**2. Theoretical Analysis**

Two analytical approaches were followed: one based on the application of analytical models taking the data from the static tensile tests, and one taking the data from dynamical mechanical tests.

#### *2.1. Analytical Predictive Model Based on Static Tests*

For rigid fillers and for ultra-short fibers composites the effect of the fiber content correlated to the fiber/matrix adhesion on the composite stress at break lies among two limits called upper and lower bound [49]. Where low adhesion exists, the load is sustained only

by the polymeric matrix and a simple expression (derived by Nicolais and Nicodemo [50]) can be written (Equation (1)):

$$
\sigma\_{\mathcal{C}} = \sigma\_m \left( 1 - 1.21 \, V\_f^{\frac{2}{3}} \right) \tag{1}
$$

In Equation (1) *σ<sup>c</sup>* and *σ<sup>m</sup>* are, respectively, the composite strength and the matrix strength while *Vf* is the filler volume fraction. This equation represents the lower bound for the prediction of the tensile strength of not well bonded particulate filled composites. On the other hand, the upper bound can be obtained considering that the filler decreases the load bearing capacity of the matrix (Equation (2)) [49,51].

$$
\sigma\_{\mathcal{C}} = \sigma\_m \left(1 - V\_f\right) \tag{2}
$$

Pukanszky et al. [52,53] provided an empirical relationship for composites strength prediction (Equation (3)) where an empirical constant, named *B*, was correlated to: the particles surface area, particles density and interfacial bonding energy. However, the *B* parameter does not provide a numerical value of the interfacial shear stress (MPa), it is able to give information about the filler/matrix adhesion. In fact, a value of *B* equal or very close to zero corresponds to poor interfacial bonding where the fillers will not carry any load.

$$
\sigma\_c = \sigma\_m \frac{1 - V\_f}{1 + 2.5V\_f} \exp\left(BV\_f\right) \tag{3}
$$

By writing Equation (3) in a linear form (Equation (4)), is obtained a linear correlation where the interaction parameter, *B*, corresponds to the slope of the Pukánszky's plot (obtained plotting the natural logarithm of Pukánszky's reduced strength, *σred*, against the volume filler fraction).

$$
\ln \sigma\_{red} = \ln \frac{\sigma\_c \left(1 + 2.5 V\_f\right)}{\sigma\_m \left(1 - V\_f\right)} = BV\_f \tag{4}
$$

Lazzeri and Phuong [54] correlated the Pukánszky's interaction B parameter with the interfacial shear stress (*IFFS*). In particular, for composites having a length below the critical length (*Lc*) the failure of the composite occurs by plastic flow of the matrix and the interfacial shear strength (*IFSS*) can be calculated with the following expression:

$$\text{tr}\left(or\ IFSS\right) = \frac{\text{2}\sigma\_m(B-2.04)}{\eta\_0 a\_r} \tag{5}$$

where *η<sup>0</sup>* is the fiber orientation factor, *ar* is the fibers aspect ratio. In the case of particulate fillers the fibers orientation factor and the fibers aspect ratio can be taken equal to one; for very short aspect ratio fibers (such as those used in this work), a randomly fiber orientation can be considered and a value *η*<sup>0</sup> = 3/8 can be taken [27,55]. It is evident, from Equation (5), that the *IFSS* is directly proportional to *B* and 2.04 is the lower limit for the *IFSS* estimation.
