# The Statistical Damage Constitutive Model of the Mechanical Properties of Alkali-Resistant Glass Fiber Reinforced Concrete

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Uniaxial Tensile Test Material

- (1)
- Alkali-resistant glass fiber

^{®}HD/alkali-resistant fiber and Anti-Crak

^{®}HP/alkali-resistant glass fiber. The performance parameters are listed in Table 1.

- (2)
- Concrete matrix materials

#### 2.2. Mix Proportion Design of Concrete Samples

#### 2.3. Preparation and Test Scheme for Concrete Samples

- (1)
- After the concrete specimens were cured, the surface moisture of the specimens was wiped off and a force transfer device was placed on both sides of the specimens.
- (2)
- The specimen force sensor was attached and the corresponding stress sensors were placed on both sides. The stress monitor was attached on the left and right sides of the specimens.
- (3)
- The test piece was continuously and uniformly loaded, and the load speed was 0.05–0.08 MPa/s. When the deformation of the test piece increased rapidly, the accelerator was stopped and adjustments were made until the test piece was destroyed. At this time, the failure load value was recorded.

## 3. Test Results and Analysis of the AR-GFRC

#### 3.1. Effect of Fiber Content on the Tensile Strength of Concrete

#### 3.2. Stress-Strain Curve of the Whole Process of the AR-GFRC

- (1)
- Elastic deformation to micro-fracture development stage (OA region)

- (2)
- Stress peak stage (AB region)

- (3)
- Progressive rupture stage (BC region)

- (4)
- Post-rupture stage (CD region)

## 4. Development of the Statistical Damage Constitutive Model for the AR-GFRC

#### 4.1. Assumptions

- (1)
- Assuming that the AR-GFRC consists of a series of isotropic elastic micro-units, each of which is an ideal elastomer with the same stiffness prior to the destruction.
- (2)
- Assuming that the micro-unit strength follows the Weibull distribution.
- (3)
- Assuming that the concrete consists of an undamaged part and a damaged part and the stress-strain relationship of the undamaged part obeys Hooke’s law.
- (4)
- Assuming that the concrete damage only occurs under axial stress, damage occurs after a load has been applied, and immediately after damage, failure occurs.

#### 4.2. Derivation of the Statistical Damage Constitutive Model

#### 4.2.1. The Constitutive Equation of the Elastic Section of Uniaxial Tensile Concrete

#### 4.2.2. The Constitutive Equation of the Inelastic Section in the Whole Process of Uniaxial Tensile Concrete

_{f}> E

_{m}and σ

_{f}> σ

_{m}. Therefore, the tensile statistical damage constitutive model of AR-GFRC is:

#### 4.3. Parameter Determination of the Statistical Damage Constitutive Model

## 5. Verification of the Statistical Damage Constitutive Model of the AR-GFRC

#### 5.1. Verification of Field Test

#### 5.1.1. Project Overview and Field Test

#### 5.1.2. On-Site Deformation Monitoring and Result Analysis

- (1)
- Stress variation in concrete lining

- (2)
- Variation of the convergence value around the tunnel

#### 5.2. Verification of the Theory and Laboratory Experiment

- (1)
- The proposed constitutive model describes the post-peak damage evolution of the AR-GFRC and the trends and characteristics of the concrete strength in the stress state. The theoretical and experimental stress-strain curves are in good agreement for the different fiber contents. The early tensile strength of the concrete was improved by the addition of alkali-resistant glass fiber.
- (2)
- As shown in Figure 10a–f and Figure 11a–f, the proposed tensile statistical damage constitutive model of AR-GFRC can accurately describe the stress-strain evolution law of the different fiber contents of alkali-resistant glass fiber concrete in this concrete tensile test. The content of the alkali-resistant glass fiber doping range was considered 0 to 1.5%, the 7-day peak strength difference between HD and HP concrete under standard curing between the experimental results and the theoretical results is small, and the 28-d peak strength shows similar patterns of change. The theoretical equation curve is consistent with the indoor test data curve in the elastic stage, peak strength, and nonlinear change segment after its peak. Different types of fibers (HD and HP) have little effect on the theoretical and test curves. This shows that the proposed AR-GFRC tensile statistical damage model is also applicable to the description of the concrete stress-strain curve of HD and HP fibers.

## 6. Discussion

## 7. Conclusions

- (1)
- A new damage variable of the AR-GFRC was defined based on the results of the uniaxial tensile test and the stress-strain curve was redefined by taking into consideration the residual strength.
- (2)
- The meso-statistical damage theory, microcontinuum theory, and composite material theory were used to develop a statistical damage constitutive equation of the AR-GFRC based on a Weibull distribution. The calculation methods for determining the mechanical and statistical parameters of the concrete have been provided, and the research results can be used to provide reference data for practical concrete engineering.
- (3)
- The constitutive theoretical curve was verified in MATLAB based on the uniaxial tensile test data of the concrete. The theoretical and experimental stress-strain curves of the AR-GFRC were in good agreement.
- (4)
- The fiber contents (0.5%, 1%, and 1.5%) and the size and shape parameters in the constitutive equation were quantified, and it was determined that the proposed model was well suited to describe the microscopic damage evolution and failure mechanism of the concrete while considering the residual strength effects. The results provide a theoretical basis and have practical application value.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Tensile curves of alkali-resistant glass fiber reinforced concrete (AR-GFRC) and ordinary concrete. (

**a**) HD 28 d curing and (

**b**) HP 28 d curing.

**Figure 7.**Variation of stress in concrete lining. (

**a**) Common concrete spray; (

**b**) AR-GFRC (1.0%HP); and (

**c**) AR-GFRC (1.0%HD).

**Figure 10.**The stress-strain curve of HD concrete under 7 d and 28 d of curing. (

**a**) 0.5%, 7 d; (

**b**) 1.0%, 7 d; (

**c**) 1.5%, 7 d; (

**d**) 0.5%, 28 d; (

**e**) 1.0%, 28 d; and (

**f**) 1.5%, 28 d.

**Figure 11.**The stress-strain curve of HP concrete under 7 d and 28 d of curing. (

**a**) 0.5%, 7 d; (

**b**) 1.0%, 7 d; (

**c**) 1.5%, 7 d; (

**d**) 0.5%, 28 d; (

**e**) 1.0%, 28 d; and (

**f**) 1.5%, 28 d.

Type | Length/mm | Equivalent Diameter/um | Fracture Strength | Elongation at Break/% | Modulus/GPa | Melting Point/°C |
---|---|---|---|---|---|---|

Anti-Crak^{®}HD | 6/12 | 14 | 1700 | 3.6 | 72 | 1580 |

Anti-Crak^{®}HP | 6/12 | 700 | 1700 | 3.6 | 72 | 1580 |

Sieve Hole Size | 31.5 | 26.5 | 16.0 | 4.75 | 2.36 |
---|---|---|---|---|---|

Actual cumulative percentage of sieve remainder/% | 0 | 0~5 | 30~70 | 90~95 | 95~100 |

Number | Name | Cement | Mineral Powder | Fly Ash | Sand | Stone | Water | Admixture | Fiber Contents (%) |
---|---|---|---|---|---|---|---|---|---|

C35 | Concrete (without fiber) | 245 | 100 | 95 | 735 | 1040 | 175 | 10.5 | 0 |

HD-35-1 | Anti-Crak^{®} (HD) Concrete | 245 | 100 | 95 | 735 | 1040 | 175 | 10.5 | 0.5 |

HD-35-2 | 245 | 100 | 95 | 735 | 1040 | 175 | 10.5 | 1 | |

HD-35-3 | 245 | 100 | 95 | 735 | 1040 | 175 | 10.5 | 1.5 | |

HP-35-1 | Anti-Crak^{®} (HP) Concrete | 245 | 100 | 95 | 735 | 1040 | 175 | 10.5 | 0.5 |

HP-35-2 | 245 | 100 | 95 | 735 | 1040 | 175 | 10.5 | 1 | |

HP-35-3 | 245 | 100 | 95 | 735 | 1040 | 175 | 10.5 | 1.5 |

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**MDPI and ACS Style**

Shi, X.; Zhang, C.; Zhou, X.
The Statistical Damage Constitutive Model of the Mechanical Properties of Alkali-Resistant Glass Fiber Reinforced Concrete. *Symmetry* **2020**, *12*, 1139.
https://doi.org/10.3390/sym12071139

**AMA Style**

Shi X, Zhang C, Zhou X.
The Statistical Damage Constitutive Model of the Mechanical Properties of Alkali-Resistant Glass Fiber Reinforced Concrete. *Symmetry*. 2020; 12(7):1139.
https://doi.org/10.3390/sym12071139

**Chicago/Turabian Style**

Shi, Xianzeng, Cong Zhang, and Xingde Zhou.
2020. "The Statistical Damage Constitutive Model of the Mechanical Properties of Alkali-Resistant Glass Fiber Reinforced Concrete" *Symmetry* 12, no. 7: 1139.
https://doi.org/10.3390/sym12071139