Accurate simulation of the cracking process caused by rust expansion of reinforced concrete (RC) structures plays an intuitive role in revealing the corrosion-induced failure mechanism. Considering the quasi-brittle fracture of concrete, the fracture phase field driven by the compressive-shear term is constructed and added to the traditional brittle fracture phase field model. The rationality of the proposed model is verified by a mixed fracture example under a shear displacement load. Then, the extended fracture phase model is applied to simulate the corrosion-induced cracking process of RC. The cracking patterns caused by non-uniform corrosion expansion are discussed for RC specimens with homogeneous macroscopically or heterogeneous with different polygonal aggregate distributions at the mesoscopic scale. Then, the effects of the protective layer on the crack propagation trajectory and cracking resistance are investigated, illustrating that the cracking angle and cracking resistance increase with the increase of the protective layer thickness, consistent with the experimental observation. Finally, the corrosion-induced cracking process of concrete specimens with large and small spacing rebars is simulated, and the interaction of multiple corrosion cracking is easily influenced by the reinforcement spacing, which increases with the decrease of the steel bar interval. These conclusions play an important role in the design of engineering anti-corrosion measures. The fracture phase field model can provide strong support for the life assessment of RC structures.

Concrete is an important building material, and its fracture characteristics play a key role in engineering safety [

The simulation research of the concrete fracture process, especially the crack initiation and propagation, has attracted great attention. The failure simulation methods used for concrete materials and structures are usually classified into two broad categories, i.e.,

For the diffuse damage approach, the damage is used to describe the crack initiation and evolution. However, if the concrete is treated as a homogeneous material, it is difficult to simulate the effect of damage localization. Based on the elastic damage model, Zhu et al. simulated the localized damage distribution by introducing the random mechanical characteristics with Weibull distribution [

Another diffuse approach simulating crack growth is the very popular phase field method [

In summary, because of the advantages of the fracture phase field method in tracking crack propagation with no requirement for explicit fracture criteria and mesh reconstruction, the fracture phase field method will be adopted and modified to simulate the corrosion-induced cracking of RC structures for the possible quasi-brittle fracture. The cracking patterns of different RC specimens macroscopically homogeneous or microscopically heterogeneous with different polygonal aggregate distributions will be explored respectively. Furthermore, the effects of protective layer thickness on the corrosion-induced cracking path will be investigated for the concrete specimens with single-, double-, or multi-steel bars.

This paper is organized as follows. In

The total potential energy for a cracking system is composed of elastic strain potential energy, dissipative potential energy generated by fracture, and external force potential energy [

Bourdin et al. [

Francfort et al. [

Without considering the kinetic energy term, the total Lagrangian energy functional can be expressed as the sum of the fracture energy of

The above energy functional integral equation is discretized by finite element method and introduces the interpolation mode of displacement and fracture phase field of element, the corresponding finite element incremental solution format can be obtained by variational extreme value of functional

To facilitate implementation in COMSOL software, the corresponding strong form of the differential equations are adopted as follows:

For the solution strategy of the coupled

To refine the cracked presentation results, a mixed fracture phase field variable

To verify the rationality of the proposed phase field model, an example of crack tracking under shear load is designed as follows: A plate with a horizontal edge crack is fixed on the bottom and a horizontal displacement ^{*} is applied to the top boundary under the normal constraint, as shown in

In the early study of the distribution characteristics of corrosion products, there was a lack of accurate tests and observation instruments. Researchers believed that the distribution of corrosion product layers around reinforcement was uniform. Based on this assumption, theoretical analysis [

After the steel bars are corroded, the molar volume of ferric hydroxide generated is larger than that of iron, resulting in expansion. The volume of corrosion products is about three to four times that of the original steel bars [

Take the reinforced concrete test specimen with a square cross-section as an example, the side length of the cross-section is 100 mm, the diameter of the steel reinforcement is 16 mm, and the thickness of the concrete protective layer is

Parameter | Symbol | Value |
---|---|---|

Density (kg/m^{3}) |
2300 | |

Elastic modulus (GPa) | 25 | |

Poisson’s ratio | 0.25 | |

Length scale (mm) | 0.2 | |

Griffith’s constant for mode I (J/m^{2}) |
5 | |

Griffith’s constant for mode II (J/m^{2}) |
90 |

To investigate the influence of the concrete protective layer on the corrosion expansion failure of reinforced concrete, three different thicknesses, i.e.,

To investigate the evolution of load exerted on concrete by rust expansion during reinforcement corrosion, the curves of the reaction force at the maximum corrosion location on the surface of reinforcement in RC samples with different cover thicknesses vs. rust expansion are extracted from the numerical results, presented as in

In general, concrete material at the mesoscale is considered to consist of three phases, i.e., randomly distributed aggregate, a hardened cement paste, and the interface transition zone (ITZ) between them. Compared with the cement paste, aggregate is almost impermeable, and hence the distribution of aggregate may directly affect the diffusion path of chloride ions. As shown in

Parameter | Aggregate | Hardened cement paste | ITZ |
---|---|---|---|

Elastic modulus (GPa) | 70 | 25 | 15 |

Tensile strength (MPa) | 8 | 4.5 | 2.5 |

Poisson’s ratio | 0.2 | 0.2 | 0.2 |

Length scale (mm) | 0.25 | 0.2 | |

Griffith’s constant for mode I (J/m^{2}) |
5 | 3.5 | |

Griffith’s constant for mode II (J/m^{2}) |
90 | 52 |

To illustrate the effect of randomly distributed aggregate, Sample A with the random distribution of mostly elongated and sharp aggregates, Sample B with three gradation random distribution of blunt polygon aggregates, and Sample C with two gradation random distribution of blunt polygon aggregates, are chosen as presented in

When it comes to the mechanism of rust expansion and fracture, the simulation results of homogeneous concrete can basically show a rough propagation path. However, due to the principle of weakest component strength, which means that the initiation of rust expansion and crack is determined by the ITZ in the concrete around the steel bar, further crack propagation will be induced by the ITZ. Therefore, the strength homogenization treatment of homogeneous concrete is not suitable, which leads to inaccurate rust crack paths.

In this section, the corrosion cracking process of reinforced concrete with different spacing arrangements of steel bars is studied. The first example is a test specimen with 110 mm spacing between two steel bars, and the diameter of both rebars is 16 mm, as shown in the left of

To investigate the multiple cracking interactions during the corrosion process, we next consider another specimen with a small spacing reinforcement arrangement. Except for reducing the spacing between reinforced bars to 50 mm, the calculated parameters, constraints, and loads of the latter example are still the same as the former, as shown in the right of

Considering the quasi-brittle fracture of concrete, the fracture phase field driven by the compressive-shear term is constructed and added to the traditional brittle fracture phase field model. The rationality of the proposed model is verified by a designed example of mixed cracking under a shear load. The fracture phase field models of RC specimens homogeneous macroscopically or heterogeneous with different polygonal aggregate distributions at the mesoscopic scale are established, and their non-uniform corrosion-induced cracking process and cracking patterns are simulated successfully. Then, the effects of protective layer thickness on the crack propagation path and cracking resistance are investigated. Finally, the corrosion-induced cracking process of concrete specimens with large and small spacing rebars is simulated, and the interaction mechanism of multiple corrosion cracking is also explored. Some conclusions are obtained as follows:

The advantages of fracture phase fields in tracking the corrosion-induced crack propagation are again demonstrated, namely, no need for explicit fracture criteria, no need to preset cracks, and no need to re-mesh.

The corrosion-induced cracking of concrete is still mainly mode I cracking because the critical fracture energy release rate of mode II is much greater than that of mode I for concrete material, which is the square magnitude of the fracture toughness ratio.

The cracking resistance of concrete increases with the increase of the protective layer thickness under the same corrosion expansive level. The thicker the concrete protective layer is, the closer the crack propagation direction is to the direction parallel to the surface of the protective layer, which confirms the findings of Bažant.

Aggregate within the concrete can hinder and delay the propagation of cracks, while the weak ITZ can induce cracks to grow or develop towards itself, which results in the corrosion-induced cracking path locally depending on the distribution, shape, and grade of aggregates. Nevertheless, the cracking pattern for heterogeneous concrete with different randomly distributed aggregates is more or less similar to the case of homogeneous concrete under the same non-uniform corrosion of steel bar.

The cracks caused by the rust expansion of adjacent reinforcements are easy to coalesce with the decrease of reinforcement spacing; that is, when the spacing of reinforcement is closer, the rust expansion cracks are easier to connect and penetrate.

It should be noted that the Phase-field model of compression shear fracture proposed in this paper is not perfect, for example, the correlation between residual shear stiffness and spherical stress has not been established, and how to properly calibrate the length scale parameters of mode I and mode II fractures. The more efficient adaptive algorithm and the more perfect fracture Phase-field model will be explored in another paper.

The financial support of the National Natural Science Foundation of China and the Fundamental Research Funds for the Central Universities is gratefully acknowledged.

This work has been partially supported by the National Natural Science Foundation of China (Qing Zhang, Nos. 11932006, U1934206, 12172121), and the Fundamental Research Funds for the Central Universities (Xin Gu, No. B210201031).

The authors confirm contribution to the paper as follows: study conception and design: Xiaozhou Xia; computational simulation: Changsheng Qin; analysis and interpretation of results: Xiaozhou Xia, Guangda Lu, Xin Gu; draft manuscript preparation: Xiaozhou Xia, Xin Gu, Guangda Lu, Qing Zhang. All authors reviewed the results and approved the final version of the manuscript.

Readers can access the data used in the study by Email xiaxiaozhou@163.com.

The authors declare that they have no conflicts of interest to report regarding the present study.