**2. Theoretical Analysis and Theoretical Background**

Creep rupture of a polymer is the result of combined events (such as viscoelastic deformation, primary and secondary bond rupture, shear yielding, crazing, void formation and growth) and fibril breakdown with intrinsic and extrinsic flaws, leading to fracture. In the case of polymer blends or composites, the interfacial strength and morphology must also be taken into account [44]. According to the creep curves (*creep strain* (*ε*) *vs. creep time* (*t*)), in polymers, four stages can be considered [45]: (I) the first stage of instantaneous deformation (ε0), (II) the second stage named primary creep (ε1), (III) the third stage named secondary or transient creep (ε2) in which the creep rate reaches a steady-state value, and the fourth stage (IV) (ε4) in which the creep rate increases abruptly and the final creep rupture occurs. Transient creep of many materials (including polymeric materials) obeys Andrade's law in which creep strain is proportional to the cube root of time according to the following equation [46–48]:

$$
\varepsilon(t) = \varepsilon(0) + bt^{\frac{1}{3}} \tag{1}
$$

where *ε*(*0*) is the time independent instantaneous elongation due to the elastic or plastic deformation of polymer once the external load is applied; b is a function of stress and temperature. The validity of the Andrade's approach can be easily verified by plotting in the region of the transient creep, *ε*(*t*) against *t* 1/3, and checking if the data align on a straight line. If the data fits the model, the *b* parameter can be thus obtained, because it is equal to the angular coefficient of the straight line.

If the *b* parameter is known, it can be related to the activation volume according to the following Eyring relationship [48]:

$$b = Q \exp\left(\frac{\gamma V \sigma}{k\_B T}\right) \tag{2}$$

where *Q* is a constant, γ is a stress concentration factor, *V* is the activation volume for the deformation process, *kB* Boltzmann's constant, and *T* the temperature. Making the assumption that in the absence of rubber particles (or other stress-concentrating additives like rigid fillers), γ = 1 and the activation volume can be easily calculated for the pure matrix. Once the activation volume is obtained, the variation in the apparent stress

concentration factor induced by the addition of the rubber particles to the matrix can be obtained following the procedure explained in [48].

In [8], it was shown that Eyring plots of log *b* against applied stress were linear for the pure polyamide (PA66), but for the rubber toughened polyamide (RTPA66), it showed a sharp increase in *d* log *b*/*dσ*, where significant dilatation begins. Matching the results of creep tests and scanning electron micrographs, it was concluded that this cavitation accelerates shear yielding in the nylon matrix. The main explanation for this behavior was correlated with the energy-balance model for cavitation combined with the modified version of Gurson's equation for dilatation at yielding. According to the energy-balance model, the critical volume deformation Δ*<sup>v</sup> <sup>c</sup>* above which a particle can cavitate can be determined by Equation (3) [18]:

$$
\Delta\_v^c = 4 \left( \frac{4\Gamma}{3K\_r D} \right)^{3/4} \tag{3}
$$

where Γ is the surface energy of the rubber, *D* the particle diameter, and *Kr* the rubber bulk modulus. When a rubber-toughened material is subjected to an external load, during the earlier stages of deformation, the hydrostatic component of the stress in the material starts to build up and, at a certain point, the biggest particles will start to cavitate and/or to debond. In this initial stage, voids will appear randomly, but their presence significantly affects the yielding and fracture behavior of polymers. If the particles cavitate, Lazzeri and Bucknall [17,18] proposed a modified version of the Gurson yield function [49] to account for the effects of cavitation on the yielding behavior of rubber-toughened polymers:

$$
\sigma\_{\varepsilon}^{2} = \sigma\_{\phi}^{2} \left[ \left( 1 - \frac{\mu \sigma\_{m}}{\sigma\_{\phi}} \right) - 2fq\_{1} \cosh \left( \frac{3q\_{2} \sigma\_{m}}{2\sigma\_{\phi}} \right) + \left( q\_{1} f \right)^{2} \right] \tag{4}
$$

where σ*<sup>φ</sup>* = σ*<sup>0</sup>* (1 − *q*1*φ*) is the effective stress at yield for a rubber-toughened polymer containing a volume fraction *φ*, when the mean normal stress σ*<sup>m</sup>* and the void content *f* are both zero. In this equation, σ*<sup>m</sup>* is the yield stress of the pure matrix at σ*<sup>m</sup>* = *f* = 0. The factors *q*<sup>1</sup> = 1.375 and *q*<sup>2</sup> = 0.927 were introduced to improve the fit between Gurson's predictions and data from numerical analysis [19]. Following dilatational yielding, the measured activation volume *Vm* increases with the volume fraction of voids, *f*, according to the following relationship:

$$V\_m = V(1+2f) \tag{5}$$

This equation shows that the presence of voids significantly affects the rate of yielding as indicated by the increase in apparent activation volume.

If the dominating micromechanical deformation process is the rubber particle debonding, the stress necessary to initiate debonding, the number of debonded particles, and the size of the voids formed can be described by the following equation proposed by Pukanszky and Voros [12]:

$$
\sigma^D = -\mathbb{C}\_1 \sigma^T + \mathbb{C}\_2 \left(\frac{W\_{AB}E}{R}\right)^{\frac{1}{2}} \tag{6}
$$

where *σ<sup>D</sup>* and *σ<sup>T</sup>* are debonding and thermal stresses, respectively, WAB is the reversible work of adhesion, and R denotes the radius of the particle. *C*<sup>1</sup> and *C*<sup>2</sup> are constants which depend on the geometry of the debonding process.

#### **3. Materials and Methods**

#### *3.1. Materials*

The materials used in this work for the binary blends production were:

• Poly(lactic) acid (PLA), trade name Luminy LX175, purchased from Total Corbion PLA. It is a biodegradable PLA, derived from natural resources, that appears as white spher-

ical pellets. According to the datasheet producer, this PLA contains approximately 4% of D-lactic acid units and can be used alone or blended with other polymers or additives for the production of suitable blends and composites. This PLA grade can be processed easily on conventional equipment for film extrusion thermoforming or fiber spinning (density: 1.24 g/cm3, melt flow index (MFI) (210 ◦C/2.16 kg): 6 g/10 min);

• Poly(butylene succinate-co-adipate) (PBSA), trade name BioPBS FD92PM, purchased from Mitsubishi Chemical Corporation, is a copolymer of succinic acid, adipic acid, and butandiol. It is a soft and flexible semicrystalline polyester that can be blended in extruder with other polymers but can be also processed by blown and cast film extrusion (density of 1.24 g/cm3, MFI (190 ◦C, 2.16 kg): 4 g/10 min).
