# Effect Range of the Material Constraint-I. Center Crack

^{*}

## Abstract

**:**

## 1. Introduction

_{2}[5], T

_{Z}[6,7,8], have been established to represent the stress fields at the crack tip under different geometry constraint conditions. In addition, the constraint effect due to material strength mismatch, which be called the material constraint, is also an important factor effects on the fracture behavior of material.

## 2. Materials and Models Design

#### 2.1. Materials

#### 2.2. Models Design

#### 2.3. Gurson-Tvergaard-Needleman (GTN) Damage Model

_{1}, q

_{2}and q

_{3}, the void nucleation parameters ε

_{Ν}, S

_{N}and f

_{N}, the initial void volume fraction f

_{0}, the critical void volume fraction f

_{C}and the final failure parameter f

_{F}. The void coalescence occurs when the void volume fraction reaches the critical value f

_{C}, and the fracture occurs when the void volume fraction reaches the final value f

_{F}. These parameters have been obtained and listed in Table 1 [26].

_{52Mb}= 16 mm is illustrated in Figure 4a, the minimum size of mesh in the crack growth region is 0.1 mm × 0.1 mm [27], as shown in Figure 4b. This typical model contains 75,872 elements and 87,849 nodes. In addition, the surface-to-surface contact (explicit) interaction type was used in the model. Moreover, the sliding formulation is finite sliding, the mechanical constraint formulation is kinematic contact method.

## 3. Results and Discussion

#### 3.1. “121” Model

_{52Mb}= 0 mm, it is the same with the homogeneous material A508; when the W

_{52Mb}= 80 mm, it is the same with the homogeneous material 52Mb. Thus, the results in Figure 5 show that for an over-matched joint, the J-resistance curve of the joint is higher than the base material. In addition, a notable phenomenon is that when the width of 52Mb is up to 8 mm, the J-resistance curves of the “121” models are same with the J-resistance curve of homogeneous material 52Mb. It means that the crack is out of the effect range of the material constraint induced by the A508/52Mb interface. In this condition, it does not matter even if the material on the outside is soft or hard. That is, when the crack locates out of the effect range of material constraint, the fracture resistance curve of the weld joint no longer influenced by the material constraint anymore. Of course, the effect range is also related to different materials and models.

_{p}= 0.1 isoline at crack tip at the same J-integral (J = 1600 kJ/m

^{2}) for different “121” models. It can be found that though the distributions of equivalent plastic strain are different for different models, but the equivalent plastic strains surrounded by ε

_{p}= 0.1 isoline are within the scope of 8 mm for all the models. When the interface is located within this scope, the J-resistance curve will be affected by the material constraint; when the interface is located outside this scope, the J-resistance curve will not be affected by material constraint. This scope is the effect range of the material constraint.

_{p}= 0.1 isoline reflect the same change rule with the J-resistance curves, as shown in Figure 7. Because the constraint is the resistance of a structure against plastic deformation, at the same J-integral (driving force) a lower plastic deformation reflects a higher constraint and a lower J-resistance curve, and vice versa. The same change rules between J-resistance curves and areas can prove each other and also reflect the change rules are related to the constraint.

_{p}= 0.1 isoline was selected here, when a small ε

_{p}value was selected, the scope will beyond the 8 mm. Therefore, there may exist a main control value or control zone. For this study, the main control value is ε

_{p}= 0.1.

#### 3.2. “123” Model

_{52Mb}= 0 mm has the lowest J-resistance curve and the model with W

_{52Mb}= 4 mm has the highest J-resistance curve. When the width of 52Mb is up to 16 mm, the J-resistance curve will not change with increasing of the 52Mb’s width.

_{52Mb}= 0 mm, the model is the same with the bimetallic welded joint with an interface crack. In this condition, the model has the lowest J-resistance curve, which shows that the interface crack in bimetallic welded joint is very dangerous. With increasing of the width of 52Mb, the J-resistance curve of the model increases. When the W

_{52Mb}= 4 mm, there exists an optimal width and the model has the highest J-resistance curve. Then, the J-resistance curves of the models decrease and remain steady at last.

_{p}= 0.1 isoline at crack tip at the same J-integral (J = 1600 kJ/m

^{2}) for different “123” models. It reflects the same change rule with the J-resistance curves. The same change rules can prove each other also.

#### 3.3. “12321” Model

_{p}= 0.1 isoline at crack tip at the same J-integral (J = 1600 kJ/m

^{2}) for different “12321” models are shown in Figure 10b, which also reflects the same change rule with the J-resistance curves.

_{52Mw}= 0 mm has the lowest J-resistance curve and the model with W

_{52Mw}= 16 mm has the highest J-resistance curve. When the width of 52Mw up to 32 mm, the J-resistance curve of the model will not change by increasing the 52Mw’s width.

_{52Mw}= 16 mm, there exists an optimal width and the model has the highest J-resistance curve. Then, the J-resistance curves of the models decrease and remain steady at last when the total width of 52Mb and 52Mw over the effect range of the material constraint.

_{p}= 0.1 isoline at crack tip at the same J-integral (J = 1600 kJ/m

^{2}) for different “12321” models are shown in Figure 11b, which also reflects the same change rule with the J-resistance curves.

#### 3.4. “12324” Model

_{52Mb}= 0 mm and W

_{52Mb}= 0.5 mm have the lowest J-resistance curves and the model with W

_{52Mb}= 2 mm has the highest J-resistance curve. When the width of 52Mb up to 8 mm, the J-resistance curve will not change by increasing the 52Mb’s width.

_{52Mb}= 2 mm, there exists an optimal width and the model has the highest J-resistance curve. Then, the J-resistance curves of the models decrease and remain steady at last when the width of 52Mb over the effect range of the material constraint.

_{p}= 0.1 isoline at crack tip at the same J-integral (J = 1600 kJ/m

^{2}) for different “12324” models are shown in Figure 12b, which also reflects the same change rule with the J-resistance curves.

_{p}= 0.1 isoline at crack tip at the same J-integral (J = 1600 kJ/m

^{2}) for different “12324” models are shown in Figure 13b, which also reflects the same change rule with the J-resistance curves.

## 4. Conclusions

- (1)
- For all the weld joints, the effect ranges of the material constraints are real. When the crack locates without the effect range of material constraint, the fracture resistance curves of the weld joints are no longer influenced by the material constraint any more.
- (2)
- The J-resistance curves of the weld joints are influenced by all the materials within the effect range, no matter whether the material is adjacent to the crack or not.
- (3)
- The areas surrounded by the ε
_{p}isoline reflect the same change rule with the J-resistance curves. The J-resistance curves of the materials are controlled by the strain fields rather than the stress fields, and there may exist a main control value or control zone.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The true stress-strain curves of four materials. Reprinted from Materials Science and Engineering A, 568, Wang, H.T.; Wang, G.Z.; Xuan, F.Z.; Liu, C.J.; Tu, S.T.; Local mechanical properties of a dissimilar metal welded joint in nuclear power systems, 108, Copyright 2013, with permission from Elsevier. [22].

**Figure 3.**Four different basic models, (

**a**) “121” model, (

**b**) “123” model, (

**c**) “12321” model and (

**d**) “12324” model.

**Figure 10.**The J-resistance curves (

**a**) and the areas surround by the ε

_{p}= 0.1 isoline (

**b**) of different “12321” models with the same W

_{52Mw}.

**Figure 11.**The J-resistance curves (

**a**) and the areas surround by the ε

_{p}= 0.1 isoline (

**b**) of different “12321” models with the same W

_{52Mb}.

**Figure 12.**The J-resistance curves (

**a**) and the areas surround by the ε

_{p}= 0.1 isoline (

**b**) of different “12324” models with the same W

_{52Mw}.

**Figure 13.**The J-resistance curves (

**a**) and the areas surround by the ε

_{p}= 0.1 isoline (

**b**) of different “12324” models with the same W

_{52Mb}.

**Table 1.**The GTN parameters of different materials. Reprinted by permission from Springer, Copyright 2017. [26]

Material | A508 | 52Mb | 52Mw | 316L |
---|---|---|---|---|

q_{1} | 1.5 | 1.5 | 1.5 | 1.5 |

q_{2} | 1 | variable | 1 | variable |

q_{3} | 2.25 | 2.25 | 2.25 | 2.25 |

ε_{Ν} | 0.3 | 0.3 | 0.3 | 0.3 |

S_{N} | 0.1 | 0.1 | 0.1 | 0.1 |

f_{N} | 0.002 | 0.002 | 0.002 | 0.002 |

f_{0} | 0.00008 | 0.000001 | 0.00015 | 0.000001 |

f_{C} | 0.04 | 0.04 | 0.04 | 0.04 |

f_{F} | 0.25 | 0.25 | 0.25 | 0.25 |

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**MDPI and ACS Style**

Yang, J.; Wang, L.
Effect Range of the Material Constraint-I. Center Crack. *Materials* **2019**, *12*, 67.
https://doi.org/10.3390/ma12010067

**AMA Style**

Yang J, Wang L.
Effect Range of the Material Constraint-I. Center Crack. *Materials*. 2019; 12(1):67.
https://doi.org/10.3390/ma12010067

**Chicago/Turabian Style**

Yang, Jie, and Lei Wang.
2019. "Effect Range of the Material Constraint-I. Center Crack" *Materials* 12, no. 1: 67.
https://doi.org/10.3390/ma12010067