# An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Original Models

#### 2.1. Original PPR Model

#### 2.2. Unloading/Reloading Relationship

## 3. Simplified PPR Traction–Separation Law

#### 3.1. Modification

#### 3.2. Path Dependence of Work-of-Separation

#### 3.2.1. Proportional Separation

#### 3.2.2. Non-Proportional Separation

#### 3.3. Mixed-Mode Bending (MMB) Test Verification

_{0}= 33.7 mm, c = 60 mm, and B = 25.4 mm.

## 4. Improved Unloading/Reloading Relationship

#### 4.1. Modification

#### 4.2. Comparison

^{2}) for model (ii) is lower than ${\varphi}_{n}$.

^{2}(${\varphi}_{n}$) at $\mathsf{\Delta}=0.016\mathrm{mm}$ to 97.0 J/m

^{2}at $\mathsf{\Delta}=0.017\mathrm{mm}$. Such an energy discontinuity is inherent to any CZM model that obeys a curved line in the reversible range and an unloading straight line when irreversibility has appeared, which was discussed in detail by Gilormini et al. [39]. Therefore, the small energy jump is accepted.

^{2}.

## 5. Application

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\Psi $ | potential function for cohesive fracture |

${\mathsf{\Delta}}_{n}$, ${\mathsf{\Delta}}_{t}$ | normal and tangential separation |

${\mathsf{\Delta}}_{n}^{max}$, ${\mathsf{\Delta}}_{t}^{max}$ | maximum normal and tangential separations in a loading history |

${\mathsf{\Delta}}_{n}^{\chi}$, ${\mathsf{\Delta}}_{t}^{\gamma}$ | state variables for maximum normal and tangential traction |

${\mathsf{\Delta}}_{n}^{i}$, ${\mathsf{\Delta}}_{t}^{i}$ | normal and tangential separations at step i |

${\varphi}_{n}$, ${\varphi}_{t}$ | mode I and mode II fracture energy |

${\Gamma}_{n}$, ${\Gamma}_{t}$ | energy constants in the PPR model |

${\delta}_{n}$, ${\delta}_{t}$ | normal and tangential final crack opening widths |

${\delta}_{n}^{peak}$, ${\delta}_{t}^{peak}$ | normal and tangential separation for peak traction |

$\alpha $,$\beta $ | shape parameter |

$m$, $n$ | exponents |

${T}_{n}$, ${T}_{t}$ | normal and tangential tractions |

${T}_{n}^{\upsilon}$, ${T}_{t}^{\upsilon}$ | normal and tangential tractions for the unloading/reloading relation |

${\sigma}_{max}$, ${\tau}_{max}$ | normal and tangential cohesive strength |

${\lambda}_{n}$, ${\lambda}_{t}$ | initial slope indicators in the PPR model |

${\overline{\mathsf{\delta}}}_{\mathrm{n}}$, ${\overline{\mathsf{\delta}}}_{\mathrm{t}}$ | normal and tangential conjugate final crack opening widths |

$\theta $ | separation angle between the path direction and tangent |

Δ | magnitude of ${\mathsf{\Delta}}_{n}={\mathsf{\Delta}}_{t}$ applied during preloading |

${\mathsf{\Delta}}_{r}$ | separation for proportional path |

${\mathsf{\Delta}}_{n,max}$, ${\mathsf{\Delta}}_{t,max}$ | maximum normal and tangential separations |

${W}_{sep}$ | work-of-separation |

${W}_{n},{W}_{t}$ | work conducted by the normal and tangential cohesive traction |

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**Figure 2.**The cohesive tractions with and ${\varphi}_{t}=200\mathrm{N}/\mathrm{m}$ (

**a**,

**b**); or ${\varphi}_{n}=200\mathrm{N}/\mathrm{m}$ and ${\varphi}_{t}=100\mathrm{N}/\mathrm{m}$ (

**c**,

**d**); ${\varphi}_{n}=200\mathrm{N}/\mathrm{m}$ and ${\varphi}_{t}=200\mathrm{N}/\mathrm{m}$ (

**e**,

**f**). Normal traction (

**a,c,e**); tangential traction (

**b,d,f**).

**Figure 4.**The cohesive tractions with ${\varphi}_{n}=100\mathrm{N}/\mathrm{m}$, ${\varphi}_{t}=200\mathrm{N}/\mathrm{m}$, ${\sigma}_{max}=40\mathrm{MPa}$, ${\tau}_{max}=30\mathrm{MPa}$, $\alpha =5$, $\beta =1.3$, ${\lambda}_{n}=0.1$, and ${\lambda}_{t}=0.2$. Normal traction (

**a**); tangential traction (

**b**).

**Figure 5.**Proportional separation path (${\mathsf{\Delta}}_{r}$) with the separation angle ($\theta $). Before separation (

**a**); after separation (

**b**).

**Figure 6.**The work-of-separation ${W}_{sep}$ (

**a**,

**b**); work conducted by the normal traction ${W}_{n}$ (

**c**,

**d**); and work conducted by the tangential traction ${W}_{t}$ (

**e**,

**f**); with respect to the change of the proportional angle $\theta $. Park–Paulino–Roesler (PPR) model (

**a,c,e**); simplified PPR traction–separation law (SPPR) model (

**b,d,f**).

**Figure 7.**Two arbitrary separation paths for the debonding analysis: (

**a**) non-proportional Path 1 and (

**b**) non-proportional Path 2.

**Figure 8.**Variation of the work-of-separation for the case of ${\varphi}_{n}<{\varphi}_{t}$ (${\varphi}_{n}=100\mathrm{N}/\mathrm{m}$ and ${\varphi}_{t}=200\mathrm{N}/\mathrm{m}$): non-proportional Path 1 (

**a**,

**b**); or non-proportional Path 2 (

**c**,

**d**). PPR model (

**a**,

**c**); SPPR model (

**b**,

**d**).

**Figure 9.**Variation of the work-of-separation for the case of ${\varphi}_{n}>{\varphi}_{t}$ (${\varphi}_{n}=200\mathrm{N}/\mathrm{m}$ and ${\varphi}_{t}=100\mathrm{N}/\mathrm{m}$): non-proportional Path 1 (

**a**,

**b**); or non-proportional Path 2 (

**c**,

**d**). PPR model (

**a**,

**c**); SPPR model (

**b**,

**d**).

**Figure 11.**Computational results for different models (

**a**) considering the same facture energy (${\varphi}_{n}={\varphi}_{t}=1\mathrm{N}/\mathrm{m}$), or (

**b**) considering different facture energies (${\varphi}_{n}=1\mathrm{N}/\mathrm{m}and{\varphi}_{t}=2\mathrm{N}/\mathrm{m}$).

**Figure 12.**Variations of the normal traction component (

**a**) and tangential component (

**b**) under mode I (mode II) loading, mixed-mode ${\mathsf{\Delta}}_{n}={\mathsf{\Delta}}_{t},$ and mixed-mode ${\mathsf{\Delta}}_{n}=0.5{\mathsf{\Delta}}_{t}$.

**Figure 13.**Dissipated energy in the process of proportional loading/unloading and mode I reloading, with the increase in the amplitude of proportional loading. Model (i) (solid line), model (ii) (dashed line), and model (iii) (dotted line).

**Figure 14.**Variations of the traction components ${T}_{n}$ (solid lines) and ${T}_{t}$ (dashed lines) during the process of proportional loading/unloading and mode I reloading, for a proportional loading amplitude of $\mathsf{\Delta}=0.016\text{}\mathrm{mm}$. Model (i) and model (iii) (

**a**); model (ii) (

**b**).

**Figure 15.**Variations of the traction components ${T}_{n}$(solid lines) and ${T}_{t}$ (dashed lines) during the process of proportional loading/unloading and mode I reloading, for a proportional loading amplitude of $\mathsf{\Delta}=0.017\text{}\mathrm{mm}$. Model (i) (

**a**); model (ii) and model (iii) (

**b**).

**Figure 16.**Variations of the traction components ${T}_{n}$ (solid lines) and ${T}_{t}$ (dashed lines) during the process of proportional loading/unloading and mode I reloading, for a proportional loading amplitude of $\mathsf{\Delta}=0.019\text{}\mathrm{mm}$ (

**a**,

**b**); or $\mathsf{\Delta}=0.020\text{}\mathrm{mm}$ (

**c**,

**d**). Model (i) (

**a,c**); model (ii) and model (iii) (

**b,d**).

**Figure 17.**Variations of the traction components ${T}_{n}$ (solid lines) and ${T}_{t}$ (dashed lines) during the process of proportional loading/unloading and mode I reloading, for a proportional loading amplitude of $\mathsf{\Delta}=0.042\text{}\mathrm{mm}$ (

**a**,

**b**); or $\mathsf{\Delta}=0.043\text{}\mathrm{mm}$ (

**c**,

**d**). Model (i) (

**a,c**); and model (ii) and model (iii) (

**b,d**).

**Figure 18.**Variations of the traction components ${T}_{n}$ (solid lines) and ${T}_{t}$ (dashed lines) during the process of proportional loading/unloading and mode I reloading, for a proportional loading amplitude of $\mathsf{\Delta}=0.098\text{}\mathrm{mm}$ (

**a**,

**b**); or $\mathsf{\Delta}=0.099\text{}\mathrm{mm}$ (

**c**,

**d**). Model (i) (

**a,c**); model (ii) and model (iii) (

**b,d**).

**Figure 20.**The finite element model of CRTS-II slab track before longitudinal connection (1/4 model).

**Figure 21.**Interface crack opening (COPEN) distribution of the slab track system as a result of temperature change (scale factor is 20).

**Figure 22.**Interface stresses of slab corner varying with time: (

**a**) normal stress, (

**b**) longitudinal shear stress, and (

**c**) lateral shear stress.

**Figure 23.**Interface normal stresses (CPRESS) between slab and CA mortar layer when interface crack happens: (

**a**) the model proposed in the paper, (

**b**) PPR model, and (

**c**) cohesive zone model in ABAQUS.

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## Share and Cite

**MDPI and ACS Style**

Zhong, Y.; Gao, L.; Cai, X.; An, B.; Zhang, Z.; Lin, J.; Qin, Y.
An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks. *Appl. Sci.* **2021**, *11*, 456.
https://doi.org/10.3390/app11010456

**AMA Style**

Zhong Y, Gao L, Cai X, An B, Zhang Z, Lin J, Qin Y.
An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks. *Applied Sciences*. 2021; 11(1):456.
https://doi.org/10.3390/app11010456

**Chicago/Turabian Style**

Zhong, Yanglong, Liang Gao, Xiaopei Cai, Bolun An, Zhihan Zhang, Janet Lin, and Ying Qin.
2021. "An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks" *Applied Sciences* 11, no. 1: 456.
https://doi.org/10.3390/app11010456