# Heat build-up and micro-tomography measurements used to describe the fatigue mechanisms and to evaluate the fatigue lifetime of elastomers

Yann Marco, Vincent Le Saux, Sylvain Calloch, Pierre Charrier

► To cite this version:

Yann Marco, Vincent Le Saux, Sylvain Calloch, Pierre Charrier. Heat build-up and micro-tomography measurements used to describe the fatigue mechanisms and to evaluate the fatigue lifetime of elastomers. European Conference on Constitutive Models for Rubber, Sep 2011, Dublin, Ireland. pp.1-6.
hal-05432581

HAL Id: hal-05432581

https://hal.science/hal-05432581v1

Submitted on 26 Dec 2025

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 Attribution 4.0 International License
# Heat build-up and micro-tomography measurements used to describe the fatigue mechanisms and to evaluate the fatigue lifetime of elastomers

Y. Marco

ENSTA Bretagne, Laboratoire LBMS (EA 4325), 2, rue François Verny, 29806 Brest Cedex 9 – France

V. Le Saux

UBS, LIMATB - Equipe ECoMatH, rue de Saint-Maudé - BP 92116 56321, Lorient Cedex - France

S. Calloch

ENSTA Bretagne, Laboratoire LBMS (EA 4325), 2, rue François Verny, 29806 Brest Cedex 9 – France

P. Charrier

Modyn Trelleborg, Zone ind. de Carquefou, BP 419, 44474 Carquefou Cedex - France

**ABSTRACT:** The aim of this paper is to investigate the ability of a recently developed protocol to identify the fatigue lifetime of elastomers. This protocol is based on thermal and micro-tomography measurements that are coupled throughout an energetic criterion. Interrupted fatigue tests are performed for several global displacements and micro-tomography measurements are used to investigate the fatigue mechanisms involved (initiation and propagation) along the fatigue tests. Moreover, we consequently use micro-structural information obtained from a X-ray computed micro-tomography investigation to evaluate the ratio of the global dissipated energy to the one related to fatigue damage. This global dissipated energy can be related to the skin thermal measurements thanks to finite element simulations using a model developed recently (Le Chenadec et al., 2007). A critical energy criterion is then applied and the results are compared to a classical Wöhler curve.

# 1 INTRODUCTION

The failure of industrial structures is mostly related to ageing and / or fatigue. Both phenomena are difficult to master because the characteristic times are long, the influent parameters are numerous and involve probabilistic aspects. Dealing with fatigue, the classical way to identify a design criterion is to build a so called “Wöhler curve”, linking the chosen parameter to the number of cycles leading to failure (break or initiation of a crack). To be reliable, this classical approach presents at least two main disadvantages: it requires long duration tests and a large number of specimens (usually a minimum of 25 specimens is needed) in order to have a good estimation of the fatigue intrinsic dispersion. These two disadvantages obviously limit the material studied and restrained the scientific study of the influence of the numerous parameters involved. To accelerate the identification of the fatigue properties is therefore a main concern for both industrial and academic partners.

The aim of this paper is to investigate the opportunity to use thermal measurements as a quick indicator of the fatigue behavior of elastomeric materials. Based on a specific protocol, this pragmatic approach is giving interesting results but suffers from two flaws. The first one is that temperature is not an intrinsic data of the material. It is therefore mandatory to link the temperature at skin to the dissipation sources. This link is presented here, using a recent dissipation model from Y. Le Chenadec et al. (2009). The second one is that energetic measure-

ments like heat sources measurements makes difficult to split the sources involved or not in the fatigue mechanisms, like viscous dissipation, for example. To get closer from the fatigue mechanisms, a micro-structural study using X-ray Computed tomography is achieved. Not only the basic initiation mechanisms but also the scenario of the fatigue damage evolution can be identified by this technique. These informations are very relevant but are still based on geometry, giving no hints on the energy dissipated. In a last part, we consequently use the microstructural information on the cavities population to evaluate the ratio of the global dissipated energy to the one related to fatigue damage. The global dissipated energy is evaluated from the thermal measurements. A critical energy criterion is then applied and the results are compared to a classical Wöhler curve. This comparison exhibits a very good correlation, opening a new field of investigation.

# 2 EXPERIMENTAL SECTION

## 2.1 Materials and samples

More than 15 different materials were tested during this study, in order to cover a wide range of industrial standard recipes. The analysis based on the thermal measurements will include all of them, while the specific study including tomography measurements will be restrained to a silica-filled Polychloroprene (CR). For all these compounds, hourglass shaped specimens (called AE2 in the following) were manufactured from a single batch in order to
ensure the reliability of mixing and moulding condi-
tions. The geometry of the specimen are classical for
fatigue investigations and are presented elsewhere
(Le Saux et al., 2010). This geometry of specimen
was chosen for several reasons:

- it is classically used to obtain Wöhler curves;

- the initiation zone is well mastered and located in
the thinner section;

- the central section is thin enough to prevent a high
temperature gradient between the skin and the core
due to the low thermal conductibility of rubber, and
also limits the rise of temperature under cyclic load-
ing.

<figure><img src="image_2.png" /><figcaption>Figure 1. Experimental set-up</figcaption></figure>

## 2.2 *Experimental campaigns*

Three experimental campaigns have been
achieved in this study: classic fatigue testing to build
the Wöhler curves of the materials studied; heat-
build up specific tests; interrupted fatigue tests for
the tomography investigation. The fatigue tests were
performed at Trelleborg Modyn laboratory on servo-
hydraulic machines described elsewhere (Ostoja-
Kuczynski et al., 2005). The experimental determi-
nation of the initiation Wöhler curve was achieved
using a end-of-life criterion based on the variation of
the effective stiffness (Ostoja-Kuczynski et al.,
2005). It has been shown that the criterion is equiva-
lent to the apparition of crack of 2 mm on the surface
of the specimen (Ostoja-Kuczynski et al., 2003). The
heat build-up tests were performed on a servo-
hydraulic test machine (Instron 1342) at ENSTA
Bretagne, at a frequency of 2 Hz and were displace-
ment controlled. This frequency was chosen to be
the same than the one used to perform the fatigue
tests, in order to be as representative as possible
from the fatigue lifetime evaluation. A heat build-up
experiment can be defined as a succession of cyclic
tests of increasing loading conditions during which
the temperature of the specimen is measured. The
number of cycles used for each loading condition is
the number of cycles needed for the temperature to
stabilize (for example, 2000 cycles at 2 Hz are suffi-

cient for rubber-like materials). For the interrupted
fatigue tests, performed at ENSTA Bretagne on the
same machine as the heat build-up tests, three ma-
croscopic maximum displacements (2, 4, and 6mm)
were chosen, associated with the respective numbers
of cycles that lead to initiation (referenced Ni in the
following and obtained from the Wöhler curve). For
each of these displacements, at least five interrupted
tests were achieved. The tests were stopped after five
cycles and after 10, 25, 50, and 100% of Ni.

## 2.3 *Thermal measurements*

The thermal measurements were performed
thanks to an infrared camera, which gives access to a
2D measurement with a high acquisition rate (50
frames/s) and a very good precision (about 30 mK).
Even if it provides only a surface measurement, AE2
specimen are thin enough to avoid a too high core to
skin temperature gradient. The infrared camera that
has been used is a Flir Systems camera (reference
Phoenix MWIR 9705) with a Stirling-cycle cooled
Indium Antimonide (InSb) Focal Plane Array (FPA).
The FPA is a 320 X 256 array of detectors digitized
on 14 bits, sensitive in the 3–5 lm spectral band. A
careful preliminary calibration operation, detailed
elsewhere (Le Saux, 2010) allows the conversion of
the thermosignal (proportional to the thermal radia-
tion) into a temperature in degree Celsius (°C).

The temperature measurement raised two major
technical problems: to take into account the large
displacements of the specimen and to define and
measure accurately a given heat build-up tempera-
ture. For each loading block, a dedicated protocol al-
lowed us to measure both the mean temperature rise
for a un-strained position, and the amplitude of the
temperature induced by thermomechanical coupl-
ings. This study focuses on the intrinsic dissipation
and will therefore use the mean temperature rise.
The study of the evolution of the temperature along a
cycle is presented in another paper of ECCMR 2011
(Le Saux et al., 2011).

## 2.4 X-ray tomography measurements

The X-ray CT device is an industrial Phoenix de-
vice used (v|tome|x L 240) with a resolution high
enough to detect the defect sizes classically meas-
ured by SEM measurements (range from 10 to 400
µm). A dedicated commercial software (myVGL
2.0) performing 3D image analysis was then used to
study the inclusions and the defects population of the
samples. The minimal size of detection for defects
was set to 4 voxels (i.e., an equivalent radius of 13
µm) to ensure an effective detection. All the speci-
mens were slightly stretched (relative displacement
of 2 mm of the inserts) to open the cavities and to
make them easier to detect. For a more detailed de-
scription, please refer to Le Saux *et al.*, 2011.
# 3 ANALYSIS FROM THE THERMAL MEASUREMENTS ONLY

## 3.1 Method calibration

At this stage of the study, we were evaluating if a quick empirical protocol could be used to evaluate quickly the fatigue lifetime (defined here as initiation after $10^6$ cycles). The first step was therefore to identify an empirical protocol on a given material, with a known fatigue lifetime.

Fig. 2 presents the temperature measurements obtained for this material. On this curves, one can see the temperatures (for strained and unstrained state along the cyclic loading) stabilized after 2000 cycles. The first striking point is that the evolution of the heat build-up curve is much smoother than for metallic materials. This was expected as no clear fatigue limit is to be seen on the Wöhler curves obtained for elastomers. Moreover, rubbers exhibit many dissipative sources (viscosity, damage, microstructural changes...).

The empirical analysis proposed to meet the $10^6$ cycles strain identified on a Wöhler curve for this material, is illustrated on figure 2 and consists in considering only the curve related to the intrinsic dissipation (square symbols on figure 2) and to draw a line from the last points.

<figure><img src="image_3.png" /><figcaption>Figure 2. Heat build-up curve and graphical analysis</figcaption></figure>

It is worth noting that the repeatability of the tests was checked for several materials, tested with small variations of the ambient temperature and for different loading histories (different numbers of loading blocks, with different increasing amplitudes).

## 3.2 Results for a wide range of elastomers

More than 15 materials were tested. The aim was to cover a wide range of materials, crystallizing, non crystallizing, unfilled, monomer or copolymers matrix with a wide range of mechanical properties (fatigue lifetime, tan delta). The protocol defined in paragraph 3.1 was then applied to analyze the heat build-up curves obtained from other materials. It was

observed (Le Saux *et al.*, 2010) from the good agreement with the values provided by the Wöhler curves that the proposed graphical analysis is giving good results.

## 3.3 Limits

This good agreement is surprising as viscosity is obviously a first order dissipation phenomenon that could be dissociated from the fatigue properties. To challenge the protocol, it was consequently applied to specific materials, with either the same fatigue limit but different tan delta, or the same tan delta but different fatigue limit. In the first case, the protocol is still giving satisfactory results, but in the second one, illustrated on figure 3, one can see that the curves are very difficult to dissociate despite very different numbers of cycles to initiation. It can therefore be concluded that even if the rough estimation analysis presented above has proven to be quite effective for several materials, the global dissipated energy could not be directly related to the fatigue damage and that the description of the damage at lower scales was mandatory. A second step is needed as well. In this section, only temperature variation is considered, which is clearly not an intrinsic material variable. It is therefore required to link the temperature measured at the skin to the dissipation sources activated in the volume. These two points are investigated in the next paragraphs.

<figure><img src="image_4.png" /><figcaption>Figure 3. Illustration of the difficulty to directly relate the fatigue lifetime to the rise of temperature.</figcaption></figure>

# 4 FROM THE TEMPERATURE MEASUREMENTS TO DISSIPATION SOURCES

## 4.1 Heat build-up model

The first step of the demarch is to solve the mechanical problem in order to compute the dissipated energy. A first way would be to use a dissipative model, giving a direct access to the dissipation sources. Nevertheless, this approach requires numerous parameters and requires an excellent reproduc-
tion of the hysteresis loop. Moreover, the computation on several cycles is needed to reach the stabilization of the behaviour, which can be time consuming for industrial components. A hyperelastic approach has been used to model the behaviour of our material and because no large temperature variation was expected, an isothermal hyperelastic model was chosen. The hyperelastic potential is identified on the stabilized behaviour of the material, with a classical Mooney-Rivlin potential.

<figure><img src="image_5.png" /><figcaption>Figure 4. Schematic 1D illustration of the cyclic and dissipated energy.</figcaption></figure>

As the hyperelastic modelling leads to no dissipation, a specific approach needs to be developed to evaluate the dissipated energy from the hyperelastic modeling. This approach requires the definition of two energetic quantities (Y. Le Chenadec et al. 2007), which are illustrated on figure 4: a cyclic energy (Ecyc) representative of the cyclic hyperelastic loading and a dissipated energy (Ediss) which corresponds to the fraction of the elastic energy converted into heat. In this study, these energies have been computed according to a model developed recently (Le Chenadec et al., 2009). The cyclic energy writes:

$$E_{cyc} = \int_{E_{\min}^{(0)}}^{E_{\max}^{(0)}} (T^{(0)} - T_{\min}^{(0)}) : dE^{(0)}$$

where $E^{(0)}$ and $T^{(0)}$ are the order 0 conjugated Hill tensor. Conjugate Hill tensors are used to define the cyclic energy in order to guarantee the objectivity of the quantity (Hill, 1968). Thanks to pure shear experiments where $E_{cyc}$ can be obtained analytically, Le Chenadec et al. (2007) showed that the dissipated energy can be related to the cyclic energy through a power law function:

$$E_{diss} = \kappa E_{cyc}^{\gamma}$$

Here, $\kappa$ and $\gamma$ will be considered as not temperature-rependant which is justified by the fact that the expected heat build-up level are lower than 20°C.

Once these quantities have been evaluated from the mechanical response of the structure, the power loss quantity, defined as:

$$P_{diss} = F_{frequency} \cdot E_{diss}$$

is then introduced as heat sources in a thermal finite element calculation thanks to user subroutines programmed in fortran 77 (Abaqus USDFLD and HETVAL subroutines). To take advantage of the difference between mechanical and thermal characteristic times (many mechanical cycles are needed to reach the steady-state temperature) and to reduce the computational costs, a strategy based on an uncoupled cyclic algorithm is adopted: the mechanical problem is first solved, a dissipated energy is then computed from the mechanical quantities and put as heat sources in a thermal calculation.

## 4.2 Parameters identification

Two kinds of parameters are to be evaluated in this step: the first one is related to the mechanical behavior and the other one to the thermal properties. The identification of the constitutive model for rubber elasticity and the material thermal parameters (volumic density, specific heat and thermal conductivity) is very classic. Otherwise, the boundary conditions and the parameters $\kappa$ and $\gamma$ needs a careful identification, which is detailed in a forthcoming paper (Le Saux et al., submitted to European Journal of Mechanics – A/Solids).

<figure><img src="image_6.png" /><figcaption>Figure 5. Illustration of the linear relation between the volumic heat sources and the skin temperatures for AE2 samples.</figcaption></figure>

## 4.3 Approach validation and relation between temperature and dissipation sources

Using the modelization and numerical tools detailed in the paragraph 5, we are now able to relate the temperature at the skin, measured in paragraph 2 to the dissipated energy in a given volume. Figure 5 plots the dissipated energy in the central section of
the sample (central section of 1 mm thick evaluated by tomography) and illustrates a linear dependence between these variables. We will take advantage of this relation in section 6 dealing with the proposal of an energetic fatigue life criterion.

## 5 X-RAY TOMOGRAPHY MEASUREMENTS

X-ray computed micro-tomography was used to investigate the fatigue damage occurring in a polychloroprene rubber. This non destructive technique allowed us to study the initiation and propagation mechanisms (Le Saux et al. 2011). We will focus here on the evolution of the defect population along the fatigue cycles and its dependency on the maximum local strain. This study revealed that the defect volumic density depends both on the number of cycles and on the maximum local strain. Figure 6 illustrates that the defect density evolves quickly during the first 10% of the fatigue test and then increases very slowly. The value obtained after a few thousands cycle is therefore representative of the final value (at the initiation stage).

<figure><img src="image_7.png" /><figcaption>Figure 6. Stabilization along the fatigue cycles of the volumic density of the cavites for several local maximum strain.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7. Evolution of the stabilized volumic density of the cavites along the local maximum strain.</figcaption></figure>

Figure 7 illustrates the dependency of the volumic density of the cavites on the maximum local strain. This plot was obtained from several samples and for several fatigue test durations and exhibits a remarkable “master curve”. We therefore know the dependency of the defect density on the maximum strain but we still do not know how the material dissipates energy.

## 6 WÖHLER CURVE PREDICTION

In this study, the maximal strain is kept as the fatigue parameter and an initiation criterion based on the cumulated dissipated energy is used. The principle of the approach is therefore very classic and simple: whatever the fatigue parameter may be, the cumulated dissipative energy needed to initiate a crack will be a constant, called here Critical Dissipated Energy (CDE). This energy based approach is close to the one proposed by Mars (2001) or Grandcoin (2008). In elastomers, the energy dissipated evolves during the fatigue tests but reaches a stabilized value after the number of cycles needed to stabilize the rise of temperature induced by the heat build-up. Once this stabilization step is achieved, the dissipated energy measured along the fatigue test is almost a constant, for a given global displacement. This is confirmed by the stabilization of the heat build-up along a fatigue test, showing that the dissipation sources are constant. It is therefore possible to write that:

$$CDE = N_i \cdot E_{\text{Fatigue,diss/cycle}}$$

with CDE the Critical Dissipated Energy considered as an intrinsic constant, Ni the number of fatigue cycles needed to reach the initiation and Efatigue,diss/cycle the energy dissipated per cycle by the fatigue mechanisms, which is a function of the maximum strain. Once the defect population and its evolution with respect to the maximum local strain are described by tomography, a first way to evaluate the energy dissipated by the fatigue mechanisms would be to sum all the energy dissipated by each defect along one fatigue cycle. This is clearly not an easy task, from both modelization and numerical points of view.

Here, we have chosen a much more phenomenological approach: we chose to relate the total dissipated energy (evaluated from the stabilized rise of temperature) to the energy dissipated by the defect population. The first hypothesis assumes that the ratio between the total dissipated energy and the one related to the fatigue mechanisms (whatever the kind of dissipation it leads to) is linearly dependent on the defect density. It would then come:

$$E_{\text{Fatigue,diss/cycle}} = A \cdot \omega_d \cdot E_{\text{diss/cycle}}$$

with A being a constant, ωd the defect volumic density in the central zone of the sample, V the considered volume and Ediss/cycle the total dissipated
energy during one fatigue cycle in that volume. The second hypothesis is that the total dissipated energy is can be evaluated from the temperature rise measured at the skin, which was proved in paragraph 4. We could consequently write that:

$$E_{\text{diss/cycle}} = B \cdot \theta$$

with B a constant parameter and θ the stabilized minimum rise of temperature in the central zone of the sample, which is dependent on the maximum imposed deformation. This global evaluation is also consistent as because the rise of temperature, measured at the skin, is an average value of the dissipation sources that are located in the volume underneath, which is also true for the defect volumic density. From the former equation, it can now be written:

$$CDE = N_i \cdot A \cdot \omega_d \cdot B \cdot \theta$$

which also writes:

$$N_i \cdot \omega_d \cdot \theta = \text{Constant}$$

for a given volume. To validate the approach, it is possible to build a lifetime curve from this last equation. The first step is to take one point of the Wöhler curve (here for ε<sub>max</sub>=30%), for a given local strain, to evaluate the constant. Then, it is possible to fit power laws from the curves giving the dependence of ω<sub>d</sub> on strain (from tomography measurements) and from the curve giving the dependence of θ on strain (from thermal measurements). The curve obtained is plotted on Fig. 8 (called Mean lifetime curve) and exhibits a very good correlation with the other experimental fatigue data.

<figure><img src="image_9.png" /><figcaption>Figure 8. Validation of the Wöhler curve prediction and illustration of the correlation between the dispersion on the population defects and the fatigue dispersion.</figcaption></figure>

As the temperature rise is very repeatable from a sample to another, it was also very tempting to try to link the dispersion observed on figure 5 (volumic density of the cavities along strain) with intrinsic fatigue dispersion. We therefore identified two power laws from figure 5, from the lowest values and the

highest values, and then plotted the curves obtained on the figure 8 (respectively upper and lower lifetime curves). The evaluation is actually very good, even if more numerous tomography data are required to be confident in the scattering range evaluated.

## 7 CONCLUSION

The present study needs of course further validation on other materials. Nevertheless, it illustrates how powerful can be the coupling of microscopic description and macroscopic energetic evaluation to understand and model the fatigue behavior of elastomeric materials

## REFERENCES

J. Grandcoin (2008). Contribution à la modélisation du comportement dissipatif des élastomères chargés: d'une modélisation micro-physiquement motivée vers la caractérisation de la fatigue. Ph.D. thesis, Université d'Aix-Marseille II.

R. Hill (1968), On constitutive inequalities for simple materials I, Journal of the Mechanics and Physics of Solids 16 229-242.

Y. Le Chenadec, Y., Stolz, C., Raoult, I. Nguyen T., M.-L., Delattre, B. Charrier, P. (2007). A novel approach to the heat build-up problem of rubber. *ECCMR V, Paris*.

Y. Le Chenadec, I. Raoult, C. Stolz, M. Nguyen-Tajan (2009). Cyclic approximation of the heat equation in finite strains for the heat build-up problem of rubber, Journal of Mechanics of Materials and Structures 4 (2) 309-318.

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.

Le Saux, V., Y. Marco, S. Calloch, P. Charrier, and D. Taveau (2011). Heat build-up problem of rubbers under cyclic loadings: experimental investigations and numerical predictions. European Journal of Mechanics - A/Solids (submitted).

Le Saux, V., Y.Marco, S. Calloch, C. Doudard, and P. Charrier (2010). Fast evaluation of the fatigue lifetime of rubber-like materials based on a heat build-up protocol and micro-tomography measurements. International Journal of Fatigue 32, 1582-1590.

Le Saux V, Marco Y, Calloch S, Charrier P. (2011) Evaluation of the fatigue defect population in an elastomer using X-ray computed micro-tomography. J Polym Eng Sci, accepted for publication.

Mars W (2001). Multiaxial fatigue of rubber. Ph.D. thesis, University of Toledo.

Ostoja-Kuczynski, E., Charrier, P., Verron, E., Marckmann, G., Gornet, L. & Chagnon, G. 2003. Crack initiation in filled natural rubber: experimental database and macroscopic observations. *ECCMR III*.

Ostoja-Kuczynski, E., Charrier, P., Verron, E., Gornet, L. & Marckmann, G. 2005. Influence of mean stress and mean strain on fatigue life of carbon black filled natural rubber. *ECCMR IV*.