Fiber tensile strength factor (FTSF)
One of the most simple and elegant models that can be used to predict the micromechanical behavior of a composite is a modified rule of mixtures (mROM). Its formulation for the tensile strength of a short fiber semialigned reinforced composite is (Lee et al., 2014, Thomason, 2002b):
In the equation, f_{c} is the compatibility factor; in the case of favorable interphases, f_{c} is supposed to be 0.2. a_{t}^{F} and a_{t}^{c} are the ultimate intrinsic tensile strength of the fiber and the tensile strength of the composite material. a_{t}m* is the tensile strength of the matrix at the breaking point of the composite and V^{F} is the fiber volume fraction.
Figure 5.4 Tensile strength of composites reinforced with a 30% of fibers (** Theoretical value of a well bonded composite with BSKP showing 900 MPa intrinsic tensile strength).
In the presented form the equation shows two unknowns, f_{c} and of. While there are experimental and analytical methods to obtain the intrinsic tensile strength, these are expensive and need laboratory equipment (Vallejos et al., 2012). In previous works, some of the authors defined a fiber tensile strength factor (FTSF), that accounts for the net contribution of the reinforcements to the final strength of the composite. Both unknowns are united and named FTSF, then the equation could be
Figure 5.5 Fiber Tensile Strength Factor.
solved, and the FTSF accounts for the slope of the regression curve between 0 and the net contributions of the fibers to the final strength of the composite against the fiber volume fraction (Reixach et al., 2013a; Lopez et al., 2012b). The rearranged version of the rule of mixtures will be:
Then, the value of FTSF will be the slope of the regression curve showed in Fig. 5.5.
The computed value for the FTSF if all the data (15—35%) are used is 116.88 MPa. If the 35% value is discarded then the value of the FTSF is 123.98 MPa. The values are clearly higher than that obtained for SGW-reinforced PP, 109.4 MPa, but lower than the values obtained for glass fiber-reinforced PP, 273.85 MPA and 427.75 MPa for the uncoupled and coupled composites, respectively.
In a recent study, some of the authors used the Kelly and Tyson modified equation (Kelly and Tyson, 1965) with the solution provided by Bowyer and Bader (1972) to compute the intrinsic tensile strength of the BSKP fibers (Granda et al., 2016a in press.). The result was 668 MPa, very similar to that of SGW, with a mean value of 617 MPa (Lopez et al., 2011), but lower than other intrinsic tensile strengths published for BSKP, near 900 MPa (Karlsson, 2007; Li, 1999). If the last value is used with Eq. (5.1) to compute the value of the coupling factor, the mean value (discarding the 35% value) is around 0.14. The value is very far from 0.2, which is considered a good to optimal value. If the coupling factor is computed using the 668 MPa value, then the result is around 0.18, slightly closer to 0.2, but
Table 5.2 Flexural properties of the BSKP-reinforced PLA composites
BSKP (%) |
V^{F} |
af (MPa) |
?f^{C} (mm) |
ajm* (MPa) |
0 |
— |
60 (0.26) |
3.2 (0.13) |
— |
15 |
0.130 |
77.05 (0.29) |
3.15 (0.14) |
59.2 |
20 |
0.174 |
79.05 (0.24) |
3.1 (0.12) |
58.2 |
25 |
0.220 |
83.65 (0.21) |
3 (0.11) |
57.3 |
30 |
0.266 |
93.65 (0.22) |
3 (0.08) |
56.4 |
35 |
0.313 |
94.8 (0.23) |
3 (0.12) |
56.4 |
also far. Consequently, the interphase between the BSKP fibers and the PLA matrix could be considered from fair to slightly good, but not optimal, leaving room for further investigation. If the mROM (Eq. 5.1) is used to compute the theoretical value of the tensile strength of a well bonded 30% BSKP-reinforced PLA composite, the result is 73.8 MPa. Fig. 5.4 shows the value as f_{c} = 0.2. The computed value is 13% higher than the experimental value.