**2. Constitutive Model**

The Gurson–Tvergaard–Needleman (GTN) model [17,30,31] is used in this study, which is on the basis of damage growth in metals due to void nucleation, growth, and coalescence. The void growth is a function of the plastic strain rate **D***<sup>P</sup>* :

$$\left(\dot{f}\right)\_{growth} = (1 - f)\mathbf{I} : \mathbf{D}^P \tag{1}$$

and the void nucleation is assumed to be strain controlled as follows:

$$\left(\dot{f}\right)\_{\text{nucleation}} = \overline{A}\dot{\overline{\varepsilon}}^P \tag{2}$$

where . *ε P* is the effective plastic strain rate, and the parameter *A* is chosen so that nucleation follows a normal distribution as suggested by Chu and Needleman [32]:

$$\overline{A} = \frac{f\_N}{S\_N \sqrt{2\pi}} exp\left[ -\frac{1}{2} \left( \frac{\mathbb{E}^p - \varepsilon\_N}{S\_N} \right)^2 \right] \tag{3}$$

here, *f<sup>N</sup>* is the volume fraction of void nucleating particles, *ε<sup>N</sup>* is the average void nucleating strain, and *S<sup>N</sup>* is the standard deviation of the void nucleating strain.

Additionally, the shear damage growth proposed by Nahshon and Hutchinson [19] is as follows: *p*

$$df\_{\text{Shear damage}} = k\_w w(\sigma\_{\text{ij}}) f \frac{S\_{\text{ij}} d\varepsilon\_{\text{ij}}^r}{\sigma\_{\text{ef}}} \tag{4}$$

*kw* is the magnitude of the damage growth rate in the pure shear test. The function *w σij* identifies the current state of stress, which is defined as *w σij* = 1.0 − 27*J*<sup>3</sup> 2*σ* 3 *eq* 2 , where *J*<sup>3</sup>

is the third invariant of the deviatoric stress matrix. The growth of existing voids and the nucleation of new voids are considered in the

evolution of void volume fraction as follows:

$$
\dot{f} = \left(\dot{f}\right)\_{\text{growth}} + \left(\dot{f}\right)\_{\text{nucleation}} + \left(\dot{f}\right)\_{\text{shear damage}}\tag{5}
$$

and the function of void volume fraction (*f* ∗ (*f*)) is defined to consider coalescence as follows:

$$f^\* = \begin{cases} f & \text{for } f \le f\_\varepsilon \\\ f\_\varepsilon + \frac{f\_u^\* - f\_\varepsilon}{f\_f - f\_\varepsilon}(f - f\_\varepsilon) & \text{for } f > f\_\varepsilon \end{cases} \tag{6}$$

where *f<sup>c</sup>* is the critical void volume fraction for coalescence and *f f* is the void volume fraction at failure. The parameter *f* ∗ *<sup>u</sup>* = <sup>1</sup> *q*1 is defined. It should be mentioned that void growth and nucleation does not happen when the stress state of an element is compressive; it may only occur in tension.

Finally, the approximate yield function to be used in which *f* ∗ is distributed randomly is as follows:

$$\Phi(\sigma, \overline{\sigma}, f) = \frac{\sigma\_{\varepsilon}^{2}}{\overline{\sigma}^{2}} + 2f^\* q\_1 \cosh\left(\frac{3q\_2 \sigma\_H}{2\overline{\sigma}}\right) - \left[1.0 + (q\_2 f^\*)^2\right] = 0\tag{7}$$

where *σ* is the macroscopic Cauchy stress tensor and *σ<sup>e</sup>* , *σH*, and *σ* are the equivalent stress, hydrostatic stress, and matrix stress, respectively. In fact, the matrix stress and equivalent stresses are damaged and undamaged stresses in the GTN model. Additionally, *q*<sup>1</sup> and *q*<sup>2</sup> are calibrated parameters.

The uniaxial elastic–plastic undamaged stress–strain curve for the matrix material is provided by the following power-law form:

$$\overline{\varepsilon} = \begin{cases} \frac{\overline{\sigma}}{\overline{\Xi}}, & \text{for } \overline{\sigma} \le \sigma\_y \\\ \frac{\sigma\_y}{\overline{\Xi}} \left( \frac{\overline{\sigma}}{\sigma\_y} \right)^n, & \text{for } \overline{\sigma} > \sigma\_y \end{cases} \tag{8}$$
