# Propagation Speed of Dynamic Mode-I Cracks in Self-Compacting Steel Fiber-Reinforced Concrete

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Program

#### 2.1. Material Characterization

#### 2.2. Three-Point Bending Tests

#### 2.2.1. Measuring the Load and Strain Rate upon Crack Initiation

#### 2.2.2. Quantification of the Energy Absorption Capacity

#### 2.2.3. Measuring the Crack Propagation Velocity Using the Strain Gage Readings

#### 2.2.4. Crack Velocity Measurement Using DIC

## 3. Results and Discussion

#### 3.1. Load and Strain Rates upon Crack Initiation

#### 3.2. The Failure Patterns, Measured Toughness Indices and Residual Strength Factors

#### 3.3. The Measured Crack Propagation Velocity Using the Strain Gages

#### 3.4. Measuring the Crack Propagation Velocity Using DIC

#### 3.5. Numerical Simulations Based on the Cohesive Theories of Fracture

#### 3.6. Comparison of the Obtained Crack Velocities

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) A beam prepared for testing using the servo-hydraulic testing machine; and (

**b**) experimental set-up using the drop-weight impact device, the strain gages were connected to the individual channels of the data acquisition system.

**Figure 3.**Determining the strain rate upon crack initiation using (

**a**) the strain history data recorded in SG5 (shown for PA2 loaded at 22 $\mathsf{\mu}$m/s); and (

**b**) a generic cohesive law.

**Figure 4.**Measuring the toughness indices using the load-deflection curve (shown for PA3 loaded at 2.2 $\mathsf{\mu}$/s and PA19 impacted at 2.66 m/s).

**Figure 5.**Typical strain-history curves of a PA specimen recorded in SG1–SG4 (loading rate: 2.66 m/s).

**Figure 6.**The experimental setup with the high-speed camera (

**left**), and the speckle pattern obtained from black/white paint using air spray (

**right**).

**Figure 7.**Typical load versus-strain curves recorded in SG1 for beams made from concrete PA (

**left**), PB (

**central**) and PC (

**right**) loaded at 22 $\mathsf{\mu}$m/s (the 1st row), 22 mm/s (the 2nd row), 1.77 m/s (the 3rd row) and 2.66 m/s (the 4th row), respectively.

**Figure 8.**Peak load and initial load (with respective standard deviations) versus concrete type for the loading rate of (

**a**) 2.2 $\mathsf{\mu}$m/s; (

**b**) 22 mm/s; (

**c**) 1.77 m/s and (

**d**) 2.66 m/s, respectively; (

**e**) the ratio between both loads; and (

**f**) the first-crack strength, ${f}_{R}$.

**Figure 9.**The strain rates ${\dot{\epsilon}}_{i}$ and ${\dot{\epsilon}}_{ic}$ upon crack initiation at the notch tip.

**Figure 10.**Typical failure modes of the PA, PB and PC beams loaded under four different loading rates.

**Figure 11.**The toughness indices, ${I}_{5}$, ${I}_{10}$, ${I}_{20}$, and the residual strength factors, ${R}_{5,10}$ and ${R}_{20,10}$.

**Figure 12.**Typical strain histories for concrete PA (at four loading rates) and PC (at the lowest and highest loading rates only).

**Figure 13.**Typical crack velocities marked (in m/s) along the main crack path for concrete PA, PB and PC loaded at four loading rates: 2.2 $\mathsf{\mu}$m/s, 22 m/s, 1.77 m/s and 2.66 m/s, respectively.

**Figure 14.**The crack propagation captured using a high-speed camera and processed using a DIC software as strain contours for the specimens PA15, PB15 and PC15 impacted at 2.66 m/s.

**Figure 15.**(

**a**) Distribution of mesh in the specimen and ligament nodes; (

**b**) evolution of the numerical and experimental reaction forces contrasted with the impact forces at loading rate of 1.77 m/s (PB12).

**Figure 16.**Comparison of the obtained crack velocities using strain gages (SG), digital image Correlation (DIC) and numerical simulations (NUM) for PA, PB and PC beams impacted at 2.66 m/s.

**Table 1.**Properties of concretes at the age of seven months (values in parentheses are the standard deviations), the longitudinal wave velocity, ${C}_{L}$, and the estimated terminal crack velocity, ${v}_{ct}$.

${\mathit{f}}_{\mathit{c}}$ (MPa) | ${\mathit{f}}_{\mathit{R}}$ (MPa) | E (GPa) | $\mathit{\nu}$ - | $\mathit{\rho}$ (kg/m${}^{3}$) | ${\mathit{C}}_{\mathit{L}}$ (m/s) | ${\mathit{v}}_{\mathit{ct}}$ (m/s) | |
---|---|---|---|---|---|---|---|

PA | 112 (1) | 8.1 (0.2) | 46.4 (3.0) | 0.18 (0.01) | 2362 (3) | 4432 | 1684 |

PB | 112 (6) | 8.5 (0.5) | 45.2 (2.0) | 0.17 (0.01) | 2376 (32) | 4362 | 1658 |

PC | 114 (3) | 11.4 (0.8) | 45.9 (3.0) | 0.17 (0.01) | 2408 (11) | 4366 | 1659 |

**Table 2.**Three-point bending tests loading rate with tested beam ID, the corresponding fiber content. The beams with glued strain gages are designed to measure ${P}_{max}$, ${P}_{ini}$, ${f}_{R}$, ${\dot{\epsilon}}_{i},{\dot{\epsilon}}_{ic}$, ${I}_{5},\phantom{\rule{3.33333pt}{0ex}}{I}_{10},\phantom{\rule{3.33333pt}{0ex}}{I}_{20},\phantom{\rule{3.33333pt}{0ex}}{R}_{5,10},\phantom{\rule{3.33333pt}{0ex}}{R}_{10,20}$ and ${V}_{SG}$, whereas the three beams PA15, PB15 and PC15 were assigned to DIC analysis to measure the crack speed.

Loading Rate (mm/s) | Beam ID | Fiber Content | Gages or DIC |
---|---|---|---|

0.0022 | PA2, PA3, PA4 | 0.51% | |

22 | PA7, PA8, PA9 | 0.51% | Strain gages |

1770 | PA12, PA13, PA14 | 0.51% | |

2660 | PA17, PA18, PA19 | 0.51% | |

2660 | PA15 | 0.51% | DIC |

0.0022 | PB2, PB3, PB4 | 0.77% | |

22 | PB7, PB8, PB9 | 0.77% | Strain gages |

1770 | PB12, PB13, PB14 | 0.77% | |

2660 | PB17, PB18, PB19 | 0.77% | |

2660 | PB15 | 0.77% | DIC |

0.0022 | PC2, PC3, PC4 | 1.23% | |

22 | PC7, PC8, PC9 | 1.23% | Strain gages |

1770 | PC12, PC13, PC14 | 1.23% | |

2660 | PC17, PC18, PC19 | 1.23% | |

2660 | PC15 | 1.23% | DIC |

**Table 3.**Measured impact load ${P}_{max}$, ${P}_{ini}$, the ratios between both, the strain rate upon the initiation of a cohesive crack ${\dot{\epsilon}}_{i}$, and the strain rate upon the formation of a stress-free crack ${\dot{\epsilon}}_{ic}$, the corresponding mean values with standard deviations shown in the parentheses are given in the fourth row.

Beam No. | Loading Rate (mm/s) | ${\mathit{P}}_{\mathit{ini}}$ (kN) | ${\mathit{P}}_{\mathit{max}}$ (kN) | ${\mathit{P}}_{\mathit{ini}}/{\mathit{P}}_{\mathit{max}}$ (%) | ${\dot{\mathit{\epsilon}}}_{\mathit{i}}$ ($\mathsf{\mu}$$\mathit{\epsilon}$/s) | ${\dot{\mathit{\epsilon}}}_{\mathit{ic}}$ ($\mathsf{\mu}$$\mathit{\epsilon}$/s) |
---|---|---|---|---|---|---|

PA2 | 11.5 | 12.6 | 91 | 48 | 950 | |

PA3 | 0.0022 | 11.1 | 12.0 | 92 | 300 | 3300 |

PA4 | 11.2 | 11.4 | 99 | 84 | 980 | |

11.3 (0.2) | 12.0 (0.6) | 94 (4) | 144 (136) | 1743 (1348) | ||

PB2 | 12.5 | 23.8 | 53 | 26 | 740 | |

PB3 | 0.0022 | 11.2 | 17.6 | 64 | 38 | 480 |

PB4 | 11.8 | 16.7 | 71 | 51 | 550 | |

11.9 (0.7) | 19.4 (3.8) | 62 (9) | 38 (13) | 590 (135) | ||

PC2 | 17.2 | 39.6 | 44 | 45 | 530 | |

PC3 | 0.0022 | 15.4 | 32.4 | 47 | 49 | 780 |

PC4 | 15.1 | 39.5 | 38 | 46 | 480 | |

15.9 (1) | 37.2 (4.1) | 43 (5) | 47 (2) | 597 (161) | ||

($\epsilon $/s) | ($\epsilon $/s) | |||||

PA7 | 13.3 | 16.0 | 83 | 1.1 | 8.5 | |

PA8 | 22 | 13.3 | 16.0 | 83 | 0.6 | 7.6 |

PA9 | 12.5 | 15.2 | 82 | 1.9 | 15.0 | |

13.0 (0.5) | 15.8 (0.5) | 83 (1) | 1.2 (0.7) | 10.0 (4.0) | ||

PB7 | 13.6 | 29.4 | 46 | 0.2 | 2.9 | |

PB8 | 22 | 13.2 | 29.1 | 45 | 0.3 | 3.1 |

PB9 | 16.4 | 27.2 | 60 | 0.3 | 2.4 | |

14.4 (2) | 28.6 (1.2) | 51 (8) | 0.3 (0.1) | 2.8 (0.4) | ||

PC7 | 19.5 | 35.2 | 56 | 0.2 | 1.4 | |

PC8 | 22 | 16.8 | 46.5 | 36 | 0.6 | 2.9 |

PC9 | 16.5 | 44.8 | 37 | 0.3 | 1.6 | |

17.6 (1.7) | 42.2 (6.1) | 43 (11) | 0.4 (0.2) | 2 (0.8) | ||

PA12 | - | 49.3 | - | 10.0 | 73.0 | |

PA13 | 1770 | 28.3 | 102.8 | 28 | 10.1 | 74.2 |

PA14 | 29.8 | 57.8 | 52 | 9.8 | 66.4 | |

29.1 (1.1) | 70.0 (28.8) | 40 (17) | 10 (0.2) | 71.3 (4.3) | ||

PB12 | 35.3 | 57.0 | 62 | 13.1 | 65.0 | |

PB13 | 1770 | 23.3 | 63.4 | 37 | 10.3 | 77.8 |

PB14 | 30.6 | 54.9 | 56 | 11.9 | 61.0 | |

29.7 (6.0) | 58.5 (4.4) | 52 (13) | 11.8 (1.4) | 67.9 (8.8) | ||

PC12 | 65.5 | 70.4 | 93 | 10.9 | 83.6 | |

PC13 | 1770 | 67.2 | 74.7 | 90 | 11.9 | 77.7 |

PC14 | 52.1 | 63.4 | 82 | 10.4 | 74.0 | |

61.6 (8.3) | 69.5 (6.0) | 88 (6) | 11.1 (0.8) | 78.4 (4.8) | ||

PA17 | 42.8 | 77.5 | 55 | 16.8 | 85.2 | |

PA18 | 2660 | 84.1 | 122.5 | 69 | 16.1 | 96.8 |

PA19 | 40.8 | 73.2 | 56 | 20.3 | 84.6 | |

55.9 (24.4) | 91.1 (27) | 60 (8) | 17.7 (2.3) | 88.9 (6.9) | ||

PB17 | 54.3 | 112.7 | 48 | 19.7 | 79.1 | |

PB18 | 2660 | 63.7 | 147.9 | 43 | 16.1 | 92.0 |

PB19 | 47.8 | 83.1 | 53 | 14.5 | 73.0 | |

55.3 (8) | 114.6 (32.4) | 50 (8) | 16.8 (2.7) | 81.4 (9.7) | ||

PC17 | 105.9 | 122.5 | 86 | 12.4 | 86.4 | |

PC18 | 2660 | 87.5 | 112.7 | 78 | 10.9 | 85.2 |

PC19 | 91.8 | 119.7 | 77 | 11.3 | 87.4 | |

95.1 (9.6) | 118.3 (5.1) | 80 (5) | 11.5 (0.8) | 86.3 (1.1) |

**Table 4.**The measured first-crack strength, ${f}_{R}$, in MPa, at four different loading-displacement rates for concrete PA, PB and PC.

Concrete Type | 2.2 ($\mathsf{\mu}$m/s) | 22 (mm/s) | 1.77 (m/s) | 2.66 (m/s) |
---|---|---|---|---|

PA | 8.1 (0.2) | 9.4 (0.3) | 20.9 (0.8) | 40.2 (17.6) |

PB | 8.5 (0.5) | 10.4 (1.3) | 21.4 (4.4) | 39.8 (5.7) |

PC | 11.4 (0.8) | 12.7 (1.2) | 44.3 (5.9) | 68.4 (6.9) |

**Table 5.**The measured toughness indices and residual strength factors with their corresponding standard deviations in the parentheses for the three types of concrete at four different loading rates.

Concrete | Loading (mm/s) | ${\mathit{I}}_{\mathbf{5}}$ | ${\mathit{I}}_{\mathbf{10}}$ | ${\mathit{I}}_{\mathbf{20}}$ | ${\mathit{R}}_{\mathbf{5},\mathbf{10}}$ | ${\mathit{R}}_{\mathbf{10},\mathbf{20}}$ |
---|---|---|---|---|---|---|

PA | 4.9 (0.2) | 10.3 (0.6) | 19.6 (1.9) | 107 (9) | 93 (13) | |

PB | 0.0022 | 6.3 (0.7) | 13.6 (2.2) | 25.3 (4.5) | 149 (30) | 116 (24) |

PC | 7.7 (0.1) | 18.2 (0.2) | 38.1 (1.2) | 210 (3) | 200 (10) | |

PA | 5.2 (0.2) | 10.8 (0.6) | 20.9 (0.8) | 113 (7) | 101 (3) | |

PB | 22 | 6.3 (0.2) | 14.9 (0.6) | 28.6 (1.5) | 170 (11) | 138 (12) |

PC | 7.8 (1.0) | 18.8 (3.7) | 40 (10.3) | 221 (57) | 212 (68) | |

PA | 4.1 (0.0) | 6.6 (0.3) | 9.4 (0.6) | 51 (5) | 27 (9) | |

PB | 1770 | 4.7 (1.7) | 8.7 (3.5) | 14.3 (6.5) | 80 (35) | 55 (31) |

PC | 4.3 (0.7) | 5.8 (1.5) | 11.4 (3.5) | 30 (17) | 56 (20) | |

PA | 5.2 (0.8) | 6.8 (2.7) | 8.6 (2.4) | 33 (39) | 18 (7) | |

PB | 2660 | 5.2 (0.3) | 7.3 (1.2) | 11.6 (1.6) | 42 (22) | 44 (4) |

PC | 3.5 (1.1) | 5.8 (1.4) | 7.7 (2.6) | 52 (10) | 20 (13) |

**Table 6.**Measured crack speed for the quasi-static loading rate of 2.2 $\mathsf{\mu}$m/s (specimens PX2, PX3 and PX4) and 22 mm/s (specimens PX7, PX8 and PX9), the corresponding mean and standard deviations are given in the fourth row of each concrete type.

Beam | ${\mathit{V}}_{\mathit{S}\mathit{G}\mathbf{12}}$ (mm/s) | ${\mathit{V}}_{\mathit{S}\mathit{G}\mathbf{23}}$ (mm/s) | ${\mathit{V}}_{\mathit{S}\mathit{G}\mathbf{34}}$ (mm/s) |
---|---|---|---|

PA2 | 2.7 | 4.9 | 3.3 |

PA3 | 8.3 | 15.6 | 0.8 |

PA4 | 5.4 | 4.4 | 6.4 |

5.5 (3.0) | 8.3 (6.0) | 3.5 (3.0) | |

PB2 | 0.85 | 0.39 | 0.25 |

PB3 | - | - | - |

PB4 | 0.94 | 1.17 | 0.32 |

0.90 (0.06) | 0.78 (0.55) | 0.28 (0.05) | |

PC2 | 0.37 | 0.41 | 0.65 |

PC3 | 0.38 | 0.53 | 0.23 |

PC4 | 1.69 | 6.51 | 0.43 |

0.81 (0.8) | 2.48 (3.5) | 0.44 (0.20) | |

Beam | ${V}_{SG12}$ (m/s) | ${V}_{SG23}$ (m/s) | ${V}_{SG34}$ (m/s) |

PA7 | 20.5 | 20.2 | 20.48 |

PA8 | 13.6 | 20.4 | 20.41 |

PA9 | 13.6 | 20.4 | 20.41 |

15.9 (4) | 20.4 (0.1) | 20.43 (0.04) | |

PB7 | 10.2 | 8.2 | 1.5 |

PB8 | 10.2 | 13.6 | 3.1 |

PB9 | - | - | - |

10.2 (0) | 10.9 (3.8) | 2.3 (1.2) | |

PC7 | 2.2 | 3.1 | 6.8 |

PC8 | 4.1 | 1.1 | 6.8 |

PC9 | 10.3 | 0.85 | 3.1 |

5.5 (4.3) | 1.7 (1.2) | 5.6 (2.1) |

**Table 7.**Measured crack velocities using the strain gages for the loading rate at 1.77 m/s (PX12, PX13 and PX14) and 2.66 m/s (PX17, PX18 and PX19), the corresponding mean and standard deviations are given in the fourth row of each concrete type. The maximum velocities in each group are marked as bold numbers.

Beam | ${\mathit{V}}_{\mathit{S}\mathit{G}\mathbf{12}}$ (m/s) | ${\mathit{V}}_{\mathit{S}\mathit{G}\mathbf{23}}$ (m/s) | ${\mathit{V}}_{\mathit{S}\mathit{G}\mathbf{34}}$ (m/s) |
---|---|---|---|

PA12 | - | - | - |

PA13 | 304 | 386 | 121 |

PA14 | 266 | 327 | 94 |

285 (27) | 357 (42) | 108 (19) | |

PB12 | 425 | 472 | 283 |

PB13 | 193 | 193 | 304 |

PB14 | 472 | 472 | 327 |

363 (149) | 379 (22) | 305 (22) | |

PC12 | 386 | 304 | 266 |

PC13 | 630 | 333 | 125 |

PC14 | 425 | 327 | 147 |

480 (131) | 321 (16) | 179 (76) | |

PA17 | 262 | 115 | 58 |

PA18 | 1063 | 354 | 150 |

PA19 | 354 | 283 | 104 |

559 (438) | 251 (123) | 104 (46) | |

PB17 | 708 | 425 | 75 |

PB18 | 425 | 708 | 531 |

PB19 | 607 | 531 | 425 |

580 (144) | 555 (143) | 344 (239) | |

PC17 | 250 | 1417 | 137 |

PC18 | 202 | 708 | 69 |

PC19 | 531 | 185 | 147 |

328 (118) | 770 (618) | 117 (43) |

**Table 8.**Measured crack velocities along the first 51 mm of crack extension, using the technique of DIC for the beams impacted at 2.66 m/s.

Beam | ${\mathit{V}}_{\mathbf{1}}$ (m/s) | ${\mathit{V}}_{\mathbf{2}}$ (m/s) | ${\mathit{V}}_{\mathbf{3}}$ (m/s) |
---|---|---|---|

PA15 | 617 | 461 | 377 |

PB15 | 654 | 503 | 409 |

PC15 | 944 | 702 | 311 |

**Table 9.**Numerical results on the crack velocity (m/s) for the loading rate of 1.77 m/s (PX12, PX13 and PX14) and 2.66 m/s (PX17, PX18 and PX19), the mean values with the standard deviation in the parentheses are given in the fourth row of each concrete type. The maximum values in each group are marked as bold numbers.

Beam | ${\mathit{V}}_{\mathbf{12}}$ | ${\mathit{V}}_{\mathbf{23}}$ | ${\mathit{V}}_{\mathbf{34}}$ | ${\mathit{V}}_{\mathbf{45}}$ | ${\mathit{V}}_{\mathbf{56}}$ | ${\mathit{V}}_{\mathbf{67}}$ | ${\mathit{V}}_{\mathbf{78}}$ | ${\mathit{V}}_{\mathbf{89}}$ | ${\mathit{V}}_{\mathbf{9},\mathbf{10}}$ |
---|---|---|---|---|---|---|---|---|---|

PA12 | 224 | 264 | 260 | 254 | 223 | 197 | 129 | 86 | 54 |

PA13 | 252 | 296 | 308 | 298 | 331 | 242 | 172 | 113 | 72 |

PA14 | 222 | 254 | 260 | 245 | 210 | 192 | 132 | 83 | 66 |

229 (20) | 271 (22) | 276 (27) | 266 (28) | 255 (66) | 211 (28) | 144 (24) | 94 (16) | 64 (9) | |

PB12 | 217 | 289 | 285 | 282 | 245 | 197 | 141 | 80 | 14 |

PB13 | 153 | 215 | 206 | 174 | 144 | 127 | 103 | 82 | 56 |

PB14 | 206 | 278 | 260 | 275 | 233 | 180 | 126 | 76 | 17 |

192 (34) | 261 (40) | 251 (40) | 244 (61) | 207 (55) | 168 (36) | 123 (20) | 79 (3) | 29 (23) | |

PC12 | 583 | 651 | 667 | 694 | 436 | 409 | 256 | 150 | 89 |

PC13 | 718 | 725 | 667 | 737 | 476 | 425 | 254 | 149 | 89 |

PC14 | 313 | 440 | 482 | 600 | 359 | 347 | 264 | 144 | 87 |

538 (207) | 605 (148) | 605 (107) | 677 (71) | 424 (59) | 394 (34) | 258 (5) | 148 (3) | 88 (1) | |

PA17 | 284 | 379 | 372 | 353 | 326 | 271 | 206 | 143 | 84 |

PA18 | 441 | 543 | 589 | 496 | 388 | 270 | 219 | 158 | 116 |

PA19 | 282 | 388 | 377 | 347 | 306 | 273 | 207 | 143 | 82 |

336 (91) | 436 (92) | 446 (124) | 399 (84) | 340 (42) | 271 (2) | 211 (7) | 148 (8) | 94 (19) | |

PB17 | 206 | 347 | 359 | 326 | 274 | 240 | 184 | 131 | 78 |

PB18 | 237 | 366 | 401 | 342 | 293 | 237 | 202 | 139 | 93 |

PB19 | 195 | 311 | 316 | 288 | 258 | 233 | 176 | 120 | 75 |

213 (22) | 341 (28) | 359 (42) | 318 (22) | 275 (18) | 237 (3) | 187 (13) | 130 (8) | 82 (9) | |

PC17 | 921 | 963 | 916 | 992 | 591 | 529 | 408 | 176 | 82 |

PC18 | 473 | 651 | 765 | 958 | 521 | 463 | 366 | 210 | 79 |

PC19 | 868 | 980 | 1016 | 1016 | 567 | 534 | 396 | 170 | 83 |

754 (244) | 865 (185) | 899 (127) | 989 (29) | 560 (36) | 509 (40) | 390 (22) | 186 (22) | 81 (2) |

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## Share and Cite

**MDPI and ACS Style**

Pan, K.; Yu, R.C.; Zhang, X.; Ruiz, G.; Wu, Z.
Propagation Speed of Dynamic Mode-I Cracks in Self-Compacting Steel Fiber-Reinforced Concrete. *Materials* **2020**, *13*, 4053.
https://doi.org/10.3390/ma13184053

**AMA Style**

Pan K, Yu RC, Zhang X, Ruiz G, Wu Z.
Propagation Speed of Dynamic Mode-I Cracks in Self-Compacting Steel Fiber-Reinforced Concrete. *Materials*. 2020; 13(18):4053.
https://doi.org/10.3390/ma13184053

**Chicago/Turabian Style**

Pan, Kaiming, Rena C. Yu, Xiaoxin Zhang, Gonzalo Ruiz, and Zhimin Wu.
2020. "Propagation Speed of Dynamic Mode-I Cracks in Self-Compacting Steel Fiber-Reinforced Concrete" *Materials* 13, no. 18: 4053.
https://doi.org/10.3390/ma13184053