# Fracture Behavior of Two Biopolymers Containing Notches: Effects of Notch Tip Plasticity
## 3 Results and Discussion
In other words, the plastic zone in the notch tip vicinity plays a vital role in assessing the fracture behavior of UNBD specimens with elastic-plastic behavior, and purely linear-elastic criteria, such as the MTS and the MS, do not provide accurate predictions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **Specimen** & **MTS** & **MS** & **EMC-MTS** & **EMC-MS** \\ \hline \multirow{3}{*}{BC} & C-1 & \(-\)17 & +1 & - & - \\  & C-2 & \(-\)15 & 0 & - & - \\  & C-4 & \(-\)19 & \(-\)9 & - & - \\ \hline DM & C-1 & \(-\)32 & \(-\)19 & 0 & 10 \\  & C-2 & \(-\)43 & \(-\)35 & \(-\)9 & 4 \\  & C-4 & \(-\)45 & \(-\)40 & \(-\)7 & \(-\)4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Discrepancies (%) between experimental results and analytical predictions. Figure 8: Variation of linear elastic tangential stress versus the notch tip radius at different distances from the notch tip and for an arbitrary load of 100 N. (**a**) Distances (D) of interest for DM (r\({}_{\rm c}\) = 0.04 mm); (**b**) Distances of interest (D) for BC material (r\({}_{\rm c}\) = 0.17 mm). In order to determine the effective plastic zone size (EPZS) at fracture onset for the DM UNBD specimens, three elastoplastic FE analyses were carried out, in which the true stress-strain curve of the DM is used as the material property. Figure 9 shows two examples of the contours of tangential stresses (at fracture onset) in UNBD specimens with notch tip radii of 1 and 4 mm, together with their corresponding EPZSs, which were calculated as 0.6, 0.9, and 1.6 mm for notch radii of 1, 2, and 4 mm respectively. To determine the type of ductile failure regime in the UNBD polymeric specimens tested, the EPZS should be compared with the effective ligament size (ELS), which is that part of the ligament encountering tensile stresses (i.e., 6 mm). The ELS can be simply obtained from a linear elastic FE analysis, and for the cases being analyzed is equal to (about) 6 mm from the notch tip. From the values obtained, it is obvious that for the notch tip radii 1, 2, and 4 mm, the plastic zone occupies 10%, 15%, and 27% of the ELS. It is then proposed that the fracture regime for the 1 mm notch tip radius is the small-scale yielding (SSY) regime, while for the notch tip radii 2 and 4 mm, it is the moderate-scale yielding (MSY) regime. This is directly related with the fact that, although the two linear-elastic criteria (MTS and MS) provide inaccurate results for the DM predictions, the accuracy becomes lower when the notch radius (and then, the EPZS) becomes larger. At the end of this section, it should be underlined that some fracture models other than the MTS and MS criteria, which are both based on the Theory of Critical Distances (TCD), have been  
Figure 9: Tangential stress distributions (MPa) around a U-notch together with the effective plastic zone size (EPZS). (**a**) Notch tip radius = 1 mm; (**b**) notch tip radius = 4 mm. recently proposed in literature. One of them is the phase-field fracture model [41]. In this model, the fracture problem is described by a phase-field method and by using the Bayesian inversion, the problem can be solved on a rather coarse mesh and the corresponding parameters can be fitted. The possibility of applying the phase-field fracture model to the notch fracture problems investigated in the present contribution can be an interesting topic for future investigations. In parallel, one may want to obtain the numerical FE results from a specific computational code rather than directly from the ABAQUS software. In this regard, different FE approaches can be employed to prepare the code (see for instance [42]).