# A New Index of Energy Dissipation Considering Time Factor under the Impact Loads

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{td}), is proposed to evaluate the energy dissipation considering the time factor. The relationships between strain rate, energy time density, and specific energy absorption are analyzed. A metric (K

_{u}) is defined to describe the degree of rock fragmentation quantitatively. The correlations of fractal dimension and K

_{u}with different impact pressures are compared. It was concluded that there is a noticeable peak point in the energy time density curve. The energy time density of the stress equilibrium point is three times that of the peak point. The energy time density declines after the peak point, then the energy consumption density tends to be stable. The linear relationship between strain rate and peak point energy time density is stronger. The new index can describe energy dissipation well under dynamic loading. In addition, the experimental results indicate that the degree of crush K

_{u}can describe the degree of crush, and the effect of fractal dimension to quantify the fracture characteristics of the rocks is less good in this test. The crushing degree of rocks increases with the increase of strain rate. Furthermore, the prediction effect of energy time density is better than that of strain rate about K

_{u}.

## 1. Introduction

^{3}through dynamic fracture tests. During the SHPB test, there is a rich variation in the rock under dynamic loading, and the energy dissipation quantified by energy consumption density has attracted widespread attention. The critical parameters for calculating energy consumption density are incident energy, transmitted energy, and reflected energy. Zheng et al. [15] analyzed the relationship between the cyclic load threshold and energy evolution law of damaged sandstone. Zhou et al. [16] found two evident sub-regions for the energy partition by using the modified split Hopkinson pressure bar to discuss the fracturing process of rock under the axial static pre-stresses varying from 0 to 75 MPa. In the research of Li [17] et al., the energy calculation was used to explain the law of energy dissipation density under different strain rates for cemented backfill. Li et al. [18] found a correlation between the various patterns of energy dissipations and the development of microcracks. In the study of Zhang et al. [19], the energy dissipation density as an important parameter of the damage variable can better express the dynamic stress–strain relationship of sandstones under real-time high temperatures. In evaluating energy, energy consumption density as a standard metric has its disadvantage in describing the time effect. Energy time density [20] was used as a new metric to study the energy dissipation in dynamic loading to solve the problem. Liu [21] considered that the energy time density and incident energy show an upward trend. Following a further study by Pan [22], it was demonstrated that energy absorption reaches its highest level when the joint angle is 45°. Li [23] found that the penetration rate of a joint can lead to the increase of energy time density. The fractal dimension is the most common index to evaluate the crushing degree of rocks under dynamic loads. The study of Wang et al. [24] proved that the higher the fractal dimension, the higher the crushing energy density of rocks, which indicates the crushing degree of rock more completely. Zhao et al. [25] developed the continuum statistical constitutive model and described the fragmentation characteristics by fractal theory. Although the fractal dimension is the essential evaluation parameter, the effectiveness of this evaluation is unsatisfactory. This is mainly because the generalization of the regularity of this approach is controversial, and the fractal dimension exhibits an insignificant linear relationship with the staining rate [24,25]. In addition, quadratic functions have been used to fit the curve between strain rate and fractal dimension [26]. In summary, energy consumption density and fractal dimension are the parameters used to evaluate the degree of fragmentation in the study of energy dissipation under dynamic loading. The above approaches have had many achievements, however, gaps still exist. For example, (1) the assessment of energy dissipation is imperfect in terms of time effects and (2) the fractal dimension cannot accurately describe the degree of rock fragmentation.

_{td}) for evaluating the energy dissipation of rock in the stress equilibrium phase under the impact load. The relationship of strain rate, stress difference, energy time density, specific energy absorption, and stress equilibrium factor were analyzed for different impact air pressure. The new metric K

_{u}can be used to quantify the fragmentation of rocks. The results show that the new index can describe energy dissipation more accurately and K

_{u}can clearly define the degree of fragmentation. The article can provide a new analysis method for SHPB tests and provide some research references.

## 2. Materials and Methods

#### 2.1. Split Hopkinson Pressure Bars (SHPB) Test System

#### 2.2. Specimen Preparation

#### 2.3. Preparation of SHPB System before Formal Tests

#### 2.4. Calculation of the Strain and Stress

_{0}and E

_{0}are the cross-sectional areas and elastic modulus of the bar. A

_{s}, C

_{0}and A

_{s}are the cross-sectional area, P-wave velocity, and length of the specimen. ${\epsilon}_{\mathrm{I}}$, ${\epsilon}_{\mathrm{R}}$ and ${\epsilon}_{\mathrm{T}}$ are the incident, reflected, and transmitted strain of the test, respectively. ${l}_{s}$ is the length of the granite specimen.

_{s}is introduced to evaluate the absorption of energy in rock, and the calculation is shown in Equation (4):

_{td}, and the calculation is shown in Equation (5):

## 3. Results and Analysis

_{s}rises significantly, and most of the absorbed energy in the rock is mainly absorbed in this stage.

^{−4}J·cm

^{−3}·μs

^{−1}. A peak point appears in the curve. The energy time is 2.28 × 10

^{−3}J·cm

^{−3}·μs

^{−1}at 185 μs. The energy time density of the stress equilibrium point is three times that of peak point. The assimilation of energy mainly happens in the stress equilibrium stage. As Figure 4 shows, there is a peak point in the curve of energy time density and energy consumption density. The peak points appear at 188 μs and 224 μs, respectively. The curve of energy time density declines after the peak point; the energy consumption density tends to be stable in this stage. Therefore, the energy time density considers the time factor under the dynamic loading and can evaluate the energy absorption in the rock better than the energy consumption density.

#### 3.1. Energy Analysis

_{td}to strain rate. As Figure 6 shows, there is a linear relationship between the equilibrium point energy time density and peak point energy time density with strain rate. The energy time density increases as the strain rate increases. The fitting equation is as follows:

^{2}of 0.97 and 0.76 for the peak point energy time density and the equilibrium point energy time density, respectively, there was general regularity which is not very high for the equilibrium point energy time density. The reason is that stress distribution is uneven, and then the energy absorption is unstable before stress equilibrium. This is similar to the discussion for Figure 4. A more significant linear relationship for the peak point is that the R

^{2}is 0.97. Therefore, the effect by which the peak point energy time density w

_{td}evaluates the energy absorption of rock under dynamic loading is better.

_{td}of different aspect ratios are shown in the following equation:

_{s}of different aspect ratios are shown in the following equation:

^{2}of 0.97, 0.96, 0.98, 0.99 and 0.99 for the energy time density and the R

^{2}of energy consumption density are 0.98, 0.88, 0.99, 0.98 and 0.98. The results show that the fitting effect of energy time density is better, and prove the importance of the time factor in valuing energy dissipation, and so the new index can connect the loaded time and energy dissipation.

#### 3.2. The Degree of Crush K_{u}

_{u}is the index of rock crushing characterized by the fracture condition of the rock under crushing scale d. In the unloaded rock, the function of gradation is ${S}_{0}$, and the ${U}_{\mathrm{e}}={U}_{\mathrm{w}}$ and the K

_{u}is 0 shows that the rock is intact under this condition. There is an ideal situation in which the rock consists of fragments with radius d, and ${U}_{\mathrm{e}}$ is 0, which means that the rock is completely broken. When the polynomial of gradation is less than $1/d$ in Equation (22), the K

_{u}is more than 1, which means the rock is excessively broken.

_{u}-D curve shows in Figure 8, the D tends to increase with the K

_{u}increasing, showing that the degree of crushing K

_{u}can describe the degree of crushing in dynamic loading. The calculation method of fractal dimension is the same as in [24].

_{u}are calculated in Table 3, and Figure 9 shows that fragmentation becomes finer as air pressure increases. Therefore, the fragmentation is finer with the strain rate higher. When the air pressure is 0.12 MPa, the fragment of the granite specimen is generally large. It is finer under the 0.24 MPa air pressure. In Table 3, it is difficult to describe the degree of crushing by the sieve sizes. For example, the proportions of >26.5 mm sieve size are 78.21% and 76.94% in the Jobs “1.2–0.12” and “1.2–0.15”, respectively. The K

_{u}can quantify the degree of crushing. To prove the stability of the K

_{u,}the curves are shown in Figure 10.

_{u}with energy time density and strain rate is more pronounced. In Table 4, the correlation coefficients of strain rate–fractal dimension and energy time density–fractal dimension are 0.31~0.79 and 0.41~0.83. In most cases, the K

_{u}can evaluate the degree of fracture characteristics well, and the energy time density achieves better prediction results than the strain rate.

## 4. Conclusions

_{u}was proposed to quantify the crushing of rocks in the test. This research provides a reference for further study of rock mechanics testing. The main conclusions are as follows:

- (1)
- The changing trend of the new index (energy time density) and energy consumption density are similar; the peak point is more prominent and appears around 180 μs. The new index is more sensitive to energy dissipation.
- (2)
- There are linear trends in the new index and energy consumption density with strain. The correlation coefficients R
^{2}of energy time density and the R^{2}of energy consumption density are 0.96~0.99 and 0.88~0.99, and the correlation between the new index (energy time density) and strain rate is substantial. - (3)
- The fractal dimension is not highly correlated with strain and energy time density in this study. The degree of crush K
_{u}can quantify fracture characteristics of the rock. The degree of crush K_{u}is 0.024 to 0.179 under the dynamic impact tests. The rock crushing degree’s evaluation effect is better than the fractal dimension. - (4)
- The K
_{u}increases as the strain rate and the energy time density increase. In most cases, the energy time density achieved better prediction results than the strain rate. The correlation coefficients R^{2}are 0.31~0.79 and 0.41~0.83, respectively.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**Specimens of granite and experiment system: (

**a**) the granite specimens, (

**b**) the granite between the incident bar and transmitted bar.

**Figure 3.**Trial experiment and dynamic stress equilibrium: (

**a**) typical stress wave pattern of a tested rock specimen; (

**b**) dynamic stress equilibrium.

**Figure 5.**The time-energy time density curves: (

**a**) aspect ratio is 0.6; (

**b**) aspect ratio is 0.8; (

**c**) aspect ratio is 1.0; (

**d**) aspect ratio is 1.2; (

**e**) aspect ratio is 1.4.

**Figure 6.**The curve of the equilibrium point and the peak point energy time density with strain rate.

**Figure 7.**Plots of the relationship between the strain rate and energy absorption: (

**a**) the peak energy time density curve; (

**b**) the energy consumption density.

**Figure 9.**Damage pattern of granite specimen at different air pressures: (

**a**) aspect ratio is 0.6; (

**b**) aspect ratio is 0.8; (

**c**) aspect ratio is 1.0; (

**d**) aspect ratio is 1.2; (

**e**) aspect ratio is 1.4.

**Figure 10.**Plots of the relationship between the strain rate and degree of fragmentation: (

**a**) the peak energy time density—K

_{u}curve; (

**b**) the strain rate—K

_{u}curve; (

**c**) the energy time density—D (

**d**) the strain rate—D.

Density (kg/m^{3}) | P-Wave Velocity (m/s) | Elastic Modulus (GPa) |
---|---|---|

2723 | 4888 | 36.68 |

Specimen ID number | Incident Energy /J | Strain Rate /s ^{−1} | Dynamic Strength /MPa | Energy Time Density in the Peak Point /J·cm ^{−3}·μs^{−l} | Energy Time Density in Equilibrium Point /J·cm ^{−3}·μs^{−l} | Specific Energy Absorption /J·cm ^{−3} |
---|---|---|---|---|---|---|

0.6–0.12 | 186.34 | 33.44 | 85.91 | 0.0026 | 0.0008 | 0.50 |

0.6–0.15 | 236.27 | 59.54 | 109.34 | 0.0050 | 0.0009 | 0.71 |

0.6–0.18 | 282.42 | 84.38 | 103.18 | 0.0079 | 0.0019 | 0.95 |

0.6–0.24 | 332.76 | 132.01 | 122.72 | 0.0125 | 0.0066 | 1.57 |

0.8–0.13 | 242.57 | 56.22 | 106.68 | 0.0043 | 0.0017 | 0.74 |

0.8–0.15 | 235.79 | 51.25 | 85.66 | 0.0056 | 0.0026 | 0.30 |

0.8–0.24 | 281.73 | 104.26 | 121.76 | 0.0088 | 0.0024 | 0.88 |

1.0–0.13 | 227.76 | 43.80 | 113.26 | 0.0024 | 0.0008 | 0.42 |

1.0–0.23 | 317.43 | 89.77 | 139.48 | 0.0056 | 0.0034 | 1.23 |

1.0–0.25 | 343.08 | 101.36 | 131.26 | 0.0067 | 0.0047 | 1.51 |

1.2–0.14 | 229.02 | 30.55 | 114.39 | 0.0018 | 0.0014 | 0.40 |

1.2–0.15 | 229.89 | 49.60 | 87.49 | 0.0034 | 0.0010 | 0.30 |

1.2–0.24 | 341.45 | 86.04 | 149.26 | 0.0047 | 0.0035 | 1.43 |

1.4–0.14 | 221.63 | 38.00 | 102.78 | 0.0023 | 0.0010 | 0.25 |

1.4–0.15 | 213.66 | 35.10 | 97.69 | 0.0021 | 0.0018 | 0.36 |

1.4–0.22 | 325.35 | 78.59 | 143.27 | 0.0038 | 0.0031 | 1.17 |

1.4–0.24 | 328.18 | 93.91 | 127.41 | 0.0046 | 0.0034 | 1.09 |

Specimen Number | >26.5 mm | 26.5 mm | 19 mm | 16 mm | 13.2 mm | 9.5 mm | 4.75 mm | <2.36 mm | D | K_{u} | Energy Time Density |
---|---|---|---|---|---|---|---|---|---|---|---|

0.6–0.12 | 74.37% | 25.63% | - | - | - | - | - | - | - | 0.016 | 0.0006 |

0.6–0.15 | 16.65% | 54.81% | 5.70% | 7.15% | 5.57% | 2.28% | 4.68% | 3.16% | 1.72 | 0.066 | 0.0014 |

0.6–0.18 | - | 42.72% | 33.80% | 12.03% | 3.16% | 2.03% | 4.62% | 1.65% | 1.39 | 0.062 | 0.0028 |

0.6–0.24 | 19.24% | 30.57% | 4.05% | 13.99% | 16.27% | 8.35% | 6.20% | 1.33% | 1.37 | 0.074 | 0.0065 |

0.8–0.12 | 63.86% | 23.27% | 2.41% | 1.80% | 0.57% | 0.24% | 2.51% | 5.35% | 2.13 | 0.065 | 0.0006 |

0.8–0.15 | 20.15% | 46.55% | 14.76% | 2.65% | 3.22% | 3.93% | 1.94% | 6.81% | 1.98 | 0.086 | 0.0011 |

0.8–0.17 | 29.90% | 10.50% | 7.00% | 16.65% | 14.24% | 14.62% | 3.74% | 3.36% | 1.64 | 0.089 | 0.0022 |

0.8–0.24 | - | 14.71% | 12.87% | 13.62% | 14.33% | 20.86% | 12.20% | 11.40% | 2.12 | 0.179 | 0.0034 |

1.0–0.09 | 90.57% | - | 3.69% | 3.61% | 0.74% | 1.12% | 0.27% | - | 0.25 | 0.024 | 0.0004 |

1.0–0.13 | 48.76% | 32.66% | 4.00% | 4.07% | 5.12% | 1.59% | 1.36% | 2.44% | 1.65 | 0.051 | 0.0006 |

1.0–0.18 | - | 24.40% | 28.47% | 11.48% | 7.64% | 8.53% | 9.81% | 9.66% | 2.07 | 0.145 | 0.0012 |

1.0–0.23 | - | 37.63% | 14.66% | 4.58% | 16.18% | 10.90% | 7.21% | 8.84% | 2.02 | 0.135 | 0.0021 |

1.2–0.12 | 78.21% | 12.13% | 3.82% | - | - | - | 3.98% | 1.86% | 1.90 | 0.047 | 0.0001 |

1.2–0.15 | 76.94% | 9.85% | 5.94% | - | - | 0.54% | 1.01% | 5.72% | 2.22 | 0.065 | 0.0007 |

1.2–0.18 | 61.43% | 29.88% | 2.91% | - | - | 2.65% | 0.41% | 2.72% | 1.83 | 0.047 | 0.0016 |

1.2–0.25 | - | 26.91% | 28.49% | 6.89% | 13.30% | 11.56% | 9.95% | 2.91% | 1.64 | 0.107 | 0.0020 |

1.4–0.12 | 89.72% | 4.70% | - | 0.94% | 1.13% | 1.42% | 0.46% | 1.64% | 1.85 | 0.037 | 0.0002 |

1.4–0.14 | 66.45% | 28.10% | - | - | - | 2.98% | 1.34% | 1.13% | 1.60 | 0.040 | 0.0004 |

1.4–0.18 | 49.97% | 43.59% | - | - | - | - | 4.11% | 2.33% | 1.89 | 0.054 | 0.0015 |

1.4–0.24 | 13.50% | 33.41% | 11.06% | 9.98% | 8.21% | 6.66% | 6.82% | 10.36% | 2.13 | 0.136 | 0.0021 |

The R^{2} of Curve | Aspect Ratio | ||||
---|---|---|---|---|---|

0.6 | 0.8 | 1.0 | 1.2 | 1.4 | |

Strain rate—K_{u} | 0.58 | 0.75 | 0.79 | 0.37 | 0.60 |

Energy time density—K_{u} | 0.46 | 0.83 | 0.72 | 0.41 | 0.74 |

Strain rate—D | 0.73 | 0.06 | 0.54 | 0.43 | 0.58 |

Energy time density—D | 0.59 | 0.01 | 0.52 | 0.44 | 0.66 |

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## Share and Cite

**MDPI and ACS Style**

Wang, X.; Guo, L.; Xu, Z.; Wang, J.; Deng, D.; Xu, J.; Hu, Z.
A New Index of Energy Dissipation Considering Time Factor under the Impact Loads. *Materials* **2022**, *15*, 1443.
https://doi.org/10.3390/ma15041443

**AMA Style**

Wang X, Guo L, Xu Z, Wang J, Deng D, Xu J, Hu Z.
A New Index of Energy Dissipation Considering Time Factor under the Impact Loads. *Materials*. 2022; 15(4):1443.
https://doi.org/10.3390/ma15041443

**Chicago/Turabian Style**

Wang, Xuesong, Lianjun Guo, Zhenyang Xu, Junxiang Wang, Ding Deng, Jinglong Xu, and Zhihang Hu.
2022. "A New Index of Energy Dissipation Considering Time Factor under the Impact Loads" *Materials* 15, no. 4: 1443.
https://doi.org/10.3390/ma15041443