Corresponding to the experiment curves, the elastic–plastic model in this paper is as follows:

$$ \bar \sigma = \left\{ {\begin{array}{*{20}{l}} {E\bar \varepsilon ,}&{\bar \sigma < {\sigma_s},} \\ {{k_0} + Q\left( {1 - {e^{ - \gamma {{\bar \varepsilon }_p}}}} \right),}&{\bar \sigma \geqslant {\sigma_s},} \end{array}} \right. $$

(1)

where \(\bar \sigma \) and *σ*_{s} are the equivalent stress and the yield stress respectively, \(\bar \varepsilon \) and \({\bar \varepsilon_{\text{p}}}\) are the equivalent strain and the equivalent plastic strain respectively, *E* is the elastic modulus, *k*_{0}, *Q* and *γ* are constants from the Voce hardening law [5], and *k*_{0} = *σ*_{s}.

To account for non–proportional loading paths, in the damage initiation model the instability variable *F* is defined as [14]:

$$ F = \int_0^{{{\bar \varepsilon }_{\text{p}}}} {\frac{{{\text{d}}{{\bar \varepsilon }_{\text{p}}}}}{{{{\bar \varepsilon }_{\text{c}}}\left( \eta \right)}}} , $$

(2)

$$\eta = \frac{{{\sigma_{\text{m}}}}}{\bar \sigma }, $$

(3)

where \({\bar \varepsilon_{\text{c}}}\left( \eta \right)\) is the critical equivalent strain related to the stress triaxiality *η*, *σ*_{m} is the mean stress.

The relationship between the critical equivalent strain and the stress triaxiality represented by the MMC criterion is defined as follows [5, 10]:

$$ {\bar \varepsilon_{\text{c}}} = {\left\{ {\frac{A}{{{c_{2{\text{c}}}}}}{f_3}\left[ {\sqrt {\frac{{1 + {c_{1{\text{c}}}}^2}}{3}} {f_1} + {c_{1{\text{c}}}}\left( {\eta + \frac{{f_2}}{3}} \right)} \right]} \right\}^{ - \frac{1}{n}}}, $$

(4)

$${f_1} = \cos \left\{ {\frac{1}{3}\arcsin \left[ { - \frac{27}{2}\eta \left( {{\eta^2} - \frac{1}{3}} \right)} \right]} \right\},$$

(5)

$$ {f_2} = \sin \left\{ {\frac{1}{3}\arcsin \left[ { - \frac{27}{2}\eta \left( {{\eta^2} - \frac{1}{3}} \right)} \right]} \right\}, $$

(6)

$$ {f_3} = {c_{3{\text{c}}}} + \frac{\sqrt 3 }{{2 - \sqrt 3 }}\left( {1 - {c_{3{\text{c}}}}} \right)\left( {\frac{1}{{f_1}} - 1} \right), $$

(7)

where *A* and *n* are the material constants related to the plastic deformation, and *c*_{1c}, *c*_{2c}, *c*_{3c} are the material constants for the damage initiation.

When the instability variable *F* in Eq. (2) reaches unity, the damage is coupled to the stress according to the GISSMO model and the coupling relationship is as follows [14]:

$$ {\bar \sigma^*} = \bar \sigma \left( {1 - {{\left( {\frac{{D - {D_{\text{c}}}}}{{1 - {D_{\text{c}}}}}} \right)}^m}} \right), $$

(8)

where \({\bar \sigma^*}\) and \(\bar \sigma \) are the equivalent stress with and without damage respectively, *D* and *D*_{c} are the current damage value and the damage value at the onset of stress–damage coupling respectively, *m* is the material constant for the effect of damage on stress. When plastic deformation occurs, the damage starts to accumulate by Eq. (9):

$$ D = \int_0^{{{\bar \varepsilon}_{\text{p}}}} {\frac{{{\text{d}}{{\bar \varepsilon }_{\text{p}}}}}{{{{\bar \varepsilon }_{\text{f}}}\left( {\eta ,{l_{\text{e}}},t} \right)}}} ,$$

(9)

where \({\bar \varepsilon}_{{\text{f}}} \left( {\eta, l_{e}, t} \right)\) is the equivalent failure strain which is related to element size *l*_{e}, thickness *t* and stress triaxiality *η*. When damage value *D* reaches unity, the material fails.

Based on the relationship between the equivalent failure strain and element size from Ref. [28], this paper introduces a calibration parameter regarding the reference element size by Eq. (10):

$$ {\bar \varepsilon_{\text{f}}} = {\bar \varepsilon_{\text{c}}}\left( \eta \right) + \left[ {{{\bar \varepsilon }_{{\text{f}}0}}\left( \eta \right) - {{\bar \varepsilon }_{\text{c}}}\left( \eta \right)} \right]{\left( {\frac{{t{l_{{\text{e}}0}}}}{{{t_0}{l_{\text{e}}}}}} \right)^r}, $$

(10)

$$ {l_{\text{e}}} = \sqrt {S}, $$

(11)

where \({\bar \varepsilon_{\text{c}}}\left( \eta \right)\) is the critical equivalent strain in Eq. (4), *l*_{e}, and *t* are the element size and the thickness of shell elements respectively, *S* is the area of shell elements, \({\bar \varepsilon_{{\text{f}}0}}\left( \eta \right)\) is the equivalent failure strain of the reference element related to triaxiality *η*, *l*_{e0} and *t*_{0} are the element size and the thickness of the reference shell element respectively, *r* is the parameter related to element size.

It’s worthwhile to mention that no matter what shape, e.g., square, rectangle and triangle, the element is, *l*_{e} represents the equivalent size as the area keeps constant.

The relationship between the equivalent failure strain and the stress triaxiality for the reference element is still characterized by the MMC criterion Eq. (4). The values of parameters *c*_{1c}, *c*_{2c} and *c*_{3c} are different from the damage initiation model and are denoted by *c*_{1f}, *c*_{2f} and *c*_{3f} now, while the values of* A* and *n* are consistent with the damage initiation model.

The calibration of the model parameters was done using a surrogate model and an intelligent optimization algorithm. The calibration flow is shown in Figure 3. Sample points are determined based on the D–optimality design criterion. And the surrogate model is a linear polynomial response surface. The optimization based on the surrogate model includes two steps: first, use the adaptive simulated annealing algorithm to find an approximate global optimum; second, starting from the solution, use the gradient–based dynamic leap–frog method to find the local optimum. Thus, the optimum is obtained. Further, the domain of parameters is adapted based on the accuracy of the previous optimum [39]. The calibration process was done in the LS–OPT.

As shown in Figure 3, the calibration includes two parts: the elastic-plastic model and the failure model. As we all know, the elastic-plastic model has no relation with specimen geometry, thus all specimens were adopted to calibrate its parameters. However, the failure model is related to element size. A general method is calibrating failure model with reference element firstly, then calibrating element size regularization parameters with other elements [14]. In this research, the shell element is used, thus two parameters, thickness and element size, need to be concerned. The element with thickness 2.5 mm and size 0.2 mm is employed as the reference element, and specimens with thickness 2.5 mm, namely specimens with notches, are adopted to calibrate failure model parameters *m*, *A*, *n*, *c*_{1c}, *c*_{2c}, *c*_{3c}, *c*_{1f}, *c*_{2f}, *c*_{3f}. Then, element size 0.3 mm, 0.5 mm are used to calibrate element size regularization parameter *r*. Since the regularization function (10) takes thickness and element size into account, the standard specimen with thickness 2 mm and notched specimens with thickness 2.5 mm are both adopted in the calibration of parameter *r*. As to the determination of element size, both efficiency and accuracy are considered, which is described with simulation results in Section 4.