# Fracture Behavior of Two Biopolymers Containing Notches: Effects of Notch Tip Plasticity
## 2 Materials and Methods
### Finite Element Analyses  
At this point, fracture mechanics analysis of cracked specimens is needed to convert the experimentally achieved fracture loads into the associated values of fracture toughness of the two biopolymers. This is usually done by means of finite element (FE) simulations and performing linear-elastic analysis. Moreover, as it is evident from Sections 2.2.1 and 2.2.2, the application of both the MTS and the MS fracture criteria to predict the LCC of UNBD samples, requires the linear-elastic stress distribution around the U-notches to be determined. Hence, in this subsection, the FE simulation and analysis of the cracked and notched specimens utilized in the experimental program is performed. The UNBD, cracked rectangular, and cracked SCB specimens were analyzed by finite element method using eight-node plane-stress quadratic elements. Regarding the boundary conditions, it should be explained that for the cracked rectangular and cracked SCB specimens tested under three-point bending, the displacements of the specimens at the bottom supports were set equal to zero in vertical direction, and the load was applied to the specimen at the top of the specimen as a concentrated compressive nodal force. For the UNBD specimens, however, the lowest node sticking to the bottom horizontal plate was fully fixed, and the concentrated compressive nodal force was applied at the opposite node located at the top of the specimen. Here, it should be noticed that, because of significant stress gradient at the crack and the notch tips, refined meshes were required to achieve accurate numerical results. In order to find the largest mesh size below which the numerical results are independent of the mesh size, a mesh sensitivity analysis was performed: the model was continuously refined until reaching a mesh size below which the numerical results are not affected by the mesh size. In this case, the final mesh size was fixed at 0.02 mm. A sample mesh pattern for the cracked rectangular and SCB specimens is shown in Figure 6.  
To obtain the fracture toughness of the two biomaterials, both the cracked rectangular and SCB specimens (two well-known accepted fracture toughness specimens) were simulated in the FE software under three-point bend loading. The experimentally achieved critical loads (average values) of the cracked specimens reported in Table 2 (145 N and 333 N for the rectangular and the SCB samples, respectively) were applied to the corresponding FE models. Then, the value of the mode I stress intensity factor (SIF) \(\mathrm{K_{I}}\) for each specimen was directly computed from the FE models. These values correspond to the fracture toughness of the two materials being analyzed, with 1.2 MPa.m\({}^{0.5}\) for the BC material and 1.32 MPa.m\({}^{0.5}\) for the DM. As shown in Figure 2, the BC material has a brittle behavior with negligible capacity for developing plastic deformations. Therefore, the value of 1.2 MPa.m\({}^{0.5}\) is considered here as the final value of its fracture toughness for this material. The DM, however, due to its capacity to develop elastic-plastic behavior (see Figure 2), experiences non-negligible plastic deformations at the crack tip neighborhood. Thus, the virtual crack extension method (VCEM) was used to modify the value of the initial fracture toughness by considering the impact of the plastic area around the crack tip. With this aim, the length of the initial crack was virtually extended. According to  
Figure 6: Examples of mesh patterns. (**a**) cracked rectangular specimens (DM); (**b**) cracked SCB specimens (BC material). this method, the radius of the plastic area around the crack tip at the crack growth onset, \(\rm r_{p}\), should first be calculated and secondly added to the initial length \(\rm a_{0}\) to obtain the virtual crack length.