*2.1. Displacement Extrapolation Technique (DET)*

The DET is based on the nodal displacement around the crack tip. The construction of quarter-point elements around the crack tip is generally needed for this procedure. Generally, the existence of the quarter-point element is essential in order to correctly represent the linear elastic singularity (1/√*r*) for stresses and strains at the crack tip. The polynomial isoparametrically representative of the singularity is typically obtained by moving the mid-side nodes adjacent to the crack tip to a quarter-length edge closer to the crack tip. Crack tip elements based on this method were separately suggested by [22,23]. In this study, the natural triangle–quarter-point element was selected as the type of crack-tip element and its configuration follows the schematic formation of the rosette around the crack-tip, as seen in Figure 2.

**Figure 2.** A quarter-point singular element around the tip of the crack.

For the calculation of stress intensity factors, the displacement extrapolation method [24] was used as follows:

$$K\_{I} = \frac{E}{3(1+\nu)(1+\kappa)}\sqrt{\frac{2\pi}{L}} \left[4(v\_{\,\,b}^{\prime} - v\_{\,\,d}^{\prime}) - \frac{(v\_{\,\,c}^{\prime} - v\_{\,\,c}^{\prime})}{2}\right] \tag{1}$$

$$K\_{II} = \frac{E}{3(1+\nu)(1+\kappa)}\sqrt{\frac{2\pi}{L}}\left[4(\boldsymbol{u}\_{\boldsymbol{b}}^{\prime} - \boldsymbol{u}\_{\boldsymbol{d}}^{\prime}) - \frac{(\boldsymbol{u}\_{\boldsymbol{c}}^{\prime} - \boldsymbol{u}\_{\boldsymbol{c}}^{\prime})}{2}\right] \tag{2}$$

where *E* is the modulus of elasticity, *ν* is the Poisson's ratio, *κ* is the elastic parameter defined by

$$\kappa = \begin{cases} 3 - 4\nu \text{ for plane strain} \\ \frac{(3-\nu)}{(1+\nu)} \text{ for plane stress} \end{cases} \tag{3}$$

and *L* is the quarter-point element length. The *u* and *v* are the displacement components in the *x'* and *y'* directions, respectively. The subscriptions represent their position, as seen in Figure 2.
