**1. Introduction**

Constraint is the resistance of a structure or specimen against plastic deformation [1]. In recent years, the constraint effect due to structure or specimen geometry have been investigated as an important factor affecting the stress distribution around a crack. Some constraint parameters, such as *T* [2], *Q* [3,4], *A2* [5], *TZ* [6–8], 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.

The material constraint was firstly demonstrated by Joch et al. [9] and Burstow et al. [10] to show how the slip-line fields were changed by altering the yield strength of the base material. Then, Zhang et al. [11] analyzed a two-material problem where the crack was located in the interface of two dissimilar materials and established a material constraint parameter *M* to consider the effect of strength mismatch on crack tip stress fields, as follows:

$$M = \frac{\sigma\_{Yw}}{\sigma\_{Yb}},$$

where the *σYw* is the yield stress of the weld material and *σYb* is the yield stress of the base material. They also proposed that the stress fields of an interface crack in a mismatched problem could be obtained using the *J-Q-M* formulation, which was derived by extending the *J-Q* theory. Betegón et al. [12] defined a procedure similar to the *J-T*, also by establishing an additional parameter *β<sup>m</sup>* that quantifies the material constraint, and a total constraint parameter *β<sup>T</sup>* was defined as follows:

$$
\beta\_T = \beta\_m \cdot \sqrt{\frac{a}{h}} + \beta\_{\mathfrak{g}'} \tag{2}
$$

where *β<sup>m</sup>* is a constraint parameter defined for the overmatched welded joints to quantify the material constraint effect on the crack tip stress fields, *β*<sup>g</sup> is a geometry parameter by means of the *T*-stress to quantify the geometry constraint, *a* is crack length and *h* is weld semi-width. Recently, the author [13–15] defined a unified constraint parameter *Ap* based on the areas surrounded by the equivalent plastic strain (*ε <sup>p</sup>*) isolines ahead of the crack tip to characterize both geometry and material constraint. The unified constraint parameter *Ap* was defined as follows:

$$A\_p = \frac{A\_{PEEQ}}{A\_{ref}},$$

where *APEEQ* is the areas surrounded by the *ε <sup>p</sup>* isolines ahead of the crack tip and *Aref* is the reference areas surrounded by the *ε <sup>p</sup>* isolines in a standard test.

Furthermore, many scholars focused their studies on the fracture behavior of bi-material affected by the material constraint. Negre et al. [16] and Samal et al. [17] investigated the altering of fracture resistance and crack path deviation in the bi-material interface region affected by the material constraint. Fan et al. [18–20] studied the *J*-resistance curves, fracture toughness, crack growth paths and stress triaxiality of bi-materials under different work-hardening mismatches. Besides, some scholars focused their studies on the fracture behavior of dissimilar metal-welded joints affected by the material constraint. Rakin et al. [21] investigated the fracture behaviors of the over-matched and under-matched high-strength low-alloyed steel weld joints. Wang et al. [22,23] studied the local fracture resistances and crack growth paths of a dissimilar metal-welded joint at different crack positions with different material constraints. Xue et al. [24] investigated the stress and strain of a micro region influenced by material yield strength mismatch at the crack tip of a dissimilar metal-welded joint. Zhu et al. [25] studied the stress fields of a crack tip affected by material constraints in a nuclear pressure steel A508-III dissimilar metal welded joint.

These studies clarified the effect of a material constraint on the fracture behaviors of welded joints, and laid the foundation for the building of accurate structure integrity assessment. Nevertheless, most studies focus their attention on the strength mismatch of both sides of the crack, such as over-match, under-match, and so on. There is another interesting and important issue, the effect range of the material constraint, which also needs to be clarified. This includes whether exists the effect zone or not, who effects it, whether the material constraint affected by the no adjacent area or not, and so on. Solving this issue is of significance in developing solid mechanics, optimizing joint design and structure integrity assessment.

Thus, in this study, different basic models which represent different single metallic-welded joints, bimetallic-welded joints and dissimilar metal-welded joints were designed. Then, the fracture resistance curves and crack tip strain fields of different models under different material constraints were calculated. Based on the results, the questions above were answered, and the effect range of the material constraint was investigated.

## **2. Materials and Models Design**
