# Analysis of Interlayer Crack Propagation and Strength Prediction of Steel Bridge Deck Asphalt Pavement Based on Extended Finite Element Method and Cohesive Zone Model (XFEM–CZM) Coupling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods and Damage Models

#### 2.1. XFEM

#### 2.2. Crack Initiation Criterion and Expansion Criterion of the Bonded Layer

#### 2.3. Interface Damage Model

- (1)
- Injury initiation criterion

- (2)
- Damage evolution

## 3. Experimental Interlayer Model of a Double Cantilever Beam with a Steel Deck Asphalt Pavement

#### 3.1. Experimental Design of Double Cantilever Beam

#### 3.1.1. Specimen Design

#### 3.1.2. Test Piece Fabrication Method

#### 3.2. Finite Element Model

^{−5}, significantly improving the model convergence without substantial impacts on the strength and crack propagation predictions. The finite element model is illustrated in Figure 5.

## 4. Numerical Validation and Parameter Discussion

#### 4.1. Numerical Verification

#### 4.2. Crack Extension and Interface Debonding Analysis

^{−5}m, the stress value triggering the initiation of crack damage was reached. The crack started to extend towards the asphalt paving layer side and penetrated a single element. With a radial displacement of 2.531 × 10

^{−4}m, the crack extended to the interface between the bonding layer and the asphalt paving layer. Subsequently, as displacement increased, strain energy accumulated at the interface, waiting for the interface strength to be reached before releasing the strain energy. This was followed by the propagation of cracks along the interface.

^{−4}m, damage began to appear at the interface; however, the crack had not reached the interface. Under the influence of the stress field at the crack tip, damage accumulated at the interface, increasing as the crack approached. At a radial displacement of 2.537 × 10

^{−4}m, the crack reached the interface, causing further damage accumulation at the interface. However, debonding had not occurred, resulting in interface layering. Subsequently, at a radial displacement of 6.789 × 10

^{−4}m, partial debonding between the bonding layer and the asphalt paving layer occurred, leading to interface layering. Evidently, similar damage also arose at the interface between the steel bridge panel and the bonding layer, as depicted in Figure 11. This aligned with real-world scenarios, wherein after debonding, the steel plate remains partially bonded to the adhesive material while partially remaining smooth.

#### 4.3. Impact of Parameters

#### 4.3.1. Influence of Initial Crack Length

#### 4.3.2. Effect of Interface Parameters

- (1)
- Influence of Interface Stiffness

^{11}, 4.069 × 10

^{11}, 5.069 × 10

^{11}, 6.069 × 10

^{11}, and 7.069 × 10

^{11}N/mm

^{3}. The results demonstrated that as interface stiffness increased, the crack propagation path within the bonding layer expanded, as depicted in Figure 17. Specifically, when the stiffness was 3.069 × 10

^{11}N/mm

^{3}, the crack traversed 14 elements to reach the interface between the bonding and asphalt layers. However, for stiffness values of 5.069 × 10

^{11}and 7.069 × 10

^{11}N/mm

^{3}, the crack penetrated 15 elements and the penetration length increased. The load–displacement curves for a steel bridge deck and asphalt paving layer with varying stiffness values are shown in Figure 18a. Figure 18b provides a comparison with the results obtained through the cohesive method.

^{11}and 6.069 × 10

^{11}N/mm

^{3}yielded a good agreement with the actual strength values and improved the computational convergence. Therefore, when simulating the propagation of Type I cracks between a steel bridge deck and asphalt paving layers, an interface stiffness value in the range of 5.069 × 10

^{11}to 6.069 × 10

^{11}N/mm

^{3}is recommended. Figure 19 depicts the predicted curves of strength and failure displacement, revealing a notable linear relationship between them.

- (2)
- Influence of Interface Strength

#### 4.3.3. Influence of Bonding Layer Thickness

#### 4.3.4. Significance Analysis

## 5. Conclusions

- Cracks originating within the bonding layer propagated in the direction of the maximum principal stress until they reached the asphalt–bonding layer interface, resulting in interface damage and the occurrence of layering. Similar interface damage was observed between the bonding layer and the steel bridge deck. As displacement loads increased, various layering phenomena manifested at these interfaces.
- Longer crack lengths within the layer led to reduced crack propagation, resulting in diminished strength and increased failure displacement. Increased interfacial stiffness widened the crack propagation path within the layer, consequently reducing strength and augmenting failure displacement. Although the interfacial strength exhibited a minor influence on the crack propagation path, it significantly impacted overall strength and interlayer failure displacement.
- It is worth noting that interface stiffness and strength had minimal effects on the crack propagation path within the layer, but exerted a significant influence on the interlayer bonding strength. Enhanced stiffness diminished the bonding layer strength and failure displacement, while an elevated interface strength fortified the bonding layer strength and augmented failure displacement.
- Variations in the thickness of the bonding layer affected the crack propagation path within the layer. An initial increase in thickness enhanced the bonding strength, but subsequent increments resulted in reduced strength. Our analysis recommended an optimal bonding layer thickness of approximately 1 mm to achieve a higher strength.
- A significance analysis underscored that changes in interface stiffness had the most substantial impact on interlayer strength and failure displacement, followed by the influence of the crack length, interface strength, and bonding layer thickness. These findings shed light on the intricate interplay of parameters that influence crack propagation and interlayer bonding behavior.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Wang, H.; Li, G. Study of factors influencing gussasphalt mixture performance. Constr. Build. Mater.
**2015**, 101, 193–200. [Google Scholar] [CrossRef] - Ma, Y.; Wang, H.; Zhao, K.; Yan, L.; Yang, D. Study of a Modified Time Hardening Model for the Creep Consolidation Effect of Asphalt Mixtures. Materials
**2022**, 15, 2710. [Google Scholar] [PubMed] - Du, H.; Ni, F.; Ma, X. Crack resistance evaluation for In-service asphalt pavements by using SCB tests of layer-core samples. J. Mater. Civ. Eng.
**2021**, 33, 04020418. [Google Scholar] [CrossRef] - Zhang, M.; Hao, P.; Li, Y. Interfacial adhesive property in “asphalt mixture-PMA copolymer-steel plate” system: Experimental and molecular dynamics simulation. Constr. Build. Mater.
**2021**, 281, 122529. [Google Scholar] - Liu, Y.; Qian, Z.; Gong, M.; Huang, Q.; Ren, H. Inter-layer residual stress analysis of steel bridge deck pavement during gussasphalt pavement paving. Constr. Build. Mater.
**2022**, 324, 126624. [Google Scholar] [CrossRef] - Xue, C.; Wan, Y.; Wang, H. Mechanics Property Analysis for A New Steel Bridge Deck Pavement with UTAC—UHPC. For. Eng.
**2020**, 36, 76–84+93. [Google Scholar] - Liu, Y.; Qian, Z.; Yin, Y.; Ren, H. Investigation on inter-layer Behaviors of a Double-Layered Heterogeneous Asphalt Pavement Structure for Steel Bridge Deck. J. Mater. Civ. Eng.
**2022**, 34, 04022062. [Google Scholar] [CrossRef] - Jin, W.; Zhao, Y.; Wang, W.; He, F. Performance evaluation and optimization of waterproof adhesive layer for concrete bridge deck in seasonal frozen region using AHP. Adv. Mater. Sci. Eng.
**2021**, 2021, 5555535. [Google Scholar] [CrossRef] - Fu, J.; Shen, A.; Yuan, Z. Properties of Different Waterproof Bonding Layer Systems for Cement Concrete Bridge Deck Pavement. Coatings
**2022**, 12, 308. [Google Scholar] [CrossRef] - Zhang, M.; Hao, P.; Men, G.; Liu, N.; Yuan, G. Research on the compatibility of waterproof layer materials and asphalt mixture for steel bridge deck. Constr. Build. Mater.
**2021**, 269, 121346. [Google Scholar] [CrossRef] - Xu, Y.; Lv, X.; Ma, C.; Liang, F.; Qi, J.; Chou, Z.; Xu, S. Shear Fatigue Performance of Epoxy Resin Waterproof Adhesive Layer on Steel Bridge Deck Pavement. Front. Mater.
**2021**, 7, 618073. [Google Scholar] [CrossRef] - Ai, C.; Huang, H.; Rahman, A.; An, S. Establishment of a new approach to optimized selection of steel bridge deck waterproof bonding materials composite system. Constr. Build. Mater.
**2020**, 264, 120269. [Google Scholar] [CrossRef] - Chen, W.; Hui, B.; Rahman, A. Inter-layer Shear Characteristics of Bridge Deck Pavement through Experimental and Numerical Analysis. Materials
**2022**, 15, 7001. [Google Scholar] [CrossRef] [PubMed] - Xu, J.; Li, N.; Xu, T. Temperature changes of interlaminar bonding layer in different seasons and effects on mechanical properties of asphalt pavement. Int. J. Pavement Res. Technol.
**2022**, 15, 589–605. [Google Scholar] [CrossRef] - Zhang, W.; Li, Q.; Wang, J.; Meng, Y.; Zhou, Z. Aging Behavior of High-Viscosity Modified Asphalt Binder Based on Infrared Spectrum Test. Materials
**2022**, 15, 2778. [Google Scholar] [CrossRef] - Ma, X.; Kan, J.; Liu, S.; Tu, M.; Wang, D. Investigation of the Effect of Filler on Cohesive Bond Strength of Asphalt Mastic Using Binder Bond Strength (BBS) Test. Coatings
**2023**, 13, 1001. [Google Scholar] [CrossRef] - Chen, Z.; Xu, W.; Zhao, J.; An, L.; Wang, F.; Du, Z.; Chen, Q. Experimental Study of the Factors Influencing the Performance of the Bonding Interface between Epoxy Asphalt Concrete Pavement and a Steel Bridge Deck. Buildings
**2022**, 12, 477. [Google Scholar] [CrossRef] - Liu, Y.; Qian, Z.; Zheng, D.; Zhang, M. Interlaminar thermal effect analysis of steel bridge deck pavement during gussasphalt mixture paving. Int. J. Pavement Eng.
**2019**, 20, 1323–1335. [Google Scholar] [CrossRef] - Graczyk, M.; Zbiciak, A.; Michalczyk, R.; Kowalewski, Ł. Numerical modelling of bubbles formation in the bridge asphalt pavement under gas pressure impact. Transp. Res. Procedia
**2016**, 14, 3925–3934. [Google Scholar] [CrossRef] - Nie, W.; Wang, D.; Sun, Y.; Xu, W.; Xiao, X. Integrated design of structure and material of epoxy asphalt mixture used in steel bridge deck pavement. Buildings
**2021**, 12, 9. [Google Scholar] [CrossRef] - Yuya, W.; Yasushi, T.; Futoshi, K.; Kazuhiro, W. Evaluation of the Effect of Interlayer Bonding Condition on the Deterioration of Asphalt Pavement. Transp. Res. Rec.
**2023**, 2677, 03611981231153649. [Google Scholar] [CrossRef] - De Santis, Y.; Pasca, D.P.; Aloisio, A.; Stenstad, A.; Mahnert, K.C. Experimental, analytical and numerical investigation on the capacity of composite glulam beams with holes. Eng. Struct.
**2023**, 285, 115995. [Google Scholar] [CrossRef] - Koord, J.; Völkerink, O.; Petersen, E.; Hühne, C. Effect of low temperature on mode I and mode II interlaminar fracture toughness of CFRP-steel hybrid laminates. Compos. Part B Eng.
**2023**, 262, 110773. [Google Scholar] [CrossRef] - Yao, X.; Li, C.; Xu, T. Interfacial adhesive behaviors between SBS modified bitumen and aggregate using molecular dynamics simulation. Surf. Interfaces
**2022**, 33, 102245. [Google Scholar] [CrossRef] - Zhou, W.; Zhang, C.; Gan, S. Crack propagation analysis and strength prediction of glued joints based on XFEM-CZM coupling method. J. Beijing Univ. Aeronaut. Astronaut.
**2020**, 46, 2121–2130. [Google Scholar] - Feng, W.; Xu, H.; Yu, H.; Li, M. Adhesive damage and defect analysis of scarf-repaired composite by combining extended finite element method and cohesive zone model. J. Compos.
**2018**, 35, 1354–1360. [Google Scholar] - Belytschko, T.; Black, T. Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng.
**1999**, 45, 601–620. [Google Scholar] [CrossRef] - Moës, N.; Dolbow, J.; Belytschko, T. A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng.
**1999**, 46, 131–150. [Google Scholar] [CrossRef] - Zhuo, Z. Extended Finite Element Method; Tsinghua University Press: Beijing, China, 2012; pp. 35–36. [Google Scholar]
- Melenk, J.M.; Babuška, I. The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Eng.
**1996**, 139, 289–314. [Google Scholar] [CrossRef] - Gao, L.; Deng, X.; Zhang, Y.; Ji, X.; Li, Q. Fracture parameters and cracking propagation of cold recycled mixture considering material heterogeneity based on extended finite element method. Materials
**2021**, 14, 1993. [Google Scholar] [CrossRef] - Heidari-Rarani, M.; Sayedain, M. Finite element modeling strategies for 2D and 3D delamination propagation in composite DCB specimens using VCCT, CZM and XFEM approaches. Theor. Appl. Fract. Mech.
**2019**, 103, 102246. [Google Scholar] [CrossRef] - Wu, S. Study on the Interface Characteristics of Steel Deck Pavement with Engineered Cementitious Composites. Ph.D. Thesis, Southeast University, Dhaka, Bangladesh, 2019. [Google Scholar]
- Qian, Z.; Liu, Y. Mechanical analysis of waterproof bonding layer on steel bridge deck under bridge-temperature-load coupling effect. J. Southeast Univ. Nat. Sci. Ed.
**2012**, 42, 729–733. [Google Scholar]

**Figure 9.**Initiation and propagation of cracks in the bonding layer: (

**a**) crack initiation, U = 4.563 × 10

^{−5}m; (

**b**) crack propagation, U = 7.766 × 10

^{−5}m; (

**c**) crack propagation, U = 1.313 × 10

^{−4}m; (

**d**) crack reaches interface, U = 2.531 × 10

^{−4}m.

**Figure 10.**Interface layering process: (

**a**) interface undamaged, U = 1.223 × 10

^{−4}m; (

**b**) interface damage occurs, U = 1.343 × 10

^{−4}m; (

**c**) interface damage occurs, U = 6.213 × 10

^{−4}m; (

**d**) interface debonding, U = 7.132 × 10

^{−4}m.

**Figure 13.**Interface layering process using the CZM method: (

**a**) interface undamaged, U = 1.223 × 10

^{−4}m; (

**b**) interface damage occurs, U = 1.343 × 10

^{−4}m; (

**c**) interface damage occurs, U = 6.213 × 10

^{−4}m; (

**d**) interface debonding, U = 7.132 × 10

^{−4}m.

**Figure 25.**Strength and Failure Displacement Prediction Curves for Different Bond Layer Thicknesses.

Materials | Length/m | Width/m | Thickness/m |
---|---|---|---|

AC | 0.8 | 0.35 | 0.066 |

Q345qD | 0.8 | 0.35 | 0.014 |

Parameters | Steel Plate | AC Layer | Bonding Layer |
---|---|---|---|

$E/\mathrm{M}\mathrm{P}\mathrm{a}$ | 210,000 | 4500 | 100 [34] |

$\upsilon $ | 0.3 | 0.25 | 0.25 |

${\sigma}_{f}/\mathrm{M}\mathrm{P}\mathrm{a}$ | 0.8 | ||

${\tau}_{f}/\mathrm{M}\mathrm{P}\mathrm{a}$ | 0.5 | ||

${G}_{IC}/\left(\mathrm{J}\xb7{\mathrm{m}}^{-2}\right)$ | 233 | ||

${G}_{IIC}/\left(\mathrm{J}\xb7{\mathrm{m}}^{-2}\right)$ | 142 |

Interface Parameters | Numerical Value |
---|---|

$K/\left(\mathrm{N}\xb7{\mathrm{m}\mathrm{m}}^{-3}\right)$ | 5.069 × 10^{11} |

$N/\mathrm{M}\mathrm{P}\mathrm{a}$ | 0.8 |

$S=T/\mathrm{M}\mathrm{P}\mathrm{a}$ | 0.5 |

${G}_{IC}/\left(\mathrm{J}\xb7{\mathrm{m}}^{-2}\right)$ | 233 |

${G}_{IIC}={G}_{IIIC}/\left(\mathrm{J}\xb7{\mathrm{m}}^{-2}\right)$ | 142 |

**Table 4.**Influence of element count on load–displacement prediction results for Type I cracks between asphalt paving layers of a steel bridge deck.

Element Count | Maximum Load/N | Displacement/mm |
---|---|---|

50,000 | 6248.321 | 0.688 |

65,000 | 5882.212 | 0.656 |

70,000 | 5652.465 | 0.628 |

75,000 | 5544.823 | 0.629 |

85,000 | 5642.330 | 0.629 |

95,000 | 5598.426 | 0.631 |

Method | Maximum Load/N | Displacement/mm | Load Error/% | Displacement Error/% |
---|---|---|---|---|

Experiment | 5589.130 | 0.713 | 0 | 0 |

VCCT | 5450.182 | 0.504 | 2.4 | 29.1 |

XFEM–VCCT | 5536.503 | 0.554 | 0.9 | 22.3 |

CZM | 5771.36 | 0.713 | 3.2 | 0.1 |

XFEM–CZM | 5544.17 | 0.688 | 0.8 | 0.3 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhu, C.; Li, W.; Wang, H.
Analysis of Interlayer Crack Propagation and Strength Prediction of Steel Bridge Deck Asphalt Pavement Based on Extended Finite Element Method and Cohesive Zone Model (XFEM–CZM) Coupling. *Coatings* **2023**, *13*, 1973.
https://doi.org/10.3390/coatings13111973

**AMA Style**

Zhu C, Li W, Wang H.
Analysis of Interlayer Crack Propagation and Strength Prediction of Steel Bridge Deck Asphalt Pavement Based on Extended Finite Element Method and Cohesive Zone Model (XFEM–CZM) Coupling. *Coatings*. 2023; 13(11):1973.
https://doi.org/10.3390/coatings13111973

**Chicago/Turabian Style**

Zhu, Chen, Weiwei Li, and Hongchang Wang.
2023. "Analysis of Interlayer Crack Propagation and Strength Prediction of Steel Bridge Deck Asphalt Pavement Based on Extended Finite Element Method and Cohesive Zone Model (XFEM–CZM) Coupling" *Coatings* 13, no. 11: 1973.
https://doi.org/10.3390/coatings13111973