# A comprehensive study on the effects of the macromolecular network structure on the fatigue properties and damage modes of natural rubber

G. Delahaye, G. Delaporte, Y. Mouslih, J. Rosselgong, Sophie M. Guillaume, B. Ruellan, I. Jeanneau, J. -B. Le Cam

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G. Delahaye, G. Delaporte, Y. Mouslih, J. Rosselgong, Sophie M. Guillaume, et al.. A comprehensive study on the effects of the macromolecular network structure on the fatigue properties and damage modes of natural rubber. International Journal of Fatigue, 2025, 201, pp.109014. 10.1016/j.ijfatigue.2025.109014. hal-05244619

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# A comprehensive study on the effects of the macromolecular network structure on the fatigue properties and damage modes of natural rubber

G. Delahaye<sup>a,b,c</sup>, G. Delaporte<sup>a,b,c</sup>, Y. Mouslih<sup>a,b,c</sup>, J. Rosselgong<sup>d ID</sup>, S.M. Guillaume<sup>d ID</sup>, B. Ruellan<sup>b,c</sup>, I. Jeanneau<sup>b,c</sup>, J.-B. Le Cam<sup>a,c ID,*</sup>

## ARTICLE INFO

### Keywords:

- Active chain density

- Uniaxial fatigue

- Fatigue lifetime reinforcement

- Natural rubber

- Carbon black filler

## ABSTRACT

The present study investigates the effects of the macromolecular network structure, i.e., the active chain density and cross-link length, on the fatigue resistance of carbon-black filled natural rubber. A wide range of vulcanization systems (conventional, semi-efficient and efficient) and conditions (time and temperature) have been considered. The active chain density was evaluated by swelling tests and using the Flory-Rehner relationship. Uniaxial fatigue tests under relaxing and non-relaxing tension loading conditions were performed, in order to characterize the fatigue resistance and the fatigue lifetime reinforcement. Moreover, a complementary post-mortem analysis at both the macro- and the microscopic scales was systematically carried out to understand the link between the damage mechanisms and the macromolecular network structure. The active chain density was found to drive the fatigue properties and damage mechanisms. Furthermore, the fatigue lifetime reinforcement and the markers of the strain-induced crystallization have been correlated. They both depend on the active chain density and the strain level. Long cross-link length enhanced the fatigue properties at low active chain density, but this effect tends to decrease as the active chain density increases for relaxing tension loadings. A *contrario*, the cross-link length was found to have negligible effect on the fatigue lifetime reinforcement and damage mechanisms. The vulcanization time showed a strong effect for long cross-link length, while no effect was observed for short cross-links. The vulcanization temperature had only slight effect on fatigue resistance for each vulcanization system studied. Finally, several diagrams were proposed to fully describe and characterize the relationship between active chain density, fatigue properties and damage mechanisms.

## 1. Introduction

Natural rubber (NR) is a highly deformable material with a remarkable fatigue resistance and good damping properties. For these reasons, it is classically employed for anti-vibratory applications, where the numerous mechanical cycles applied induce fatigue damage. The fatigue properties of NR has been intensively studied since the pioneering work due to Cadwell et al. [1], who highlighted the significant fatigue lifetime reinforcement when non-relaxing cyclic loading conditions are applied [2–4].<fnref n="1" /> Since the 2000s, fatigue of rubbers has been reviewed several times [5–7], and the main factors that govern the fatigue properties have been identified. Among them, ageing [8] and mechanical loading conditions [2] were found the most impacting. The

elastomer type, the vulcanization system and fillers were also identified as parameters of primary importance. During the vulcanization process, rubber macromolecules are chemically bonded by cross-links to form a network: it is generally denoted as the macromolecular network, or rubber network. In the case of NR (cis-1,4-polyisoprene), sulfur is generally used as the cross-linking agent. The parts of macromolecules between cross-links are denoted “active chains” and are characterized by the active chain density. Usually, sulfur is associated with an accelerator, which controls the vulcanization reaction initiation. The sulfur content [S], the accelerator content [A] and [A]/[S] ratio determine the macromolecular network structure, i.e., the cross-link length and active chain density. It is worth noting that the accelerator

<footnote><sup>a</sup> Univ. Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251, F-35042, Rennes, France</footnote>

<footnote><sup>b</sup> Contitech AVS France, 24 rue Nicolas Joseph Cugnot, F-35000, Rennes, France</footnote>

<footnote><sup>c</sup> ELAST-D<sup>2</sup> Joint Research Lab Continental/UR/CNRS. Campus de Beaulieu, Bat 10B, F-35042, Rennes, France</footnote>

<footnote><sup>d</sup> Univ. Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes) - UMR 6226, F-35042, Rennes, France</footnote>

<footnote>* Corresponding author at: Univ. Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251, F-35042, Rennes, France.
E-mail address: jean-benoit.lecam@univ-rennes.fr (J.-B. Le Cam).</footnote>

<footnote n="1">¹ It should be noted that the term fatigue lifetime reinforcement relates here to the enhancement of lifetime observed for non-relaxing loading conditions of NR.</footnote>
<table><caption>**Table 1**
Classification of the vulcanization system proposed by Quirk [10]. Accelerator [A] and sulfur [S] contents are given in parts per hundred of rubber (phr), in weight. For the sake of clarity, CV, SEV and EV are given in black, blue and red, respectively.</caption><thead><tr><th>Vulcanization system</th><th>[A]</th><th>[S]</th><th>[A]/[S]</th></tr></thead><tbody><tr><td>Conventional (CV)</td><td>0.4 - 3.5</td><td>2 - 3.5</td><td>0.1 - 0.6</td></tr><tr><td>Semi-efficient (SEV)</td><td>1.2 - 2.4</td><td>1 - 1.7</td><td>0.7 - 2.5</td></tr><tr><td>Efficient (EV)</td><td>2 - 5</td><td>0.4 - 0.8</td><td>2.5 - 12</td></tr></tbody></table>

type also affects the macromolecular network formed [9]. In practice, a low [A]/[S] ratio leads mainly to polysulphide (long) while a high value gives mono- and-disulphide (short) cross-links. A distinction is generally made between conventional (CV), semi-efficient (SEV) and efficient (EV) vulcanization systems, leading mainly to long, both short and long, and mainly short cross-links, respectively [10]. Table 1 recalls the vulcanization system classification according to the range of [A] and [S] contents as well as the range of [A]/[S] ratio.

For a given vulcanization system, i.e., given sulfur [S] and accelerator [A] contents, the vulcanization conditions, namely the time and the temperature of vulcanization, also influence the macromolecular network structure. On the one hand, a high vulcanization temperature reduces the active chain density as well as the cross-links length by a desulfurization process [11–14]. On the other hand, the vulcanization time directly controls the cross-linking, and thereby triggers the active chain density and cross-link length [15–19]. The active chain density and cross-link length significantly affect the physical, chemical and mechanical properties of rubbers. For instance, the ageing resistance is enhanced by short cross-links [20–22] and higher active chain density increases static stiffness [15,23]. To go even further, the strain-induced crystallization (SIC) ability, which is usually considered as responsible for the fatigue resistance especially under non-relaxing loading conditions, is closely related to the macromolecular network [24–30]. Moreover, in the early 1950s, Dogadkin and Tarasova were among the first to address the effect of the cross-links length on the fatigue resistance, for unfilled NR and styrene butadiene rubber (SBR) vulcanized with various vulcanization systems [31]. The authors showed that the longer the cross-links, the higher the fatigue resistance at room temperature, while the reverse situation occurs at elevated temperatures. Years later, Yanyo demonstrated that polysulphide cross-links led to a higher crack growth resistance than monosulphide and carbon-carbon ones [32], which confirmed the results of Dogadkin and Tarasova. For the author, longer cross-link increases the active chain mobility. This would allow these to be cyclically stretched and relaxed without breaking. Other authors explained these results by the ability of the cross-links to break and rearrange [5,33]. According to Beatty, contrarily to polysulphide cross-links, monosulphide ones cannot rearrange themselves at the ultimate local strain, and therefore break [5]. This explanation is close to the one proposed by Bhowmick, who states that both the rearrangement of the cross-links and their scission, before the active chains break, dissipate the strain energy [33]. For Lake and Lindley, the explanation lies in the flexible nature of these long cross-links. This would be responsible for stresses relieving, which increases the energy required to break the active chains of the macromolecular network [34]. Chain slipping phenomenon has also been proposed to explain this increased resistance to failure [35]. For Chan, polysulphide cross-links increase the fatigue resistance by at least a factor of two or three compared to shorter cross-links, for relaxing loading conditions [36]. Nevertheless, the author mentioned that the temperature and the strain level have a significant effect on this behavior. As shown by Beatty, the sulfur content, i.e., the active chain density, governs the crack growth resistance for SBR [5]. Chan mentioned that an excessive active chain density increases the sensitivity of NR to fatigue. Hence, this implies the existence of an optimal active chain density for fatigue application [36], as also suggested by Mars and Fatemi [6]. More recently, Grasland showed that for CV systems, the crack growth rate depends on the active chain density [37]. Finally, the

vulcanization conditions, i.e., the time and temperature, also affect the fatigue resistance. For SBR, it was shown that increasing the vulcanization time beyond a certain value drastically reduces the crack growth resistance [5]. Furthermore, it seems that there is an optimum vulcanization time for the fatigue life, that would correspond to a sufficiently high stiffness and hysteresis [6]. However, for carbon black (CB) filled nitrile butadiene rubber (NBR), Gehling et al. measured that increasing the vulcanization time above an optimal state leads to a decrease in crack growth resistance [38]. The authors also demonstrated that the higher the vulcanization temperature, the higher the crack growth resistance. Finally, CB content and grade impact the fatigue resistance of NR [39–41].

To summarize, numerous results have been reported in the literature. Many parameters were found to affect the relationship between the macromolecular network and the fatigue properties, namely the curing time and temperature, the vulcanization system type, the fillers content and grade and the rubber matrix type. Nevertheless, these results were obtained with very different compounds and vulcanization conditions. To the best of our knowledge, no study investigated the effect of these parameters, especially a wide range of active chain densities and cross-link lengths on the fatigue resistance of the same elastomer filled with the same filler type and content.

In the present work, a wide range of vulcanization systems (accelerator, sulfur and accelerator-to-sulfur ratio) and conditions (time, temperature) have been considered for the same CB-filled NR. The fatigue properties have been characterized under relaxing and non-relaxing tensile loading conditions. Systematic post-mortem analyses have been performed to understand the relationship between the fatigue damage mechanisms, fatigue resistance and the macromolecular network structure.

Thereafter, Section 2 presents the experimental set-up: the materials of the study and the specimens geometry, the swelling tests, the fatigue tests facilities and the fatigue loading conditions applied. The section closes with the end-of-life criterion, the prediction of the mechanical state and the choice of the damage predictor. Results are reported and discussed in Section 3. Concluding remarks and perspective close the paper.

## 2. Experimental section

This section presents the material formulations, the active chain density evaluation from swelling tests, the fatigue tests and the fractographic analyses.

### 2.1. Materials

The material considered here is a NR (mainly composed of cis-1,4-polyisoprene) of a SMR (Standard Malaysian Rubber) grade. Components are given in part per hundred of rubber (phr), in weight. All formulations contain 10 phr of zinc oxide (ZnO), 2 phr of plasticizer, 2 phr of stearic acid and 3 phr of antioxidants, respectively. Each material was filled with 20 phr of N330 HAF (High Abrasion Furnace) CB grade. The vulcanization system is composed of sulfur and N-cyclohexyl-2-benzothiazole sulfenamide (CBS) as accelerator. Their contents were varied to modify the macromolecular network formed. First, five [A]/[S] ratios were defined to cover a large range of cross-link lengths. It corresponds to 0.2 and 0.6 for CV, 1.5 and 2.5 for SEV
<table><caption>**Table 2**
Material formulations.</caption><thead><tr><th>Designation</th><th>Vulcanization system</th><th>[A] (phr)</th><th>[S] (phr)</th><th>[A]/[S] (-)</th><th>CB (phr)</th></tr></thead><tbody><tr><td>N R<sup>2</sup><sub>0.2</sub></td><td>CV</td><td>0.4</td><td>2</td><td>0.2</td><td>20</td></tr><tr><td>N R<sup>4</sup><sub>0.2</sub></td><td>CV</td><td>0.8</td><td>4</td><td>0.2</td><td>20</td></tr><tr><td>N R<sup>6</sup><sub>0.2</sub></td><td>CV</td><td>1.2</td><td>6</td><td>0.2</td><td>20</td></tr><tr><td>N R<sup>2</sup><sub>0.6</sub></td><td>CV</td><td>1.2</td><td>2</td><td>0.6</td><td>20</td></tr><tr><td>N R<sup>4</sup><sub>0.6</sub></td><td>CV</td><td>2.4</td><td>4</td><td>0.6</td><td>20</td></tr><tr><td>N R<sup>6</sup><sub>0.6</sub></td><td>CV</td><td>3.6</td><td>6</td><td>0.6</td><td>20</td></tr><tr><td>N R<sup>0.8</sup><sub>1.5</sub></td><td>SEV</td><td>1.2</td><td>0.8</td><td>1.5</td><td>20</td></tr><tr><td>N R<sup>1.2</sup><sub>1.5</sub></td><td>SEV</td><td>1.8</td><td>1.2</td><td>1.5</td><td>20</td></tr><tr><td>N R<sup>1.6</sup><sub>1.5</sub></td><td>SEV</td><td>2.4</td><td>1.6</td><td>1.5</td><td>20</td></tr><tr><td>N R<sup>0.8</sup><sub>2.5</sub></td><td>SEV</td><td>2</td><td>0.8</td><td>2.5</td><td>20</td></tr><tr><td>N R<sup>1.2</sup><sub>2.5</sub></td><td>SEV</td><td>3</td><td>1.2</td><td>2.5</td><td>20</td></tr><tr><td>N R<sup>1.6</sup><sub>2.5</sub></td><td>SEV</td><td>4</td><td>1.6</td><td>2.5</td><td>20</td></tr><tr><td>N R<sup>0.8</sup><sub>4</sub></td><td>EV</td><td>3.2</td><td>0.8</td><td>4</td><td>20</td></tr></tbody></table>

<table><caption>**Table 3**
Description of the vulcanization conditions investigated for N R<sup>4</sup><sub>0.2</sub>, N R<sup>1.2</sup><sub>1.5</sub> and N R<sup>0.8</sup><sub>4</sub>.</caption><thead><tr><th>Condition N°</th><th>t<sub>vulc</sub> (-)</th><th>T<sub>vulc</sub> (°C)</th></tr></thead><tbody><tr><td>0</td><td>t<sub>98</sub></td><td>160</td></tr><tr><td>1</td><td>t<sub>80</sub></td><td>160</td></tr><tr><td>2</td><td>t<sub>98</sub> +40% of time</td><td>160</td></tr><tr><td>3</td><td>t<sub>98</sub></td><td>150</td></tr><tr><td>4</td><td>t<sub>98</sub></td><td>170</td></tr></tbody></table>

and 4 for EV. Then, for a fixed [A]/[S] ratio, [A] and [S] were varied in order to reach various active chain densities for a same cross-link length. For the sake of clarity, CV, SEV and EV will be respectively denoted in black, blue and red in the following. These recipies are in agreement with the standard distinction introduced by Quirk and presented in Table 1 [10].

Table 2 details the different formulations. They are denoted N R<sup>1</sup><sub>J</sub>, with I referring to the sulfur content [S] and J to the [A]/[S] ratio, respectively.

The specimens were obtained by a two-step mixing process. The NR gum and adjuvants were first incorporated into an internal mixer. The sulfur and accelerator were then added in an external mixing, in an open-roll mill, at a lower temperature below 100 °C in order to prevent early cross-linking reaction from occurring. The vulcanization conditions, i.e., time and temperature, are hereafter denoted t<sub>vulc</sub> and T<sub>vulc</sub>, respectively. The specimens were molded at a temperature of 160 °C, at a time corresponding to t<sub>98</sub>,<fnref n="2" /> determined with a Rubber Process Analyzer 1000 from Alpha Technologies. For N R<sup>4</sup><sub>0.2</sub>, N R<sup>1.2</sup><sub>1.5</sub> and N R<sup>0.8</sup><sub>4</sub>, additional curing temperatures of 150 °C and 170 °C were used. As the rheologic behavior changes with temperature, the vulcanization times were adjusted to reach t<sub>98</sub>. Under- and over-vulcanization were also investigated for these three materials.<fnref n="3" /> The vulcanization times correspond to t<sub>80</sub> and t<sub>98</sub> +40% of time. These conditions are numbered 0 to 4 and are recalled in Table 3. In the following, when the vulcanization condition number is not mentioned, results refer to materials cured for condition N°0 in Table 3, i.e., t<sub>98</sub> and 160 °C.

## 2.2. Active chain density evaluation

The objective of evaluating the active chain density, denoted v<sub>ac</sub> in the following, is to provide a quantitative information on the macro-molecular network. v<sub>ac</sub> was evaluated from swelling test and weight measurements. The procedure applied is fully detailed in [43]. It can be summed-up as follows. The swelling experiments are made by using circular specimens cut with a punch in a 2 mm thick and 0.4 g initial length sheet. Such small specimens are preferred since they can swell up to several times their original volume. The reported values are the average of three specimens. The specimens are beforehand weighed before being immersed for 72 h in a closed vial of toluene, HPLC (High Performance Liquid Chromatography) grade. The weight is measured with a model XT-220A (Precisa scale) balance. Then, specimens are released, and a paper cloth is used to dry the solvent excess from their surface. The weight at swollen state is then measured. In order to evaporate toluene trapped in the specimens, they are dried in a vacuum-oven until a constant weight is obtained, typically for 24 h at 60 °C. The dried specimens are finally weighed. These tests were performed at ambient temperature, without isolating the vial from the light. UV light is known to cause a chemical degradation of NR and could affects the swelling measurements [44]. However, by looking at the swelling kinetics of Candau on similar materials, the degradation happened only at a relatively long time, around 300 h [45]. Since it is considerably longer than the chosen immersion time, UV exposure is assumed to have no effect.

In practice, for a given initial weight, the higher the weight at swollen state, the lower the v<sub>ac</sub> value. The Flory-Rehner relationship, for an affine network assumption and a Flory-Huggins interaction χ = 0.39, as well as a correction for the volume of insoluble components, were used to calculate v<sub>ac</sub>. This is fully explained in [43]. v<sub>ac</sub> has been determined for each NR used in the present study. In order to present the results obtained for this large number of materials, v<sub>ac</sub> values are reported with respect to the sulfur [S] and accelerator [A] contents and their ratios [A]/[S], as first introduced by Delahaye et al. [43]. It is depicted in Fig. 1, for materials cured at t<sub>98</sub> corresponding to condition N°0 in Table 3. In this figure, the X and Y axes represent the sulfur [S] and accelerator [A] contents, respectively. [A]/[S] ratios are constant along the half lines starting from the origin (0,0). Each material is therefore mapped according to its vulcanization system. v<sub>ac</sub> is characterized by the corresponding color level. The associated values are summarized in Table 4.

The effect of the vulcanization conditions on v<sub>ac</sub> are summarized in Table 5. It should be noted that the vulcanization time and temperature were found to have a pronounced impact on v<sub>ac</sub> for N R<sup>4</sup><sub>0.2</sub>. These effects on the fatigue properties are discussed in Section 3.

<footnote n="2">² t<sub>X</sub> corresponds to the time required to reach X% of the maximum torque measured by a rheometer, during vulcanization.</footnote>

<footnote n="3">³ For the over-vulcanized materials, since the torque kinetics strongly depends on the vulcanization system, it would have been trivial to set a time corresponding to a torque value. For instance, CV system tends to rapidly reverse while for EV system, a torque plateau is formed, and it only decreases slightly. Consequently, it is preferred to use an additional period of time, defined as 40% of the total time corresponding to t<sub>98</sub>. For more details on the effect of the vulcanization system on the vulcanization kinetics, the reader can refer to Refs [9,16,42].</footnote>
<table><caption>**Table 4**
Summary of $v_{ac}$. Values are given in $\times 10^4$ mol/cm$^3$.</caption><thead><tr><th>Formulation</th><th>$N R_{0.2}^2$</th><th>$N R_{0.2}^4$</th><th>$N R_{0.2}^6$</th><th>$N R_{0.6}^2$</th><th>$N R_{0.6}^4$</th><th>$N R_{0.6}^6$</th></tr></thead><tbody><tr><td>$v_{ac}$</td><td>0.95</td><td>1.53</td><td>2.28</td><td>1.41</td><td>2.27</td><td>3.20</td></tr><tr><td>$N R_{1.5}^{0.8}$</td><td>$N R_{1.5}^{1.2}$</td><td>$N R_{1.5}^{1.6}$</td><td>$N R_{2.5}^{0.8}$</td><td>$N R_{2.5}^{1.2}$</td><td>$N R_{2.5}^{1.6}$</td><td>$N R_{4}^{0.8}$</td></tr><tr><td>0.95</td><td>1.24</td><td>1.49</td><td>1.11</td><td>1.38</td><td>1.71</td><td>1.29</td></tr></tbody></table>

<table><caption>**Table 5**
Effect of the vulcanization conditions on $v_{ac}$ for $N R_{0.2}^4$, $N R_{1.5}^{1.2}$ and $N R_{4}^{0.8}$.</caption><thead><tr><th>Condition N°</th><th>t<sub>vule</sub> (–)</th><th>T<sub>vule</sub> (°C)</th><th>N R<sup>4</sup><sub>0.2</sub></th><th>N R<sup>1.2</sup><sub>1.5</sub></th><th>N R<sup>0.8</sup><sub>4</sub></th></tr></thead><tbody><tr><td>1</td><td>t<sub>90</sub></td><td>160</td><td>1.61</td><td>1.23</td><td>1.34</td></tr><tr><td>2</td><td>t<sub>98</sub> +40% of time</td><td>160</td><td>1.43</td><td>1.22</td><td>1.31</td></tr><tr><td>3</td><td>t<sub>98</sub></td><td>150</td><td>1.70</td><td>1.34</td><td>1.38</td></tr><tr><td>4</td><td>t<sub>98</sub></td><td>170</td><td>1.63</td><td>1.20</td><td>1.30</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Vulcanization system mapping with respect to accelerator [A] and sulfur [S] contents. The qualitative color scale gives the value of $v_{ac}$, according to the representation first introduced in [43].</figcaption></figure>

## 2.3. Fatigue tests

### 2.3.1. Fatigue test facilities and specimens

The fatigue tests were carried out using an uniaxial MTS Landmark machine, as illustrated in Fig. 2(a). 8 AE40 axisymmetric specimens were used for each fatigue test, which is of great benefit since it drastically reduces the fatigue campaign duration and accounts better for the inherent fatigue life dispersion. It corresponds to “Diabolo” specimen with a radius of curvature equal to 40 mm. The height of the specimen is 32 mm, the diameters at half height and close to the metallic insert are 20 and 32 mm, respectively. This volumetric specimen is preferred to a planar specimen since it is more representative of the volume of matter involved in the industrial rubber parts. The stress and strain are concentrated at half of its height [46], but complex multiaxial deformation occurs in the vicinity of metallic inserts [47]. The reaction force was measured for all the specimens, which enabled to determine the fatigue life of each specimen, independently. More details are given in Section 2.3.3.

### 2.3.2. Experimental procedure

The fatigue tests were conducted under uniaxial tension loading conditions at room temperature. They were prescribed under displacement, denoted U. In regard to the large number of formulations investigated in this study, two formulations were tested simultaneously, with 4 Diabolo specimens of each. Two different loading ratios, defined in terms of displacement as $R_U = \frac{U_{min}}{U_{max}}$, were applied: $R_U = 0$ and $R_U = 0.25$. They correspond respectively to repeated (Fig. 2(b)) or fully relaxing tension [47], and non-relaxing tension (Fig. 2(c)) loadings. For

the sake of clarity, $R_U = 0$ is denoted relaxing tension loadings in the following. For each loading ratio, 3 maximum displacements $U_{max}$ were imposed. The tests were carried out at a maximum frequency of 4 Hz in order to limit the self-heating at the specimen's surface and to prevent the material from further damage due to additional thermal degradation. These conditions were found to limit self-heating to 15 °C at the specimen's surface [46], in the case of similar filled NR. The testing conditions are summarized in Table 6. It has to be mentioned that other maximum displacements were applied for formulations presenting a higher fatigue life at $R_U = 0.25$. In this case, higher displacements were used to reach fatigue life lower than $10^6$ cycles. In practice, the tests were conducted until failure, or stopped at $1.7 \times 10^6$ cycles.

### 2.3.3. End-of-life criterion

The end-of-life criterion can be defined as the loss of the specimen mechanical properties, which generally coincides with the appearance of the crack at its surface. It is similar to the one used by Ruellan [46]. In practice, it corresponds to the development of a crack which length is lower than 5 mm [4,46]. Fig. 3 depicts the evolution of the maximal reaction force during a fatigue test with a $N R_{0.2}^4$ specimen, and for the loading condition n°1, i.e., relaxing tension at a maximal strain equal to 60%. Three steps are observed in the evolution of the reaction force during the test: (i) a slight decrease in the force at the beginning of the test corresponding to the stress softening, (ii) the stabilized maximal force state, which corresponds to the major part of the lifetime of the Diabolo specimen and (iii) the drop of stiffness. The end-of-life $N_i$ is defined as the last cycle of step 2, i.e., before the drop of stiffness. It corresponds to the number of cycles at the end of the plateau, when the first order derivative of the maximal reaction force $F_{max}$ is no longer constant.

### 2.3.4. Representation of the results

Two approaches are generally used to represent the fatigue life of rubber, both inherited from the fatigue analysis of metallic materials: (i) a two-dimensional graph that gives the evolution of the fatigue life with respect to a fatigue predictor [8,39,48]. It is also referred to as “endurance curve”, “Wöhler curve” or “S-N” curve in the literature, and by (ii) a three-dimensional diagram, which maps the fatigue tests according to the mean and amplitude of the loading applied, and called the Haigh diagram. The fatigue life is given as a third axis and iso-lifetime curves are used to investigate the reinforcement under non-relaxing loadings [2,4,46,49–51]. Both representations are used in the present study. The endurance curve is used to address the effect of the formulation for $R_U = 0$ while the Haigh diagram is chosen to investigate the fatigue lifetime reinforcement for $R_U = 0.25$. Since the formulations of the study strongly differ from one to another and lead to different mechanical behaviors, using an appropriate predictor is of major importance to analyze the fatigue results. To do so, it must be ensured the mechanical state of the Diabolo is correctly predicted. This is addressed in Section 2.3.5.
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** (a) Home-made apparatus used in this study, with an example of a Diabolo specimen. The cyclic time-displacement curve is given for (b) repeated ($R_U$ = 0) and (c) non-relaxing ($R_U$ = 0.25) tension loadings.</figcaption></figure>

<table><caption>**Table 6**
Summary of the fatigue tests performed.</caption><thead><tr><th>Condition n°</th><th>R<sub>U</sub> (-)</th><th>U<sub>max</sub> (mm)</th><th>U<sub>min</sub> (mm)</th><th>ε<sub>max</sub> (-)</th><th>ε<sub>min</sub> (-)</th><th>Frequency (Hz)</th></tr></thead><tbody><tr><td>1</td><td>0</td><td>11.3</td><td>0</td><td>60</td><td>0</td><td>4</td></tr><tr><td>2</td><td>0</td><td>16.9</td><td>0</td><td>90</td><td>0</td><td>4</td></tr><tr><td>3</td><td>0</td><td>23.3</td><td>0</td><td>120</td><td>0</td><td>4</td></tr><tr><td>4</td><td>0.25</td><td>16.9</td><td>4.6</td><td>90</td><td>22.5</td><td>4</td></tr><tr><td>5</td><td>0.25</td><td>36.8</td><td>8.3</td><td>172</td><td>43</td><td>2</td></tr><tr><td>6</td><td>0.25</td><td>45.2</td><td>9.6</td><td>200</td><td>50</td><td>2</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Evolution of the maximal reaction force over the number of cycles, for relaxing tension loading at $ε_{max}$ = 60%, and for $N R_{0.2}^4$.</figcaption></figure>

### 2.3.5. Prediction of the mechanical state

The choice of an appropriate model for predicting the mechanical state is of major importance to interpret the fatigue results. Finite element analysis (FEA) is used to calculate the mechanical state at any point of the Diabolo. The methodology proposed by Mouslih et al. [4] is applied here. It consists in taking into account the heterogeneous accommodation of the specimen with the Ogden-Roxburgh model. As shown in [4], the heterogeneity of the accommodation has a much more important effect on the prediction than multiaxial effects as the mechanical state is very close to the uniaxial tension in the AE40 specimen under tension. The model consists in describing the first load

with a purely hyperelastic law, the second order Ogden law [52], and the further cycles with the Ogden-Roxburgh law [53] to account for the stress softening. The formalism of the model, the identification method from a uniaxial tensile test, as well as the validation of the approach, are detailed in [4]. Fig. 4 shows the experimental and predicted stress-strain curves for $N R_{0.2}^2$, $N R_{0.2}^4$, and $N R_{0.2}^6$, which are representative of the $v_{ac}$ range measured on the formulations used in the present study. The nominal stress, or the first Piola-Kirchhoff stress, is defined as the ratio of the force over the initial cross-section area. It is denoted $π$ in the following. Note that these tests were performed with the same specimen geometry as the one used by Ruellan [46] and Mouslih [4]. The specimens are cut in a pure shear specimen by using the water-jet technique. The cross-section of this specimen is equal to $8 \times 2 \text{ mm}^2$, with a length between the cylindrical extremities equal to 23 mm. As shown in Figs. 4(a) and (b), this model accounts well for the Mullins effect, which is of great interest since the Diabolo specimens soften, see Fig. 3.

This model was used to calculate the stress-strain field in the Diabolo specimens. The predicted force was compared to the experimental one to validate the model. Fig. 5 gives the correlation between the experimental and the predicted reaction forces for each formulation. In these graphs, the predicted and experimental reaction forces are given in X and Y axes, respectively. A perfect prediction is given by the dotted straight line representing $x = y$. Since all data are within a 10% scatter band, the constitutive model is considered representative of the formulations behavior, and therefore validated for the range of $v_{ac}$ studied. Thus, the calculated stress and strain can be used to represent the lifetime.

### 2.3.6. Choice of the damage predictor

In the case of uniaxial fatigue loading, different continuum quantities were used in the literature such as the maximum stress $σ_{max}$, the maximum strain $ε_{max}$ and the strain energy density $W_{strain}$ [54]. It is to be noted that these mechanical quantities unify the fatigue results
<figure><img src="image_7.png" /><figcaption>Fig. 4. Prediction of the (a) 1st and (b) 5th load for uniaxial tension, for $NR_{0.2}^2$, $NR_{0.2}^4$ and $NR_{0.2}^6$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Correlation between the measured and predicted reaction force (a) $R_U = 0$ and (b) $R_U = 0.25$. The black dashed line stands for perfect prediction. The red dashed lines represent ± 10% error.</figcaption></figure>

in the case of proportional loading [54]. $\sigma_{max}$, $\epsilon_{max}$ and $W_{strain}$ were calculated by FEA, and then compared in order to choose the damage predictor that will be used in this study. The comparison is again made on $NR_{0.2}^2$, $NR_{0.2}^4$ and $NR_{0.2}^6$ as they cover a large $v_{ac}$ range, and consequently very different stiffnesses. Fig. 6 presents the fatigue life of the three materials with respect to (a) the maximal Cauchy stress $\sigma_{max}$, (b) the maximal strain $\epsilon_{max}$ and (c) the strain energy density $W_{strain}$ for relaxing tension loading conditions. The maximum stress and strain energy density show similar results and led to similar trends. However, the endurance curve plotted with respect to the maximal strain shows strong differences. Indeed, $NR_{0.2}^2$ is found to be greatly superior to $NR_{0.2}^4$ and $NR_{0.2}^6$. As the tests are prescribed under displacement and the materials' stiffness are different, the responses in terms of stresses differ. This is highlighted by the red dashed line in Fig. 6(b) and the red circles in Figs. 6(a) and (c). Here, the maximum Cauchy stress is used to compare the fatigue response of the materials.

## 2.4. Microscopy

Microscopy was used to analyze the failure surface of the fatigue-tested Diabolos to provide additional information on the damage mechanisms and the effect of the materials. The fatigue damage mechanisms of NR under both relaxing and non-relaxing tension were already well-established, and highlight the strong influence of SIC [3,4,50,51,55]. The present post-mortem analysis study does not aim to revisit the damage mechanisms of NR, but rather to investigate the extent to which the SIC markers form in relation to the macromolecular network structure, i.e., $v_{ac}$ and cross-link length. The experimental procedure is first described before standard damage mechanisms are presented.

<caption>Table 7</caption>

Summary of the analyzed elements on the fracture surfaces of the Diabolo specimens.

<table><thead><tr><th>Macroscopic scale</th><th>Microscopic scale</th></tr></thead><tbody><tr><td rowspan="7">Damage modes</td><td>Number of initiations</td></tr><tr><td>Position of the defect relative to the injection point</td></tr><tr><td>Defect-surface distance</td></tr><tr><td>Defect length</td></tr><tr><td>Stable propagation area</td></tr><tr><td>Presence of fatigue striations and their regime</td></tr><tr><td>Presence of cones and their location</td></tr><tr><td>Final ligament surface area</td></tr><tr><td>Presence of cavitations</td></tr></tbody></table>

### 2.4.1. Experimental procedure

A wide field ZEISS Axio Zoom V16 optical microscope was used to analyze and observe the fracture surface features. Each Diabolo specimen was analyzed and various features were studied at both the macro- and microscopic scales. Table 7 summarizes the different elements to be observed. The damage mechanisms considered for the analysis of these elements are presented hereafter in Section 2.4.2. It should be noted that the stable propagation area, presence of fatigue striations, presence and location of cones, final ligament surface area and presence of cavitations were used to compare the fracture surfaces.

A JEOL IT 300 LA SEM was used to analyze some failure surface with a higher magnification. It is coupled with a EDSX probe in order to quantify the atoms present at the specimen surface. Secondary and back scattered electrons were used for imaging and composition analysis, respectively.
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Endurance curves for the $NR_{0.2}^2$, $NR_{0.2}^4$ and $NR_{0.2}^6$ by using (a) the maximal stress $\sigma_{max}$, (b) the maximal strain $\epsilon_{max}$ and (c) the strain energy density $W_{strain}$ as a damage predictor.</figcaption></figure>

### 2.4.2. Damage mechanisms to be observed

The damage modes of NR under fatigue loading conditions are classified according to the classification introduced by Le Cam et al. [47], and then used and further enriched by Ruellan et al. [50]. This classification covers damages observed at both macro- and microscopic scales.

At the macroscopic scale. Three main macroscopic damage modes are encountered. They are given in Fig. 7. They can be described as follows:

* **Damage mode 1:** the failure occurs perpendicularly to the stretching direction, in the middle of the specimen (Fig. 7(a)). It was generally observed for relaxing loading conditions [47],

* **Damage mode 2:** it involves multiple surface cracks at half height of the Diabolo specimen, propagating through the volume with successive bifurcations. It is exclusively observed when the applied load is non-relaxing, suggesting that the reinforcement due to SIC is activated (Fig. 7(b)),

* **Damage mode 3:** it corresponds to a cohesive failure under the inserts, where the hydrostatic pressure is high. This damage mode is exclusively observed under highly non-relaxing conditions (Fig. 7(c)).

At the microscopic scale. At this scale, the crack initiates and propagates, revealing different patterns that differ according to the damage mode.

* **Damage mode 1:** for relaxing loading conditions, the crack initiates near the surface, either due to the presence of a particle part of the formulation (i.e., CB or ZnO) that concentrates stresses [2, 56–58], or due to a geometric singularity such as a joint plane. The crack then propagates elliptically until it reaches the surfaces,

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** The tree main damage modes encountered at the macroscopic scale.</figcaption></figure>

thereby causing alteration of the mechanical properties. The stable propagation phase characterized by a rather smooth surface is followed by the unstable one, during which wrenchings and small Regime 1 fatigue striations are formed, depending on the strain level. When the stress exceeds the stress at break, the Diabolo fails monotonously, by forming a very smooth final ligament. This is
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Schematic representation of a fracture surface of a filled NR Diabolo specimen tested under relaxing fatigue loading conditions at $R_U$ = 0. Both optical (the smallest magnifications) and SEM images are displayed.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Schematic representation of a fracture surface of a filled NR Diabolo specimen tested under non-relaxing fatigue loading conditions at $R_U$ = 0.25. Both optical (the smallest magnifications) and SEM images are displayed.</figcaption></figure>

summarized in Fig. 8. It should be recalled that the wrenchings and the fatigue striations are patterns attributed to SIC (see [59] for the wrenchings and [55] for the fatigue striations). These patterns are observed even though the loading is relaxing, i.e., without any fatigue lifetime reinforcement.

* **Damage mode 2:** under non-relaxing loading conditions, multiple initiations occur near the surface but do not propagate perpendicular to the direction of strain, contrarily to Damage mode 1, because of the high reinforcement due to SIC.<fnref n="4" /> Instead, the crack propagates along the path of least energy, almost parallel to the strain direction. This mode of propagation leads to the formation of cones, further reflecting the strong activation of SIC. The crack then propagates from the main cone, similar to Damage mode 1 by successively forming wrenchings, well-formed fatigue striations according to Regime 2 [46] with lamellar and triangular shapes [55], and the final ligament. This is illustrated in Fig. 9.

* **Damage mode 3:** in the case of highly non-relaxing loading conditions, the specimen middle section remains undamaged due

to the high reinforcement, which prevents any crack from developing. The damage zone is shifted under the inserts where high hydrostatic pressures occur [47,61]. This leads to the formation of cavitations that are close together and have jagged edges, creating a similar appearance to blooming flowers. This is depicted in Fig. 10. It is worth noting that the competition between crack branching and crack below metallic inserts was observed by Le Cam et al. [47].

# 3. Results and discussion

## 3.1. Relaxing tension loadings

First of all, it should be recalled that results are plotted in terms of endurance curves representing the fatigue predictor, i.e., the Cauchy stress $\sigma$, and the mean fatigue life $N_i$. Experimental points for each materials are given in a SM file.

### 3.1.1. Effect of the vulcanization system on the fatigue resistance

As discussed in the Introduction section, the accelerator [A] and sulfur [S] contents drive the macromolecular network structure, i.e., $v_{ac}$ and cross-link length. The effect of these two parameters on the fatigue response are here independently addressed.

<footnote n="4">⁴ It is noteworthy that neither fatigue striations nor cones were observed on SBR, which is non-crystallizable [60]. These features can therefore be considered as markers of crystallization.</footnote>
Damage mode 3

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Schematic representation of a fracture surface of a filled NR Diabolo specimen tested under highly non-relaxing fatigue loading conditions at $R_u = 0.25$. Both optical (the smallest magnification) and SEM images are displayed.</figcaption></figure>

**Active chain density effect.** Fig. 11 presents the endurance curves for NR with [A]/[S] of (a) 0.2 and (b) 0.6 for CV systems, and (c) 1.5 and (d) 2.5 for SEV ones. For each endurance curve, the effect of the $v_{ac}$ is shown through three materials. First, Fig. 11(a) shows the endurance curves for materials with long cross-links (CV), with [A]/[S]=0.2. Raising the value of higher $v_{ac}$ results in an increase in stress. This is also observed for the other [A]/[S] ratios, and is due to the increase in stiffness caused by the higher $v_{ac}$ [62]. Even though, the fatigue life decreases as $v_{ac}$ increases, it seems that they follow a similar trend. Experimental data was fitted with a unique power law with a correlation coefficient $R^2 = 0.89$ (see Figure S1 in the SM file). This suggests that $v_{ac}$ has a little effect on the fatigue life for [A]/[S]=0.2. Then, again for CV, but with [A]/[S]=0.6 (Fig. 11(b)), a comparable fatigue response was obtained for $NR^{0.2}$ and $NR^{0.6}$, traducing only minor effect of $v_{ac}$ on the fatigue life. However, for $NR^{0.6}$ presenting the highest $v_{ac}$, the fatigue life is drastically reduced in comparison to the $NR^{0.6}$ and $NR^{0.6}$. This could be interpreted by the network mobility restrictions caused by too many active chains. To continue, the endurance curves for [A]/[S]=1.5 are presented in Fig. 11(c). A larger $v_{ac}$ increases the stress as well as the fatigue life. In particular, above $0.95 \times 10^4$ mol/cm$^3$, the fatigue resistance is significantly enhanced. This suggests that for these materials, a higher $v_{ac}$ and consequently a higher stiffness is truly efficient to enhance fatigue resistance. However, it is worth noting that the range of $v_{ac}$ for [A]/[S]=1.5 is significantly lower than the ones for [A]/[S]=0.2 and 0.6. Finally, for the SEV with [A]/[S]=2.5 (Fig. 11(d)), a similar fatigue response is obtained for $NR^{1.2}$ and $NR^{1.6}$, which is significantly better than the $NR^{0.8}$ having a lower $v_{ac}$. This suggests that for this type of SEV, a certain stiffness is required to maximize the fatigue resistance: above $1.11 \times 10^4$ mol/cm$^3$, the fatigue resistance is drastically improved. Nevertheless, there is no gain in fatigue resistance with the higher $v_{ac}$ of $1.71 \times 10^4$ mol/cm$^3$, which suggests two effects: (i) a threshold for which the network is no longer able to improve fatigue resistance, and (ii) an optimum of $v_{ac}$, the value being between $NR^{1.2}$ and $NR^{1.6}$.

<caption>**Table 8**</caption>

Summary of the groups used to evaluate the effect of the cross-link length. Values of $v_{ac}$ are given in $\times 10^4$ mol/cm$^3$.

<table><thead><tr><th>Group n°</th><th>Formulations</th><th>Range of v<sub>ac</sub></th></tr></thead><tbody><tr><td>1</td><td>NR<sup>2</sup><sub>0.2</sub> ⇌ NR<sup>0.8</sup><sub>1.5</sub></td><td>0.95</td></tr><tr><td>2</td><td>NR<sup>1.2</sup><sub>1.5</sub> ⇌ NR<sup>0.8</sup><sub>1.5</sub></td><td>1.24–1.29</td></tr><tr><td>3</td><td>NR<sup>2</sup><sub>0.6</sub> ⇌ NR<sup>1.2</sup><sub>2.5</sub> ⇌ NR<sup>1.6</sup><sub>1.5</sub></td><td>1.38–1.49</td></tr><tr><td>4</td><td>NR<sup>4</sup><sub>0.2</sub> ⇌ NR<sup>1.6</sup><sub>1.5</sub></td><td>1.49–1.53</td></tr></tbody></table>

To sum-up, $v_{ac}$ was found to strongly affect the fatigue properties of filled NR, by impacting the mechanical response. However, the effect of $v_{ac}$ on the fatigue life differed depending on the [A]/[S] ratio, i.e., the cross-link length. For example, increasing the $v_{ac}$ and consequently the stiffness is of benefit for the filled NR with an [A]/[S] ratio of 1.5, while it was not the case for the filled NR with an [A]/[S] ratio of 0.6. It is noteworthy that the $v_{ac}$ ranges varied between the different [A]/[S] studied, making the comparison difficult. Finally, this suggests three possibilities: (i) there is an optimum value of $v_{ac}$ as mentioned in [6,42,63], (ii) the cross-link length affects the fatigue response, or (iii) the both. To further investigate this question, we next focus on the effect of the cross-link length on the fatigue properties.

**Cross-link length effect.** The effect of the cross-link length on the fatigue properties was evaluated based on a comparison of filled NR having similar $v_{ac}$ and different [A]/[S] ratios. Four groups of filled NR were defined in order to cover a wide range of $v_{ac}$. It is to be noted that results from these comparisons need to be analyzed with caution, since the $v_{ac}$ of each group (i) is not constant (except for Group n°1), and (ii) they are based on a measurement that implies an experimental error. These couples are presented in Table 8.

Fig. 12 presents the endurance curves of the four material groups listed in Table 8. The individual fatigue life is also reported for the sake of clarity.
<figure><img src="image_14.png" /><figcaption>Fig. 11. Endurance curves for [A]/[S] ratio of (a) 0.2, (b) 0.6, (c) 1.5 and (d) 2.5, for different $v_{ac}$.</figcaption></figure>

First, Fig. 12(a) compares the endurance curves of $NR_{0.2}^2$ and $NR_{1.5}^{0.8}$ having low $v_{ac}$. The longer cross-link length of $NR_{0.2}^2$ significantly increases the fatigue life, and confirms that polysulphide cross-links improve the fatigue life [32,35,36] in comparison to shorter mono- and-disulphide cross-links. Hence, CV systems are better than SEV ones for a low $v_{ac}$. Then, Fig. 12(b) presents the endurance curves of $NR_{1.5}^{1.2}$ and $NR_4^{0.8}$. The SEV system returns a better fatigue resistance than the EV one, which indicates, once more, that longer cross-links can enhance the fatigue resistance. This effect is less pronounced than with CV system in Fig. 12(a). To continue, Fig. 12(c) depicts that $NR_{0.6}^2$, having mainly polysulphide cross-links, is less resistant to fatigue than the $NR_{1.5}^{1.6}$ and $NR_{2.5}^{1.2}$. This suggests for this range of $v_{ac}$ that longer cross-links do not improve the fatigue resistance, contrarily to what was observed for lower $v_{ac}$. This raises the question of whether $v_{ac}$ has a stronger effect than cross-links length on the fatigue life improvement, for this range of $v_{ac}$. Finally, Fig. 12(d) evidences that comparable fatigue life was obtained between CV and SEV with $v_{ac}$ around $1.50 \times 10^4$ mol/cm$^3$. Thus, it can be concluded CV system is no longer better than the SEV, for this $v_{ac}$ value. This may indicate that the beneficial effect of long cross-links on the network mobility is reduced at high $v_{ac}$.

To sum-up, the effect of the cross-link length depends on $v_{ac}$. Low $v_{ac}$ combined to polysulphide cross-links lead to significantly higher fatigue life. This confirms the results previously reported in the literature [32, 35,36] and could be explained by structural rearrangements, without being able to establish the exact phenomenon in question. Increasing $v_{ac}$ will decrease this effect until it completely disappears. This might be related to possible mobility restrictions overcoming the benefits of polysulphide cross-links.

### 3.1.2. Effect of the vulcanization conditions on the fatigue resistance

Vulcanization time and temperature have an effect on the macro-molecular network structure, driven by $v_{ac}$ and cross-link length. As

reviewed by Akiba and Hashim [16], CV systems are more prone to reversion than SEV and EV ones, which have shorter cross-links and are much more thermally stable. Thus, a vulcanization carried out for too long is known to reduce the mechanical properties [16]. This can be explained by (i) the formation of elastically inactive structures, such as “cyclic sulphides” [17] that decreases $v_{ac}$ and (ii) NR chains modifications causing the macromolecules to lose their stereoregularity due to the change from cis to trans configurations [42]. These effects are presented in Figs. 13 and 14, which sum-up the effect of vulcanization time and temperature on the fatigue response, respectively.

**Vulcanization time.** Fig. 13 presents the endurance curves for (a) the $NR_{0.2}^4$, (b) the $NR_{1.5}^{1.2}$ and (c) the $NR_4^{0.8}$, corresponding to CV, SEV and EV, respectively. Fig. 13(a) shows that for $NR_{0.2}^4$, the stress response as well as the fatigue life significantly decrease after over-curing. This illustrates the effect of the reversion, which is known to decrease $v_{ac}$ [14] as well as the cross-link length [11,12]. This suggests that even with a decrease in $v_{ac}$, i.e., less energy available for crack growth, over-curing involves other phenomena that significantly reduce the fatigue life. To continue, SEV systems (see Fig. 13(b)) show no significant dependence to the over-vulcanization, as endurance curves are similar between optimally and over-vulcanized materials. However, the fatigue life is significantly enhanced in the case of under-vulcanization. Since $v_{ac}$ is relatively similar for all of them, the better fatigue resistance of SEV systems when under-vulcanized could be related to the higher cross-link length, or to the formation of inactive structures above $t_{80}$ such as pendant cross-links or cyclic cross-links. This result correlates well with the fact that SEV systems are more resistant than CV ones to long time vulcanization, with only a few changes in $v_{ac}$ [16]. No effect on the fatigue resistance above $t_{98}$ was observed. For EV systems (see Fig. 13(c)), the endurance curves for the three vulcanization conditions are superimposed. This is expected as EV systems are much more thermally stable than CV [14]. This indicates that $v_{ac}$ does not evolve
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** The effect of the cross-link length on the fatigue response, in which four groups of NR are compared: group (a) n°1 (b) n°2 (c) n°3 and (d) n°4 in Table 8, respectively.</figcaption></figure>

when additional thermal energy is provided. Furthermore, the fact that the under-vulcanized EV is similar to the others correlates well with the fact that polysulphide cross-links decomposed early in the vulcanization process of EV systems [16]. As a result, lifetime cannot be improved by under-vulcanization, but this material will not suffer from over-vulcanization.

**Vulcanization temperature.** Fig. 14 presents the endurance curves for (a) the $NR_{0.2}^4$, (b) the $NR_{1.5}^{1,2}$ and (c) the $NR_4^{0,8}$, corresponding to CV, SEV and EV, respectively. The vulcanization temperature changed $v_{ac}$ for the CV system $NR_{0.2}^4$ (See Table 5), the lowest $v_{ac}$ was unexpectedly obtained at 160 °C (Fig. 14(a)). Consequently, $v_{ac}$ and stresses for CV vulcanized at 150 and 170 °C are higher than at 160 °C, especially at high strain. The corresponding fatigue life is slightly higher, but this is not significant. To continue on the endurance curves for SEV and EV systems (Figs. 14(b) and (c)), the same conclusions can be addressed as the endurance curves are superimposed. As long cross-links are much less thermally stable, this shows that network modifications have led to differences in stiffness and lifetime. To sum-up, the vulcanization temperature has no significant effect on the fatigue resistance as long as the vulcanization time is adjusted accordingly, i.e., equivalent to $t_{98}$. It should be noted that more precautions are needed for the CV system. This result differs from that of Gehling et al. who found a higher crack growth resistance for CB filled NBR vulcanized at 170 °C, with a EV system [38]. This suggests that the effect of vulcanization temperature is different for this type of rubber.

### 3.1.3. Damage modes analysis

The Diabolo tested under relaxing tension, i.e., for $R_U = 0$ only, failed under Damage mode 1. Some of the materials have been chosen to illustrate the changes induced by the vulcanization system, namely $v_{ac}$ and the cross-link length. Fractographies from optical microscope

are presented in this section. For the sake of clarity, since the results regarding the effect of vulcanization conditions do not provide any truly significant insights into fracture surfaces, they are not presented here.<fnref n="5" />

**Effect of the active chain density on the damage mechanisms.** Fig. 15 presents representative failure surfaces of $NR_{0.5}^{0,8}$, $NR_{2.5}^{1,2}$ and $NR_{2.5}^{1,6}$ tested at 60, 90 and 120%. The Figure is constructed as follows: (i) $v_{ac}$ increases through column from the left to the right, and (ii) the strain level increases through row, from the top to the bottom.

For each failure surface, the crack propagates from the left to the right, showing a wide stable propagation surface, wrenchings and a smooth final ligament surface. Increasing the strain level enlarges the final ligament surface area, indicating an earlier catastrophic failure as a higher strain level induces more energy used for crack growth. Also, wrenchings are significantly more inclined. For $NR_{2.5}^{0,8}$ and $NR_{2.5}^{1,2}$, having the same $v_{ac}$, fatigue striations were observed at 120% of strain. This indicates that SIC markers are promoted by a specific range of $v_{ac}$, and suggests that SIC depends on $v_{ac}$ [64]. By increasing $v_{ac}$, the final ligament surface widens. This effect is much more pronounced at high strain levels traducing a catastrophic failure. This could be explained by the fact that a high $v_{ac}$ (i) implies a high stiffness and provide a high strain energy quantity for crack growth, (ii) but drastically reduces the macromolecular network mobility and consequently its ability to dissipate this energy. To go further, increasing $v_{ac}$ leads to an increase and then a decrease of fatigue striations number, suggesting that there

<footnote n="5">⁵ The results obtained by varying the vulcanization time and temperature has no significant effect; however, they do confirm the observations made under optimal conditions. For instance, for non-relaxing loading conditions, $NR_4^{0,8}$ exhibited a competition between Damage modes 2 and 3 regardless of the vulcanization temperature.</footnote>
<figure><img src="image_16.png" /><figcaption>Fig. 13. Effect of the vulcanization time on the fatigue response for the (a) $NR_{0.2}^4$ (CV) (b) $NR_{1.5}^{1.2}$ (SEV) and (c) $NR_{4}^{0.8}$ (EV), respectively.</figcaption></figure>

<table><caption>**Table 9**
Summary of the three pairs used to evaluate the effect of the cross-link length on the damage mechanisms. Values of $v_{ac}$ are given in $\times 10^4$ mol/cm$^3$.</caption><thead><tr><th>Pair n°</th><th>Formulations</th><th>Range of v<sub>ac</sub></th></tr></thead><tbody><tr><td>1</td><td>$NR_{0.2}^6$ &harr; $NR_{0.6}^4$</td><td>2.27-2.28</td></tr><tr><td>2</td><td>$NR_{0.2}^4$ &harr; $NR_{1.5}^{1.6}$</td><td>1.49-1.53</td></tr><tr><td>3</td><td>$NR_{1.5}^{1.2}$ &harr; $NR_{4}^{0.8}$</td><td>1.24-1.29</td></tr></tbody></table>

may be an $v_{ac}$ maximizing SIC. This is in a strong analogy with what was found by Chenal et al. [24] under quasi-static loadings. As a summary, it has been shown that both $v_{ac}$ and strain level have an effect on the fracture surface, which is consistent with the previous results obtained in Section 3.1.

**Effect of the cross-link length on the damage mechanisms.** The effect of the cross-link length on the fracture surface is addressed by comparing materials with similar $v_{ac}$ and different [A]/[S]. Table 9 summarizes the materials to be compared. Fig. 16 presents a comparison of materials having close $v_{ac}$ value: (a) $NR_{0.2}^6$ is compared with (b) $NR_{0.6}^4$, (c) $NR_{0.2}^4$ is compared with (d) $NR_{1.5}^{1.6}$ and (e) $NR_{1.5}^{1.2}$ is compared with (f) $NR_{4}^{0.8}$. Since the fractographic images evidenced comparable changes at 60, 90 and 120% of maximum strain, only the fracture surfaces obtained at 120% are shown.

Figs. 16(a) and (b) are both CV materials of [A]/[S] ratio of 0.2 and 0.6, respectively. Clearly, fracture surfaces are very similar with a tiny

stable propagation zone and a very large smooth surface, characteristic of a catastrophic failure. This suggests that their high $v_{ac}$ drives the damage mechanisms. Since they are both CV systems, this does not exclude a potential effect of the cross-link length. To continue with the second pair (Figs. 16(c) for CV and (d) for SEV), no major difference was observed. Indeed, both fracture surfaces consist of a stable propagation surface, wrenchings, and a smooth area characteristic of a catastrophic failure. Finally, results revealed no significant effect of the cross-link length on the fracture surface for last Pair n°3 (Figs. 16(e) and (f)). They exhibit similar stable propagation surface and wrenchings, presence of fatigue striations, and a comparable final ligament surface. To sum-up, it can be concluded that the cross-link length has no significant effect on the fracture surface.

### 3.1.4. Summary

In this section dedicated to the fatigue results under relaxing tension loadings, the effect of the vulcanization system and conditions was studied. $v_{ac}$ was found to drive the fatigue properties and damage mechanisms. Clearly, a sufficient stiffness must be achieved to maximize the fatigue resistance, although it should not be too high, as this would leads to reduced lifetime and catastrophic failure. Moreover, the effect of the cross-link length on damage mechanisms was found to be negligible, suggesting that stiffness drives the damage modes. However, the cross-link length showed a slight effect on the fatigue resistance which depends on $v_{ac}$. Notably, no effect of cross-link length on fatigue resistance was found for high $v_{ac}$, but low $v_{ac}$ has proved to
<figure><img src="image_17.png" /><figcaption>Fig. 14. Effect of the vulcanization temperature on the fatigue response for the (a) $NR_{0.2}^4$ (CV) (b) $NR_{1.2}^{1.2}$ (SEV) and (c) $NR_4^{0.8}$ (EV), respectively.</figcaption></figure>

be significantly improved by longer cross-link length. In terms of the vulcanization conditions, it was shown that vulcanization temperature had no effect on fatigue resistance when vulcanized at the appropriate optimum time. However, increasing the vulcanization time tends to decrease the fatigue resistance for CV systems. It is to be noted that for both CV and SEV, under-vulcanized Diabolo specimens showed better fatigue resistance, while EV showed no difference in lifetime. In summary, vulcanization systems and conditions provide a wide range of parameters to drive the macromolecular network structure and, thereby, the fatigue resistance of filled NR.

## 3.2. Non-relaxing tension loadings

As above-mentioned, non-relaxing loading conditions lead to a fatigue lifetime reinforcement. This reinforcement is generally represented by the Haigh diagram, as a typical “V shape” of iso-lifetime is observed. As only two R ratios superior or equal to zero have been tested, the “V-shape” does not appear in the following Haigh diagrams. Thus, the slopes shown below correspond to the level of fatigue lifetime reinforcement, the higher the slope the higher the reinforcement. It is also worth noting that since the tests were prescribed under displacement, the R ratio corresponding to the relaxing loading is negative in a plot in stress due to the residual strain induced. In this section, interpolated iso-lifetimes of $10^4$ (…), $10^5$ (- -) and $10^6$ (-) cycles are plotted with respect to the amplitude $\sigma_{amp}$ and mean $\sigma_{mean}$ Cauchy stresses. They are both calculated by using $\sigma_{min}$ and $\sigma_{max}$: $\sigma_{amp} = 0.5 \times (\sigma_{max} - \sigma_{min})$ and $\sigma_{mean} = 0.5 \times (\sigma_{max} + \sigma_{min})$, respectively.

### 3.2.1. Effect of the vulcanization system on the fatigue lifetime reinforcement

*Active chain density effect.* Fig. 17 presents the Haigh diagram for NR with [A]/[S] of (a) 0.2, (b) 0.6, (c) 1.5 and (d) 2.5. For each Haigh diagram, the effect of the $v_{ac}$ is shown through three materials.

The fatigue lifetime reinforcement can be discussed with Fig. 17(a) for CV, i.e., $NR_{0.2}^2$, $NR_{0.2}^4$ and $NR_{0.2}^6$. The slope of each iso-lifetimes displayed are different, which means that $v_{ac}$ has a strong effect on the fatigue lifetime reinforcement magnitude. Clearly, increasing $v_{ac}$ decreases the slope, suggesting that a too much $v_{ac}$ is detrimental for CV of [A]/[S]=0.2. To continue with CV, $NR_{0.6}^2$, $NR_{0.6}^4$ and $NR_{0.6}^6$ gave very different slopes (Fig. 17(b)). Thus, results are consistent with the previous ones (Fig. 17(a)): $v_{ac}$ drives the slope of the iso-lifetimes, with low $v_{ac}$ values leading to the highest fatigue lifetime reinforcement. A different trend is obtained for SEV with [A]/[S]=1.5 (Fig. 17(c)). These materials are highly reinforced, with the higher the $v_{ac}$ the higher the fatigue lifetime reinforcement. This suggests that the cross-link length may have an effect, or as the $v_{ac}$ range is different, it may have a different impact on the one previously observed. Finally, Fig. 17(d) shows a higher slope for $NR_{1.5}^{1.2}$, corresponding to a higher fatigue lifetime reinforcement. For higher and lower $v_{ac}$, the fatigue lifetime reinforcement is significantly decreased. This indicates that $v_{ac}$ drives the fatigue lifetime reinforcement for [A]/[S]=2.5.

To sum-up, $v_{ac}$ was found to have a strong effect on the fatigue lifetime reinforcement. Since (i) the effects were different depending on [A]/[S], and (ii) considering that they have different range of $v_{ac}$,
<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Fractographies obtained with an optical microscope of $NR_{0.2}^{0.8}$, $NR_{1.5}^{1.2}$ and $NR_{2.5}^{1.6}$ tested in fatigue under relaxing tension loadings. The effect of the maximal strain for $NR_{2.5}^{0.8}$ at (a) 60, (d) 90 and (g) 120%, $NR_{2.5}^{1.2}$ at (b) 60, (e) 90 and (h) 120% and $NR_{2.5}^{1.6}$ at (c) 60, (f) 90 and (i) 120% is given.</figcaption></figure>

this does not exclude the existence of an optimum $v_{ac}$ range maximizing the reinforcement. However, it is difficult to analyze and generalize the effect of $v_{ac}$ on the fatigue lifetime reinforcement. This is further addressed in Section 3.3.

**Cross-link length effect.** Fig. 18 presents the effect of the cross-link length on the fatigue lifetime reinforcement. Fig. 18(a) shows the fatigue lifetime reinforcement for Group n°1. The slope of each iso-lifetime displayed are very similar indicating a comparable fatigue lifetime reinforcement. Similarly, Group n°2 and 3 (Fig. 18(b) and (c)) showed no difference in lifetime trends, suggesting that the cross-link length does not affect the fatigue lifetime reinforcement for these ranges of $v_{ac}$. However, Group n°4 gave higher slope for SEV than CV systems (Fig. 18(d)). This is surprising, and could be due to excessive $v_{ac}$ inhibiting the SIC for CV. Without going into further explanation of these results, it should be noted that the CV system is less reinforced.

To sum-up, the cross-link length has a negligible effect on the fatigue lifetime reinforcement, further demonstrating the strong effect of $v_{ac}$ on fatigue response. Nevertheless, we note that $NR_{0.2}^{4}$ showed a difference with $NR_{1.5}^{1.6}$.

### 3.2.2. Effect of the vulcanization conditions on the fatigue lifetime reinforcement

The effect of the vulcanization conditions on the fatigue lifetime reinforcement are independently addressed. It is recalled that vulcanization conditions are changed accordingly to Table 3.

**Vulcanization time.** Fig. 19 presents the effect of the vulcanization time on fatigue response for (a) $NR_{0.2}^{4}$ (b) $NR_{1.5}^{1.2}$ and (c) $NR_{4}^{0.8}$. Fig. 19(a) illutsrates the effect of the vulcanization time on fatigue response for $NR_{0.2}^{4}$. For a too long time (N°2), a decrease in the slope is obtained, indicating a lower fatigue lifetime reinforcement. On the other hand, under-vulcanized $NR_{0.2}^{4}$ (N°1) presents a higher slope and consequently a higher fatigue lifetime reinforcement. This suggests

that under-vulcanization significantly enhance the fatigue lifetime reinforcement for CV system. However, for $10^4$ cycles it is noticeable that the optimal $NR_{0.2}^{4}$ (N°0) was slightly more reinforced. Figs. 19(b) and (c) shows the results obtained for $NR_{1.5}^{1.2}$ and $NR_{4}^{0.8}$, respectively. No significant effect of the vulcanization time on the fatigue lifetime reinforcement was observed. This could be due to the fact that SEV and EV systems are less thermally sensitive, resulting in few structural changes.

**Vulcanization temperature.** Fig. 20 presents the effect of the vulcanization temperature on fatigue response for (a) $NR_{0.2}^{4}$ (b) $NR_{1.5}^{1.2}$ and (c) $NR_{4}^{0.8}$. Fig. 20(a) shows the effect of the vulcanization temperature for $NR_{0.2}^{4}$. A vulcanization temperature of 170 °C exhibits a slightly higher slope than the two other temperatures. Nevertheless, it appears that slopes of $10^4$ cycles iso-lifetimes decrease drastically for 150 °C and 170 °C. It is noteworthy that they have much higher $v_{ac}$ than 160 °C. This could induce a $v_{ac}$ threshold, and above this threshold a decrease in fatigue response at high stress is obtained. Fig. 20(b) presents a fatigue response for $NR_{1.5}^{1.2}$ vulcanized at 150 °C, 160 °C and 170 °C. As comparable results were obtained, this suggests that there is no effect of the vulcanization temperature for SEV system. The same results were obtained for each EV system (Fig. 20(c)). This is again correlated with the better thermal resistance of short cross-links, i.e., SEV and EV systems.

### 3.2.3. Damage modes analysis

For non-relaxing loading, three damage modes are observed, namely damage modes 1, 2 and 3 as defined in Section 2.4.2. These damage modes are used in order to investigate the effect of $v_{ac}$ on the damage mechanisms. For the sake of clarity, the same three NR presented in Section 3.1 in Fig. 15 are used again. Moreover, since most of the Diabolo specimens did not fail during the fatigue tests at maximum strain of 90%, no result at this strain level will be presented.
<figure><img src="image_19.png" /><figcaption>**Fig. 16.** Fractographies obtained with an optical microscope of the NR pairs used to evaluate the effect of the cross-link length on the damage mechanisms: (a) $NRR_{0.2}^6$, (b) $NRR_{0.6}^4$, (c) $NRR_{0.2}^4$, (d) $NRR_{1.5}^{1.6}$, (e) $NRR_{1.5}^{1.2}$ and (f) $NRR_4^{0.8}$. These correspond to fracture surfaces tested at 120% strain.</figcaption></figure>

*Effect of the active chain density on the damage mechanisms.* Fig. 21 depicts the effect of maximum strain on the fracture surfaces of $NRR_{2.5}^{0.8}$, $NRR_{2.5}^{1.2}$, and $NRR_{2.5}^{1.6}$.

Figs. 21(a), (b) and (c) present the fracture surfaces obtained at 172% of maximum strain. For the lower $v_{ac}$, Damage mode 2 was obtained, with wrenchings, cones along the edge and well-defined fatigue striations, and a slight final ligament. For $v_{ac}$ of $1.38 \times 10^4$ mol/cm$^3$, the same damage mode was recorded, but with a more prominent final ligament. Then, for the highest $v_{ac}$, i.e., $NRR_{2.5}^{1.6}$, an almost unique smooth surface characteristic of a catastrophic failure in Damage mode 1 is observed. Furthermore, no SIC marker was observed, which questions the effect of SIC and the ability to crystallize for this material with a high $v_{ac}$. Subsequently, the second row, corresponding to maximum of 200% strain (Fig. 21(d), (e) and (f)), exhibited some differences. For $NRR_{2.5}^{0.8}$ the only difference is a larger final ligament. However, $NRR_{2.5}^{1.2}$ was altered by Damage mode 3 with cavitations under the metallic inserts, due to a higher reinforcement than the others. This suggests that $v_{ac}$ is strongly correlated with the fatigue resistance. It is perhaps worth emphasizing that $NRR_{2.5}^{1.6}$ (Fig. 21(e)) is almost solely composed of a wide and smooth surface, characteristic of a catastrophic failure. This could be due to an excessively high $v_{ac}$. More importantly, analogous outcomes were observed for [A]/[S] ratios of 0.2, 0.6, and 1.5.

To sum-up, as for relaxing tension loadings, both $v_{ac}$ and strain level influence the fracture surface. Smooth fracture surfaces are found in the case of high $v_{ac}$ while low $v_{ac}$ enables SIC markers, i.e., fatigue striations and cones as well as cavitation to form. This questions once more the relationship between SIC and $v_{ac}$. This topic is fully addressed in Section 3.3.

*Effect of the cross-link length on the damage mechanisms.* The effect of the cross-link length on the fracture surface is evaluated by comparing the same materials as those presented in Table 9, with similar $v_{ac}$ but different [A]/[S]. Since fractographic images showed similar changes at 172%, and 200% of maximum strain, the 172% strain condition will be used as a reference example hereafter. Fig. 22 presents a comparison of materials with close values of $v_{ac}$. Similarly to relaxing tension loadings, Pair n°1 gave a comparable fracture surface with a tiny stable propagation area and a very large smooth surface, characteristic of a catastrophic failure. For this high $v_{ac}$, it suggests that the cross-link length is not a first order parameter driving the damage mechanisms. Continuing with Pair n°2, there is no significant difference between the two surfaces. Indeed, both fracture surfaces consist of wrenchings, and a consequent smooth area characteristic of a catastrophic failure. Finally, no differences were obtained with Pair n°3. Both exhibited wrenchings, fatigue striations and cones along the edge. From these observations it is concluded that the cross-link length has no significant effect on the fracture surface.

### 3.3. Discussion

The relationship between $v_{ac}$, the fatigue response and the damage modes is discussed in this section. A new representation based on the clustering of the different formulations regarding their $v_{ac}$ is here proposed. Fatigue response and damage modes are independently addressed in Sections 3.3.1 and 3.3.2, respectively. Finally, a comparison between them is performed in Section 3.3.3.
<figure><img src="image_20.png" /><figcaption>Fig. 17. Effect of $v_{ac}$ on the fatigue lifetime reinforcement for [A]/[S] of (a) 0.2 (b) 0.6 (c) 1.5 and (d) 2.5.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 18. Effect of the cross-link length on the fatigue lifetime reinforcement for Groups (a) n°1 (b) n°2 (c) n°3 and (d) n°4 defined in Table 8 in part I.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Fig. 19. Effect of the vulcanization time on the fatigue lifetime reinforcement for (a) $NR_{0.2}^4$ (CV) (b) $NR_{1.5}^{1.2}$ (SEV) and (c) $NR_{4}^{0.8}$ (EV).</figcaption></figure>

<caption>Table 10</caption>

Formulations categorization according to the fatigue life for relaxing tension loadings.

<table><thead><tr><th>Category</th><th>Range of v<sub>ac</sub></th><th>Formulations</th><th>Power law</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>1</td><td>0.95–1.20</td><td>NR<sup>2</sup><sub>0.2</sub>, NR<sup>0.8</sup><sub>2.5</sub></td><td>σ<sub>max</sub> = 34.836 × N<sup>-0.262</sup></td><td>0.95</td></tr><tr><td>2</td><td>1.20–1.30</td><td>NR<sup>1.2</sup><sub>1.5</sub>, NR<sup>0.8</sup><sub>4</sub></td><td>σ<sub>max</sub> = 167.76 × N<sup>-0.391</sup></td><td>0.94</td></tr><tr><td>3</td><td>1.30–1.70</td><td>NR<sup>1.6</sup><sub>1.5</sub>, NR<sup>1.2</sup><sub>2.5</sub>, NR<sup>2</sup><sub>0.6</sub>, NR<sup>4</sup><sub>0.2</sub>, NR<sup>1.6</sup><sub>2.5</sub></td><td>σ<sub>max</sub> = 75.518 × N<sup>-0.309</sup></td><td>0.97</td></tr><tr><td>4</td><td>2.20–2.30</td><td>NR<sup>0.8</sup><sub>0.2</sub>, NR<sup>4</sup><sub>0.6</sub></td><td>σ<sub>max</sub> = 24.534 × N<sup>-0.206</sup></td><td>0.99</td></tr><tr><td>5</td><td>0.95 and &gt; 3.20</td><td>NR<sup>0.8</sup><sub>1.5</sub>, NR<sup>0.6</sup><sub>4</sub></td><td>σ<sub>max</sub> = 18.701 × N<sup>-0.228</sup></td><td>0.98</td></tr></tbody></table>

### 3.3.1. Effect of the active chain density on the fatigue life

Relaxing tension loading conditions. The different materials were clus- tered into groups with respect to their fatigue resistance. Five categories were defined as they minimize the regression coefficient of the fitted endurance curves. The materials categories are summarized in Table 10.

Fig. 23 presents the (a) endurance curves of each category (b) summarized in the [A]/[S] diagram. Categories 4 and 5 correspond to the highest and the lowest fatigue resistance, respectively. Surprisingly, the best fatigue resistance is reached for $2.27 \times 10^4$ mol/cm$^3$, while it drastically drops above $3.20 \times 10^4$ mol/cm$^3$ for category 5. This demonstrates that a high $v_{ac}$ increases the number of active chains contributing to the network strength, and therefore increases the strain energy required to break the active chains [65]. However, an excessive $v_{ac}$ will lead to too much stiffness and stress, resulting in too much energy input for crack growth [5]. This would cause premature rupture. Thus, this suggests the existence of an optimum value of $v_{ac}$ maximizing the fatigue resistance. The fatigue resistance and the stress response

decrease progressively, from category 4 to 1 corresponding to 2.30 to $0.95 \times 10^4$ mol/cm$^3$, respectively. This is consistent with what reported Mars and Fatemi: “There is an optimum crosslink density, which results in the longest fatigue life.” [6]. It is worth emphasizing that $NR_{1.5}^{0.8}$ (encircled in red in Fig. 23 (b)), which has a $v_{ac}$ corresponding to $0.95 \times 10^4$ mol/cm$^3$, does not follow this trend. As demonstrated in Section 3.1, a low $v_{ac}$ combined with short cross-links is responsible for a lower fatigue life. To continue, the two endurance curves for Categories 4 and 5 are almost parallel while those for Categories 1, 2 and 3 are much higher by comparison. This could be due to a stiffening effect that depends on the level of strain. According to the authors, since SIC is driven by the strain level, it could be related to this stiffening effect [66]. It is noteworthy that some studies from the literature can be used to further discuss our results. For instance, Tosaka et al. found an optimum $v_{ac}$ maximizing SIC around $1.50 \times 10^4$ mol/cm$^3$ [67,68], which is very close to the values of Categories 2 and 3. Chenal et al. reported an optimum $v_{ac}$ of $1.20 \times 10^4$ mol/cm$^3$ [24], which corresponds to Category 1.
<figure><img src="image_23.png" /><figcaption>Fig. 20. Effect of the vulcanization temperature on the fatigue lifetime reinforcement for (a) $NR_{0.2}^4$ (CV) (b) $NR_{1.5}^{1.2}$ (SEV) and (c) $NR_4^{0.8}$ (EV).</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 21. Fractographies obtained with an optical microscope of $NR_{2.5}^{0.8}$, $NR_{2.5}^{1.2}$ and $NR_{2.5}^{1.6}$ tested in fatigue under non-relaxing loading conditions. The effect of the maximal strain for $NR_{2.5}^{0.8}$ at (a) 172% and (d) 200%, $NR_{2.5}^{1.2}$ at (b) 172% and (e) 200% and $NR_{2.5}^{1.6}$ at (c) 172% and (f) 200% is given.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Fig. 22. Fractographies obtained with an optical microscope of (a) $NR_{0.2}^6$, (b) $NR_{0.6}^4$, (c) $NR_{0.2}^4$, (d) $NR_{1.5}^{1.6}$, (e) $NR_{1.5}^{1.2}$ and (f) $NR_4^{0.8}$. Diabolos were tested under non-relaxing loading conditions at 172% of maximum strain.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 23. Clustering of the vulcanization system depending on their fatigue resistance: (a) endurance curves and (b) cartography of $v_{ac}$ categories in the [A]/[S] diagram.</figcaption></figure>

**Non-relaxing tension loading conditions.** Formulations were clustered into groups regarding their fatigue lifetime reinforcement. Fig. 24(a) presents the $10^4$ cycles iso-lifetime associated with these groups. These iso-lifetimes correspond to a given reference formulation providing an example of the reinforcement corresponding to its $v_{ac}$ range. The categories are given in Table 11.

Categories 1, 2 and 3 returned a very similar reinforcement as they have the same slopes. In particular, ranges of 1.20–1.50 ×$10^4$ mol/cm$^3$ demonstrates a slightly higher fatigue lifetime reinforcement. This indicates that $v_{ac}$ drives the fatigue lifetime reinforcement, therefore suggesting better ability to crystallize for these categories. This is in good agreement with $v_{ac}$ ranges that maximized SIC, according to

the results of Tosaka et al. [67,68]. By increasing $v_{ac}$ above 1.50 × $10^4$ mol/cm$^3$, which corresponds to Category 4, this fatigue lifetime reinforcement is much less pronounced. It weakens further above 1.70 ×$10^4$ mol/cm$^3$ for Category 5 and almost disappears for Category 6. Hence, $v_{ac}$ showed a strong correlation with the fatigue lifetime reinforcement, suggesting that a too high $v_{ac}$ decreases the ability to crystallize and the fatigue lifetime reinforcement. It is to be noted that the most interesting materials differ from the results found at $R_U = 0$, i.e., 0.95–1.50 ×$10^4$ mol/cm$^3$ vs 1.20–1.70 ×$10^4$ mol/cm$^3$ at $R_U = 0.25$ and $R_U = 0$, respectively.
<table><caption>**Table 11**
Formulation categorization according to the fatigue lifetime reinforcement for non-relaxing tension loadings.</caption><thead><tr><th>Category</th><th>Range of v<sub>ac</sub></th><th>Formulations</th><th>Equation</th></tr></thead><tbody><tr><td>1</td><td>0.95-1.20</td><td>N R<sup>2</sup><sub>0.2</sub>, N R<sup>0.8</sup><sub>1.5</sub>, N R<sup>0.8</sup><sub>2.5</sub></td><td>σ<sub>amp</sub> = 0.6472 × σ<sub>mean</sub><sup>0.3913</sup></td></tr><tr><td>2</td><td>1.20-1.30</td><td>N R<sup>1.2</sup><sub>1.5</sub>, N R<sup>0.8</sup><sub>4</sub></td><td>σ<sub>amp</sub> = 0.6886 × σ<sub>mean</sub><sup>0.363</sup></td></tr><tr><td>3</td><td>1.30-1.50</td><td>N R<sup>1.6</sup><sub>1.5</sub>, N R<sup>1.2</sup><sub>2.5</sub>, N R<sup>2</sup><sub>0.6</sub></td><td>σ<sub>amp</sub> = 0.682 × σ<sub>mean</sub><sup>0.4397</sup></td></tr><tr><td>4</td><td>1.50-1.70</td><td>N R<sup>4</sup><sub>0.2</sub>, N R<sup>1.6</sup><sub>2.5</sub></td><td>σ<sub>amp</sub> = 0.4518 × σ<sub>mean</sub><sup>0.4965</sup></td></tr><tr><td>5</td><td>1.70-2.30</td><td>N R<sup>6</sup><sub>0.2</sub>, N R<sup>4</sup><sub>0.6</sub></td><td>σ<sub>amp</sub> = 0.2195 × σ<sub>mean</sub><sup>0.9849</sup></td></tr><tr><td>6</td><td>3.20</td><td>N R<sup>6</sup><sub>0.6</sub></td><td>σ<sub>amp</sub> = 0.0913 × σ<sub>mean</sub><sup>0.7262</sup></td></tr></tbody></table>

<figure><img src="image_27.png" /><figcaption>**Fig. 24.** Effect of $v_{ac}$ on the fatigue resistance. In (a) the fatigue response was classified with respect to $v_{ac}$, leading to five categories of responses. In (b) these categories are recalled in terms of the vulcanization system.</figcaption></figure>

<table><caption>**Table 12**
Formulation categorization according to the damage mechanisms, for relaxing tension loadings at $\epsilon_{max}=120\%$.</caption><thead><tr><th>Category</th><th>Range of v<sub>ac</sub></th><th>Formulations</th><th>Damage mode</th></tr></thead><tbody><tr><td>1</td><td>0.95-1.20</td><td>N R<sup>2</sup><sub>0.2</sub>, N R<sup>0.8</sup><sub>1.5</sub></td><td>1</td></tr><tr><td>2</td><td>1.20-1.50</td><td>N R<sup>0.8</sup><sub>2.5</sub>, N R<sup>1.2</sup><sub>1.5</sub>, N R<sup>0.8</sup><sub>4</sub>, N R<sup>1.2</sup><sub>2.5</sub>, N R<sup>2</sup><sub>0.6</sub>, N R<sup>1.6</sup><sub>1.5</sub></td><td>1</td></tr><tr><td>3</td><td>1.50-1.70</td><td>N R<sup>4</sup><sub>0.2</sub></td><td>1</td></tr><tr><td>4</td><td>1.70-3.20</td><td>N R<sup>4</sup><sub>0.6</sub>, N R<sup>6</sup><sub>0.2</sub>, N R<sup>6</sup><sub>0.6</sub>, N R<sup>1.6</sup><sub>2.5</sub></td><td>1</td></tr></tbody></table>

<table><caption>**Table 13**
Formulation categorization according to the damage mechanisms, for non-relaxing loadings at $\epsilon_{max}=172\%$.</caption><thead><tr><th>Category</th><th>Range of v<sub>ac</sub></th><th>Formulations</th><th>Damage mode</th></tr></thead><tbody><tr><td>1</td><td>0.95-1.25</td><td>N R<sup>2</sup><sub>0.2</sub>, N R<sup>0.8</sup><sub>1.5</sub>, N R<sup>0.8</sup><sub>2.5</sub>, N R<sup>1.2</sup><sub>1.5</sub></td><td>2</td></tr><tr><td>2</td><td>1.25-1.50</td><td>N R<sup>0.8</sup><sub>4</sub>, N R<sup>1.2</sup><sub>2.5</sub>, N R<sup>1.6</sup><sub>1.5</sub></td><td>2</td></tr><tr><td>3</td><td>1.50-1.55</td><td>N R<sup>4</sup><sub>0.2</sub></td><td>2</td></tr><tr><td>4</td><td>1.55-3.20</td><td>N R<sup>1.6</sup><sub>2.5</sub>, N R<sup>4</sup><sub>0.6</sub>, N R<sup>6</sup><sub>0.2</sub></td><td>1</td></tr><tr><td>No failure</td><td>1.41</td><td>N R<sup>2</sup><sub>0.6</sub></td><td>-</td></tr><tr><td>Not tested</td><td>3.20</td><td>N R<sup>6</sup><sub>0.6</sub></td><td>-</td></tr></tbody></table>

### 3.3.2. Effect of the active chain density on the damage modes

**Relaxing tension loading conditions.** In Section 3.1, $v_{ac}$ was found to be the main parameter driving the damage mechanisms. The fracture surfaces were clustered into 4 categories regarding the size of the stable and unstable crack propagation zones, the presence and size of striations and the size of the final ligament. The material categories are given in Table 12, as well as the corresponding $v_{ac}$ and damage mode. Figs. 25(a) and (b) presents characteristic failure surfaces of materials in Categories 1, 2, 3 and 4, respectively, for a test carried out at $\epsilon_{max} = 120\%$ and for $R_U = 0$. It has to be noted that similar conclusions were drawn at other strain levels.

First, the size of the stable crack propagation zone increases with $v_{ac}$. However, this trend is not followed for $v_{ac}$ above $1.70 \times 10^4 \text{ mol/cm}^3$. Since the transition between the stable and unstable crack propagation zone is correlated with the resistance to crack growth, this goes in the sense that increasing $v_{ac}$ increases the fatigue resistance. Then, the size of the final ligament increases with $v_{ac}$. It almost constitutes the whole fracture surface for filled NR in Category 4. This could be explained by

the too high $v_{ac}$ and stiffening, which leads to catastrophic failure. The fatigue striations were only observed for $v_{ac}$ from 0.95 and to $1.50 \times 10^4 \text{ mol/cm}^3$. This suggests that this range of $v_{ac}$ presents good ability to crystallize. Above $1.50 \times 10^4 \text{ mol/cm}^3$, fatigue striations completely disappeared. It has to be noted that the damage mechanism is fully driven and evolves continuously with $v_{ac}$, which is in opposition with the fatigue life categorization (Table 10) for which materials having $v_{ac}$ below 0.95 and above $2.30 \times 10^4 \text{ mol/cm}^3$ had comparable fatigue resistance. This evidences that the post-mortem analysis, and fatigue lifetime have to be put into perspectives and compared systematically.

**Non-relaxing tension loading conditions.** As above-mentioned, the damage modes under non-relaxing loading conditions evolve with the strain level. At 90% of maximum strain, there are solely two groups: (i) those that do not break and (ii) those that experience early catastrophic failure above $1.70 \times 10^4 \text{ mol/cm}^3$. There is no in-between, supporting the possibility that a threshold exists, above which fatigue properties are reduced. Fig. 26(a) (1, 2, 3 and 4) illustrates the four failure surfaces identified for each category at 172% of strain, which are given in Table
<figure><img src="image_28.png" /><figcaption>**Fig. 25.** Effect of $v_{ac}$ on the damage mechanisms, for relaxing tension loading condition at $\epsilon_{max}=120\%$. Fracture surfaces obtained with an optical microscope are given in (a) and correspond to the groups numbered 1 to 4, represented in (b).</figcaption></figure>

13. In addition, Fig. 27(a) (1, 2, 3 and 4) displays the four characteristic failure surfaces at 200% of strain. Categories are given in Table 14.

At 172% of strain, fracture surfaces of Categories 1 and 2 were similar and composed of cones at the edges of the Diabolo specimen, well-defined fatigue striations, rough wrenchings, and a small final ligament. Therefore, from 0.95 and to $1.50 \times 10^4$ mol/cm$^3$, fatigue resistance is very high. By further increasing $v_{ac}$, fatigue striations disappeared for Category 3. Indeed, failure surfaces are composed solely of rough wrenchings and a larger smooth area similar to a final ligament. This suggests a lower fatigue lifetime reinforcement. Finally, Category 4 consists of materials of $v_{ac}$ ranging from $1.55$ to $3.20 \times 10^4$ mol/cm$^3$, with fracture surfaces composed solely of a completely smooth surface, indicative of a catastrophic failure. For these materials, the fracture surface is comparable to those obtained for relaxing tension loading conditions, which suggests no significant differences in fatigue life, i.e., no fatigue lifetime reinforcement. Finally, for highly non-relaxing loading conditions (i.e., $R_U = 0.25$ and 200% strain), the behavior of the materials continue to evolve, leading to a new classification, given in Table 14.

At 200% of strain, for Category 1, i.e., $v_{ac}$ ranging between 0.95 and $3.20 \times 10^4$ mol/cm$^3$, cones at the edges of the Diabolo specimen,

well-defined fatigue striations, rough wrenchings, and a small final ligament were obtained. This corresponds to a significant fatigue lifetime reinforcement. On the other hand, Category 2 solely includes the $NR^{0.8}$ with $v_{ac}$ equal to $1.29 \times 10^4$ mol/cm$^3$. For this material, both Damage modes 2 and 3 were reported. Category 3, with $v_{ac}$ between 1.20 and $1.50 \times 10^4$ mol/cm$^3$, shows a fracture shifted under the metallic inserts as indicated by the presence of cavitations, traducing a high reinforcement in the center of the Diabolo specimens. Then, Category 4 with $v_{ac}$ ranging from $1.50$ to $1.70 \times 10^4$ mol/cm$^3$ presents fracture surfaces composed of some wrenchings and a very large smooth surface, characteristic of a catastrophic failure. Thus, a too high $v_{ac}$ shows the absence of SIC markers. Finally, there is also three materials not tested, with $v_{ac}$ ranging from $1.70$ to $3.20 \times 10^4$ mol/cm$^3$ ($NR^{0.4}$, $NR^{0.6}$, and $NR^{0.2}$). The white color is used for those materials that cannot be classified.

### 3.3.3. Relationship between the active chain density, fatigue response and damage mechanisms

**Relaxing tension loading conditions.** The categorization of the effects of the $v_{ac}$ ranges on fatigue previously made in Sections 3.3.1 and 3.3.2 are here compared and discussed. At high $v_{ac}$, active chains
<figure><img src="image_29.png" /><figcaption>**Fig. 26.** Effect of $v_{ac}$ on damage mechanisms, for non-relaxing loadings at $\epsilon_{max}=172\%$. Fracture surfaces obtained with an optical microscope are given in (a) and correspond to the groups numbered 1 to 4, represented in (b).</figcaption></figure>

<table><caption>**Table 14**
Formulation categorization according to the damage mechanisms, for non-relaxing loadings at $\epsilon_{max}=200\%$.</caption><thead><tr><th>Category</th><th>Range of v<sub>ac</sub></th><th>Formulations</th><th>Damage mode</th></tr></thead><tbody><tr><td>1</td><td>0.95–1.20</td><td>N R<sup>2</sup><sub>0.2</sub>, N R<sup>0.8</sup><sub>1.5</sub>, N R<sup>0.8</sup><sub>2.5</sub></td><td>2</td></tr><tr><td>2</td><td>1.29</td><td>N R<sup>0.8</sup><sub>4</sub></td><td>2 & 3</td></tr><tr><td>3</td><td>1.20–1.50</td><td>N R<sup>1.2</sup><sub>1.5</sub>, N R<sup>1.2</sup><sub>2.5</sub>, N R<sup>2</sup><sub>0.6</sub>, N R<sup>1.6</sup><sub>1.5</sub></td><td>3</td></tr><tr><td>4</td><td>1.50–1.70</td><td>N R<sup>1.6</sup><sub>2.5</sub>, N R<sup>4</sup><sub>0.2</sub></td><td>1</td></tr><tr><td>Not tested</td><td>1.70–3.20</td><td>N R<sup>4</sup><sub>0.6</sub>, N R<sup>6</sup><sub>0.2</sub>, N R<sup>6</sup><sub>0.6</sub></td><td>—</td></tr></tbody></table>

are restricted, which would result in the network losing its ability to dissipate the strain energy, as discussed in [6,15,69,70]. In addition, network rearrangements are limited, which may lead to higher local stress concentrations' and promote fatigue damage. At high $v_{ac}$, which is typically the case for $N R_{0.6}^6$ having a $v_{ac}$ of $3.20 \times 10^4$ mol/cm³, poor fatigue resistance were obtained. Surprisingly, even though they have a high $v_{ac}$, but slightly lower than previously, i.e., $2.30 \times 10^4$ mol/cm³, and a smooth fracture surfaces, $N R_{0.2}^6$ and $N R_{0.6}^4$ showed much better fatigue response. This raises the question of a threshold effect of $v_{ac}$ on fatigue resistance. This also suggests that fracture surface mechanisms are governed by stress, and should be carefully and systematically analyzed with respect to endurance curves. To continue, SIC seems

activated only for $v_{ac}$ between 0.95 and $1.50 \times 10^4$ mol/cm³, as evidenced by SIC markers, i.e., fatigue striations of Regime 1. Furthermore, this $v_{ac}$ range shows endurance curves with a higher slope, depending on the maximal strain reached. These materials with low $v_{ac}$ are of great interest since they offer significant fatigue resistance at high strains. This demonstrates the benefits of SIC in relaxing loading conditions. More importantly, it is to be recalled that the range of $v_{ac}$ for these materials is close to optimal $v_{ac}$ values maximizing SIC, a priori ranging from 1.20 to $1.50 \times 10^4$ mol/cm³ [24,25,67,68].

**Non-relaxing tension loading conditions.** As discussed in Le Cam and Toussaint, high solicitation level at the crack tip locally increases the crystallinity, preventing the crack from propagating in a direction
<figure><img src="image_30.png" /><figcaption>**Fig. 27.** Effect of $v_{ac}$ on the damage mechanisms, for non-relaxing loadings at $\epsilon_{max}=200\%$. Fracture surfaces obtained with an optical microscope are given in (a) and corresponds to the groups numbered 1 to 4, represented in (b).</figcaption></figure>

perpendicular to the applied load and therefore causing the crack to bifurcate [55]. As a result, the fracture surfaces experience simultaneous tearing, sliding and relaxation along the highly crystallized tip, leading to the formation of fatigue striations. This is typically encountered below $1.20 \times 10^4 \text{ mol/cm}^3$, as evidenced by the fatigue striations of Regime 1. This indicates that these low $v_{ac}$ values are of benefits to SIC and thereby to the fatigue resistance since they also present a significant fatigue lifetime reinforcement. To continue, for $v_{ac}$ ranges between $1.20$ and $1.50 \times 10^4 \text{ mol/cm}^3$, cavitations under the metallic inserts were indicative of a reinforcement so significant that failure no longer occurs in the middle of the Diabolo specimen. This demonstrates the higher level of reinforcement possible from a failure analysis point of view. For this range of $v_{ac}$, the highest fatigue lifetime reinforcement was also evidenced in the Haigh diagram. This proves that SIC, as evidenced by the SIC markers, provides a fatigue lifetime reinforcement, i.e., a higher fatigue life that depends on $v_{ac}$. By further increasing $v_{ac}$ above $1.50 \times 10^4 \text{ mol/cm}^3$, the fatigue lifetime reinforcement strongly decreased and SIC markers totally disappeared. This is consistent with the literature results since an excessively high active chain density results in poor SIC ability [25].

Nevertheless, this raises the following question: is there a threshold for which the macromolecular network mobility and thereby the fatigue resistance are strongly decreased? To discuss on this issue, it is necessary to recall that in the case of relaxing loadings, a $v_{ac}$ of $2.30 \times 10^4 \text{ mol/cm}^3$ provided the best fatigue resistance. On the contrary, for non-relaxing loadings, this is an excessively high $v_{ac}$ leading to a low fatigue lifetime reinforcement. This suggests that the physical phenomena driving the fatigue lifetime for NR in both relaxing and non-relaxing loading conditions are either different or/and affected by the mean stress. Considering that (i) $v_{ac}$ drives both stiffness and SIC ability, and that (ii) since the range of $v_{ac}$ maximizing the fatigue lifetime is shifted toward lower $v_{ac}$ for non-relaxing in comparison to relaxing loading conditions, then competing effects between stiffness and SIC may occur. On the one hand, as highlighted in 3.1, it is necessary to ensure a sufficient stiffness to provide significant fatigue lifetime. This encourages fatigue resistance up to a limit of $v_{ac}$ for which too much strain energy is provided for crack growth. In the case of non-relaxing loadings having a higher mean stress, this $v_{ac}$ threshold is decreased as much more energy is provided: a high $v_{ac}$ and stiffness is no longer beneficial. On the other hand, a too high $v_{ac}$ would decrease the SIC ability. This has no significant effects on the lifetime
in the case of relaxing loadings, excepted for a range of $v_{ac}$ presenting
endurance curves with high slopes and SIC markers. Nevertheless, this
is of primary importance in the case of non-relaxing loadings, as it
seems to compensate for the high mean stress, and drastically enhanced
the lifetime. The direct consequence is that, for a low $v_{ac}$, a significant
fatigue lifetime reinforcement is achieved.

If a threshold exists, it would result in the limit between a too
high $v_{ac}$ providing both a too high stiffness and a limited ability to
crystallize. Thus, above this threshold, due to the high mean stress
encountered and the non-sufficient SIC effect, a catastrophic failure
occurs. Regarding our results, a $v_{ac}$ comprising between 1.30 and 1.50×
$10^4$ mol/cm$^3$ is a good compromise to maximize fatigue resistance for
both relaxing and non-relaxing loading conditions.

## 4. Conclusions and perspectives

This study explores the effect of the macromolecular network struc-
ture on the fatigue response of CB filled NR. Different active chain
densities and cross-link lengths were studied, by varying vulcanization
conditions and system. The active chain density of each material was
evaluated by using swelling tests and the Flory-Rehner relationship.
Fatigue tests at two different loading ratios and post-mortem analyses
were performed in order to study the effect of the macromolecular net-
work on the fatigue resistance and the fatigue lifetime reinforcement.
For relaxing tension loading conditions, fatigue resistance was found to
depend on the active chain density. A minimum active chain density is
necessary to ensure good fatigue properties. In addition, a certain range
of materials showed a fatigue response strongly dependent on the strain
level. However, it was found that increasing the active chain density
leads to an increase in the final ligament area on the fracture surface
which suggests catastrophic failure. For non-relaxing tension loading
conditions, it was found that both fatigue lifetime reinforcement and
SIC markers are related to the active chain density. There is an optimum
range of active chain density maximizing both the SIC markers, and
the fatigue lifetime reinforcement. For highly cross-linked materials,
SIC markers disappeared, and the fatigue lifetime reinforcement was
weakened. In conclusion, this study provides the first relationship
between the macromolecular network structure, the fatigue proper-
ties and the damage mechanisms. In perspective, these results raise
additional fundamental questions: does the range of $v_{ac}$ maximizing
the fatigue lifetime reinforcement and SIC markers also maximize the
crystallinity level and/or rate? The authors also wondered about the
effects of other formulations, i.e., (i) with different filler types and
contents, (ii) with different NR grades and (iii) with changes in the
non-rubber components.

CRediT authorship contribution statement

**G. Delahaye:** Writing – original draft, Methodology, Data cura-
tion, Conceptualization. **G. Delaporte:** Writing – original draft, Inves-
tigation, Formal analysis, Data curation. **Y. Mouslih:** Methodology.
**J. Rosselgong:** Resources, Methodology. **S.M. Guillaume:** Writing –
review & editing, Resources, Methodology. **B. Ruellan:** Writing – orig-
inal draft, Supervision, Funding acquisition. **I. Jeanneau:** Validation,
Supervision, Resources. **J.-B. Le Cam:** Writing – review & editing, Writ-
ing – original draft, Validation, Supervision, Project administration,
Investigation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.

Acknowledgment

The authors thank the Contech AVS France company for support-
ing this work and for fruitful discussions. The authors thank also the
National Center for Scientific Research (MRCT-CNRS and MI-CNRS),
Rennes Metropole and Région Bretagne for supporting this work fi-
nancially. SEM images were performed at CMEBA facility (ScanMAT,
University of Rennes) which received a financial support from the
European Union (CPER-FEDER 2007-2014).

**Appendix A. Supplementary data**

Supplementary material related to this article can be found online
at https://doi.org/10.1016/j.ijfatigue.2025.109014.

Data availability

Data will be made available on request.

References

[1] Cadwell SM, Merrill RA, Sloman CM, Yost FL. Dynamic fatigue life of rubber. Rubber Chem Technol 1940;13:304-15.

[2] Saintier N, Cailletaud G, Piques R. Multiaxial fatigue life prediction for a natural rubber. Int J Fatigue 2006;28:530-9.

[3] Ruellan B, Le Cam J-B, Jeanneau I, Canévet F, Mortier F, Robin E. Fatigue of natural rubber under different temperatures. Int J Fatigue 2019;124:544-57.

[4] Mouslih Y, Le Cam J-B, Ruellan B, Jeanneau I, Canévet F. Fatigue life reinforcement of carbon black filled natural rubber under non-relaxing torsion loadings and comparison with non-relaxing tension loadings. Fatigue Fract Eng Mater Struct 2024;47:1620-37.

[5] Beatty JR. Fatigue of rubber. Rubber Chem Technol 1964;37:1341-64.

[6] Mars WV, Fatemi A. Factors that affect the fatigue life of rubber: a literature survey. Rubber Chem Technol 2004;77:391-412.

[7] Tee YL, Loo MS, Andriyana A. Recent advances on fatigue of rubber after the literature survey by mars and fatemi in 2002 and 2004. Int J Fatigue 2018;110:115-29.

[8] Broudin M. Vieillissement thermo-oxydatif d'un élastomère industriel pour applications automobiles antivibratoires: caractérisations, compréhension, outils de dimensionnement [Ph.D. thesis], Brest; 2017.

[9] Heideman G, Datta RN, Noordermeer JWM, Van Baarle B. Activators in accelerated sulfur vulcanization. Rubber Chem Technol 2004;77:512-41.

[10] Quirk RP. Overview of curing and crosslinking of elastomers. Prog Rubber Plast Technol 1988;4:31-45.

[11] Loo CT. High temperature vulcanization of elastomers: 2. network structures in conventional sulphenamide-sulphur natural rubber vulcanizates. Polymer 1974;15:357-65.

[12] Loo CT. High temperature vulcanization of elastomers: 3. network structure of efficiently vulcanized natural rubber mixes. Polymer 1974;15:729-37.

[13] Fan RL, Zhang Y, Li F, Zhang YX, Sun K, Fan YZ. Effect of high-temperature curing on the crosslink structures and dynamic mechanical properties of gum and n330-filled natural rubber vulcanizates. Polym Test 2001;20:925-36.

[14] Mukhopadhyay R, De SK, Chakraborty SN. Effect of vulcanization temperature and vulcanization systems on the structure and properties of natural rubber vulcanizates. Polymer 1977;18:1243-9.

[15] Coran AY. Vulcanization. Elsevier; 1994, p. 339-85, chapter 7.

[16] Akiba M, Hashim AS. Vulcanization and crosslinking in elastomers. Prog Polym Sci 1997;22:475-521.

[17] Kruželák J, Šykora R, Hudec I. Sulphur and peroxide vulcanisation of rubber compounds-overview. Chem Pap 2016;70:1533-55.

[18] Datta RN. Rubber curing systems, vol. 12, iSmithers Rapra Publishing; 2002.

[19] Bornstein D, Pazur RJ. The sulfur reversion process in natural rubber in terms of crosslink density and crosslink density distribution. Polym Test 2020;88:106524.

[20] Mathew NM, De SK. Thermo-oxidative ageing and its effect on the network structure and fracture mode of natural rubber vulcanizates. Polymer 1983;24:1042-54.

[21] Rohana Yahya YS, Azura AR, Ahmad Z. Effect of curing systems on thermal degradation behaviour of natural rubber (smr cv 60). J Phys Sci 2011;22:1-14.

[22] Choi S-S. Influence of rubber composition on change of crosslink density of rubber vulcanizates with ev cure system by thermal aging. J Appl Polym Sci 2000;75:1378-84.

[23] Studebaker ML, Beatty JR. Effects of compounding on dynamic mechanical properties of rubber. Rubber Chem Technol 1974;47:803-24.

[24] Chenal J-M, Chazeau L, Guy L, Bomal Y, Gauthier C. Molecular weight between physical entanglements in natural rubber: A critical parameter during strain-induced crystallization. Polymer 2007;48:1042-6.
[25] Trabelsi S, Albouy P-A, Rault J. Crystallization and melting processes in vulcanized stretched natural rubber. Macromolecules 2003;36:7624-39.

[26] Tosaka M, Murakami S, Poompradub S, Kohjiya S, Ikeda Y, Toki S, et al. Orientation and crystallization of natural rubber network as revealed by waxd using synchrotron radiation. Macromolecules 2004;37:3299-309.

[27] Sainumsai W, Toki S, Amnuaypornsri S, Nimpaiboon A, Sakdapipanchi J, Rong L, et al. Dependence of the onset of strain-induced crystallization of natural rubber and its synthetic analogue on crosslink and entanglement by using synchrotron x-ray. Rubber Chem Technol 2017;90:728-42.

[28] Sainumsai W, Suchiva K, Toki S. Influence of sulphur crosslink type on the strain-induced crystallization of natural rubber vulcanizates during uniaxial stretching by in situ waxd using a synchrotron radiation. Mater Today: Proc 2019;17:1539-48.

[29] Chenal J-M, Gauthier C, Chazeau L, Guy L, Bomal Y. Parameters governing strain induced crystallization in filled natural rubber. Polymer 2007;48:6893-901.

[30] Trabelsi S, Albouy P-A, Rault J. Effective local deformation in stretched filled rubber. Macromolecules 2003;36:9093-9.

[31] Dogadkin BA, Tarasova ZN. Vulcanization structures and their influence on the heat resistance and fatigue of rubber. Rubber Chem Technol 1954;27:883-98.

[32] Yanyo LC. Effect of crosslink type on the fracture of natural rubber vulcanizates. In: Structural integrity. Springer; 1989, p. 103-10.

[33] Bhowmick AK, Gent AN, Pulford CTR. Tear strength of elastomers under threshold conditions. Rubber Chem Technol 1983;56:226-32.

[34] Lake GJ, Lindley PB. The mechanical fatigue limit for rubber. J Appl Polym Sci 1965;9:1233-51.

[35] Cox WL, Parks CR. Effect of curing systems on fatigue of natural rubber vulcanizates. Rubber Chem Technol 1966;39:785-97.

[36] Chan BL, Elliott DJ, Holley M, Smith JF. The influence of curing systems on the properties of natural rubber. J Polym Sci: Polym Symp 1974;48:61-86.

[37] Grasland F. Vieillissement du caoutchouc naturel par thermo-oxydation: Etudes de ses conséquences sur la cristallisation sous déformation, la fissuration et la rupture [Ph.D. thesis], Université de Lyon; 2018.

[38] Gehling T, Kerschbaumer RC, Pinter G, Fasching MA, Schwarz T, Schieppati J. Investigating the impact of manufacturing conditions on crack growth rate in nitrile butadiene rubber for enhanced service life - part 01. Results Eng 2023;20:101484.

[39] Masquelier I. Influence de la formulation sur les propriétés en fatigue d'élastomères industriels [Ph.D. thesis], Brest; 2014.

[40] Nie Y, Wang B, Huang G, Qu L, Zhang P, Weng G, et al. Relationship between the material properties and fatigue crack-growth characteristics of natural rubber filled with different carbon blacks. J Appl Polym Sci 2010;117:3441-7.

[41] Bhowmick AK. Threshold fracture of elastomers. Polym Rev 1988;28:339-70.

[42] Coran AY. Chemistry of the vulcanization and protection of elastomers: a review of the achievements. J Appl Polym Sci 2003;87:24-30.

[43] Delahaye G, Redon A, Ruellan B, Jeanneau I, Rosselgong J, Guillaume SM, et al. A comprehensive review on active chain density evaluation from swelling and insights for better accounting for insoluble particles. J Appl Polym Sci 2024;141.

[44] Valentin JL, Carretero-Gonzalez J, Mora-Barrantes I, Chasse W, Saalwachter K. Uncertainties in the determination of cross-link density by equilibrium swelling experiments in natural rubber. Macromolecules 2008;41:4717-29.

[45] Candau N. Compréhension des mécanismes de cristallisation sous tension des élastomères en conditions quasi-statiques et dynamiques [Ph.D. thesis], INSA de Lyon; 2014.

[46] Ruellan B. Fatigue of natural rubber at different temperatures: reinforcement due to strain-induced crystallization and modelling the non-linear damage evolution [Ph.D. thesis], University of Rennes 1; 2019.

[47] Le Cam J, Huneau B, Verron E. Description of fatigue damage in carbon black filled natural rubber. Fatigue Fract Eng Mater Struct 2008;31:1031-8.

[48] Neuhaus C, Lion A, Johlitz Me, Heuler P, Barkhoff M, Duisen F. Fatigue behaviour of an elastomer under consideration of ageing effects. Int J Fatigue 2017;104:72-80.

[49] Saintier N. Fatigue multiaxiale dans un élastomère de type NR chargé : mécanismes d'endommagement et critère local d'amorçage de fissure [Ph.D. thesis], Ecole Nationale Supérieure des Mines de Paris; 2001.

[50] Ruellan B, Le Cam J-B, Robin E, Jeanneau I, Canévet F, Mauvoisin G, et al. Fatigue crack growth in natural rubber: The role of sic investigated through post-mortem analysis of fatigue striations. Eng Fract Mech 2018;201:353-65.

[51] Mouslih Y, Le Cam J-B, Ruellan B, Jeanneau I, Canevet F. Temperature effects on the lifetime reinforcement due to strain-induced crystallization in carbon black filled natural rubber under non-relaxing torsion: comparison with non-relaxing tensile loadings. Mech Mater 2024;196:105045.

[52] Ogden RW. Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc Lond 1972;326:565-84.

[53] Ogden RW, Roxburgh DG. A pseudo-elastic model for the mullins effect in filled rubber. Proc R Soc Lond A 1999;455:2861-77.

[54] Le Cam J-B. Endommagement en fatigue des élastomères [Ph.D. thesis], Nantes; 2005.

[55] Le Cam J-B, Toussaint E. The mechanism of fatigue crack growth in rubbers under severe loading: the effect of stress-induced crystallization. Macromolecules 2010;43:4708-14.

[56] Le Cam J-B, Huneau B, Verron E. Fatigue damage in carbon black filled natural rubber under uni- and multiaxial loading conditions. Int J Fatigue 2013;52:82-94.

[57] Bennani A. Elaboration, comportement et durée de vie en fatigue du caoutchouc naturel renforcé de silice [Ph.D. thesis], École Nationale Supérieure des Mines de Paris; 2006.

[58] Le Gorju-Jago K. Fatigue life of rubber components: 3d damage evolution from x-ray computed microtomography. In: Constitutive models for rubber v. 2007.

[59] Le Cam J-B, Huneau B, Verron E, Gornet L. Mechanism of fatigue crack growth in carbon black filled natural rubber. Macromolecules 2004;37:5011-7.

[60] Le Cam J-B, Verron E, Huneau B, Gornet L, Pérocheau F. Micro-mechanism of fatigue crack growth: comparison between carbon black filled nr and sbr. In: Constitutive models for rubber IV. 2005.

[61] Le Gorju-Jago K, Bathias C. Fatigue initiation and propagation in natural and synthetic rubbers. Int J Fatigue 2002;24:85-92.

[62] Treloar LRG. The elasticity and related properties of rubbers. Rep Progr Phys 1973;36(755).

[63] Gehling T, Schieppati J, Balasooriya W, Kerschbaumer RC, Pinter G. Fatigue behavior of elastomeric components: A review of the key factors of rubber formulation, manufacturing, and service conditions. Polym Rev 2023;63:763-804.

[64] Huneau B. Strain-induced crystallization of natural rubber: a review of x-ray diffraction investigations. Rubber Chem Technol 2011;84:425-52.

[65] Lake GJ, Thomas AG. The strength of highly elastic materials. Proc R Soc Lond Ser A. Math Phys Sci 1967;300:108-19.

[66] Toki S, Fujimaki T, Okuyama M. Strain-induced crystallization of natural rubber as detected real-time by wide-angle x-ray diffraction technique. Polymer 2000;41:5423-9.

[67] Tosaka M, Kawakami D, Senoo K, Kohjiya S, Ikeda Y, Toki S, et al. Crystallization and stress relaxation in highly stretched samples of natural rubber and its synthetic analogue. Macromolecules 2006;39:5100-5.

[68] Tosaka M, Kohjiya S, Ikeda Y, Toki S, Hsiao BS. Molecular orientation and stress relaxation during strain-induced crystallization of vulcanized natural rubber. Polym J 2010;42:474-81.

[69] Gent AN. Engineering with rubber: how to design rubber components. Munich; New York: Hanser Publishers; New York: Distributed in the United States of America and in Canada by Oxford University Press; 1992.

[70] Zhao J, Ghebremeskel GN. A review of some of the factors affecting fracture and fatigue in sbr and br vulcanizates. Rubber Chem Technol 2001;74:409-27.