**3. Theoretical Analysis**

During the lab scale investigation, different analytical models were applied on the HSP/PLA based composites to estimate the fiber/matrix adhesion and to predict the tensile strength trend as a function of the HSPs volumetric content. The addition of rigid particles into a polymeric matrix can affect the strength in two ways. The tensile strength prediction of particulate filled composites is not easy because it is affected by different parameters, such as interface adhesion, stress concentration, and defect size/spatial fillers distribution [36].

For particulate fillers and for fibers with low aspect ratio, the prediction of the tensile strength can be expressed quantitatively by the following equation, proposed by Pukánszky [37]:

$$
\sigma\_c = \sigma\_m \left[ \frac{1 - V\_f}{1 + 2.5V\_f} \right] \exp\left(BV\_f\right) \tag{2}
$$

where, *σ<sup>c</sup>* and *σ<sup>m</sup>* are the stress at break of the composite and matrix, respectively, while *Vf* is the volume fiber fraction. The term in square bracket is correlated to a decrement of the tensile strength of the composite caused by the fillers addition that reduce the load-bearing cross-section of the composite. The parameter *B* is an interaction parameter that takes into account the efficiency of the stress transmission between the matrix and the filler and can be indirectly correlated to the filler/matrix adhesion [38]. Simplifying Equation (2), a linear correlation can be obtained (Equation (3)) in which the B parameter is found as the slope of the natural logarithm of reduced strength (*σred*) against the volume filler fraction.

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

For particulate fillers, in the case that the stress cannot be transferred from the matrix to the filler and the final composite tensile strength is determined from the effective sectional area of the load-bearing matrix, the tensile strength of the composites lies between an upper and lower bound [36]. Based on the hypothesis that poor adhesion exists between the filler and the polymer and the load is sustained completely by the polymer matrix, the following equation (Equation (4)) formulated by Nicolais and Nicodemo [39] gives the lower-bound strength of the composite.

$$
\sigma\_c = \sigma\_m \left( 1 - 1.21 V\_f^{\frac{2}{3}} \right) \tag{4}
$$

The upper bound is immediately obtained as follows (Equation (5)):

$$
\sigma\_{\mathcal{C}} = \sigma\_m \left( 1 - V\_f \right) \tag{5}
$$

Equation (5) generally has been considered as an ideal unattainable upper bound since, in addition to a matrix area reduction, critical effects are also induced by the filler particles in the system, with a further decrease of the composite strength.
