# A Size and Boundary Effects Model for Quasi-Brittle Fracture

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## Abstract

**:**

## 1. Introduction

## 2. Size and Boundary Effects

#### 2.1. Size Effect Induced by Specimen Sizes

#### 2.2. Boundary Effect Induced by Crack Sizes

## 3. Proposed Failure Model

#### 3.1. Derivative of the Energy Release Rate

#### 3.2. Asymptotic Model

#### 3.3. Proposed Failure Model for Geometrically Similar Specimens

## 4. Model Validations

#### 4.1. Concrete Experiments

#### 4.2. Limestone Experiments

#### 4.3. Hardened Cement Paste Experiments

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

BEM | Boundary Effect Model |

C-TPB | Cracked Three-Point Bending |

FPZ | Fracture Process Zone |

LEFM | Linear Elastic Fracture Mechanics |

SEL | Size Effect Law |

SiC | Silicon Carbide |

TCDs | Theories of Critical Distance |

USEL | Universal Size Effect Law |

## Appendix A. Comparison of the Proposed Model with the Hu–Duan Boundary Effect Model

**Figure A2.**The ratio of the nominal strength predicted by the strength mechanism and tensile strength ${\mathsf{\sigma}}_{\mathrm{N}}^{\mathrm{strength}}/{f}_{\mathrm{t}}$ for various crack to height ratios.

**Figure A3.**The ratio of nominal strength predicted by BEM and the proposed model and tensile strength ${\mathsf{\sigma}}_{\mathrm{N}}/{f}_{\mathrm{t}}$ for various crack to height ratios.

**Figure A4.**Percentages of the difference of the nominal strength given by the boundary effect model and the proposed model.

## Appendix B. Comparison of the Proposed Model with Type 2 SEL

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**Figure 1.**(

**a**) Similar cracked structures and (

**b**) Size Effect Law (SEL) bridging the failure mechanisms of material strength and Linear Elastic Fracture Mechanics (LEFM).

**Figure 2.**Nominal strengths versus crack length in SiC (modified from [12]).

**Figure 4.**Correction factors $A\left(\mathsf{\alpha}\right)$ and $H\left(\mathsf{\alpha}\right)$ versus the crack to height ratio $\mathsf{\alpha}$.

**Figure 5.**(

**a**) Beam with equivalent loading and equivalent crack length ${a}_{\mathrm{e}}$ and (

**b**) equivalent crack length ${a}_{\mathrm{e}}$ versus real crack length a.

**Figure 7.**Nominal strength to tensile strength ratios versus equivalent crack length ${a}_{\mathrm{e}}$.

**Figure 8.**Nominal strengths versus the crack to height ratio $\mathsf{\alpha}$ for various beam heights.

**Figure 10.**Model predictions of failure load versus beam height comparing with Type 2 SEL predictions, boundary effect model predictions and the test results of concrete.

**Figure 13.**Model predictions of failure load versus beam height comparing with Type 2 SEL predictions, boundary effect model predictions and test results of limestone.

**Figure 14.**Model predictions of failure load versus beam height comparing with Type 2 SEL predictions, boundary effect model predictions and test results of hardened cement paste. The error bars indicate the standard deviations of the experimental results.

Specimen Dimensions (mm) [32] | ${\mathit{a}}_{\mathbf{e}}$ (mm) | ${\mathit{P}}_{\mathbf{max}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathbf{N}\right)$ [32] | ${\mathsf{\sigma}}_{\mathbf{N}}$ (MPa) |
---|---|---|---|

$152.4\times 38.1\times 25.4$ | 7.32 | 366.53 | 2.27 |

$304.8\times 76.2\times 25.4$ | 14.63 | 721.28 | 2.24 |

$609.6\times 152.4\times 25.4$ | 29.27 | 1065.79 | 1.65 |

$914.4\times 228.6\times 25.4$ | 43.90 | 1759.72 | 1.82 |

$1219.2\times 304.8\times 25.4$ | 58.53 | 2179.63 | 1.69 |

$1524\times 381\times 25.4$ | 73.16 | 2288.61 | 1.42 |

$1828.8\times 457.2\times 25.4$ | 87.80 | 2470.99 | 1.28 |

$2133.6\times 533.4\times 25.4$ | 102.43 | 3113.76 | 1.38 |

Specimen Dimensions (mm) [34] | ${\mathit{a}}_{\mathbf{e}}$ (mm) | ${\mathit{P}}_{\mathbf{max}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathbf{N}\right)$ [34] | ${\mathsf{\sigma}}_{\mathbf{N}}$ (MPa) |
---|---|---|---|

$52\times 13\times 13$ | 2.35 | 78 | 2.77 |

82 | 2.91 | ||

85 | 3.02 | ||

$100\times 25\times 13$ | 4.52 | 134 | 2.47 |

140 | 2.58 | ||

140 | 2.58 | ||

$204\times 51\times 13$ | 9.22 | 238 | 2.15 |

243 | 2.20 | ||

243 | 2.20 | ||

$408\times 102\times 13$ | 18.44 | 394 | 1.78 |

405 | 1.83 | ||

418 | 1.89 |

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**MDPI and ACS Style**

Gao, X.; Koval, G.; Chazallon, C.
A Size and Boundary Effects Model for Quasi-Brittle Fracture. *Materials* **2016**, *9*, 1030.
https://doi.org/10.3390/ma9121030

**AMA Style**

Gao X, Koval G, Chazallon C.
A Size and Boundary Effects Model for Quasi-Brittle Fracture. *Materials*. 2016; 9(12):1030.
https://doi.org/10.3390/ma9121030

**Chicago/Turabian Style**

Gao, Xiaofeng, Georg Koval, and Cyrille Chazallon.
2016. "A Size and Boundary Effects Model for Quasi-Brittle Fracture" *Materials* 9, no. 12: 1030.
https://doi.org/10.3390/ma9121030