## 2 Background and Numerical Procedures  
### Background and Numerical Procedures  
It has been well known that the nucleation, growth, and coalescence of microvoids lead to the ductile crack growth in metals. In recent decades, large amounts of investigations for elastic-plastic materials incorporating void mechanisms have been made in developing the constitutive models and the widely accepted one seems to be the Gurson-Tvergaard-Needleman (GTN) model, which is firstly introduced by [PERSON] [24] and then modified by [PERSON] and [PERSON] [25; 26; 27]. The yield function of the GTN model keeps to the following form:  
\[\varphi\Big{(}\sigma_{ij},f^{\star},\sigma_{y}\Big{)}\ =\ \frac{\sigma_{e} {}^{2}}{\sigma_{y}^{2}}\ +\ 2q_{1}f^{\star}\cosh\left(\frac{3q_{2}\sigma_{m}}{2\sigma_{y}}\right)\ -\ 1\ -\ q_{1}f^{\star 2} \tag{1}\]  
in which \(\sigma_{y}\) is the flow stress of the matrix material, and \(f^{\star}\) is the effective void volume fraction, \(\sigma_{e}\) is the von Mises stress, and \(\sigma_{m}\) is the mean normal stress component. Constants \(q_{1}\) and \(q_{2}\), as introduced by [PERSON], were used to modify the [PERSON] model.  
little connection with \(M\). Afterwards, on the base of these findings, a so-called \(J\)-\(Q\)-\(M\) formulation was presented:  
\[\sigma_{ij}\approx\sigma_{ij}^{M=0,T=0}+Q\sigma_{0}\delta_{ij}+M\sigma_{0}f_{ij} (\theta+12\beta,n) \tag{4}\]  
where the \(Q\) parameter is used to rank the effect of geometry constraint on the crack-tip constraint. Additionally, the \(M\) value describes the material mismatch. In the former formula, \(n\) and \(\sigma_{0}\) are the hardening exponent and yield stress of the reference material, \(f_{ij}\) represents the angular function, which only depends on the reference material, \(\beta=1\) for under-match and \(\beta=0\) for over match and even match [14]. Equation (4) applies to the mismatch and reference materials, and it is much fitter for the later.  
In this study, the CTOD was used as the fracture parameter. [PERSON] [37] has shown that the \(J\) and CTOD, for a given elastic-plastic material, are uniquely linked. Nevertheless, it is advantageous to calculate the crack driving force. The crack tip stress field with and without strength mismatch can be written in terms of CTOD:  
\[\begin{array}{l}\sigma_{ij}^{My=1.0}=\sigma_{ij}^{Ref}+Q\sigma_{0}\delta_{ij }\\ \sigma_{ij}^{My\
eq 1.0}=\sigma_{ij}^{My=1.0}+\Delta\sigma_{ij}^{M}\end{array} \tag{5}\]  
where \(\Delta\sigma_{ij}^{M}\) denotes the difference between stress fields that are caused by the strength mismatch. The reference solution \(\sigma_{ij}^{Ref}\) can be obtained from the HRR solution or a modified boundary layer (MBL) model under small-scale yielding. In the current work, the MBL with \(T=0\) was used. Mismatch ratio \(M_{y}\) was defined as \(M_{y}=\sigma_{\text{ys}}^{\text{WM}}/\sigma_{\text{ys}}^{\text{BM}}\), where \(\sigma_{\text{ys}}^{\text{WM}}\) and \(\sigma_{\text{ys}}^{\text{BM}}\) represent the yield stress of the weld metal and base metal.  
The following definition of \(Q\) and \(M\) has been obtained because of the use of CTOD as crack driving force [5; 6; 14], according to the former formulas:  
\[\begin{array}{l}Q=\frac{\sigma_{ij}-\sigma_{ij}^{Ref}}{\sigma_{0}}=\frac{ \sigma_{22}-\sigma_{22}\text{Ref}}{\sigma_{0}}\\ M=\frac{\Delta\sigma_{ij}^{M}}{\sigma_{0}}=\frac{\sigma_{22}^{My\
eq 1.0}-\sigma_{ 22}^{My\to 1.0}}{\sigma_{0}}\end{array},\ \theta=0,\quad 1\leq\frac{r}{ \delta}\leq 5 \tag{6}\]  
where \(\sigma_{ij}\) is the opening stress component of interest, \(\sigma_{ij}^{Ref}\) is the reference stress component that was obtained by MBL model solution with \(T=0\), \(r\) is the distance from the crack tip along the crack plane. Additionally, in present work, the focus is placed on the stress field (\(\sigma_{22}\)) perpendicular to the direction of crack growth at the crack front (\(\theta=0\), \(r/\delta=3\)).