*2.2. Analytical Predicitve Model Based on Dynamic Tests*

The interfacial strength between the polymeric matrix and the different volume content of wheat bran added, can be tracked using the so called "adhesion factor" (*A*). The damping factor (*tanδ*), obtainable from DMTA tests, is an indicator of the material molecular motions. Thanks to its estimation it is possible to have an idea of the fiber-matrix interfacial bonding. The adhesion parameter was introduced by Kubat et al. [56] starting on the assumption that the mechanical loss factor (*tanδc*) of the composite can be given as follows (Equation (6)):

$$
tan\delta\_c = V\_f \tan \delta\_f + V\_i \tan \delta\_i + V\_m \tan \delta\_m \tag{6}
$$

where the subscripts *f*, *i*, and *m* subscripts denote the filler, the interphase, and matrix; while *Vf* is the volume filler fraction. Making the assumption that the volume fraction of the interphase is rather small (*δ<sup>f</sup>* ≈ 0), its contribution can be neglected and Equation (6) can be rearranged as follows [57] (Equation (7)):

$$\frac{\tan \delta\_{\mathcal{C}}}{\tan \delta\_{m}} \approx \left(1 - V\_{f}\right)(1 + A) \tag{7}$$

with the "*A*" term, called adhesion factor. Equation (7) can be rewritten in order to explicit the adhesion factor as follows (Equation (8)):

$$A = \frac{1}{1 - V\_f} \frac{\tan \delta\_c}{\tan \delta\_m} - 1 \tag{8}$$

Calculating the adhesion factor from DMTA experiments, it is possible to interpret the interactions between the fillers and the polymeric matrix. Generally, a low value of the adhesion factor is an indication of good adhesion (or high degree of interaction between the two phases) due to the reduction of the macromolecular mobility induced by the presence of the filler [58].
