# Numerical Simulation on Size Effect of Fracture Toughness of Concrete Based on Mesomechanics

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## Abstract

**:**

_{max}) of concrete was predicted using the numerical model. Based on the simulating results, the influence of specimen size of WS and 3-p-b tests on the fracture parameters was analyzed. It was demonstrated that when the specimen size was large enough, the fracture toughness (K

_{IC}) value obtained by the linear elastic fracture mechanics formula was independent of the specimen size. Meanwhile, the improved boundary effect model (BEM) was employed to study the tensile strength (f

_{t}) and fracture toughness of concrete using the mesofracture numerical model. A discrete value of β = 1.0–1.4 was a sufficient approximation to determine the f

_{t}and K

_{IC}values of concrete.

## 1. Introduction

_{IC}value no longer has size effects [4]. The similar conclusion drawn by Bazant [5] is that the mechanical parameters measured for specimens in a certain size range had size effects. That is to say, when the size of the specimens was small enough or large enough, the mechanical properties and fracture parameters remained unchanged. In view of this experimental phenomenon, Weibull [8,9] considered that the size dependence was due to the increase in the probability of encountering low-strength material elements with increasing structure sizes. Based on the law of extreme strength distribution, a size effect statistical theory was proposed. However, this theory was limited to structures that fail at the beginning of macro-cracks and small structures that only cause negligible stress redistribution in the fracture process zone when they fail. Subsequently, based on the energy theory, Bazant et al. [5] proposed a size effect theory of fracture mechanics for geometrically similar specimens with a size range of approximately 1:20 notches. Based on the concept of fractals, Carpinteri et al. [6] established a multi-fractal size effect law that reflected the unstable cracking of structures when the size range was approximately 1:10. Hu [7] concluded that the initial crack and ligament depth should be far from the specimen boundary to reflect the true mechanical parameters of a material, independent of size. Based on this, the boundary effect theory was established. In addition to this basic theory, indirect size effects such as the boundary layer effect [10], time-dependent size effects caused by diffusion phenomenon [11], and time-dependent size effects with respect to a material’s constitutive relationship have also been included [12]. Based on fractal theory, Huang et al. [13] established a fractal model for the size effect of the fracture energy of concrete. Huang [14] analyzed the existing theory of strength size laws and the phenomenon of size effects of fracture parameters and determined the strength size effects of different types of concrete and a research method for small size specimens with a brittleness index. With developments concerning the size effect model and boundary effect model [15,16,17] in recent years, the mechanism and scale law of concrete strength have become clearer.

## 2. Numerical Model and Examples

#### 2.1. Generation of Concrete Mesostructure

_{max}), as shown in Figure 1a,b.

_{1}is the number of ITZ defect units and N

_{2}is the total number of ITZ units.

_{x}and F

_{y}applied in the X, Y directions were calculated by Equation (2).

#### 2.2. Constitutive Relations and Failure Criteria

_{f}is the peak strain, ε

_{m}is the ultimate strain, σ

_{m}is the residual strength, f

_{t}is the tensile strength and n is the softening coefficient, assigned a value of one.

#### 2.3. Determination of Property Parameters

_{m}is the elastic modulus of mortar, f

_{tp}is the pure tensile strength of mortar, f

_{cm}is the compressive strength of mortar and c/w is the cement–water ratio of mortar.

#### 2.4. Determination of Mesh Size

_{max}changes when mesh size varies. Combined with the results of the experiment in [26], when the mesh size was in the range from 0.5 to 2 mm, the simulating results were relatively stable and reasonable. In this paper, the mesh size of the meso structure was 1 mm, and then it gradually increased to 5 mm in the macro structure.

## 3. Results and Discussion

#### 3.1. Cracking Process

_{max}, the concrete specimen of 3-p-b cracked from the initial notch tip, and when the load reached 17% of P

_{max}, the concrete specimen of WS cracked from the initial notch tip—thereby, the first microcracking or strain localization occurred and the inner microcracks accumulated [38] in both types. As the load continued to increase, cracks gradually expanded and converged with the crack at the crack tip, forming a macroscopic visible main crack at the crack tip in the fracture process zone, as shown in Figure 4b and Figure 5b [39]. Subsequently, the cracks steadily expanded as the external load gradually increases to the peak load. Then, cracks expanded unsteadily and the load began to decrease, ultimately causing the concrete fracture failure with increasing load. It could be concluded that the primary crack of WS and 3-p-b specimens propagates along the surface of coarse aggregates, so the shape, distribution, and particle size of coarse aggregates played important roles in the concrete fracture process [40]. These fracture process simulation results agreed well with the experimental results of [37,41].

#### 3.2. Peak Stress

_{L}was calculated accordingly by Equation (7) [42]. The Load-Crack Mouth Opening Displacement (P-CMOD) curves of numerical simulation results and test results of WS and 3-p-b specimens with a dimension of 200 mm in height are shown in Figure 6. Here, the values of specimen parameters, such as the ratio of the initial notch to the height of the specimen (a/W), specimen height (W), and specimen thickness (B) were calculated according to the numerical model, which was determined using Xu’s experimental data [26]. From the results presented in Figure 6 and Table 3 and Table 4, it can be seen that the simulation results are more brittle. That may be due to the elastic constitutive relation of meso materials. However, the simulation results have good discreteness and agree well with the experimental results before the peak point. To analyze the influence of specimen size on proportional ultimate strength and find the independent size of concrete strength, the fracture process of concrete specimens with heights larger than 1200 mm was further simulated, as shown in Table 3 and Table 4.

_{L}and the crack length decreased with increasing concrete specimen dimensions for specimens of both types. However, the growth of 3-p-b specimens was slower than that of WS specimens.

#### 3.3. Fracture Toughness

_{IC}of concrete increases with increasing of specimen size, and when the specimen size is large enough, the change in K

_{IC}with size is not noticeable [44,45,46]. To verify that K

_{IC}varies with size, as well as to find the scale-independent fracture parameters of concrete, the above experimental results and numerical simulation results were substituted into Equation (8) [47,48], and the fracture toughness of concrete was calculated and is shown in Table 5. According to the results, the variation of fracture toughness K

_{IC}with specimen height W was drawn and is shown in Figure 7. It can be concluded that the fracture toughness increases with the increase in specimen size. When the height of the specimen reaches 600 mm (W/d

_{max}= 60), the trend line is basically completely within the dashed lines, the fracture toughness no longer changes significantly and is approximately 1.10 MPa·m

^{1/2}, independent of the size of the specimen.

_{max}is the peak load, W is the beam height, B is the beam thickness and a is the initial notch length.

#### 3.4. Discussion of the Size Effect on Fracture Toughness and Tensile Strength

_{max}) to determine the size-independent tensile strength f

_{t}and fracture toughness K

_{IC}, shown as Equation (9), which is a more suitable application for the numerical model. The peak loads (P

_{max}) obtained from the above numerical model, the experiments of [30] and the parameters involved can be used with this equation to determine the tensile strength of concrete specimens.

_{t}and K

_{IC}. a

_{e}is the equivalent crack length, fully determined by the specimen size, type and a, which can be described as a

_{e}= B(α)a, where B(α) is the geometric shape parameter for 3-p-b specimens [15,22,23,56,57].

_{max}depends on the maximum size of coarse aggregates, and a discrete coefficient (β) can be introduced to describe the relationship between Δa and the maximum diameter of coarse aggregates (d

_{max}), shown as Equation (13).

_{IC}and tensile strength f

_{t}independent of size can be regressed by measuring the peak load P

_{max}of various size specimens. The fitting results of tensile strength and fracture toughness under various conditions of ∆a are shown in Figure 8. When β ranges from 0.6 to 3.2, the tensile strength (f

_{t}) and fracture toughness (K

_{IC}) vary from 2.44 to 4.01 MPa and 1.06 to 1.35 MPa·m

^{1/2}, respectively. It is known that the ratio of concrete tensile strength to compressive strength is approximately 1/8–1/12, and the compressive strength in [26] was 29.56 MPa. It can, thus, be inferred that the value of f

_{t}varies from 2.46 to 3.70 MPa. Meanwhile, the fracture toughness of concrete (K

_{IC}) is relatively stable at approximately 1.10 MPa·m

^{1/2}. It can be concluded that the value of β is 1.0–1.4, and reasonable tensile strength f

_{t}and fracture toughness K

_{IC}can be obtained for small aggregate fracture specimens. The results agree well with those of [53], which were reached based on concrete fracture experiments with various specimen types. In conclusion, the model can be used to simulate the fracture tests of small size concrete under different conditions. Based on this, the fracture toughness and tensile strength, independent of size, can be further determined by the boundary effect theory.

## 4. Conclusions

- (1)
- The numerical model of concrete mesofracture, considering initial defects, can simulate the fracture process and predict the peak load of concrete, so it is suitable for determining concrete fracture parameters and tensile strength;
- (2)
- Based on the above mesofracture numerical model, when the height of a concrete specimen reaches 600 mm (W/d
_{max}= 60), the fracture toughness K_{IC}calculated from P_{max}and the initial notch length according to the linear elastic fracture mechanics formula is independent of the specimen size; - (3)
- The tensile strength (f
_{t}) and the fracture toughness (K_{IC}) which are independent in specimens of concrete can be obtained by the application of the mesofracture numerical model and the BEM. This property can be well expressed by ∆a at peak load (P_{max}), and the relationship between ∆a and the maximum aggregate diameter (d_{max}) can be established by introduced a discrete coefficient (β). Discrete values of β range from 1.0 to 1.4 are a sufficient approximation to predict the f_{t}and K_{IC}values of concrete.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Diagram of cracks in fracture process zone (FPZ) of three-point bending (3-p-b) specimens with different mesh sizes (blue elements for coarse aggregates and red color for failure elements in interfacial transition zone (ITZ) and mortar).

**Figure 8.**Tensile strength (f

_{t}) of simultaneous numerical simulation specimens with various values of β.

Microscopic Components | Elastic Modulus (GPa) | Strength (MPa) | Poisson’s Ratio | Volume Content (%) |
---|---|---|---|---|

Aggregate | 60 | — | 0.168 | 45 |

Mortar | 18 | 2.5 | 0.2 | — |

Interface | 5.4 | 0.83 | 0.2 | — |

Concrete | 20 | 1.0 | 0.168 | — |

Interface initial defects | 5 × 10^{−6} | — | — | 30 |

Mesh Size (mm) | Element Number | Simulation Results P_{max} (kN) | Relative Error of P_{max} /% |
---|---|---|---|

0.5 | 67458 | 6.58 | 4.1 |

1 | 19403 | 6.84 | 0.2 |

2 | 8530 | 7.02 | 2.3 |

3 | 5467 | 9.08 | 32.3 |

5 | 4119 | 10.24 | 49.3 |

Specimen Number | a/W | W (mm) | B (mm) | Test Result f_{L} (MPa) | Simulation Results f_{L} (MPa) | Relative Error (%) |
---|---|---|---|---|---|---|

I-W200-1 | 0.46 | 200 | 200 | 0.91 | 0.94 | 0.0 |

I-W200-2 | 0.47 | 200 | 200 | 0.92 | 0.89 | |

I-W200-3 | 0.47 | 200 | 200 | 0.88 | 0.88 | |

Average | 0.90 | 0.90 | ||||

I-W400-1 | 0.45 | 400 | 200 | 0.70 | 0.73 | 5.0 |

I-W400-2 | 0.46 | 400 | 198 | 0.93 | 0.82 | |

I-W400-3 | 0.46 | 400 | 199 | 0.78 | 0.96 | |

Average | 0.80 | 0.84 | ||||

I-W600-1 | 0.46 | 600 | 193 | 0.79 | 0.76 | 3.9 |

I-W600-2 | 0.46 | 599 | 200 | 0.73 | 0.71 | |

I-W600-3 | 0.46 | 600 | 193 | 0.74 | 0.70 | |

I-W600-5 | 0.46 | 599 | 200 | 0.78 | 0.73 | |

Average | 0.76 | 0.73 | ||||

I-W800-1 | 0.45 | 799 | 196 | 0.59 | 0.64 | 8.7 |

I-W800-2 | 0.46 | 800 | 194 | 0.75 | 0.62 | |

I-W800-4 | 0.46 | 798 | 200 | 0.70 | 0.60 | |

I-W800-5 | 0.46 | 801 | 200 | 0.71 | 0.66 | |

Average | 0.69 | 0.63 | ||||

I-W1000-1 | 0.45 | 997 | 200 | 0.55 | 0.53 | 5.0 |

I-W1000-3 | 0.45 | 997 | 200 | 0.58 | 0.54 | |

I-W1000-4 | 0.45 | 999 | 196 | 0.65 | 0.70 | |

I-W1000-5 | 0.45 | 1000 | 200 | 0.60 | 0.51 | |

Average | 0.60 | 0.57 | ||||

I-W1200-0 | 0.45 | 1198 | 200 | 0.51 | 0.50 | 5.6 |

I-W1200-1 | 0.46 | 1200 | 201 | 0.52 | 0.52 | |

I-W1200-2 | 0.45 | 1200 | 200 | 0.58 | 0.50 | |

Average | 0.54 | 0.51 | ||||

I-W1500-1 | 0.5 | 1500 | 200 | — | 0.44 | — |

I-W1500-2 | 0.5 | 1500 | 200 | — | 0.44 | |

I-W1500-3 | 0.5 | 1500 | 200 | — | 0.43 | |

Average | 0.44 | |||||

I-W2000-1 | 0.5 | 2000 | 200 | — | 0.40 | — |

I-W2000-2 | 0.5 | 2000 | 200 | — | 0.37 | |

I-W2000-3 | 0.5 | 2000 | 200 | — | 0.38 | |

Average | 0.38 |

Specimen Number | a/W | W (mm) | B (mm) | Test Result f_{L} (MPa) | Simulation Results f_{L} (MPa) | Relative Error (%) |
---|---|---|---|---|---|---|

II-T200-2 | 0.47 | 200 | 200 | 3.84 | 4.03 | 8.9 |

II-T200-3 | 0.48 | 200 | 200 | 3.46 | 4.06 | |

II-T200-4 | 0.46 | 199 | 199 | 3.47 | 3.65 | |

Average | 3.59 | 3.91 | ||||

II-T300-1 | 0.47 | 298 | 200 | 2.85 | 2.99 | 2.7 |

II-T300-2 | 0.47 | 297 | 195 | 3.10 | 3.14 | |

II-T300-3 | 0.46 | 298 | 198 | 2.75 | 2.90 | |

II-T300-4 | 0.48 | 298 | 200 | 3.14 | 3.14 | |

II-T300-5 | 0.49 | 298 | 200 | 3.23 | 3.27 | |

Average | 3.01 | 3.09 | ||||

II-T400-1 | 0.46 | 401 | 199 | 2.86 | 2.76 | 0.7 |

II-T400-2 | 0.46 | 400 | 196 | 2.88 | 2.79 | |

II-T400-3 | 0.47 | 396 | 199 | 2.84 | 2.92 | |

II-T400-4 | 0.47 | 396 | 197 | 2.75 | 2.78 | |

Average | 2.83 | 2.81 | ||||

II-T500-1 | 0.53 | 499 | 200 | 2.68 | 2.76 | 5.6 |

II-T500-2 | 0.46 | 500 | 196 | 2.50 | 2.50 | |

II-T500-3 | 0.55 | 500 | 198 | 2.39 | 2.72 | |

Average | 2.52 | 2.66 | ||||

II-T600-1 | 0.5 | 600 | 200 | — | 2.40 | — |

II-T600-2 | 0.5 | 600 | 200 | — | 2.54 | |

II-T600-3 | 0.5 | 600 | 200 | — | 2.58 | |

Average | 2.51 | |||||

II-T800-1 | 0.5 | 800 | 200 | — | 2.36 | — |

II-T800-2 | 0.5 | 800 | 200 | — | 2.30 | |

II-T800-3 | 0.5 | 800 | 200 | — | 2.31 | |

Average | 2.32 |

Specimen Number | K_{IC}/(MPa·m^{1/2}) | Average | Test K _{IC}/(MPa·m^{1/2}) | Test Average |
---|---|---|---|---|

I-W200-1 | 0.7238 | 0.7329 | 0.7331 | 0.7236 |

I-W200-2 | 0.7433 | 0.7337 | ||

I-W200-3 | 0.7316 | 0.7039 | ||

I-W400-1 | 0.8376 | 0.9553 | 0.8099 | 0.9161 |

I-W400-2 | 0.9318 | 1.0553 | ||

I-W400-3 | 1.0964 | 0.883 | ||

I-W600-1 | 1.0655 | 1.0111 | 1.1039 | 1.0607 |

I-W600-2 | 0.9860 | 1.0153 | ||

I-W600-3 | 0.9776 | 1.0333 | ||

I-W600-5 | 1.0153 | 1.0901 | ||

I-W800-1 | 1.0326 | 1.0140 | 0.9596 | 1.1062 |

I-W800-2 | 0.9965 | 1.2062 | ||

I-W800-4 | 0.9662 | 1.1212 | ||

I-W800-5 | 1.0607 | 1.1377 | ||

I-W1000-1 | 0.9655 | 1.0383 | 0.9993 | 1.0846 |

I-W1000-3 | 0.9862 | 1.053 | ||

I-W1000-4 | 1.2689 | 1.187 | ||

I-W1000-5 | 0.9325 | 1.0992 | ||

I-W1200-0 | 0.9948 | 1.0085 | 1.0166 | 1.0657 |

I-W1200-1 | 1.0334 | 1.0183 | ||

I-W1200-2 | 0.9973 | 1.1622 | ||

I-W1500-1 | 1.1118 | 1.0984 | — | — |

I-W1500-2 | 1.1097 | — | ||

I-W1500-3 | 1.0734 | — | ||

I-W2000-1 | 1.1495 | 1.1065 | — | — |

I-W2000-2 | 1.0702 | — | ||

I-W2000-3 | 1.0999 | — |

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## Share and Cite

**MDPI and ACS Style**

Wang, J.; Wu, Q.; Guan, J.; Zhang, P.; Fang, H.; Hu, S.
Numerical Simulation on Size Effect of Fracture Toughness of Concrete Based on Mesomechanics. *Materials* **2020**, *13*, 1370.
https://doi.org/10.3390/ma13061370

**AMA Style**

Wang J, Wu Q, Guan J, Zhang P, Fang H, Hu S.
Numerical Simulation on Size Effect of Fracture Toughness of Concrete Based on Mesomechanics. *Materials*. 2020; 13(6):1370.
https://doi.org/10.3390/ma13061370

**Chicago/Turabian Style**

Wang, Juan, Qianqian Wu, Junfeng Guan, Peng Zhang, Hongyuan Fang, and Shaowei Hu.
2020. "Numerical Simulation on Size Effect of Fracture Toughness of Concrete Based on Mesomechanics" *Materials* 13, no. 6: 1370.
https://doi.org/10.3390/ma13061370