# Evaluation of the fatigue properties of elastomers based on thermo-mechanical characterization coupled to micro-structural measurements

Yann Marco, Isaure Masquelier, Vincent Le Saux, Bertrand Huneau, Pierre Charrier

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

## To cite this version:

Yann Marco, Isaure Masquelier, Vincent Le Saux, Bertrand Huneau, Pierre Charrier. Evaluation of the fatigue properties of elastomers based on thermo-mechanical characterization coupled to micro-structural measurements. Ninth European Conference on Constitutive Models for Rubber (ECCMR 2015), 2015, Prague, Czech Republic. pp.403-410, 10.1201/b18701-70. hal-05263285

HAL Id: hal-05263285

https://hal.science/hal-05263285v1

Submitted on 17 Sep 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Evaluation of the fatigue properties of elastomers based on thermo-mechanical characterization coupled to micro-structural measurements

Y. Marco, I. Masquelier & V. Le Saux

ENSTA Bretagne—Laboratoire Brestois de Mécanique et des Systèmes, Brest, France

B. Huneau

Ecole Centrale de Nantes, Institut de Recherche en Génie Civil et Mécanique (GeM), UMR CNRS 6183, Nantes, France

P. Charrier

Trelleborg Vibracoustic, CAE Durability Prediction Department, Nantes, France

**ABSTRACT:** In former papers a protocol using thermal measurements to predict the fatigue lifetime throughout an energy based criterion proved to afford a very efficient prediction of the deterministic Wöhler curve with only one sample, within less than one day. The present study presents two sections. In the first one, the method is challenged on many industrial compounds and a very good agreement is found for most of the considered materials. Then, an energetic criterion, using a single parameter is suggested, that couples the evolution of the defects population along the fatigue cycles obtained from a wide campaign of interrupted fatigue tests, to the dissipation curves. In this last case, the evaluation is not fast anymore as the microstructural analysis requires some time, but the very good agreement between the predicted fatigue curve and the mean experimental curve, illustrates the strong physical meaning of an approach based on dissipated energy.

## 1 INTRODUCTION

The industrial development of elastomeric compounds optimized for fatigue applications remains a very challenging task. This is induced by the combination of a very wide panel of variable parameters (ingredients, processing parameters during mixing, injection and curing) to heavy test campaigns, requiring numerous samples and long durations because low frequency is required to avoid any coupling to heat build up. Speeding up the fatigue characterization is therefore a crucial industrial need to assess quickly newly developed compounds. This is also a precious tool to enlarge the scope of the parameters tested (environment, mechanical loading, process parameters) in order to improve the understanding of the behaviour of these materials under fatigue solicitations. The understanding and prediction of the fatigue behaviour involve both the description of the thermo-mechanical response and of the microstructure (Marco et al. 2013). These aspects are usually considered separately macroscopic thermo-mechanic criteria (Mars and Fatemi, 2004), and microscopic observations of the defects population (Legorju-Jago 2012, Huneau et al. 2013, Le Saux et al. 2011)

and of the damage mechanisms (Gent and Park 1984, Le Cam et al. 2013, Le Saux et al. 2011, Masquelier 2014). This comes from the fact that bridging the thermo-mechanical data at macro and micro scales clearly remains a difficult challenge, both for modelling and experimental observations, despite recent advances (Omnes et al. 2008, Masquelier 2014).

The aim of this study is to base the analysis on the dissipated energy, which seems a very promising fatigue criterion (Mars and Fatemi 2004, Le Saux et al. 2010, Ayoub et al. 2012). In a former paper (Le Saux et al. 2011) a so called heat build-up protocol has been developed in which the temperature fields are converted into fields of dissipated energy (Marco et al. 2014, Masquelier et al. 2015). The deterministic Wöhler curve can then predicted throughout an energy based criterion using only one sample, within less than one day (Le Saux et al. 2011). This study presents two sections. In the first one, the method is challenged on many industrial compounds based on filled Natural Rubber (NR), varying the amount and types of Carbon Blacks (CB). In the second part an energetic criterion, using a single parameter is suggested, that couples the evolution of the defects population along the
fatigue cycles, obtained from a wide campaign of interrupted fatigue tests, to the dissipation curves. This last case is not focusing anymore on speeding the characterization but on understanding the energetic contribution of fatigue defects and illustrating the strong physical meaning of an approach based on dissipated energy.

## 2 EXPERIMENTAL SETTINGS

### 2.1 Materials and samples

The materials used in this study are NR compounds filled with various amounts of carbon blacks (0, 20, 43, 58 phr) keeping a same type of CB (N550) or various types of CB (N326, 550, 372) for the same amount of 43 phr. More data on the recipes can be found in Masquelier (2014). The heat build-up experiments as well as the fatigue tests are achieved on classical hourglass-shaped samples (see Fig. 1b).

### 2.2 Experimental devices

The heat build-up tests are achieved on a Bose electro-dynamical testing machine equipped with a 3.2kN cell force and are displacement controlled (see Fig. 1a). The thermal acquisition is performed thanks to a FLIR SC7600-BB infrared camera with a lens so-called “G1” providing an analyzed zone identical to the Focal Plane Array detector, (7mm*5mm here). The spatial resolution is of 15 µm. After a specific calibration detailed elsewhere (Masquelier, 2014), a thermal resolution of about 10mK is obtained for differential measurements. Fatigue tests are performed by TrelleborgVibracoustic on a specific testing bench. The fatigue results are plotted according to the maximum principal strain in the median section. This maximum strain is reached at the surface of the sample and calculated by FEA.

<figure><img src="image_3.png" /><figcaption>Figure 1. Experimental devices and samples geometry.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2. Mechanical protocol.</figcaption></figure>

### 2.3 Experimental protocol for heat build-up tests

The experimental protocol is illustrated on the Figure 2. After an accommodation stage to limit the Mullins effect, the sample is submitted to a sinusoidal loading between 0 and the maximal displacement (for example, 12mm induces a maximal local strain of about 227%) and then kept at the maximal deformation during the cooling. A thermal acquisition using a rate of 30 images/s is performed during the loading step. This protocol is repeated for several loading amplitudes. For the last amplitude tested, the cyclic loading is led to failure. A special attention is paid to the experimental conditions in order to limit as much as possible the influence of the environment on the measurements (see Fig. 1a).

### 2.4 Experimental protocol for interrupted fatigue tests and investigation of defects populations

The following protocol is used for the interrupted fatigue tests: each sample is subjected to one of the three strain levels (60%, 100% and 200%) and to one of the three percentages of the amount of cycles leading to initiation (20% Ni, 40% Ni and 60% Ni) with Ni the mean values obtained from the fatigue campaign. The defects population is observed on the surface of the samples from SEM observations (Huneau et al. 2015). This protocol adapted from previous studies (Le Saux et al. 2010) is chosen to avoid the cumulative damage induced by SEM observations but also to provide a statistical analysis.

## 3 FROM TEMPERATURE TO DISSIPATED ENERGY

The evaluation of the thermal sources from the temperature measurements requires solving the heat equation (Lemaître & Chaboche 2004).

$$ \rho C_T + \text{div}(\vec{q}) = \Delta + r + T + C_E \quad (1) $$
where $\rho$ is the density, $C$ the specific heat, $T$ the temperature, $q$ the thermal flux, $\Delta$ the intrinsic dissipation, $r$ the contribution of external radiation, $C_T$ the couplings of the internal variables to the temperature and $C_E$ the thermo-elastic couplings.

In recent papers (Marco et al. 2014, Masquerier et al. 2015) an experimental approach has been developed and validated. More details can be found in these papers and only the main ideas and hypothesis are recalled here.

First, the temperature variations are considered to be small enough (under 1°C) to neglect the couplings of the internal variables to temperature. Second, the evolution of temperature $\theta$ rather than the temperature is considered, which allows for compensating the external radiation contribution, assumed constant for two close instants. Third, an average evaluation is done over a mechanical cycle in order to neglect the contributions of thermo-elastic coupling and to consider the mean dissipated energy $\Delta$ over a cycle (time dependant parameter). It can be convenient to alternatively consider the mean dissipated energy per cycle $\Delta^*$, using the frequency $fr$.

Equation 1 therefore becomes:

$$ \rho C \dot{\theta} + i v(\vec{q}) = \tilde{\Delta} = fr \cdot \Delta^* \quad (2) $$

Finally, in order to neglect the diffusion, the principle is to consider two very close instants at the beginning of the cyclic loading (La Rosa & Risitano 2000) (Fig. 3). The characteristic time to reach the thermal equilibrium of our sample is evaluated from the cooling step to about 160s. Considering two consecutive cycles at 2Hz therefore leads to a ratio over 300 between the time considered to evaluate the temperature rate for two successive identical mechanical states and the characteristic thermal

<figure><img src="image_5.png" /><figcaption>Figure 3. Evolution of the mean temperature for cyclic loading.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4. Heat build-up curves for NR with 43phr (CB550).</figcaption></figure>

time. The heat diffusion is consequently considered negligible between the configurations considered and Equation 2 becomes:

$$ C \dot{\theta} = \tilde{\Delta} = ir \cdot \Delta^* \quad (3) $$

For each cyclic loading, the mean dissipated energy per cycle can be evaluated by considering the two first successive cycles:

$$ \Delta^* = \frac{\rho c}{fr} \cdot \frac{(\theta_2 - \theta_1)}{(t_2 - t_1)} \quad (4) $$

As explained in section 2.3 several block of loading with increasing displacement amplitudes are performed. For each loading block, the dissipated energy is evaluated on the central area of the sample. It is therefore possible to plot a heat build-up curve, relating this energy to the maximum principal strain (computed from a FE simulation). For the last amplitude tested, the cyclic loading is led to failure. Figure 4 shows the curves obtained for two samples. A good repeatability is observed, both for the dissipation curve and for the lifetimes obtained for the last block.

# 4 FAST PREDICTION OF THE FATIGUE PROPERTIES

## 4.1 Description of the approach on NR43-N550 compound

### 4.1.1 Determination of the energetic criterion

Now that the local strain is related to the dissipated energy, the idea is to use a energy based criterion to relate the dissipated energy to the fatigue lifetime (number of failure to initiation
$N_i$, defined by a given rate of stiffness drop, see Ostoja-Kuczynski et al., 2003). In this first part of the paper, a classical 2 parameters criterion is used:

$$ \Delta \cdot N^b = C \qquad (5) $$

To identify the 2 parameters $b$ and $C$, two pairs $(\Delta^*, N)$ are necessary. The first pair is coming straightforward from the last block led to failure (called $(\Delta_R^*, N_R)$ on Fig. 5). It is worth noticing that the possible damage induced by the former steps is limited thanks to the very low number of cycles performed for each blocks. The second pair is of course less easy to identify as the idea is to avoid performing fatigue test at low amplitudes, to reduce the characterization time. The way suggested here is purely empirical. From the shape of the curve, it is possible to distinguish three regimes, identified with straight lines on Figure 5. It is then assumed that the transition from the first to the second regime is obtained for a strain amplitude (and a dissipated energy) that leads to a lifetime of $10^6$ cycles. This assumption is fully empirical and pragmatic. The second pair needed $(\Delta^{*10^6}, N^{*10^6})$ is then obtained thanks to a graphical evaluation (see Fig. 5). Finally, the parameters of the energy based criterion are identified analytically.

### 4.1.2 Comparison to the fatigue curve

A math relation between the dissipated energy and the strain amplitude is identified in order to plot the curve predicted for the lifetime versus the strain amplitude. As illustrated on Figure 6, the correlation between the curve predicted and the mean Wöhler curve obtained from a classical fatigue campaign is fairly good.

<figure><img src="image_7.png" /><figcaption>Figure 5. Determination of three regimes on the heat build-up curve of NR43-N550.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6. Comparison of the fatigue curve predicted to the experimental values obtained from classical fatigue tests.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7. Heat build-up curves. NR with various phr (CB550).</figcaption></figure>

## 4.2 Application to other compounds

### 4.2.1 Variation of the CB content

NR compounds filled with the same type of CB (N550) and various amounts of CB (0, 20, 43, 58 phr) were tested. Figure 7 illustrates the heat build-up curves obtained. An energy based criterion is identified for each compound following the protocol explained previously. Figure 8 and 9 show the very good agreement observed between the fatigue lifetimes predicted and the ones measured from classical tests.

### 4.2.2 Variation of the CB type

NR compounds filled with the same content of CB (43 phr) and various types of CB (326, 550, 772) were tested. Figure 10 illustrates the heat build-up curves obtained. An energy based criterion is identified for each compound following the protocol explained previously. Figure 8 and 9 show the very good agreement observed between the fatigue lifetimes predicted and the ones measured from classical tests.
<figure><img src="image_10.png" /><figcaption>Figure 8. Fatigue curves predicted and experimental values.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 9. Fatigue cycles predicted vs experimental values.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 10. Heat build-up curves. NR with various CB (43 phr).</figcaption></figure>

rion is identified for each compound following the protocol explained previously. Figures 11 and 12 show the very good agreement observed between the fatigue lifetimes predicted and the ones measured from classical tests.

<figure><img src="image_13.png" /><figcaption>Figure 11. Fatigue curves predicted and experimental values.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12. Fatigue cycles predicted vs experimental values.</figcaption></figure>

# 5 SECOND FATIGUE CRITERION

## 5.1 Association of the evolution of defects density to the dissipated energy measured by heat build-up

As explained in section 4.1.1, the analysis of the heat build-up curve requires the determination of a threshold, which is disputable as the elastomeric materials do not show any fatigue limit. To overcome this hypothesis, an elegant way is to combine micro-structural data to dissipation measurements (see Le Saux et al. 2010 and Marco et al., 2013). The basic idea is first, to consider that for each cycle, the material is dissipating some energy $E_{diss,f}$ related to fatigue mechanisms and dependant on the local strain and on the cycle considered. Then that the material fails when a critical value $E_D$ of this energy is cumulated:

$$E_D = \int_0^{Ni} E_{iss,f}(\varepsilon, N) dN \quad (6)$$
As $E_{diss,f}$ is clearly difficult to evaluate, the next step is to express that this energy related to fatigue mechanisms is a fraction of the total dissipated energy per cycle $\Delta^*$ with a ratio proportional to the defects density $\bar{\omega}d(\epsilon, N)$:

$$diss,f (\epsilon, N) = \Delta^*(\epsilon \cdot \bar{\omega}d(\epsilon, N)) \quad (7)$$

## 5.2 Description of the defect population evolution

The description of the defect population is evaluated thanks to SEM measurements achieved on samples submitted to interrupted fatigue tests. The details on the protocol and on the defects natures are given in a companion paper (Huneau et al., 2015). It is worth noting that $\bar{\omega}d(\epsilon, N)$ is in this case a surface density as the fatigue initiation occurs exclusively on this area. Figure 13 illustrates the evolution of this defects density along the fatigue cycles for several maximum principal strains.

One can see from Figure 13 that strain is the first order factor for the evolution of the defects’ surface density. A linear evolution with the number of cycles can then be suggested as an approximation, with a slope $K(\epsilon)$ depending strongly on the strain (see Fig. 14). $\bar{\omega}d(\epsilon, N)$ therefore writes:

$$\bar{\omega}d(\epsilon, N) = K(\epsilon) \cdot N \quad (8)$$

### 5.2.1 Challenge of the criterion

By replacing $\bar{\omega}d(\epsilon, N)$ in Equation 6 the energetic criterion finally writes:

$$D = \Delta^*(\epsilon) \cdot K(\epsilon \cdot \int_0^{Ni} N \cdot d\epsilon) = \Delta^*(\epsilon) \cdot K(\epsilon \cdot \frac{Ni^2}{2}) \quad (9)$$

It is important to underline that $E_D$ is the only parameter to be identified, as $\bar{\omega}d(\epsilon, N)$ and $\Delta^*(\epsilon)$ are measured. The value of $E_D$ is evaluated for three

<figure><img src="image_15.png" /><figcaption>Figure 13. Evolution of the defects surface density with the fatigue cycles obtained from measurements on samples submitted to interrupted fatigue tests.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 14. Determination of the dependency of the slope of the kinetic of creation of surface defects on the local strain.</figcaption></figure>

<table><caption>Table 1. Evaluation of the critical cumulated energy $E_D$.</caption><thead><tr><th>ε</th><th>N<sub>i</sub></th><th>K (ε)</th><th>Δ (ε)</th><th>E<sub>D</sub></th></tr></thead><tbody><tr><td>90%</td><td>250000</td><td>1.43 · 10<sup>-4</sup></td><td>83</td><td>3.71 · 10<sup>8</sup></td></tr><tr><td>140%</td><td>58000</td><td>1.28 · 10<sup>-3</sup></td><td>167.4</td><td>3.59 · 10<sup>8</sup></td></tr><tr><td>190%</td><td>22000</td><td>5.79 · 10<sup>-3</sup></td><td>277.3</td><td>3.88 · 10<sup>8</sup></td></tr></tbody></table>

<figure><img src="image_17.png" /><figcaption>Figure 15. Fatigue curve predicted and experimental values.</figcaption></figure>

different values of local maximum strain and seems actually very close to a constant (see Table 1).

Another way to check the validity of the criterion is to use the value of $E_D$ obtained for a given strain and to plot the fatigue curve predicted. Figure 15 is comparing this curve to the experimental values and a very good agreement is indeed exhibited.

# 6 CONCLUSION

Two approaches to predict the fatigue curves of elastomeric materials are presented in this paper.
The first one relies exclusively on the heat build-up curves and provides a very efficient prediction of the fatigue curve (1 sample, 1 day) but requires a graphical threshold analysis. This very fast approach, with an energy based criterion using two parameters, is validated on several NR compounds with various CB amounts and types. The second energy based criterion requires a single parameter and associates heat build-up curves to the evolution of the defects population along the fatigue tests. In this last case, the evaluation is not fast anymore as the micro-structural analysis requires some time, but no empirical rule is needed and the very good agreement between the predicted fatigue curve and the experimental curve supports the strong physical meaning of the approach based on dissipated energy.

## ACKNOWLEDGEMENT

The authors thank the ANR for supporting the PROFEM project (ANR-2010-RMNP-010-01) and the partners (TrelleborgVibracoustic, GeM, UBS and LRCCP).

## REFERENCES

Ayoub, G. M. Naït-Abdelaziz, F. Zaïri, J.M. Gloaguen, P. Charrier. 2012. Mechanics of Materials.

Gent, A.N. & B. Park 1984. Journal of Materials Science.

Huneau, B., I. Masquelier, Y. Marco, O. Brzokewicz & P. Charrier 2013. *In Constitutive Models for Rubber VIII*.

Huneau B., Masquelier I., Y. Marco, V. Le Saux, P. Charrier 2015. *Constitutive Models for Rubber IX*.

Lemaître & Chaboche. 2004. Mécanique des matériaux solides, Dunod.

La Rosa G. and Risitano A. 2000. International Journal of Fatigue.

Le Cam, J.-B., B. Huneau & E. Verron. 2013. *International Journal of Fatigue*.

Le Gorju Jago, K. 2012. Rubber Chemistry and Technology.

Le Saux V.Y. Marco, S. Calloch, C. Doudard & P. Charrier 2010. *International Journal of Fatigue*.

Le Saux, V. Y. Marco, S. Calloch & P. Charrier 2011. *Polymer Engineering and Science*.

Marco Y., I. Masquelier, V. Le Saux, S. Calloch, B. Huneau, P. Charrier, 2013. *In Constitutive Models for Rubber VIII*.

Marco Y., V. Le Saux, L. Jégou, A. Launay, L. Serrano, I. Raoult, S. Calloch, 2014. International Journal of Fatigue.

Mars W.V. and Fatemi A. 2004. Rubber Chemistry and Technology.

Masquelier I., 2014. PhD thesis, Université de Bretagne Occidentale.

Masquelier I., Y. Marco, V. Le Saux, S. Calloch, P. Charrier. 2015. Mechanics of Materials.

Omnès B., S. Thuillier, P. Pilvin, Y. Grohens, S. Gillet. 2008. Composites: Part A.

Ostoja-Kuczynski, E., P. Charrier, E. Verron, G. Marckmann, L. Gornet & G. Chagnon. 2003. *In: Constitutive Models for Rubber III*.