# Temperature effects on the lifetime reinforcement due to strain-induced crystallization in carbon black filled natural rubber under non-relaxing torsion: with tensile

Yasser Mouslih, Jean-Benoit Le Cam, Benoit Ruellan, Isabelle Jeanneau,
Frederic Canevet

► To cite this version:

Yasser Mouslih, Jean-Benoit Le Cam, Benoit Ruellan, Isabelle Jeanneau, Frederic Canevet. Temperature effects on the lifetime reinforcement due to strain-induced crystallization in carbon black filled natural rubber under non-relaxing torsion: with tensile. Mechanics of Materials, 2024, 196, pp.105045. 10.1016/j.mechmat.2024.105045. hal-04651455

HAL Id: hal-04651455

https://hal.science/hal-04651455v1

Submitted on 16 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Research paper

# 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

Yasser Mouslih a,b,c, Jean-Benoît Le Cam a,c,*, Benoît Ruellan b,c, Isabelle Jeanneau b,c, Frédéric Canevet b,c

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

**Keywords:**

Natural rubber

Lifetime reinforcement

Temperature

Torsion

Strain-induced crystallization (SIC)

Heterogeneous accommodation

## ABSTRACT

This paper investigates the lifetime reinforcement at different temperatures of carbon black filled natural rubber under non-relaxing torsion. Torsion fatigue tests with different angle ratios have been carried out with axisymmetric-shaped specimens. Finite element analysis was used to predict the local mechanical state by taking into account temperature effects and stress softening heterogeneities. It has been shown that when the temperature of the test is increased, the mechanical response does no longer stabilize, contrary to what is observed when the test is carried out at ambient temperature. A strategy has been proposed to evaluate an equivalent loading ratio, in order to compute the minimum loading as the Cauchy stress normal to the crack plane for the minimum torsion angle applied and to represent the results in the Haigh diagram. It was found that the lifetime reinforcement under torsion loading decreases when the temperature is increased to 90 °C, but is still observed. The lifetime reinforcement under torsion at 90 °C was of the same amplitude as the one obtained with the same material under uniaxial tension at the same temperature. The study has been completed by post-mortem analysis at the macro- and the microscale in order to characterize damage mechanisms at 90 °C.

## 1. Introduction

The pioneering work on fatigue of natural rubber (NR) was carried out by Cadwell et al. (1940). The authors observed a singular phenomenon when the loading is non-relaxing: the lifetime strongly reinforces. This reinforcement is generally assumed to be due to the ability of the material to crystallize under stretch, as in the case of non-crystallizable rubber, the fatigue life decreases when increasing the maximum loading whatever the loading case (Fielding, 1943; Lindley, 1974; Beatty, 1964; Abraham et al., 2005).

Since its discovery in 1925 by Katz et al. (1925), the strain-induced crystallization (SIC) in NR has been extensively studied under quasi-static loadings (Toki et al., 2000; Trabelsi et al., 2003; Chenal et al., 2007), and under relaxing and non-relaxing fatigue loadings (Beurrot-Borgarino et al., 2013). Nevertheless, the physical origin of the lifetime reinforcement, and more particularly how SIC should contribute to, is still the subject of vivid debates. Most results have been obtained

under uniaxial fatigue loadings. To better meet the needs of industrial applications, several studies have been dedicated to multiaxial fatigue (Roberts and Benzies, 1977; Robisson, 2000; Poisson, 2012; Ayoub et al., 2011, 2014; Svensson, 1981; Xie, 1995; Mars and Fatemi, 2005; Bennani, 2006; Verron et al., 2006; Verron and Andriyana, 2008; Harbour et al., 2008; Luo, 2022; Hottin et al., 2023; André et al., 1999; Saintier, 2001; Mars, 2001; Le Cam et al., 2013), but they were performed at ambient temperature. Only a few studies explored the temperature effects on the lifetime reinforcement (Bathias et al., 1998; Le Chenadec, 2008; Ruellan et al., 2019; Warneboldt et al., 2022). Ruellan et al. (2019) conducted an exhaustive fatigue campaign at different temperatures (23, 90 and 110 °C) that demonstrated that the lifetime reinforcement decreases with temperature and disappears completely at 110 °C. Similar conclusions were found under non-relaxing tension loading for different temperatures by Warneboldt et al. (2022). The author applied a relaxed torsion with a pretension to keep

<footnote>a Université de Rennes, Institut de Physique de Rennes, UMR 6251 CNRS/Université de Rennes, Campus de Beaulieu, Bât. 10B, 35042 Rennes Cedex, France</footnote>

<footnote>b ContiTech AVS France, 24 rue Nicolas Josephugnot, 35043 Rennes, France</footnote>

<footnote>c ELAST-D®, Joint Research Laboratory ContiTech AVS France - Université de Rennes - CNRS, Rennes, France</footnote>

<footnote>* Corresponding author at: Université de Rennes, Institut de Physique de Rennes, UMR 6251 CNRS/Université de Rennes, Campus de Beaulieu, Bât. 10B, 35042 Rennes Cedex, France.</footnote>

<footnote>E-mail address: jean-benoit.lecam@univ-rennes1.fr (J.-B. Le Cam).</footnote>
the specimen under non-relaxing state during all the fatigue life under different temperatures. The question of non-relaxing torsion, i.e. the minimum torsion angle is positive, and of a quantitative comparison with uniaxial tension was only addressed at 23 °C in our previous work (Mouslih et al., 2024). A methodology was proposed in order to plot the first Haigh diagram under torsion and we demonstrated that the lifetime reinforcement under torsion is of the same amplitude of that one obtained under uniaxial tension. Nevertheless, the effect of temperature on these results, more especially on the reinforcement under non-relaxing torsion, has not yet been studied. This is the first aim of the present paper: to investigate how this lifetime reinforcement under torsion loading at 23 °C could evolve at high temperatures. Assuming that the lifetime reinforcement might be present under non-relaxing torsion at 90 °C, the question would be then to define in which extent this reinforcement is comparable to that observed under uniaxial tension.

To complete the analysis of the lifetime reinforcement, generally, post-mortem investigations are performed at the macro and micro-scales (Le Cam et al., 2004; Saintier et al., 2006a; Le Cam and Toussaint, 2010; Saintier et al., 2011; Le Cam et al., 2013, 2014; Ruellan et al., 2018). Typically, SIC markers, which are specific features appearing on the fracture surfaces, have been identified as the signature of SIC in NR, since they do not form in non-crystallizing elastomers. The temperature effects on damage mechanisms in uniaxial fatigue was studied by Ruellan et al. (2019). No SIC marker is present at 90 °C, while a lifetime reinforcement is still observed, and the fracture surfaces under uniaxial tension are comparable to that of non-crystallizing elastomers, such as SBR (Le Cam et al., 2014). Similar investigations have been carried out in the present paper for non-relaxing torsion at 90 °C. This paper is organized as follows: Section 2 provides details about the experimental conditions applied. Section 3 outlines the methodology to determine the local mechanical quantities, strain and stress that evolves from cycle to cycle at high temperatures. Section 4 introduces the fatigue predictor based on the critical plane approach for calculating the amplitude and the mean value of the loading, and the equivalent loading ratio at 90 °C. In Section 5, the results are presented and analyzed using the Haigh diagram.

## 2. Experimental conditions

### 2.1. Material and specimen geometry

The material considered here is a natural rubber (cis-1,4-polyisoprene) filled with carbon black and vulcanized with sulfur. It is denoted NR in the following. Its chemical composition is the same as in Ruellan et al. (2019) and Mouslih et al. (2024): 100 phr<fnref n="1" /> of NR, 20 to 30 phr of carbon black, 1.5 phr of sulfur, 10 phr of zinc oxide along with 2 phr of stearic acid and with 2 to 4 phr of accelerators. An axisymmetric specimen geometry denoted AE5 was used for the fatigue tests under torsion loadings. It is presented with its dimensions in Fig. 1(a). It was obtained by injection molding. The number 5 corresponds to the radius of curvature in mm. The results obtained under torsion have been compared with those obtained under tension with the AE40 specimen used in Ruellan et al. (2019) (the specimen's dimensions are given in Fig. 1(b)).

### 2.2. Loading conditions

Torsion fatigue tests were performed using an electrical Instron E3000 tension-torsion testing machine equipped with a 3 kN force cell and a 25 N.m torque cell (see Fig. 1(c)).

A thermal chamber (see Fig. 1(c)) enabled us to conduct the fatigue tests at 90 °C. The heat control system is done with a dedicated Eurotherm 3208 temperature controller mounted to the chamber.

<figure><img src="image_3.png" /><figcaption>(a) Dimensions of the AE5 specimen in mm.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) Dimensions of the AE40 specimen in mm.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) AE5 specimen in the thermal chamber.</figcaption></figure>

Fig. 1. Specimen geometry and the fatigue machine equipped with thermal chamber.

Temperature is monitored and controlled by a Type N thermocouple inside the chamber. The chamber uses a forced-convection principle in which hot air is circulated around the specimen. Heating is achieved by passing air over an electrically heated element. Each test has been repeated three to four times to take the fatigue dispersion into account.

The specimens were placed between the grips inside the thermal chamber for 30 min at 90 °C to ensure enough time for the specimen thermalization. The compression force due to thermal expansion and

<footnote n="1">¹ Part per hundred of rubber in weight.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 2. The duration for temperature and thermal expansion stabilization.</figcaption></figure>

<table><caption>**Table 1**
Torsion loading conditions.</caption><thead><tr><th>Loading type</th><th>R<sub>torque<sub>in</sub></sub></th><th>R<sub>θ</sub></th><th>θ<sub>min</sub> (°)</th><th>θ<sub>max</sub> (°)</th><th>f (Hz)</th></tr></thead><tbody><tr><td rowspan="10">Torsion</td><td rowspan="3">-0.04</td><td>0</td><td>0</td><td>64</td><td>3</td></tr><tr><td>0</td><td>0</td><td>116</td><td>1.5</td></tr><tr><td>0</td><td>0</td><td>148</td><td>1.5</td></tr><tr><td rowspan="5">0.3</td><td>0.16</td><td>11</td><td>72</td><td>3.5</td></tr><tr><td>0.17</td><td>16</td><td>98</td><td>3</td></tr><tr><td>0.19</td><td>22</td><td>116</td><td>2</td></tr><tr><td>0.19</td><td>27</td><td>148</td><td>2</td></tr><tr><td>0.26</td><td>50</td><td>187</td><td>1.5</td></tr><tr><td rowspan="5">0.41</td><td>0.42</td><td>41</td><td>98</td><td>1.5</td></tr><tr><td>0.41</td><td>48</td><td>116</td><td>2</td></tr><tr><td>0.38</td><td>58</td><td>148</td><td>1.5</td></tr><tr><td>0.36</td><td>67</td><td>186</td><td>1.5</td></tr><tr><td>0.55</td><td>80</td><td>148</td><td>3.5</td></tr></tbody></table>

the temperature at the surface of the specimen are measured, their respective evolutions are given in Fig. 2. After the thermal stabilization, the position of the grips is checked and regulated in such a way that the compression force along the specimen's axis is cancelled. Thus, the specimen is in a free loading state just before running torsion tests.

The loading conditions are given in Table 1. The tests were performed under prescribed rotation angle $\theta$ and corresponding angle ratio $R_{\theta}$. In order to investigate the lifetime reinforcement, different loading ratios $R_{torque}$, which are defined as the ratio between the minimum and the maximum torque, were used. Positive values correspond to non-relaxing loadings. At 90 °C, the torque ratio is not constant, so only the initial torque ratio $R_{torque_{ini}}$ is given in the table. As for torsion fatigue study performed in Mouslih et al. (2024), torsion is ensured by letting free axial displacement, since the compressive reaction force induced by torsion has an effect on the results.

The loading frequency was chosen between 1 and 4 Hz in order to limit the self-heating, that was ranging from 1.4 to 2.5 °C for all torsion fatigue tests at 90 °C. Further details are given in Section 5.

## 2.3. End-of-life criterion

The end-of-life criterion under torsion loadings at 90 °C is defined from the evolution of the maximum reaction torque. As for torsion fatigue at 23 °C, the global reaction torque knows the same stages during the fatigue test at 90 °C, but main differences have to be highlighted compared to fatigue test at 23 °C: a significant drop of the global reaction torque is observed in the first cycles, and to a

lesser extent for the second stage where the reaction torque does not stabilize for the rest of the fatigue life. The end of this stage can be associated with macroscopic crack initiation followed by a crack propagation phase. As shown in Fig. 3, the crack propagation can be gradual for low loading levels (see Fig. 3(a)) or very fast (one cycle) for high loading levels (see Fig. 3(b)). It should be noted that the reaction torque drops more brutally compared to all the tests at 23 °C. The end-of-life at 90 °C is defined by the sharp drop of the global reaction torque, which is correlated with the initiation of a macroscopic crack. The corresponding number of cycles is denoted $N_i$.

## 2.4. Optical microscopy

The fracture surface images were recorded with a ZEISS Axioscope microscope equipped with two lenses. The first lens is a PlanApo Z 1.0x FWD 60, with a magnification range from x7 to x112, the second lens is a PlanNeoFluar Z 2.3x FWD 10.6, with a magnification range from x16 to x256. Such optical microscopy analysis enables us to investigate damage mechanisms at the microscopic scale and to characterize patterns related to SIC. The microscope was used to measure crack orientation for each fatigue test.

# 3. Prediction of the mechanical state at high temperature

For torsion tests carried out at 23 °C, the mechanical response (the torque vs. angle) stabilizes at a given number of cycles, as shown in Fig. 4. The construction of fatigue diagrams is based on the calculation of local mechanical quantities, typically the strain or the stress, considered as representative of the behavior during the main part of the fatigue life, i.e. during the plateau where the mechanical response is stabilized. At 90 °C, the mechanical response is no longer stabilized, which may put into question the classical method used so far at 23 °C (Mouslih et al., 2024). The present section presents the method we used to determine the local mechanical state under torsion fatigue loading at 90 °C.

First of all, our last paper on non-relaxing fatigue torsion at ambient temperature demonstrated that the prediction of the mechanical state is significantly improved by taking into account the heterogeneous accommodation with the Ogden-Roxburgh model for fatigue tests at 23 °C (Mouslih et al., 2024). The same approach is employed at 90 °C, since the stress softening is determined by the maximum loading applied as given in Fig. 5 for the same initial torque ratio. The model consists of describing the first load with a purely hyperelastic law, the second order Ogden (1972) law, and the further cycles with the Ogden and Roxburgh (1999) law to account for the stress softening. The formalism of the model and the identification method used at 23 °C are detailed in Mouslih et al. (2024). However, some key aspects are recalled in the present paper.
<figure><img src="image_7.png" /><figcaption>Fig. 3. Highlighting the stiffness drop at 90 °C under low and high torsion loading levels.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. Highlighting the non-stabilized behavior at 90 °C.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Highlighting the maximum strain dependency of the global reaction torque at 90 °C for the same angle ratio.</figcaption></figure>

## 3.1. Modeling the material behavior

### 3.1.1. Experimental observations: temperature effects

The strategy involved in the material identification presented in the next section is based on accounting for the temperature effects on the mechanical response of the first load under tension and compression tests, and under fatigue loading. Given that the model comprises the second order Ogden model to describe the first load and the Ogden-Roxburgh law to account for the stress softening in the further cycles, typically the stiffness reached at the end of life, temperature effects are analyzed in regard of these two aspects as follows:

* first load: it is analyzed from tension and compression tests conducted using simple tension specimen and Goodrich specimen respectively. It is well known that the temperature increases stiffness due to entropic effects (Anthony et al., 1942; Treloar, 1975), and in the same time decreases stiffness due to viscosity. A cancellation effect in the case of the present study is observed in Fig. 6(a), resulting in an equivalent mechanical response for both temperatures for a given range of strain. However, a stress stiffening is observed at 23 °C for the highest loading levels due to SIC. In the case of compression and under the strain range applied, the mechanical response under compression loading seems to be temperature independent as shown in Fig. 6(b).

* stiffness at the end of life under fatigue loading: a comparable extent of softening at the end of life is reached for both temperatures 23 and 90 °C. This result is highlighted by plotting the global reaction force/torque at the end of life for both temperatures in terms of the maximum displacement (U)/angle (θ) applied under tension and torsion loading using diabolo specimens, denoted AE40 and AE5 specimens respectively as shown in Fig. 7 for all fatigue tests.

To sum up, the temperature has an effect on the first load used for identifying the constitutive parameters of the hyperelastic part of the model (i.e. hyperelasticity coupled with Ogden-Roxburgh). However, temperature effect on the stiffness reached at the end-of-life can be considered as negligible, since a comparable extent of softening is reached between 23 and 90 °C.

### 3.1.2. Identification strategy used for predicting the mechanical response at 90 °C

For the hyperelastic law, the constitutive parameters have been identified using the first load from tensile tests conducted at 90 °C, as shown in Fig. 8(a), using a simple tension specimen, and compression test, as shown in Fig. 6(b), using Goodrich specimen. So the stiffening effect classically observed at large deformations under tensile test at 23 °C is not considered in the model used in this paper. The same tests conditions as the ones used in our previous paper (Mouslih et al., 2024) are employed here.

The softening is modeled with the Ogden-Roxburgh law, whose parameter values are those identified at 23 °C in Mouslih et al. (2024) are employed in the present paper. The parameters are identified using cyclic loadings at different levels of strain of homogeneous tests as uniaxial and compression tests. Two factors has motivated this choice: (i) as demonstrated in the previous section, the stiffness reached at the end of life is similar between 23 and 90 °C under both uniaxial tension with AE40 and torsion loading with AE5 (as shown in Fig. 7), and (ii) because of the unsteady mechanical response at 90 °C, making it non consistent to determine a specific number of cycles in the cyclic tests for
<figure><img src="image_10.png" /><figcaption>(a) Tensile tests at 23 and 90°C.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(b) compression tests at 23 and 90°C.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(a) Global reaction force in AE40 specimen.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b) Global reaction torque in AE5 specimen.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(a) Prediction of tensile test at 90°C with a hyperelastic law at 90°C.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) Prediction of compression test with a hyperelastic law at 90°C.</figcaption></figure>

<caption>Fig. 8. The experimental data used in the identification of the hyperelastic law for the first load at 90 °C.</caption>

Ogden-Roxburgh parameter identification, as depicted in Fig. 4. More precisely, whatever the number of cycles chosen in the identification method for cyclic tests, the mechanical response will continuously decrease, and the parameters of Ogden-Roxburgh law will not be able to accurately predict the stiffness drop at the end of life in fatigue tests at 90 °C. Thus, it is challenging to identify the parameters of Ogden-Roxburgh using cyclic tests at 90 °C.

To sum up, the model used to predict the mechanical state at the end of life in fatigue tests at 90 °C is identified partially at 90 °C. The first load is identified at 90 °C and the stress softening is predicted using Ogden-Roxburgh parameters from cyclic tests at 23 °C. This approach enabled us to predict well the reaction force and torque at the end-of-life, meaning that the softening seems to be not significantly affected by the temperature. If this were not the case, the stress softening predicted
<figure><img src="image_16.png" /><figcaption>Fig. 9. Correlation of the predicted and measured global reaction force and torque in tension with AE40 and torsion with AE5 specimen respectively using the parameters identified at 23 °C.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 10. Correlation between the predicted and measured global reaction forces and torque in tension with AE40 and torsion with AE5 specimen respectively using the parameters identified at 90 °C.</figcaption></figure>

by the Ogden–Roxburgh parameters would have to be determined from cyclic tests at 90 °C.

## 3.2. Consequences of taking the temperature effect into account on the prediction of the global mechanical response

The parameters identified in the previous work (Mouslih et al., 2024) at 23 °C have been used to predict the F-U (force-displacement) and C-θ (torque-angle) responses under uniaxial tension and torsion respectively at 90 °C. The correlation between the numerical prediction and the experimental force/torque are given in Fig. 9. The experimental reaction force/torque are obtained from the fatigue tests at 90 °C at the end of life in Fig. 3. It is clearly observed that the constitutive parameters identified at 23 °C fail to predict all fatigue tests in tension and torsion at 90 °C within an error range of 10%, particularly for the highest loadings.

Despite this misestimation, it is clearly observed that the law identified at 23 °C is able to predict the global mechanical response under fatigue loading at 90 °C for the majority of tests, typically the lowest loadings, but with a lesser accuracy.

Now, the model identified at 90 °C is used to predict the F-U and C-θ responses under uniaxial tension and torsion fatigue with AE40 and AE5, respectively. The results are presented in Figs. 10(a) and 10(b). These figures show that the prediction is significantly improved by taking into account the effects of temperature, especially for the highest loading levels where the stiffening classically observed at large deformations at 23 °C is pronounced.

The correlations presented in Fig. 10 validate the constitutive parameters identified at 90 °C, which is of a primary importance to be able to determine correctly the local mechanical state and then relevantly evaluate the reinforcement at high temperature and compare it with the one obtained at 23 °C for tension and torsion fatigue.

# 4. Methodology for building the Haigh diagram

## 4.1. Predictor calculation

To compare the lifetime reinforcement under non-relaxing torsion between 23 and 90 °C, the methodology defined in our previous work is used here to plot the Haigh diagram under non-relaxing torsion at 90 °C.

The construction of the Haigh diagram in torsion, as defined in Mouslih et al. (2024), involves the definition of two parameters: the amplitude and the mean value of the fatigue predictor. The predictor chosen in Mouslih et al. (2024) and employed here is the Cauchy stress normal to the crack plane, it is denoted by $\sigma_n$ and expressed as follows:

$$ \sigma_n = \frac{n' \cdot \sigma \cdot n}{\bar{n}} \quad (1) $$

with $\bar{n}$ the vector normal to the crack plane, $\sigma$ the Cauchy stress tensor and $t$ the indication that the vector is transposed. This predictor is chosen to calculate the minimum value of stress as it is not given by the principal Cauchy stress. Torsion induces material rotation, which corresponds to the rotation of vectors attached to the material during
<figure><img src="image_18.png" /><figcaption>Fig. 11. Highlighting the consequences of the non-stabilized maximum and minimum torque on the torque ratio in torsion fatigue.</figcaption></figure>

torsion test. The rotation of the principal stress axis is very different from the material rotation, more precisely the rotation of planes of cracks within the material. Therefore, the Cauchy stress normal to the crack for the minimum stress has been computed with a UVARM subroutine in Abaqus. Considering that for the highest global loading applied, the maximum stress is given by the maximal principal Cauchy stress, the amplitude and the mean values of the stress are calculated using the maximum principal Cauchy stress $\sigma_{Imax} = \sigma_{nmax}$ as it is normal to the crack plane for the highest torsion angle, and the minimum Cauchy stress normal to the crack plane $\sigma_{nmin}$. The constitutive model used to calculate the predictor is the second order Ogden coupled with Ogden-Roxburgh as described in Section 3.1.

## 4.2. Equivalent loading ratio

Generally, amplitude and mean values of the loading are calculated from the stabilized mechanical response, as at 23 °C the loading ratio is stabilized after several thousands of cycles. However, at 90 °C the mechanical response (maximum and minimum torque) does not stabilize anymore as shown in Fig. 11(a), which leads to a variable loading ratio in terms of the global mechanical response $R_C = \frac{C_{min}}{C_{max}}$, as shown in Fig. 11(b). Therefore, a special focus should be given to the method used to compute the minimum loading (i.e. the loading ratio). This variable loading ratio has a decreasing trend with respect to the number of cycles, which implies that the non-relaxing effect for a given test maybe lost with the number of cycles. Therefore, neither the loading ratio during end-of-life, nor during the beginning or mid-life, can be considered as representative of the loading ratio experienced during fatigue life. The evolution of the loading ratio ($R_F/R_C$), for uniaxial tension or torsion respectively, with respect to the number of cycles at different maximum angles and for different initial loading ratios is given in Fig. 12. It is observed that a master curve is obtained relating the loading ratio ($R_F/R_C$) to the number of cycles whatever the maximum displacement/angle applied for a given initial loading ratio. These results are of paramount importance, since the loading ratio is fully determined by the number of cycles for a given initial loading ratio applied whatever the maximum displacement/angle applied. Hence, predicting the loading ratios is possible. In this paper, an equivalent loading ratio is proposed, which is representative of the non-relaxing effect of fatigue tests at 90 °C. Therefore, the equivalent loading ratio for a given fatigue test is computed by integration of the loading ratio throughout all the fatigue life. In practice, the area under the loading ratio has been determined by numerical integration (trapezoidal rule) and has been divided by the corresponding number of cycles. Using this approach, the loading ratio history during each fatigue test can be

taken into account. Moreover, the master curve allows us to calculate the equivalent loading ratio for a test that hypothetically lasted for a certain number of cycles. Hence, the equivalent loading ratio of the isolifetime curves ($N_1, N_2$ and $N_3$ as defined in Fig. 14) are determined by integration of the master curve from 0 to $N_1$, to $N_2$ and to $N_3$ cycles respectively.

## 5. Results and discussion

In this section, the evolution of the lifetime reinforcement under torsion loading for different temperatures is presented and compared to the results obtained at 23 °C in Mouslih et al. (2024), then, with uniaxial fatigue at 90 °C. The goal is four-fold:

* to investigate the consequences of the non-stabilized mechanical response on the loading ratio at 90 °C,

* to answer the question about the temperature effects on the lifetime reinforcement under torsion fatigue loadings,

* to determine in which extent the lifetime reinforcement under torsion at 90 °C, is comparable to the already observed lifetime reinforcement under uniaxial fatigue at 90 °C, if present,

* to investigate the physical origins of the lifetime reinforcement from self-heating (thermal signature of SIC, if possible) and damage characterization (presence of SIC markers).

### 5.1. Lifetime reinforcement at 90 °C

Classically the Haigh diagram is built only in the case of uniaxial loading for characterizing the lifetime reinforcement for crystallizable rubbers (André et al., 1999; Saintier et al., 2006b; Le Cam, 2005; Ruellan et al., 2019). The first Haigh diagram under torsion loading has been proposed in Mouslih et al. (2024) at 23 °C thanks to the critical plane approach presented in Section 4.1. For building the Haigh diagram at 90 °C, the same approach is used; the amplitude and the mean values are computed using the normal stress to the crack plane. Here, the minimum normal stress is defined by the equivalent loading ratio determined by the master curves. The relevancy of this approach is validated by accurately predicting the orientation of cracks in the undeformed configuration for each fatigue tests at 90 °C. The Fig. 13 demonstrates that the critical plane predicts the orientation of the cracks whatever the initial torque ratios applied (R1, R2 and R3). The figure gives different measurements performed on different specimens for each maximum angle applied. It should be noted that the measurements were possible for different loading ratios since the fracture surfaces did not present important friction for non-relaxing torsion tests compared to that at 23 °C.
<figure><img src="image_19.png" /><figcaption>(a) $R_{initial1}$</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(c) $R_{initial3}$</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(b) $R_{initial2}$</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 13.** Prediction of the crack orientation for different loading ratios R1, R2 and R3 at 90 °C.</figcaption></figure>

### 5.1.1. High diagram for torsion loading at 90 °C

The iso-lifetime curves in torsion at 90 °C are given in red lines in Fig. 14. The experimental points under torsion at 23 °C are not presented in the figure for clarity, but the iso-lifetime curves are plotted for the sake of comparison. They are obtained from the procedure described in the previous section. The maximum normal stress to the crack plane (i.e. the maximum principal Cauchy stress) is obtained from fitting the S-N curves by a power law similar to the description given in Mouslih et al. (2024). The minimum normal stress to the crack plane,

determined from the critical plane, is defined by the equivalent loading ratio obtained by integration of the master curves in Fig. 12 from 0 to $N_1$, $N_2$ and $N_3$.

The result in Fig. 14 demonstrates that the fatigue lifetime reinforcement under non-relaxing torsion loading is less pronounced at 90 °C compared to the one obtained at 23 °C. The loading ratio is defined here in terms of stress as follows: $R_σ = \frac{\sigma_{r,\min}}{\sigma_{f,\max}}$, with $\sigma_{f,\max}$ the maximum first principal Cauchy stress during the cycle, and $\sigma_{r,\min}$ the minimum normal stress to the crack plane.

Similarly to what is observed at 23 °C, two parts can be distinguished in the iso-lifetime curves under torsion fatigue at 90 °C: a first part where the curve slope is negative and a second part where it is positive. At 23 °C, the slope sign changes from positive $R_σ$ values, meaning that the reinforcement is activated as soon as the loading is non-relaxing. At 90 °C, the reinforcement is activated from higher values of $R_σ$ (approximately 0.125) for the lowest lifetimes (curves $N_2$ and $N_3$).

In order to compare the results obtained under torsion with tension loading conditions at the same temperature (90 °C), results obtained by Ruellan et al. (2019) are reported in green lines in Fig. 15. The experimental points under tension at 90 °C are not presented in Fig. 15 for clarity, but the iso-lifetime curves are plotted for comparison.

The result evidences that the lifetime reinforcement under non-relaxing torsion at 90 °C is of the same amplitude to the already observed reinforcement under uniaxial fatigue at 90 °C.

The result found under non-relaxing torsion at different temperatures is comparable with the conclusion of Ruellan et al. (2019) about the temperature effect on the lifetime reinforcement under uniaxial tension. This finding suggests that the mechanisms behind the lifetime
<figure><img src="image_23.png" /><figcaption>**Fig. 14.** The Haigh diagram under torsion at 23 and 90 °C. $N_i$ are the iso-lifetime curves that lay between $10^4$ and $10^6$ cycles, with $N_1 > N_2 > N_3$. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. 15.** The Haigh diagram under torsion and tension at 90 °C. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

reinforcement is less active at higher temperatures. Since SIC is generally considered as being responsible for the lifetime reinforcement, and knowing that SIC is less active at high temperatures, it seems to indicate that SIC still plays a role on the lifetime reinforcement at 90 °C. Moreover, the lifetime reinforcement is similar between tension and torsion at 90 °C. The same conclusion is found at 23 °C suggesting that crystallization reinforces similarly under uniaxial tension and torsion whatever the temperature applied.

To sum up, two results are found: (i) the lifetime reinforcement is less pronounced at 90 °C under non-relaxing torsion, and (ii) it is of a similar magnitude as for uniaxial fatigue at 90 °C.

## 5.2. Self-heating analysis

Similarly to the investigation conducted at 23 °C in Mouslih et al. (2024) and Ruellan et al. (2019), the self-heating is measured for the tests at 90 °C in the present study. The temperature measurements have been performed for each fatigue test at the specimen surface with a pyrometer at the specimen center, i.e. where the cracks initiate. Then, the self-heating is mapped in the pseudo-Haigh diagram in Fig. 16. In this diagram, we can see that the self-heating is significantly low at 90 °C and limited to (2.5 °C) compared to fatigue tests at 23 °C (10 °C) as shown in Mouslih et al. (2024). Therefore, the prescribed temperature of 90 °C is approximately the temperature experienced by the specimens for all fatigue tests, and no significant superimposition of temperature resulting from self-heating is added. Moreover, self-heating increases with the maximum stress, and no change is observed under non-relaxing torsion. Similarly to 23 °C, the relevancy of self-heating quantity, for the present material and tests conditions, cannot be considered to complete the lifetime reinforcement analysis in torsion fatigue at 90 °C, i.e. the self-heating values cannot be linked to the number of cycles at the end of life.

It is noteworthy that self-heating observed at 90 °C is lower compared to 23 °C, as it aligns with the experiments of Le Saux (2010). The author showed that self-heating depends on the initial temperature of the specimen (i.e. a higher specimen temperature corresponds to lower self-heating).

## 5.3. Damage characterization at the macro- and microscale

The fracture surface investigations under torsion loading carried out at 23 °C in Mouslih et al. (2024) confirmed the implication of SIC on the fatigue lifetime reinforcement. This is in good agreement with what is observed in the case of uniaxial tension (Ruellan et al., 2019). In this paper, the same fractographic observations are conducted in order to relate temperature effects on fracture surfaces to the lifetime reinforcement under torsion loading. Therefore, damage mechanisms at the macro- and the microscale are studied at 90 °C to investigate the physical origins of the lifetime reinforcement.

### 5.3.1. Macroscopic scale

#### Damage modes

Damage modes observed at the macroscopic scales are described and related to the loading case using the pseudo-Haigh diagram given in Fig. 17. Two damage modes were observed for all torsion fatigue tests performed at 90 °C:

* Damage mode 1: it was observed in the case of small torsion angles. It corresponds to a failure in a specific direction, with a change of the plane of propagation leading to a sawtooth-shaped pattern. It should be noted that the less the maximum angle applied, the more the sawtooth are numerous.

* Damage mode 2: it corresponds to a failure occurring on a well determined plane, as classically reported in torsion loading at 23 °C by many authors (Harbour et al., 2008; Le Cam et al., 2013). This damage mode is similar to Damage mode 1 under torsion at 23 °C in Mouslih et al. (2024). A transition from Damage mode 1 to Damage mode 2 is described in the following.

The evolution of damage modes with respect to the maximum rotation angle at 90 °C can be explained by the structural effects. Damage mode 1 is expected to follow only one propagation plane as for the case of Damage mode 2. Due to the well determined orientation of cracks observed under torsion loading, the cracks move away from the center of the specimen while propagating and the available energy is not sufficient anymore to sustain the crack propagation in the same plane. Once the maximum torsion reaches a sufficient level (i.e. in the case of the loadings applied, the sawtooth-shaped cracks disappear
<figure><img src="image_25.png" /><figcaption>Fig. 16. Pseudo-Haigh diagram in terms of self-heating at for torsion fatigue at 90 °C. The value of the heating is reported in the circles.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 17. Macroscopic damage modes mapped Pseudo-Haigh diagram for torsion fatigue at 90 °C.</figcaption></figure>

starting from an angle of 148°), the Damage mode 2 is observed with a single plane of propagation.

According to these observations under torsion loadings at 90 °C, no significant change is revealed under the non-relaxing loadings. Hence, damage modes at the macroscopic scale are driven solely by the maximum loading. These conclusions are generally comparable to the ones drawn under uniaxial tension loadings (Ruellan et al., 2019).

### 5.3.2. Microscopic scale

The different damage modes identified at the macroscopic scale at 90 °C are analyzed at the microscopic scale. The aim here is to relate SIC with the lifetime reinforcement, which is still observed even at 90 °C, through fracture surface analyses. At 23 °C, SIC markers (wrenchings and striations defined in the works of Le Cam et al. (2004)

and Le Cam and Toussaint (2010) are also observed under torsion loading). Here, the temperature effect is investigated. The microscopic damage analysis is given for each damage mode:

* Damage mode 1: (Fig. 18(a)). For low torsion angles, the fracture surfaces at the microscopic scale are slightly chaotic due to friction. Despite this, we can clearly distinguish the change of propagation plane corresponding to sawtooth patterns. It should be noted that the propagation phase for low solicitations lasted for tens of thousands of cycles, as shown in Fig. 3(a), depending on the maximum torsion angle. The more the torsion angle is high (without exceeding those applied in Damage mode 2), the less the propagation phase lasts and the less sawtooth-shaped cracks form. These remarks highlight multicracking mechanisms with a change
<figure><img src="image_27.png" /><figcaption>(a) Low torsion loading: mode 1.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>(b) High torsion loading: mode 2.</figcaption></figure>

<caption>Fig. 18. Fracture surfaces at the microscopic scale at 90 °C.</caption>

of propagation plane, which is totally driven by the maximum
loading applied.

* Damage mode 2: (Fig. 18(b)). For the highest torsion level, a single crack is observed. The crack initiated around a defect and propagated into the bulk, similarly to torsion fatigue at 23 °C. However, it forms a fully flat surface at 90 °C, as for the case of non-crystallizable elastomers, such as SBR (Le Cam et al., 2014). It should be noted that this propagation phase is accompanied with a brutal drop of the reaction torque as shown in Fig. 3(b) in one cycle.

Since SIC is less active at elevated temperature, it may explain the
absence of SIC markers under torsion loading at 90 °C. Nevertheless, a
lifetime reinforcement under non-relaxing torsion is still present even
at 90 °C. Therefore, the same conclusion as under uniaxial tension
at 90 °C are found in the case of torsion fatigue. SIC and/or chain
alignment are sufficient enough to reinforce at 90 °C, but not to form
SIC markers.

**6. Conclusion**

This work has first highlighted that when the test temperature is increased, the mechanical behavior under fatigue loadings does not meet a steady state anymore (unlike fatigue tests at ambient temperature). As a result, the loading ratio, the amplitude and the mean values of loading become variable. In this study, a strategy was proposed to establish a loading ratio representative of the loading case. This enabled us to analyze the lifetime reinforcement in torsion at 90 °C in the Haigh diagram, and to compare it with uniaxial tension at the same temperature. The local mechanical fields were predicted using FEA with a model taking into account the effects of stiffness drop to account for the heterogeneous accommodation in the specimens. The Haigh diagram was built in the case of torsion loadings at 90 °C thanks to the critical plane approach and the strategy of computing the equivalent loading ratio at 90 °C. Three main results have been obtained:

(i) similarly to uniaxial tension, the lifetime reinforcement in torsion is still present and decreases with increasing temperature,

(ii) the lifetime reinforcement is equal between torsion and uniaxial tension at 90 °C, the same result is found when comparing torsion to tension at 23 °C. These results suggest that SIC and/or chain alignment reinforce in the same manner under torsion and tension whatever the temperature applied,

(iii) damage analysis at the macro- and microscale highlighted two damage modes; sawtooth-shaped cracks under low torsion angles, and

a large single crack for higher solicitations. Damage at 90 °C is found
to be driven by the maximum loading applied similar to uniaxial
tension (Ruellan et al., 2019). At the microscopic scale, the fracture
surface is chaotic due to the change of propagation plane for low
solicitations, and perfectly flat for higher solicitations with a very fast
propagation phase as observed in non-crystallizable rubbers, such as
SBR (Le Cam et al., 2014). Even if a lifetime reinforcement is still
present at 90 °C, no SIC markers were observed.

These results are of a paramount importance for predicting the
fatigue life in the case of non-relaxing solicitations at high temperatures
for loading cases combining uniaxial tension and torsion, which are
relevant for numerous applications.

CRediT authorship contribution statement

**Yasser Mouslih:** Writing – review & editing, Writing – original draft, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Jean-Benoît Le Cam:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Formal analysis, Conceptualization. **Benoît Ruellan:** Writing – review & editing, Supervision, Conceptualization. **Isabelle Jeanneau:** Visualization, Formal analysis. **Frédéric Canevet:** Writing – review & editing, Validation, Supervision.

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.

Data availability

Data will be made available on request.

Acknowledgments

The authors thank the ContiTech 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 finan-
cially.
## References

Abraham, F., Alshuth, T., Jerrams, S., 2005. The effect of minimum stress and stress amplitude on the fatigue life of non strain crystallising elastomers. Mater. Des. 26 (3), 239–245.

André, N., Cailletaud, G., Piques, R., 1999. Haigh diagram for fatigue crack initiation prediction of natural rubber components. Kautschuk Gummi Kunststoffe 52 (2), 120–123.

Anthony, R.L., Caston, E., Guth, R.H., 1942. Equations of state for natural and synthetic rubber-like materials. i. unaccelerated natural soft rubber. J. Phys. Chem. 46 (8), 826–840.

Ayoub, G., Naït-Abdelaziz, M., Zaïri, F., 2014. Multiaxial fatigue life predictors for rubbers: Application of recent developments to a carbon-filled sbr. Int. J. Fatigue 66, 168–176.

Ayoub, G., Naït-Abdelaziz, M., Zaïri, F., Gloaguen, J.-M., Charrier, P., 2011. A continuum damage model for the high-cycle fatigue life prediction of styrene-butadiene rubber under multiaxial loading. Int. J. Solids Struct. 48 (18), 2458–2466.

Bathias, C., Houel, P., Berete, Y.N., Legorju, K., 1998. Damage characterization of elastomeric composites using x-ray attenuation. In: Progress in Durability Analysis of Composite Systems, Reifsnider & Cardon, Balkema. pp. 103–109.

Beatty, J.R., 1964. Fatigue of rubber. Rubber Chem. Technol. 37 (5), 1341–1364.

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

Beurrot-Borgarino, S., Huneau, B., Verron, E., Rublon, P., 2013. Strain-induced crystallization of carbon black-filled natural rubber during fatigue measured by in situ synchrotron x-ray diffraction. Int. J. Fatigue 47, 1–7.

Cadwell, S.M., Merrill, R.A., Sloman, C.M., Yost, F.L., 1940. Dynamic fatigue life of rubber. Rubber Chem. Technol. 13 (2), 304–315.

Chenal, J.-M., Gauthier, C., Chazeau, L., Guy, L., Bomal, Y., 2007. Parameters governing strain induced crystallization in filled natural rubber. Polymer 48 (23), 6893–6901.

Fielding, J.H., 1943. Flex life and crystallization of synthetic rubber. Ind. Eng. Chem. 35 (12), 1259–1261.

Harbour, R.J., Fatemi, A., Mars, W.V., 2008. Fatigue life analysis and predictions for nr and sbr under variable amplitude and multiaxial loading conditions. Int. J. Fatigue 30 (7), 1231–1247.

Hottin, A., Abdelaziz, M.N., Talha, A., Charrier, P., 2023. Continuum damage mechanics to predict rubber fatigue life under multiaxial loadings. Int. J. Fatigue 170, 107559.

Katz, D., et al., 1925. Der aufbau der tastwelt.

Le Cam, J.-B., 2005. Endommagement en fatigue des élastomères. (Ph.D. thesis). Ecole Centrale de Nantes (ECN); Université de Nantes.

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

Le Cam, J.-B., Huneau, B., Verron, E., 2014. Failure analysis of carbon black filled styrene butadiene rubber under fatigue loading conditions. Plast. Rubber Compos. 43 (6), 187–191.

Le Cam, J.-B., Huneau, B., Verron, E., Gornet, L., 2004. Mechanism of fatigue crack growth in carbon black filled natural rubber. Macromolecules 37 (13), 5011–5017.

Le Cam, J.-B., Toussaint, E., 2010. The mechanism of fatigue crack growth in rubbers under severe loading: the effect of stress-induced crystallization. Macromolecules 43 (10), 4708–4714.

Le Chenadec, Y., 2008. Autoéchauffement, fatigue thermomécanique des élastomères (Ph.D. thesis). Ecole Polytechnique X.

Le Saux, V., 2010. Fatigue et vieillissement des élastomères en environnements marin et thermique: de la caractérisation accélérée au calcul de structure. (Ph.D. thesis). Université de Bretagne occidentale-Brest.

Lindley, P.B., 1974. Non-relaxing crack growth and fatigue in a non-crystallizing rubber. Rubber Chem. Technol. 47 (5), 1253–1264.

Luo, R.K., 2022. Effective strain criterion under multimode and multiaxial loadings-a rubber s-n curve with the scatter-band factor of 1.6 from 90 fatigue cases. Express Polym. Lett. 16 (2), 130–141.

Mars, W.V., 2001. Multiaxial Fatigue of Rubber. The University of Toledo.

Mars, W.V., Fatemi, A., 2005. Multiaxial fatigue of rubber: Part ii: Experimental observations and life predictions. Fatigue Fract. Eng. Mater. Struct. 28 (6), 523–538.

Moushli, Y., Le Cam, J.-B., Ruellan, B., Jeanneau, I., Canévet, F., 2024. 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. 47 (5), 1620–1637.

Ogden, R.W., 1972. Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 326 (1567), 565–584.

Ogden, R.W., Roxburgh, D.G., 1999. A pseudo-elastic model for the mullins effect in filled rubber. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 455 (1988), 2861–2877.

Poisson, J.-L., 2012. Détermination d'un critère de fatigue multiaxial appliqué à un élastomère synthétique. (Ph.D. thesis). Université François Rabelais-Tours.

Roberts, B.J., Benzies, J.B., 1977. The relationship between uniaxial and equibiaxial fatigue in gum and carbon black filled vulcanizates. Proc. Rubbercon 77 (2), 1–13.

Robisson, A., 2000. Comportement visco-hyperélastique endommageable d'élastomères SBR et PU: prevision de la durée de vie en fatigue (Ph.D. thesis). Ecole des Mines de Paris.

Ruellan, B., Le Cam, J.-B., Jeanneau, I., Canévet, F., Mortier, F., Robin, E., 2019. Fatigue of natural rubber under different temperatures. Int. J. Fatigue 124, 544–557.

Ruellan, B., Le Cam, J.-B., Robin, E., Jeanneau, I., Canévet, F., Mauvoisin, G., Loison, D., 2018. Fatigue crack growth in natural rubber: The role of sic investigated through post-mortem analysis of fatigue striations. Eng. Fract. Mech. 201, 353–365.

Saintier, N., 2001. Fatigue multiaxiale dans un élastomère de type NR chargé: mécanismes d'endommagement et critère local d'amorçage de fissure. (Ph.D. thesis).

Saintier, N., Cailletaud, G., Piques, R., 2006a. Crack initiation and propagation under multiaxial fatigue in a natural rubber. Int. J. Fatigue 28 (1), 61–72.

Saintier, N., Cailletaud, G., Piques, R., 2006b. Multiaxial fatigue life prediction for a natural rubber. Int. J. Fatigue 28 (5-6), 530–539.

Saintier, N., Cailletaud, G., Piques, R., 2011. Cyclic loadings and crystallization of natural rubber: An explanation of fatigue crack propagation reinforcement under a positive loading ratio. Mater. Sci. Eng. A 528 (3), 1078–1086.

Svensson, S., 1981. Testing methods for fatigue properties of rubber materials and vibration isolators. Polym. Test. 2 (3), 161–174.

Toki, S., Fujimaki, T., Okuyama, M., 2000. Strain-induced crystallization of natural rubber as detected real-time by wide-angle x-ray diffraction technique. Polymer 41 (14), 5423–5429.

Trabelsi, S., Albouy, P.-A., Rault, J., 2003. Effective local deformation in stretched filled rubber. Macromolecules 36 (24), 9093–9099.

Treloar, L.R.G., 1975. The physics of rubber elasticity.

Verron, E., Andriyana, A., 2008. Definition of a new predictor for multiaxial fatigue crack nucleation in rubber. J. Mech. Phys. Solids 56 (2), 417–443.

Verron, E., Le Cam, J., Gornet, L., 2006. A multiaxial criterion for crack nucleation in rubber. Mech. Res. Commun. 33 (4), 493–498.

Warneboldt, I., Marco, Y., Charrier, P., Hervouet, W., Champy, C., Raoul, I., Le Saux, V., Szmytka, F., 2022. Experimental investigations about complex non-relaxing fatigue loads for carbon-black filled natural rubber. Int. J. Fatigue 156, 106696.

Xie, J., 1995. Etude de la fatigue et de la rupture des assemblages collés composite-élastomère. (Ph.D. thesis). Châtenay-Malabry, Ecole centrale de Paris.