# Fracture Behavior of Two Biopolymers Containing Notches: Effects of Notch Tip Plasticity
## 2 Materials and Methods
### Finite Element Analyses
The modified crack length (\(\rm a_{m}\)) can be calculated as follows:  
\[\rm a_{m}=a_{0}+r_{p}, \tag{5}\]  
Therefore, the cracked rectangular specimen was simulated with the modified crack length and the new value of \(\rm K_{I}\) was computed by using the average fracture load of the cracked rectangular specimens (i.e., 145 N). As previously mentioned, this value is equal to the modified fracture toughness of the DM. The value of \(\rm r_{p}\) was estimated by using the [PERSON]'s theory [40] presented in Equation (6):  
\[\rm r_{p}=(I/\pi)\times(K_{IC}/\sigma_{y})^{2}, \tag{6}\]  
The fracture toughness \(\rm K_{IC}\) and the yield strength \(\rm\sigma_{y}\) are equal to 1.32 MPa.m\({}^{0.5}\) and 34.7 MPa, respectively, resulting in an \(\rm r_{p}\) value of 0.46 mm. Finally, the simulation of a rectangular sample with an 8.46 mm initial crack length (the resulting value of \(\rm a_{m}\)), provides a modified fracture toughness (\(\rm K_{IC}\)*) for the DM equal to = 1.41 MPa.m\({}^{0.5}\). Considering the values of \(\rm K_{IC}\) (1.2 MPa.m\({}^{0.5}\)) and \(\rm\sigma_{u}\) (36.6 MPa) of the BC material, and using Equations (1) and (2), the values of \(\rm r_{c}\) and \(\rm d_{c}\) are 0.17 mm and 0.68 mm, respectively. Likewise, considering the values of \(\rm K_{IC}\) (1.32 MPa.m\({}^{0.5}\)) and \(\rm\sigma_{u}\) (48.7 MPa) for the DM, the values of the critical distances \(\rm r_{c}\) and \(\rm d_{c}\) are 0.12 mm and 0.48 mm, respectively. With all this, an arbitrary external load of 100 N was applied to the FE models of UNBD specimens, and the linear elastic tangential stress (\(\rm\sigma_{0\theta}\)) at the critical distance in front of the notch tip (i.e., \(\rm r_{c}\) = 0.17 mm for BC material and \(\rm r_{c}\) = 0.12 mm for the DM), was computed. After that, by using the failure concept of the MTS fracture criterion, the fracture loads of the UNBD specimens were theoretically estimated. At fracture, the tangential stress at the critical distance for the BC and DM specimens should reach 36.6 MPa and 48.7 MPa, respectively. Therefore, it is possible to simply estimate the fracture loads (\(\rm P_{f}\)) of the UNBD specimens using Equations (7) and (8):  
\[\rm P_{f}=(36.6/\sigma_{0\theta})\times 100\ N\ for\ BC\ material, \tag{7}\]  
\[\rm P_{f}=(48.7/\sigma_{0\theta})\times 100\ N\ for\ DM, \tag{8}\]  
The fracture loads can also be predicted according to the MS criterion. The difference is that the mean value of tangential stress over the critical distance \(\rm d_{c}\) ((\(\rm\sigma_{0\theta}\) mean) must be used, leading to:  
\[\rm P_{f}=(36.6/\sigma_{0\theta}\) mean)\times 100\ N\ for\ BC\ material, \tag{9}\]  
\[\rm P_{f}=(48.7/\sigma_{0\theta}\) mean)\times 100\ N\ for\ DM, \tag{10}\]  
Equations (8) and (10) present the fracture load predictions of the DM UNBD specimens assuming that the behavior of this material is linear elastic. However, due to the elastic-plastic behavior of the DM, such Equations should be modified using the EMC. This means that the critical distances \(\rm r_{c}\) and \(\rm d_{c}\) should be recalculated, and also that the critical stress value (\(\rm\sigma_{c}\)) should equate to the tensile strength of the equivalent elastic material (\(\rm\sigma_{f}\)*), instead of using the ultimate tensile strength of the real material (DM). The area under the true stress-strain curve until necking (i.e., the SED value derived from Figure 2b) was equal to 1.33 MPa. Using Equation (4), the value of \(\rm\sigma_{f}\)* was 87.4 MPa. Thus, using \(\rm K_{IC}\)* = 1.41 MPa.m\({}^{0.5}\) and \(\rm\sigma_{f}\)* = 87.4 MPa in Equations (1) and (2), the \(\rm r_{c}\) and \(\rm d_{c}\) values for the EMC-MTS and EMC-MS coupled criteria were 0.04 mm and 0.16 mm, respectively. Finally, the fracture loads by the EMC-MTS and EMC-MS criteria can be calculated according to Equations (11) and (12).