2.3.6. Tear Fatigue Analyzer

The fracture mechanics behavior of the rubber-silica composites was investigated using an Instron Electroplus E1000 (Darmstadt, Germany). Dumbbell-shaped pure shear specimens with length *L*0 = 10 mm, thickness *B* = 1 mm, and width *W* = 80 mm (see Figure 2) were used to measure the crack

propagation in these rubbers. The notch for crack initialization was made by hand for a length *a*0 of 25 mm on one side of the pure shear sample, which generates a crack of the length *a*, and is expected to be adequately long related to the length (*L*0) of the specimen. The crack propagation measurements were performed at room temperature (23 ◦C) using 1 N pre-force. Details of the experiments can be found in literature [25–28]. Rivlin and Thomas [29] stated that the tearing energy is the specific parameter of fatigue crack propagation in elastomers, which is defined as *T* = <sup>−</sup>(<sup>δ</sup>*W*/δ*<sup>A</sup>*)*<sup>L</sup>*. Hereby, *W* is the elastic strain energy and *A* is the interfacial area of the propagated crack. Based on the assumption of Rivlin and Thomas [29], the pure shear specimen can be divided into four regions (see Figure 2). The elastomer in area *A*1 is not strained. In region *D*, complicated deformation stress and strain fields occur due to the tip of the crack. In the *C*-region, pure shear deformation can be found and there is a region designated as *A*, where the edge effects occur. The deformation state in region *D* does not change but shift parallel in the crack growth direction, when the crack length increases by *da*. Then, the volume of the *C*-region is equal *L*0 × *B* × *da* and is, therefore, smaller and the stored elastic energy in this volume is released. Samples with pure shear geometry are used because, hereby, the tearing energy *Tp* does not depend on the crack length.

$$T\_P = w \cdot L\_0 \tag{2}$$

Hereby, *w* is the elastic pure shear strain-energy density, measured from the unloading cycle of the stress-strain curve of an un-notched sample. Two un-notched pure shear specimens with diverse widths (*W*1 = 80 mm and *W*2 = 60 mm) were employed for evaluating the pure shear elastic strain energy density *w* to eliminate the energy contribution of the edge effect region.

$$w = \frac{lL\_1 - lL\_2}{(W\_1 - W\_2) \cdot L\_0 \cdot B} \tag{3}$$

where *U*1 and *U*2 are the elastic strain energies for the two different sample widths *W*1 and *W*2, respectively.

**Figure 2.** Schematic representation of a pure shear sample with depiction of the regions having different stress conditions (according to Reference [25]).

Paris and Erdogan [30] describe the correlation between a stable fatigue crack propagation rate da/dn and the tearing energy by a power law equation.

$$
\alpha \, da/dn = \beta \cdot T^m \text{ for } T\_1 \le T \le T\_{c\prime} \tag{4}
$$

βand *m* are, hereby, constants of the material. *T*1 represents the beginning of the stable crack growth regime and *Tc* is the critical tearing energy, which characterizes the changeover to an unstable regime.
