## 3 Results and Discussion
### Calibration of Critical Stress and Critical Distances  
The distribution of linear-elastic stresses around keyhole notches was numerically obtained by executing the above-described FE analyses. The compressive stress distributions around some of the keyhole notches are represented in Figure 9. It is evident in the figure that, in agreement with the experimental observations, at the critical conditions, the horizontal edges of the key-hole notches contact each other, with the area of contact depending directly on the notch tip radius, the magnitude of the applied load, and the material being analyzed. Interestingly, Figure 9c illustrates a full contact between the notch edges for the GPPS specimen weakened by a keyhole notch of 4 mm tip radius. Figure 8: Examples of load-displacement curves obtained in key-hole notched specimens under compression: (**a**) PMMA, \(\rho\) = 0.5 mm; (**b**) PMMA, \(\rho\) = 2 mm; (**c**) GPPS, \(\rho\) = 0.5 mm; (**d**) GPPS, \(\rho\) = 2 mm. As mentioned in Section 2, under compressive loading conditions, the fracture toughness K\({}_{\text{lc}}\) becomes physically meaningless. Since the critical distances of PMMA and GPPS are necessary for the theoretical predictions derived from both the MTS and the MS criteria, and since they cannot be computed from Equations (4) and (5), an alternative experimental calibration must be performed. In what follows, one alternative approach is applied, which can be considered as an extension of the theory of critical distances (TCD) [48] from the positive mode I loading to negative mode I loading. The approach starts with the FE models of the specimens weakened by keyhole notches, with notch tip radii of 0.5 and 4 mm, and subjected to the mean values of the experimentally recorded fracture loads. These two specimens have the maximum and the minimum stress concentrations, respectively, of all the specimens being analyzed. After that, on the bisector line of the keyhole notch, and starting from the notch tip, a straight path  
Figure 9: Examples of stress distributions (MPa) for keyhole notches: (**a**) PMMA, \(\rho\) = 0.5 mm, P\({}_{1}\) = 16636 N; (**b**) GPPS, \(\rho\) = 0.5 mm, P\({}_{1}\) = 7360 N; (**c**) GPPS, \(\rho\) = 4 mm, P\({}_{2}\) =9186 N.  
is established. Regarding the MTS criterion, the variations of the compressive stresses along the distance from the notch tip are obtained for both PMMA and GPPS specimens. This results in having two stress-distance curves for each material (one for the notch tip radius of 0.5 mm, and the other for the 4 mm notch radius). Then, these stress-distance curves are drawn in a single graph, as shown in Figure 10. The coordinates of the point at which the curves intersect each other are considered to be the critical stress \(\sigma_{c}\) (y component) and the critical distance \(r_{c}\) (x component). The resulting values are reported in Table 3.  
Concerning the MS criterion, the procedure is very similar, but the stress-distance curves are drawn based on the mean compressive stresses computed by the integral method. First, several distances are chosen along the bisector line of the notch, and the corresponding mean stresses are calculated. Secondly, each distance and its corresponding mean stress are represented by a single point in a mean stress-distance plot, and by connecting these points, a curve is obtained. Again, the obtained curves for 0.5 mm and 4 mm notch tip radii are plotted in a single figure and the intercept point of the two curves is defined as the compressive critical stress \(\sigma_{c}\) and the critical distance \(d_{c}\). The mean stress-distance plots for PMMA and GPPS polymers are shown in Figure 11, and the values of the critical stress and critical distance derived from the MS criterion are gathered in Table 3.  
By examining Table 3, it can be observed that, for PMMA, \(\mathrm{d_{c}}\) is very close to \(4r_{c}\). For GPPS, however, \(\mathrm{d_{c}}\) is approximately \(4.8r_{c}\), which is 20% larger than \(d_{c}=4r_{c}\) derived from Equations (4) and (5) in tensile conditions.