# Fracture Behavior of Two Biopolymers Containing Notches: Effects of Notch Tip Plasticity
## 2 Materials and Methods
### Materials, Testing Specimens, and Experimental Results
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material Property** & **BC** & **DM** \\ \hline Elastic modulus, E (GPa) & 3.02 & 2.80 \\ Poisson’s ratio, \(\upsilon\) & 0.3 & 0.3 \\ Ultimate tensile strength, \(\sigma_{\text{u}}\) (MPa) & 36.6 & 48.7 \\ Yield strength, \(\sigma_{\text{y}}\) (MPa) & N/A & 34.7 \\ Fracture toughness, K\({}_{\text{IC}}\) (MPa.m\({}^{0.5}\)) & 1.2 & 1.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties (tensile and fracture) of BC material and DM. Figure 1: Geometries of the different specimens. (**a**) U-notched Brazilian disk (UNBD) specimen; (**b**) cracked rectangular specimen (dental material, DM); (**c**) semicircular bend (SCB) specimen (bone cement (BC) material); (**d**) standard tensile test specimen. Figure 2: Examples of load-displacement curves obtained in the tensile tests. (**a**) BC material; (**b**) DM. Three point bending (3 PB) fracture tests were also performed on the SCB (BC) and cracked rectangular (DM) specimens in order to determine the fracture toughness. This conversion can be made by calculating the value of the mode I stress intensity factor (SIF) associated with the critical load using the finite element (FE) analysis, which is actually equal to the fracture toughness of the material. The fracture loads of the BC material (UNBD and cracked SCB specimens) and the DM (UNBD and cracked rectangular specimens) are gathered in Table 2. In these tables, C indicates the compression tests on the UNBD specimens. Meanwhile, SCB and R denote the bending tests performed on the cracked specimens. Further, the numbers 1, 2, and 4 correspond to the radius of the notch tip, and numbers 8 and 10 indicate the length of the pre-existing cracks. Finally, P\({}_{1}\) (i = 1, 2, 3) and P\({}_{m}\) denote the fracture loads of the different tests performed and the resulting mean values. Beyond the fracture analyses performed below on the notched specimens, Table 1 gathers the resulting values of fracture toughness (K\({}_{\text{IC}}\)) derived from the bending tests (SCB-10 and R-8). As an example, the load-displacement curves of 2 tests performed on UNBD specimens are shown in Figure 3.  
It can be seen that, regardless of the brittle or ductile behavior observed in the tensile tests, the fracture curves in both materials are basically linear up to the final fracture. In other words, no significant nonlinear portion is observed in the curves of the cracked and UNBD specimens made of DM, in contrast to the tensile curve of this material (see Figure 2b). This is caused by the fact that the plastic deformations in the notch/crack tip vicinity are localized, while the displacement shown in the curve corresponds to the cross-head of the test machine, which is not very sensitive to moderate amounts of plasticity located at the defect tip. Accordingly, the displacement of the cross-head cannot capture properly the localized plastic deformations of the notch/crack tip. However, in the tensile specimens without stress concentration, the displacement of the cross-head is almost the same as that of the gage length and hence, it is directly affected by the bulk plastic deformations in the whole cross-section. Consequently, it should be noticed that a linear load-displacement curve in a cracked/notched specimen does not necessarily mean that the material of the specimen is developing perfectly linear elastic behavior, unless the measured displacement corresponds to a point affected by the crack/notch tip ductile behavior (if any).