# Re-Distribution of Welding Residual Stress in Fatigue Crack Propagation Considering Elastic–Plastic Behavior

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## Abstract

**:**

_{res}of the MT specimen is deduced. The stress ratio effect on K

_{res}during fatigue crack propagation is analyzed and a good agreement between experimental and numerical results is achieved.

## 1. Introduction

_{res}is used to quantify the effect of welding-induced residual stress (WRS) on crack propagation behavior. It is considered in calculating the effective stress intensity factor range $\mathit{\Delta}$K

_{eff}. Currently, there are two mainstream methods for the calculation of K

_{res}: the weight function method (WFM) and the finite element method (FEM). The WFM can calculate K

_{res}quickly based on the assumption of linear elasticity. At the same time, FEM can consider the effect of the material’s plasticity, due to which it can obtain more accurate results.

_{eff}of a T-shaped welded joint under the influence of WRS re-distribution using a 2D finite element method. Lee et al. [5] analyzed the variation in WRS during crack propagation, and found that the results were more accurate when considering the re-distribution behavior. Terada [6] proposed the WRS re-distribution rule for butt welding based on the superposition principle and ideal residual stress distribution rule, which were in good agreement with experimental results. Xu [7] studied the crack propagation behavior of the butt-welded plate and found that the WRS was constantly released during crack propagation, with a decreasing impact on the crack propagation behavior.

_{res}

_{,}and stress ratio R based on elastic–plastic simulation results. Finally, this study presented a polynomial prediction model to describe K

_{res}by loading ratio R and crack size a/W. All conclusions are drawn in Section 5.

## 2. Experimental Procedure

#### 2.1. Material and Main Dimensions of Specimens

#### 2.2. Test Setup

_{B}= 0). For butt specimens with pre-cracks, only longitudinal residual stress distribution on the crack extension line was measured. The measurement paths and local coordinate systems on the specimen are illustrated in Figure 2. Residual stress data will be presented in Section 2.3.1.

#### 2.3. Experimental Results and Discussion

#### 2.3.1. Distribution of Initial Welding Residual Stress

_{y}for all specimens are given in this section. Firstly, the difference in the longitudinal initial WRS distribution of measurement paths of butt specimens without pre-cracks is discussed, as shown in Figure 4. The positions where the tension–compression stress converts or reaches peak value are close, and the deviation of initial WRS results of each measurement path was kept within 20 MPa. This phenomenon indicates that the stress distribution of WRS on the upper and lower surfaces is similar.

_{y,0}= 131 MPa and c = 8.94 are the initial WRS distribution parameters obtained from X-ray measurement results.

_{y}decreases to 0, and there is no significant difference between distribution functions and test data. The peaks of the two are also very close to each other, which indicates the reliability of the measurements.

#### 2.3.2. Welding Residual Stress Re-Distribution Calculation

_{y}(x) is the initial longitudinal WRS value calculated by Equation (2), σ

_{y,re}(x) is the longitudinal WRS re-distribution value, and a is the current crack length. It should be noted that both Equations (1) and (3) are named as Terada rules, Equation (1) is used to describe the distribution of the initial WRS and Equation (3) is used to describe the re-distribution of the WRS. To simplify the calculation, Terada [6] proposed a numerical integration approximation method:

^{2}− a

^{2})

^{1/2}at the crack tip, the accuracy of the Terada rule is not satisfying near the crack tip.

_{y}on the crack extension line:

#### 2.3.3. Impact of Cyclic Loading on Residual Stress Distribution

## 3. Finite Element Analysis

#### 3.1. Import of Initial Welding Residual Stress

#### 3.2. Residual Stress Intensity Factor K_{res} by XFEM

- Import the initial stress distribution into the FE model (in Section 3.1).
- Calculate the stress–strain value for the first crack increment step under cyclic loadings, although all elements’ numbers will be kept the same following the initial increment step.
- Update the crack length according to the experimental data.
- Read the stress–strain state of each element under the previous crack length and assign it to the current model.
- Extract the K
_{res}at the crack tip of each increment using J-integral.

_{res}, the external load has been unloaded to 0.

#### 3.3. Effect of Cyclic Loadings on the Simulation Results

_{res}.

#### 3.4. Simulation Results and Comparison

## 4. Residual Stress Intensity Factor K_{res}

#### 4.1. Comparison between WFM and FEM

_{res}. The K

_{res}calculation equation for WFM is as follows,

_{y}(x) distribution function. They are the Terada distribution function, the Tada and Paris distribution function, as shown in the previous Section 2.3.1, and a segmented approximation function (SAF) distribution based on measured data, which can be expressed as

_{res}-2a/W curve is calculated and summarized based on the elastic–plastic FE simulation in Section 3.2, which is compared with the calculation results of WFM above, as shown in Figure 15.

- The magnitudes of K
_{res}predicted using WFM and the FE method are similar, all in the range of −4~12 MPa·m^{1/2}. At the same time, WFM neglects the effect of plastic deformation on residual stress near the crack tip, resulting in a different trend. - With the crack length increase, K
_{res}calculated using all methods shows a decreasing trend, except for that using the Terada distribution. Combined with the initial WRS distribution in Figure 5, it is speculated that the increase in K_{res}may be related to the overestimation of peak compressive stress when using the Terada distribution. The K_{res}curve predicted using Terada distribution increased near 2a/W = 0.55, where the initial residual stress is precisely located near the right side of the compressive stress peak, raising K_{res}corresponding to the release of compressive stress during crack propagation. - The calculation results of WFM depend on the choice of the initial distribution function. Weight function adopting initial distribution SAF, like that in the FE model, obtains the closest results to the simulation method. At the same time, there is a significant difference between the results of the Terada and Tada and Paris functions. This phenomenon indicates that when predicting K
_{res}, the accuracy significantly depends on the initial distribution. - With the increase in 2a/W, there is a significant difference between WFM and simulation results. It can be seen from Figure 15 that the trend of the K
_{res}-2a/W curves of the WFM tends to be stable no matter what initial distribution is used. The influence of the elastic–plastic behavior of the material causes this phenomenon. When the local stress is close to the fracture limit, a significant yield area occurs on the specimen, resulting in its re-distribution behavior being entirely dominated by plasticity. Crack tips and surfaces are subjected to the compression of the plastic zone after unloading, and the K value extracted by the J-integral gradually decreases. However, WFM is based on linear elasticity assumption, considering that the residual ligament of the specimen is too short to lead to the general release of residual stress. In addition, the influence of plastic re-distribution of residual stress on crack tip is not considered in WFM. Therefore, WFM cannot predict the rapid decrease in K_{res}when the specimen approaches the fracture toughness. - The K
_{res}-2a/W curves calculated using the simulation considering the elastic–plastic response of materials show a significant stress ratio effect, which WFM cannot predict.

#### 4.2. The Stress Ratio Effect of K_{res}

_{res}-2a/W curve calculated using the simulation method shifts downward, while the variation trend of the K

_{res}-2a/W curve at different stress ratios is the same. Therefore, it can be considered that the intercept of the K

_{res}-2a/W curve is related to the stress ratio R.

_{res}-2a/W curve shows a strong linear relationship with stress ratio R. Therefore, it is assumed that the K

_{res}-2a/W curve is linear with the decreasing offset degree of stress ratio R in this study. The K

_{res}-2a/W curve is fitted using a cubic polynomial based on the calculation results of the FEM. The fitting model is expressed as follows:

_{res}calculated by FEM. In general, the prediction deviation of the fitted model is acceptable due to the small magnitude of K

_{res}. Therefore, for the MT specimens in this study, it is proven that the fitted model predicts K

_{res}well.

## 5. Conclusions

- The accuracy of the weight function method depends on the selection of the initial distribution function. Meanwhile, the stress ratio effect of K
_{res}cannot be predicted using the weight function method. - Terada’s re-distribution rule overestimates the increment of longitudinal residual stresses, and it is not suitable for predicting the re-distribution of residual stresses in marine high-tensile steel.
- Compared with the Terada re-distribution rule and static simulation method with initial distribution, the elastic–plastic continuous simulation method considering load timing is more consistent with the experimental measurements, which indicates that the plastic re-distribution behavior should be considered in the simulation of fatigue crack propagation.
- The K
_{res}values calculated using elastic–plastic continuous simulation procedure indicate that K_{res}is affected by the crack size a/W, and exhibits a noticeable stress ratio effect when the applied stress amplitude is kept constant, which is not reflected by WFM. - The K
_{res}fitting model related to a/W considering the stress ratio effect is established, and the results are in good agreement with the finite element calculation results, which can better reflect the WRS plastic re-distribution behavior.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | current crack length |

a_{0} | initial crack length |

E | Young’s modulus |

h | specimen length |

R | loading/stress ratio |

W | specimen width |

v | Poisson’s ratio |

CT | compact tension (specimen) |

FE(M) | finite element (method) |

LEFM | linear elastic fracture mechanics |

MT | middle tension (specimen) |

SIF | stress intensity factor |

WFM | weight function method |

WRS | welding residual stress |

XFEM | extended finite element method |

C_{1}, C_{2}, γ_{1}, γ_{2} | the Chaboche combined hardening model constants |

K_{res} | residual stress intensity factor |

ΔK_{eff} | effective stress intensity factor range |

σ_{0} | initial yield strength |

Δσ_{nominal} | nominal stress range |

σ_{y} | initial longitudinal welding residual stress |

σ_{y,0} | the maximum welding residual stress at the center line of the welding |

σ_{y,res} | longitudinal welding residual stress |

Δσ_{y,res} | change in longitudinal residual stress |

σ_{ys} | yield strength |

σ_{u} | ultimate tension strength |

h(a,x) | weight function |

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**Figure 5.**Initial WRS of butt specimen without pre-cracks found using X-ray and distribution function.

**Figure 8.**Variation in longitudinal residual stress of the MT specimens without pre-cracks with the number of cyclic loadings.

**Figure 13.**Influence of fatigue load cycles on residual stress: (

**a**) different numbers of cycles; (

**b**) different crack lengths.

E/GPa | v | σ_{ys}/MPa | σ_{u}/MPa | σ_{0}/MPa | C_{1} | γ_{1} | C_{2} | γ_{2} |
---|---|---|---|---|---|---|---|---|

219 | 0.35 | 635 | 688 | 565 | 66,500 | 1485 | 950 | 14.25 |

Specimen Number | Δσ_{nominal}/MPa | R | Control Group Number |
---|---|---|---|

R − 0.1 | 100 | −0.1 | - |

R0.1 | 100 | 0.1 | R0.1-B |

R0.3 | 100 | 0.3 | R0.3-B |

R0.5 | 100 | 0.5 | R0.5-B |

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**MDPI and ACS Style**

Xia, Y.; Yue, J.; Lei, J.; Yang, K.; Garbatov, Y.
Re-Distribution of Welding Residual Stress in Fatigue Crack Propagation Considering Elastic–Plastic Behavior. *J. Mar. Sci. Eng.* **2023**, *11*, 2378.
https://doi.org/10.3390/jmse11122378

**AMA Style**

Xia Y, Yue J, Lei J, Yang K, Garbatov Y.
Re-Distribution of Welding Residual Stress in Fatigue Crack Propagation Considering Elastic–Plastic Behavior. *Journal of Marine Science and Engineering*. 2023; 11(12):2378.
https://doi.org/10.3390/jmse11122378

**Chicago/Turabian Style**

Xia, Yuxuan, Jingxia Yue, Jiankang Lei, Ke Yang, and Yordan Garbatov.
2023. "Re-Distribution of Welding Residual Stress in Fatigue Crack Propagation Considering Elastic–Plastic Behavior" *Journal of Marine Science and Engineering* 11, no. 12: 2378.
https://doi.org/10.3390/jmse11122378