect of filling mode and hole angle on typical stress thresholds of sandstone specimens.

**Table 1.**Some essential physical and mechanical properties of the tested sandstone and inclusions [26].

Properties | Sandstone | Type I | Type II |
---|---|---|---|

Density ρ (kg/m^{3}) | 2377.71 | 1875.53 | 2267.29 |

Longitudinal wave velocity V_{p} (m/s) | 2973.11 | 3193.40 | 4393.91 |

Poisson ratio v | 0.14 | 0.21 | 0.25 |

Young modulus E_{t} (GPa) | 16.68 | 6.24 | 10.79 |

Uniaxial compressive strength σ_{c} (MPa) | 69.17 | 8.68 | 35.41 |

Brazilian tensile strength σ_{t} (MPa) | 5.29 | 0.86 | 2.03 |

Fracture toughness K_{IC} (MPa·m^{1/2}) | 0.93 | -- | -- |

Specimen No. | W/mm | H/mm | T/mm | α/° | σ_{c}/MPa | ε_{c}/10^{−3} | E_{t}/GPa |
---|---|---|---|---|---|---|---|

Intact-1 | 60.54 | 120.77 | 30.75 | -- | 76.64 | 6.64 | 12.86 |

Intact-2 | 60.56 | 120.65 | 30.80 | -- | 76.94 | 6.18 | 14.00 |

Intact-3 | 60.07 | 120.71 | 30.12 | -- | 71.50 | 6.46 | 12.37 |

Sq-0-A-1 | 60.72 | 120.45 | 30.46 | 0 | 53.90 | 5.54 | 10.97 |

Sq-0-A-2 | 60.27 | 120.22 | 30.02 | 0 | 53.29 | 5.51 | 10.89 |

Sq-0-B-1 | 60.41 | 120.35 | 30.18 | 0 | 54.55 | 5.75 | 11.45 |

Sq-0-B-2 | 60.70 | 120.17 | 30.43 | 0 | 57.94 | 5.69 | 11.19 |

Sq-0-C-1 | 60.66 | 120.38 | 30.12 | 0 | 59.58 | 6.31 | 11.83 |

Sq-0-C-2 | 60.58 | 120.14 | 30.11 | 0 | 61.16 | 5.83 | 12.09 |

Sq-45-A-1 | 60.54 | 119.60 | 29.96 | 45 | 46.49 | 5.11 | 9.94 |

Sq-45-A-2 | 60.53 | 120.13 | 30.40 | 45 | 45.95 | 4.94 | 9.86 |

Sq-45-B-1 | 60.80 | 120.26 | 29.86 | 45 | 47.71 | 4.90 | 10.33 |

Sq-45-B-2 | 60.83 | 120.29 | 30.54 | 45 | 48.41 | 4.92 | 10.36 |

Sq-45-C-1 | 60.64 | 119.56 | 30.19 | 45 | 54.52 | 5.45 | 11.38 |

Sq-45-C-2 | 60.45 | 120.16 | 29.74 | 45 | 51.40 | 5.26 | 10.90 |

Specimen No. | 10 MPa | 20 MPa | 30 MPa | 40 MPa | ||||
---|---|---|---|---|---|---|---|---|

ΔD_{x}/mm | ΔD_{y}/mm | ΔD_{x}/mm | ΔD_{y}/mm | ΔD_{x}/mm | ΔD_{y}/mm | ΔD_{x}/mm | ΔD_{y}/mm | |

Sq-0-A | 0.0082 | −0.0397 | 0.0262 | −0.0775 | 0.0468 | −0.1194 | 0.0762 | −0.1600 |

Sq-0-B | 0.0054 | −0.0204 | 0.0159 | −0.0412 | 0.0360 | −0.0687 | 0.0635 | −0.1051 |

Sq-0-C | 0.0003 | −0.0139 | 0.0117 | −0.0341 | 0.0293 | −0.0539 | 0.0508 | −0.0856 |

Sq-45-A | 0.0248 | −0.0482 | 0.0556 | −0.0926 | 0.0944 | −0.1392 | 0.1533 | −0.1951 |

Sq-45-B | 0.0219 | −0.0350 | 0.0503 | −0.0756 | 0.0923 | −0.1213 | 0.1508 | −0.1723 |

Sq-45-C | 0.0017 | −0.0124 | 0.0150 | −0.0397 | 0.0424 | −0.0760 | 0.0882 | −0.1228 |

**Table 4.**Crack stress thresholds and corresponding stress levels of the specimens under uniaxial compression.

Specimen No. | σ_{c}/MPa | Crack Closure | Crack Initiation | Crack Damage | |||||
---|---|---|---|---|---|---|---|---|---|

σ_{cc}/MPa | σ_{cc/}σ_{c} | σ′_{ci}/MPa | σ′_{ci/}σ_{c} | σ_{ci}/MPa | σ_{ci/}σ_{c} | σ_{cd}/MPa | σ_{cd/}σ_{c} | ||

Int-1 | 76.48 | 19.41 | 0.25 | -- | -- | 38.14 | 0.50 | 58.89 | 0.77 |

Int-2 | 76.96 | 20.04 | 0.26 | -- | -- | 37.01 | 0.48 | 59.29 | 0.77 |

Int-3 | 71.50 | 20.55 | 0.29 | -- | -- | 34.23 | 0.48 | 54.57 | 0.76 |

Average | 75.03 | 20.00 | 0.27 | -- | -- | 36.46 | 0.49 | 57.58 | 0.77 |

Sq-0-A-1 | 53.73 | 11.94 | 0.22 | -- | -- | 19.16 | 0.36 | 40.71 | 0.76 |

Sq-0-A-2 | 53.64 | 12.33 | 0.23 | -- | -- | 20.08 | 0.38 | 40.19 | 0.75 |

Average | 53.60 | 12.14 | 0.23 | -- | -- | 19.62 | 0.37 | 40.45 | 0.76 |

Sq-0-B-1 | 54.44 | 13.42 | 0.25 | 15.75 | 0.29 | 33.32 | 0.61 | 41.42 | 0.76 |

Sq-0-B-2 | 57.71 | 14.03 | 0.24 | 16.32 | 0.28 | 36.68 | 0.63 | 43.48 | 0.75 |

Average | 56.25 | 13.73 | 0.25 | 16.04 | 0.29 | 35.00 | 0.62 | 42.45 | 0.76 |

Sq-0-C-1 | 59.52 | 14.51 | 0.24 | 22.75 | 0.38 | 41.00 | 0.69 | 45.36 | 0.76 |

Sq-0-C-2 | 61.05 | 15.8 | 0.26 | 23.52 | 0.39 | 41.36 | 0.68 | 46.29 | 0.76 |

Average | 60.37 | 15.16 | 0.25 | 23.14 | 0.39 | 41.18 | 0.69 | 45.83 | 0.76 |

Sq-45-A-1 | 46.49 | 12.15 | 0.26 | -- | -- | 20.15 | 0.43 | 35.24 | 0.76 |

Sq-45-A-2 | 45.95 | 12.14 | 0.26 | -- | -- | 19.76 | 0.43 | 36.97 | 0.81 |

Average | 46.22 | 12.15 | 0.26 | -- | -- | 19.96 | 0.43 | 36.11 | 0.79 |

Sq-45-B-1 | 47.71 | 13.56 | 0.28 | 16.06 | 0.34 | 24.84 | 0.52 | 37.52 | 0.79 |

Sq-45-B-2 | 48.41 | 12.19 | 0.25 | 17.77 | 0.37 | 26.94 | 0.56 | 38.94 | 0.80 |

Average | 48.06 | 12.88 | 0.27 | 16.92 | 0.36 | 25.89 | 0.54 | 38.23 | 0.80 |

Sq-45-C-1 | 54.52 | 14.81 | 0.27 | 23.83 | 0.44 | 31.53 | 0.58 | 42.14 | 0.77 |

Sq-45-C-2 | 51.40 | 15.05 | 0.29 | 21.10 | 0.41 | 29.25 | 0.57 | 38.66 | 0.75 |

Average | 52.96 | 14.93 | 0.28 | 22.47 | 0.43 | 30.39 | 0.58 | 40.40 | 0.76 |

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## Share and Cite

**MDPI and ACS Style**

Zhu, Q.; Li, D.
Experimental Investigation on Crack Behavior and Stress Thresholds of Sandstone Containing a Square Inclusion under Uniaxial Compression. *Appl. Sci.* **2020**, *10*, 7621.
https://doi.org/10.3390/app10217621

**AMA Style**

Zhu Q, Li D.
Experimental Investigation on Crack Behavior and Stress Thresholds of Sandstone Containing a Square Inclusion under Uniaxial Compression. *Applied Sciences*. 2020; 10(21):7621.
https://doi.org/10.3390/app10217621

**Chicago/Turabian Style**

Zhu, Quanqi, and Diyuan Li.
2020. "Experimental Investigation on Crack Behavior and Stress Thresholds of Sandstone Containing a Square Inclusion under Uniaxial Compression" *Applied Sciences* 10, no. 21: 7621.
https://doi.org/10.3390/app10217621