# Thermomechanical and calorimetric characterization of crack tip zones under large heterogeneous deformations, Part II: Full strain, calorimetric and strain-induced crystallinity fields at the crack tip of an unfilled natural rubber

J. -B. Le Cam, Jules Trubert, S. Charles, R. Fernandes, V. Lemos

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

## To cite this version:

J. -B. Le Cam, Jules Trubert, S. Charles, R. Fernandes, V. Lemos. Thermomechanical and calorimetric characterization of crack tip zones under large heterogeneous deformations, Part II: Full strain, calorimetric and strain-induced crystallinity fields at the crack tip of an unfilled natural rubber. Polymer, 2025, 341, pp.129183. 10.1016/j.polymer.2025.129183. hal-05422948

HAL Id: hal-05422948

https://hal.science/hal-05422948v1

Submitted on 7 Jan 2026

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 CC BY 4.0 - Attribution - International License
<figure><img src="image_3.png" /></figure>

# Thermomechanical and calorimetric characterization of crack tip zones under large heterogeneous deformations, Part II: Full strain, calorimetric and strain-induced crystallinity fields at the crack tip of an unfilled natural rubber

J.-B. Le Cam<sup>ID*</sup>, J. Trubert<sup>ID</sup>, S. Charlès, R. Fernandes, V. Lemos<sup>ID</sup>

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

## ARTICLE INFO

**Keywords:**

Crack tip

Natural rubber

Strain-induced crystallinity

## ABSTRACT

In the first part of this paper, a methodology has been developed for tracking moving material points within infrared images, by coupling full kinematic and thermal field measurements. The kinematic field is determined with the Digital Image Correlation (DIC) technique and the temperature field is measured with an infrared (IR) camera. This methodology enables us to provide simultaneously the strain field, the thermal field, the heat source (heat power density) field and, if applicable, the crystallinity field under large heterogeneous deformations. Its metrological performances have been fully characterized. The technique is the only one providing all these results from the same test, which avoids bias due to the repetition of the test with different specimens, different testing machines and differences in the boundary conditions applied. In this second part, the methodology is applied to a stretched crack tip in an unfilled natural rubber specimen. The crack tip is observed after several hundred of mechanical cycles applied to the notched specimen. The quantitative analysis of the different fields and profiles plotted (along the crack lips and behind the crack tip) enables us to characterize and to compare the different gradients in terms of strain, temperature, heat source and crystallinity. Quantitative analysis of the results supports the assumption that under the conditions of the experiment, the crack tip is the most stretched zone but not the most crystallized one, compared to the zones on both sides of it. The results also suggest that cavitation and/or decohesion might locally occur at the crack tip and frustrate the crystallization.

## 1. Introduction

Improving the mechanical strength of elastomers and better predicting their durability requires an understanding of how the material behaves at the crack tip. Many studies have been devoted to this field, in the case of filled and unfilled rubbers, crystallizing or not, under different loading and thermal conditions. The reader can refer to seminal studies such as [1–7]. Most studies measure the velocity of the crack propagation, which is a macroscopic and global quantity.

For this reason, recent years have seen a proliferation of studies using many other techniques listed as follows:

- scanning electron microscopy (SEM) to study the crack propagation mechanisms by observing the fracture surfaces [8] or the surface creation zone at the crack tip [9–11],

- X-ray diffraction to determine the crystallinity in several zones of the crack tip, whether under quasi-static loading [10,12–14] or dynamic loading [15]. X-ray can also be used to evaluate the cavitation at the crack tip, see for instance [14],

- X-ray micro-tomography to evaluate the effect of reinforcing charges in the crack propagation mechanism [16],

- IR thermography to evaluate spatial temperature gradients at the crack tip [17–20],

- DIC to characterize the strain concentration and state [14,18,19, 21].

These different techniques enable us better identifying which phenomena take place in the zone of influence of the crack (strain-induced crystallization (SIC), surface generation, nano-cavitation, cavity formation, crack growth from particles decohesion, etc.). Note that these phenomena are highly dependent on the formulation of the material, the process conditions, the aging state of the material and the type of loading. The main issue is that these phenomena are investigated with different techniques and tests and different formulations of natural rubber, which are not systematically provided in the papers. Furthermore, the metrological aspects of the various techniques used, such as

<footnote>* Corresponding author.
E-mail address: jean-benoit.lecam@univ-rennes1.fr (J.-B. Le Cam).</footnote>
<table><caption>**Table 1**
Chemical composition of the natural rubber used.</caption><thead><tr><td>Ingredient</td><td>NR</td></tr></thead><tbody><tr><td>NR (phr)</td><td>100</td></tr><tr><td>Zinc oxide (phr)</td><td>1.5</td></tr><tr><td>Sulphur (phr)</td><td>1.2</td></tr><tr><td>Accelerator (phr)</td><td>1.9</td></tr><tr><td>Stearic acid (phr)</td><td>2</td></tr><tr><td>Antioxidant (phr)</td><td>3</td></tr></tbody></table>

the level of measurement error, are not consistently included in the quantitative analysis of the results.

Since the early 2010s, our group has been developing alternative techniques that allow us to gather more data from a single test. We have started to combine infrared and visible measurements in large heterogeneous deformations, which allows us to reconstruct heat source field by determining the temperature evolution at any point. Our objective was to develop a new multi-physics analysis tool covering centimetric and sub-millimetric scales to investigate the kinematic, thermal and calorimetric behaviors of crack tip zones. We started applying our technique to elastomer crack tips in Samaca Martinez's work [18,19]. In 2018, we developed a methodology for determining crystallinity from heat sources. This enabled us to characterize crystallinity fields at crack tips as soon as 2019 as part of Charles's PhD thesis [22], thus enriching the results obtained by Samaca-Martinez. Recently, such an approach was successfully adopted in [23–25] for investigating strain-induced crystallinity at the crack tip of filled and unfilled NR. The authors focused on the effect of stress, strain state and level on crystallinity levels and gradients. To further exploit the results quantitatively in the case of very large heterogeneous deformations (maximal principal stretch equal to 5 and greater), we had first to fully characterize the technique metrology, i.e. to evaluate the resolution of the different measurements. This had never been done before. The metrological characterization is presented in Part 1 of the present paper.

In this second part, the technique is applied to investigate the behavior of the crack tip zone of an unfilled natural rubber specimen submitted to cyclic loadings. Section 2 presents the experimental setup, the material and specimen geometry as well as the protocoled used for creating the crack and for applying the mechanical cycles. In this section, the full field techniques and the heat source reconstruction are briefly presented as they are precisely detailed in Part I. The method for evaluating the crystallinity is, on the other hand, described in details. Section 2.6 presents the results in terms of kinematic fields, i.e., the strain state and level, of thermal and calorimetric fields, and of crystallinity field. The results are analyzed in relation with results recently obtained in the literature. Concluding remarks close the paper.

## 2. Experimental set-up

### 2.1. Material and specimen geometry

In the present study, an unfilled natural rubber is considered. It is vulcanized with 1.2 phr<fnref n="1" /> of sulphur and is denoted NR in the following. It has been provided by The “Manufacture Française des Pneumatiques Michelin”. Its chemical composition is provided in Table 1. It is the same material as NR₁.₂ studied in [26]. The mass density ρ was equal to 948.2 kg m⁻³, the heat capacity C is given as a function of temperature as 749 + 3.5T J/(kg K) (provided in [26]), where T is the absolute temperature in K. The thermal conductivity is set at 0.16 W m⁻<fnref n="1" /> K⁻<fnref n="1" /> and the fusion enthalpy ΔH at 62 J cm⁻³. The specimen geometry is given in Fig. 1. It corresponds to a pure shear (planar tension) specimen.

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Specimen geometry.</figcaption></figure>

### 2.2. Loading conditions

The specimen has been stretched vertically with the biaxial testing machine appearing in Fig. 2. It is the same as the one used in Part I for the metrological characterization of the method. In order to better visualize the specimen into the grips, the apparatus for reducing and keeping constant the external radiations during the tests<fnref n="2" /> has been removed a well as the components for limiting metallic part radiations (the principle is given in Fig. 1 in Part I).

Before performing the measurements, a 14 mm crack has beforehand been introduced with a razor blade and 1000 tensile cycles were applied under a prescribed total displacement equal to 20 mm (a global stretch λg in the tensile direction equal to 2.54) and a loading rate equal to 200 mm/min. This led to the natural propagation of the crack (about 6 mm). The crack velocity was not measured during the 1000 cycles. We relied on Rublon's protocol, detailed in his PhD thesis (RuBlon, 2013), to assume that the crack tip was blunted and that the crack velocity was stabilized. Rublon performed his fatigue tests using the same type of material as in the present study (unfilled (and filled) NR, vulcanized with 1.6 phr of sulphur). In his protocol, a notch approximately 20 mm long is made on one side of the crack using a cutter blade. Several hundred cycles are then performed at the maximum displacement at which the test will be carried out, in order to blunt the crack tip by transforming the very sharp incision into a fatigue crack and to stabilize the crack propagation speed (in the present study, 1000 cycles are performed). Rublon considers that crack propagation of at least 1 mm validates entry into the established velocity regime and blunting of the crack tip. In the present study, the propagation length was 6 mm. This value of crack growth per cycle (6 10⁶ nm) for an energy release rate around 7000 J/m<fnref n="2" /> corresponds to the value obtained by Rublon. Once the 1000 cycles have been completed, a painting is sprayed onto the specimen surface and 10 cycles are reconnected under the same loading conditions. The test is then stopped until the specimen is thermalized. The kinematic and thermal measurements are started and a last cycle is performed.

### 2.3. Full kinematic and thermal field measurements

Visible images are stored with the CCD camera described in Part I. The spatial resolution was equal to 13 µm/px. The kinematic field was obtained by the DIC technique. The software used was 7D [27]. The grid step was set to 4 pixels and the subset size to 10 pixels. Therefore, the conditions of the full kinematic measurement are in accordance with the conditions tested in Part I. From the displacement field, the principal in-plane stretches λ₁ and λ₁₁ are determined.

Temperature measurements are performed by using the FLIR (Tele-dyne FLIR LLC) IR camera presented in Part I. The only difference lies in the use of a 12 mm extension ring in order to improve the spatial resolution of the thermal field (50 µm/px). The integration time was equal to 2700 µs. The thermal resolution, or noise equivalent temperature difference (NETD), is equal to 20 mK for a temperature

<footnote n="1">¹ parts per hundred parts of rubber in weight.</footnote>

<footnote n="2">² more precisely during thermal measurements by means of an IR camera.</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 2. Overview of the experimental setup. In order to better visualize the setup, the two cameras have been moved back (they are closer during the measurement). Also, the protections against changes in external radiations have been removed.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Mechanical response obtained after 10 accommodation cycles. The global stretch reached is equal to 2.54. The loading rate is 200 mm/min.</figcaption></figure>

range between 5 and 40 °C. The IR camera is switched on several hours before the test in order to stabilize its internal temperature. The calibration of camera detectors is then performed with a black body using a one-point NUC procedure. The emissivity of the material is set to 0.94. Further information on the determination of the emissivity is provided in [28]. The cameras are each placed at about 160 mm far from the specimen and are triggered at 20 Hz.

## 2.4. Heat source field reconstruction

The reconstruction procedure has previously been presented in Part I. Here, we briefly recall the main steps. The heat source fields are reconstructed from the temperature fields by using the two dimensional formulation of the heat diffusion equation. In the case where the heat conduction is isotropic, the material incompressible, and the temperature constant over the specimen thickness, it writes as follows:

$$ \rho_0 C \left( \theta + \frac{\theta}{\tau_{2D}} \right) - \mathrm{Div}_{2D}(k_0 C^{-1} \mathrm{Grad}_{2D} \theta) = \hat{S}, \quad (1) $$

where $\theta(X, Y, t) = T(X, Y, t) - T_{ref}(X, Y)$. It should be noted that $T_{ref}$ is the ambient temperature if it remains constant over the test. $\tau_{2D}$ is denoted $\tau$ and is first measured in the undeformed state ($\tau_0$). It is then corrected according to its dependency on the stretch and the multiaxiality state in the case of incompressible materials as follows [28]:

$$ \tau = \tau(\lambda_I, B) = \tau_0 \lambda_I^{-B-1}, \quad (2) $$

where $B = \frac{\ln\lambda_{11}}{\ln\lambda_I}$ is the biaxiality ratio, which is equal to 1, 0 and -0.5 for equi-biaxial tension, pure shear, and uniaxial tension, respectively.

**Remark #1:** two assumptions have been done to establish the two dimensional formulation of the heat diffusion equation. First, it is assumed that no temperature gradient occurs over the specimen thickness. This assumption has been validated with the calculation of the Biot number for the same thickness in a natural rubber in [28]. Second, it is assumed that the heat conduction is isotropic. For the highest global stretches applied, this assumption should be strong. However, the subject must be recontextualized: there are few uses of surface calorimetry approaches under stretch in the literature, even fewer for large heterogeneous deformations. Generally, it is considered that the heat conduction is negligible, regardless to the spatial resolution of the thermal field and the coefficient of thermal conductivity. Only the heat convection term is calculated in the heat diffusion equation. The first attempt to take heat conduction into account when reconstructing the heat source was made by [18,19]. They considered the heat conduction to be isotropic. This assumption has also been considered more recently in [28]. So, this hypothesis is strong, but we believe that taking the effects of heat conduction into account, even when considering it to be isotropic, is a significant advance.

## 2.5. Determination of the strain-induced crystallinity field

Strain-induced crystallinity is classically evaluated with the X-ray diffraction (XRD) technique, which provides the crystallinity level as well as information of paramount importance about crystallites sizes and orientations. For determining heterogeneous crystallinity fields, a scan with the X-ray spot is required. Consequently, the crystallinity is not obtained simultaneously from one measurement point to another.
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Thermal field obtained with the IR camera. The images issued from the ResearchIR software by FLIR are not post-processed. Figures (a) to (d) ((e) to (i)) are images stored during loading (unloading). At the end of the loading, the crack tip exhibits two zones of higher temperature (see the red arrows and the profile plotted). During the unloading, these two zones cool down first (see the blue arrows). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

Moreover, the spot should have to be moved according to the motion of the measurement point in order to characterize the crystallinity evolution at a given point, which appears complicated. Due to these issues, we cannot use XRD for investigating heterogeneous SIC fields. The calorimetric method developed in [29] and validated in [30] overcomes this limitation, since the full crystallinity field can be deduced from the temperature field, easily obtained using IR thermography. The method

has been validated by comparing with results obtained by the XRD technique. In brief, the method consists in predicting the heat source due to thermoelastic couplings only during tensile tests at constant stretch rate. The easiest way to do so is to fit the heat source curve before SIC occurs with a Neo-Hookean-like model (see for instance [31]). The heat source due to SIC is then deduced by subtracting the heat source predicted for the thermodynamical cycle considered, from the total heat
<figure><img src="image_8.png" /><figcaption>Fig. 5. Oversampled DIC grid deformation during loading. Access the movie “Correlation grid 12-May-2021.avi” giving the DIC grip deformation during the mechanical cycle via the link https://data.mendeley.com/datasets/m9g9s4pcst/1.</figcaption></figure>

source measured. The equivalent temperature variation $T_{cryst}$ can then be determined from the integration of the heat source due to SIC over time, and assuming that it is equal to 0 before the crystallization starts. Last, the crystallinity $\chi$, which is the volume fraction of crystallites, is deduced from $T_{cryst}$ and the fusion enthalpy $\Delta H_{cryst}$ as follows:

$$ \chi(t) = \frac{\rho C_p T_{cryst}(t)}{\Delta H_{cryst}}. \qquad (3) $$

**Remark #2:** an important remark has to be done for non-constant stretch rates during stretching. This is typically the case in the crack type zone. As the heat source evolves proportionally to the stretch rate, the heat source prediction must be done accordingly (see for instance the heat source deriving from the Neo-Hookean form in [32], Eq. (25)). If strong multiaxial effects occur, they are taken into account in the prediction with the biaxiality coefficient. If the biaxiality coefficient is itself non-constant, Eq. (11) in [28] has to be used. In the present paper, the general form of the heat source prediction is applied to account for the different situations.

**Remark #3:** crystallinity can be determined from the calorimetric response only. This is not possible from the thermal response that requires a measure (or a prediction) of the stress.

## 2.6. Results

In this section, the results in terms of the raw IR images, the kinematic field and their coupling for calculating the heat source and the

<figure><img src="image_9.png" /><figcaption>Fig. 6. Oversampling principle.</figcaption></figure>

crystallinity fields are successively presented. The corresponding films can be accessed by following the links in the figure captions. For the reader information, the mechanical response of the cracked specimen obtained for the last cycle, i.e. the one for which the kinematic and thermal fields were stored, is provided in Fig. 3. The hysteresis loop in the stress-stretch curve was found very small and the residual stretch negligible. In unfilled NR, it has been shown that the hysteresis loop is due to SIC [33]. In the present case, the main part of the specimen is submitted to a global stretch equal to 2.54, which is smaller than the SIC onset under pure shear (planar tension), see for instance [34] where the SIC onset is around 4 (this is also the case under uniaxial tension [35,36]). Even though crystallization occurs at the crack tip, the crystallizing zone is too small compared to the rest of the specimen to affect the global mechanical response and to induce a hysteresis loop.

### 2.6.1. Analyzing the raw IR images

In this section, we present first the thermal fields obtained from the IR camera. Fig. 4 gives IR images of the zone surrounding the crack tip stored during the loading and the unloading. At the beginning of the loading (Fig. 4(a)), the temperature is the highest at the crack tip. The temperature gradient is due to heat production, which is higher in the most stretched zones, and to the diffusion of this heat. Heat conduction and convection therefore smooth the temperature gradient. This is the reason why identifying the zone of the highest heat production by using a calorimetric approach provides information intrinsic to the thermomechanical behavior of the zone of influence of the crack (see Section 2.6.4).<fnref n="3" /> For higher global stretches (Figs. 4(b and c)), two zones of higher temperature are distinguished and grow at the crack tip (see the magnification of Fig. 4(c) in Fig. 4(d) and the temperature profile plotted). Discussing on these highest temperature zones and their origin requires determining the strain field (for the thermoelastic and softening effects if any) as well as the heat source field (for the exothermal peak of crystallization and the crystallinity) at the crack tip. This has been done in the following.

During the unloading, the two zones identified during the loading cool down first (see the two blue arrows in Figs. 4(f and g)). This cannot be explained by the entropic elasticity only, as the stretch level is expected to be the highest at the crack tip. This is fully addressed in Section 2.6.2. Furthermore, SIC should take place around the crack tip and not more on both sides of it. The fact that the cooling is higher seems to indicate that it was the most crystallized zones. It should be noted that a zone remains hot behind the coldest one (see Fig. 4(h)). It corresponds to the accumulation of heat coming from the crack tip by heat conduction. This is well highlighted in the calorimetric responses discussed in the following. This illustrates well that under non-adiabatic

<footnote n="3">³ Classically, the zone of influence of the crack is considered as the zone where the strain gradient takes place, but this zone is smaller than the zone where the temperature gradient takes place, due to conductive effects. This may have an effect on thermosensitive phenomena out of the strain gradient zone, but inside the temperature gradient one.</footnote>
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Maximum principal stretch field mapped for intermediate (a) and maximum (b) values of the global stretch applied during loading. Access the movie "Maximal principal stretch 30-Apr-2021.avi" giving the maximum principal stretch field during the mechanical cycle via the link https://data.mendeley.com/datasets/3ymcjp78xn/1.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Biaxiality coefficient *B* mapped for intermediate (a) and maximum (b) values of the global stretch applied during loading. It is equal to 1, 0 and -0.5 for equi-biaxial tension, pure shear, and uniaxial tension, respectively. Access the movie "Biaxiality field 30-Apr-2021.avi" giving the biaxiality field during the mechanical cycle via the link https://data.mendeley.com/datasets/jhxd98p3js/1.</figcaption></figure>

conditions, the temperature is not intrinsic to the thermomechanical behavior and the thermal zone of influence of the crack is of importance as the following will highlight that it may strongly differ from the mechanical one, see for instance Fig. 4(h) corresponding to the nearly undeformed state. It should be noted that no significant intrinsic dissipation is produced as the crack tip is colder than the rest of the specimen instead of being warmer.

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Thermal fields for intermediate (a) and maximum (b) values of the global stretch applied during the loading and for an intermediate value (c) during the unloading (in °C). Note that in Fig. 4, each image is anamorphosed (the scale is adjusted according to the minimal and maximal values) in order to highlight the asymmetry in the temperature distribution. In the present figure, images are not anamorphosed in order to better relativize the temperature variations at different global stretches. This is why this asymmetry in the temperature distribution is less easy to observe, especially at the maximum global stretch applied. To go further, the temperature has been plotted along different profiles in Section 2.6.6. Access the movie "Temperature field 30-Apr-2021.avi" giving the temperature field during the mechanical cycle via the link https://data.mendeley.com/datasets/t26jxyzrgj/1.</figcaption></figure>

### 2.6.2. Kinematic measurements

Fig. 5 presents the correlation grid in projection onto the specimen surface in the undeformed, intermediate and maximum values of the global stretch applied. First of all, the deformation of the correlation grid illustrates the very large and heterogeneous deformation of the crack tip. This highlights the necessity of using a DIC algorithm taking into account the large deformation of the Zone Of Interest (ZOI). Furthermore, the higher the stretch, the higher the distance between points followed in the IR images, which alters the spatial resolution. Moreover, even though this does not affect the first two terms of the left member of the heat equation, this alters the accuracy of the heat conduction term. To limit this effect, it was not possible to decrease more the pattern size, even by changing the test preparation and processing parameters (speckle, camera distance, lighting, interpolant, etc.). Therefore, the number of measurement points has been artificially increased by over-sampling of the DIC grid (see Fig. 6). The oversampled point positions are obtained by using a bilinear interpolation at each node of the kinematic grid in projection in the IR image. In the following, the oversampled grid will be used in order to plot the
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Heat source fields (in W/m³) for intermediate (a) and maximum (b) values of the global stretch applied during the loading and for an intermediate value (c) during the unloading. Access the movie "Heat source field 30-Apr-2021.avi" giving the heat source field during the mechanical cycle via the link https://data.mendeley.com/datasets/jgmgm3t43t/1.</figcaption></figure>

different fields. Fig. 7 presents the maximum principal stretch field at the maximum global stretch (Fig. 7(b)) and at an intermediate stretch level (Fig. 7(a)). As expected, a strong gradient is induced by the crack. Note that the maximum stretch in the first principal direction is higher than 5 in the crack tip, i.e. a stretch superior to the SIC onset reported in many papers for this material [12–14,26]. The biaxiality coefficient has been mapped in Fig. 8. It shows that the crack lips are mainly under uniaxial tension and the rest of the zone under pure shear, the crack tip being in an intermediary strain state. This is in a good agreement with the kinematic measurement made in the filled NR in [18] and in [14].

### 2.6.3. Coupling full kinematic and thermal fields

The methodology presented in Part I has been applied to track the points in the infrared images and to determine the calorimetric response of the specimen surface. In order to limit the noise amplification effect to the temporal derivative of the temperature variation in the heat diffusion equation, a temporal filtering of the temperatures has been carried out with a Savitzky-Golay finite impulse response (FIR) smoothing filter [37] of polynomial order 3 and frame length 11 applied to the temperature over the time. Fig. 9 presents the thermal field obtained for the mid- and maximum values of the global stretch applied during the loading and its mid-value during the unloading. These maps fit well with the maps corresponding to the raw IR images in Fig. 4. The maximum temperature correspond to the raw data and the minimum temperature is slightly different because the size of the zone corresponding to the thermal and the kinematic fields is different.

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Crystallinity fields (in %) for intermediate (a) and maximum (b) values of the global stretch applied during the loading. Access the movie "Cristallinity field 04-May-2021.avi" giving the crystallinity field during the loading via the link https://data.mendeley.com/datasets/njnkpjjdsk/1.</figcaption></figure>

### 2.6.4. Heat source field reconstruction

The heat diffusion equation (Eq. (1)) is applied to the lagrangian temperature field. It should be noted that the divergence operator in the diffusion term $\text{Div}_{2D}(k_0C^{-1}\text{Grad}_{2D}\theta)$ is calculated with the finite difference method. Fig. 10 presents the results obtained. As expected, a heat source concentration is observed at the crack tip, which is in a good agreement with other studies presenting heat source field around a crack tip in rubber-like material [17,18]. This is explained by the fact that it is the most stretched area, the heat production due to entropic elasticity is therefore higher than in other areas and crystallization takes place as the SIC onset is exceeded [10,12–14]. Nevertheless, the resolution (thermal and spatial) of the thermal measurement in these previous calorimetric studies was not enough to detect unexpected gradient in the heat source field at the crack tip if any. Typically, our measurements show that the zones of highest heat source appear on both sides of the tip (see the white arrows), with a larger zone on the top lip side. This is confirmed by the fact that during unloading these zones are those of highest heat absorption. It should be noted that the highest global stretch applied, heat source gradient is limited to a zone smaller than the one where thermal gradient takes place, which highlights once more that the temperature is not intrinsic to the thermomechanical behavior if heat diffusion occurs. To finish, the fact that the heat sources are higher on both sides of the tip cannot be explained by the kinematic field obtained. The only possibility should be a higher crystallinity. This is investigated in the next paragraph as our technique enables us determining the crystallinity field.

### 2.6.5. Crystallinity field

The crystallinity field has been determined during the test by applying the methodology described in Section 2.5. It is presented in Figs. 11(a) and (b) for the same global stretches of the previous maps during loading. It appears clearly that the crack tip zone crystallizes over a zone of about 1 mm in width for the maximum global stretch applied. The maximum crystallinity is 4.35 %. SIC takes place at the crack tip but also along the crack lips. Moreover, the crack tip is not the most crystallized zone compared to the zones on both sides of it. This is not the case for lower global stretches at which SIC took place. This explains the heat source maps obtained in Fig. 10. As the crystallinity
<figure><img src="image_15.png" /><figcaption>Fig. 12. Two profiles have been chosen to plot the data; profile “Crack” along the crack lips and horizontal profile “Mid” in depth from the tip.</figcaption></figure>

depends on the stretch level and such dissymmetry is not observed in the stretch distribution, this should mean that other phenomena take place at high stretches at the tip of the crack. In the recent past, many studies have shown that nano-cavitation and cavitation takes place at the crack tip of elastomers. The effects of fillers and the SIC itself on nano-cavitation and cavitation onset have led to contradictory results in the literature and is therefore still the subject of debate. The reader can further refer to the following papers: [14,38–41]. In the present case, this phenomenon could explain our observations: cavitation and/or decohesion occur at the crack tip and frustrate SIC. This was observed by [41] in an unfilled NR under fatigue loading. The authors stated that “along the main direction of backwards crack, at the beginning, massive cavities or other kinds of broken structures appear in front of the initial crack tip”. In addition, fatigue loadings promote decohesion between micro-particles (zinc oxides for instance) and the rubber matrix at the fatigue crack tip (see for instance the results in [9]*). SIC frustration due to cavitation and/or decohesion could therefore explain why the most crystallized zones are located on both side of the crack tip in the unfilled NR we tested.

### 2.6.6. *Quantitative analysis*

In order to further analyze the data, the different quantities have been plotted along two profiles given in Fig. 12: profile “Crack” along the crack lips and horizontal profile “Mid” in depth from the tip. The plots are presented in Fig. 13 for the maximal global stretch. This figure enables us comparing the stretch and the crystallinity variations along the profiles and the zone where their gradients take place. Examining profiles “Mid” (Figs. 13(a to d)) and “Crack” (Figs. 13(e to h)), the following results are drawn:

- the maximal principal stretch gradient extends to a zone of 2 mm in width from the tip (see Fig. 13(a)). Its maximal value is equal to 5, but neighboring pixels can reach 6, as shown in Fig. 13(e),

- the maximal principal stretch at which SIC starts is equal to 3 (see Fig. 13(a)),

- the crystallinity gradient extends to a zone of 1 mm in width from the tip (see Fig. 13(a)),

- the maximal crystallinity reaches at the crack tip is 3%, where the maximum principal stretch is obtained (see Figs. 13(d) and (e)),

- the maximal crystallinities reached in the specimen are located on both sides of the crack tip; 4.35% on one side and 3.50% on the other side. This confirms and quantifies what has already been observed in the crystallinity maps,

- the zone along the crack tip where the crystallinity is lower (3%, see profile “Crack”) is about 2 mm long (see the dotted line in Figs. 13(d) and (h)).

The plots carried out along the two profiles are clearly complementary to the fields. They provide quantitative information to precisely characterize the gradients of the different quantities considered and to support the realistic idea that decohesion and/or cavitation are among phenomena that could explain why SIC is frustrated at the crack tip, i.e. in the most stretched zone. As a summary, a typical illustration of phenomena involved and their spatial distribution in the zone of influence of the crack is given in Fig. 14.

## 3. Conclusion

The methodology presented in the first part of the present paper enables us determining full kinematic, thermal, heat source and crystallinity fields in the case of large heterogeneous strains, and with a high resolution. In the present part, the methodology is applied to a stretched crack tip in an unfilled natural rubber. The crack tip observed has been obtained after crack propagation under fatigue loading. The strain and the heat source fields have been established in the crack tip zone. From the calorimetric response, the crystallinity field has been mapped. In addition, the different quantities have along two profiles, one along the crack tip, another from the tip in the crack propagation direction. Results show that under the loading conditions considered, the crack tip is not the most crystallizing zone compared to the lips under uniaxial tension on both sides of it. It was shown that the strain

<footnote n="4">⁴ It should be noted that [9] investigated in-situ the tearing resistance under the exposition to the electron beam during Scanning Microscopy Analysis (SEM) of different zones at the stretched fatigue crack tip of a carbon black filled natural rubber. SEM observations highlighted that the crack tip is composed of elliptical plate zones delimited by ligaments. Ligaments resisted tearing while elliptical zones no. The tearing of the elliptical zones revealed cavities containing zinc oxides. Authors concluded that the ligaments were the most crystallized zones. This is therefore consistent with the present paper: cavity formation and/or growth is damage that frustrates SIC.</footnote>
<figure><img src="image_16.png" /><figcaption>Fig. 13. Maximal principal stretch, temperature, heat source and crystallinity along two profiles given in Fig. 12: profile “Crack” along the crack lips and horizontal profile “Mid” in depth from the tip. The profiles are given for the maximal global stretch applied.</figcaption></figure>

state at the crack tip does not explain this phenomenon and that cavity formation and/or growth at the very crack tip could be a realistic explanation for SIC frustration.

This method proposed offers a powerful tool to study the pro- cesses of large deformations and damage in elastomers, by providing simultaneously these fields. Undergoing studies in our lab are focused
<figure><img src="image_17.png" /><figcaption>Fig. 14. A schematic view of phenomena involved in the zone of influence of cracks.</figcaption></figure>

on the determination of intrinsic dissipation field and on the effect of loading signal polarity (relaxing and non-relaxing loadings typically) on the macromolecular network at the crack tip of filled and unfilled natural rubbers.

### CRediT authorship contribution statement

**J.-B. Le Cam:** Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Funding acquisition, Conceptualization. **J. Trubert:** Validation, Methodology, Formal analysis, Data curation. **S. Charlès:** Validation, Software, Methodology, Formal analysis, Data curation, Conceptualization. **R. Fernandes:** Formal analysis, Data curation. **V. Lemos:** Formal analysis, Data curation.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

Dr E. Robin is acknowledged for providing the tracking point algorithm. Dr E. Robin, Dr M. Miroir, M. M. Le Fur and M. V. Burgaud are acknowledged for the development of the home-made biaxial testing machine. The authors acknowledge the “Manufacture Française des pneumatiques Michelin” for providing the specimens. Dr P. Vacher is acknowledged for providing the digital image correlation software 7D.

### Data availability

Data will be made available on request.

### References

[1] R.S. Rivlin, A.G. Thomas, Rupture of rubber. I. Characteristic energy tearing, J. Polym. Sci. 10 (1953) 291–318.

[2] T. Smith, Dependence of the ultimate properties of a gr-s rubber on strain rate and temperature, J. Polym. Sci. 32 (1958) 99–113.

[3] G. Lake, P. Lindley, Cut growth and fatigue of rubbers. ii. experiments on a noncrystallizing rubber, J. Appl. Polym. Sci. 8 (1964) 707–721.

[4] P. Lindley, Energy for crack growth in model rubber components, J. Strain Anal. 7 (1972) 132–140.

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

[6] A. Thomas, The development of fracture mechanics for elastomers, Rubber. Chem. Technol. 67 (1994) 50–67.

[7] H. Greensmith, Rupture of rubber. X. The change in stored energy on making a small cut in a test piece held in simple extension, J. Polym. Sci. 7 (1963) 993–1002.

[8] A. Bhowmick, S. De, Fractography of Rubbery Materials, Springer Netherlands, 1991.

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

[10] S. Beurrot, B. Huneau, E. Verron, In situ sem study of fatigue crack growth mechanism in carbon black-filled natural rubber, J. Appl. Polym. Sci. 117 (2010) 1260–1269.

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

[12] S. Trabelsi, P.-A. Albouy, J. Rault, Stress-induced crystallization around a crack tip in natural rubber, Macromolecules 35 (2002) 10054–10061.

[13] P. Rublon, B. Huneau, E. Verron, N. Saintier, Beurrot, C. Mocuta, D. Thiaudière, D. Berghezan, Multiaxial deformation and strain-induced crystallization around a fatigue crack in natural rubber, Eng. Fract. Mech. 123 (2014) 59–69.

[14] Q. Demassieux, Structural Changes in the Process Zone of a Cyclic Fatigue Crack in Filled Natural Rubber (Ph.D. thesis), 2016, Paris VI.

[15] K. Bruening, K. Schneider, S.V. Roth, G. Heinrich, Strain-induced crystallization around a crack tip in natural rubber under dynamic load, Polymer 54 (2013) 6200–6205.

[16] J. Kallungal, L. Chazeau, J.-M. Chenal, J. Adrien, E. Maire, C. Barrès, B. Cantaloube, P. Heuillet, F. Wilde, J. Moosmann, T. Weitkamp, Crack propagation in filled elastomers: 3d study of mechanisms involving the filler agglomerates, Eng. Fract. Mech. 274 (2022) 108771.

[17] T. Pottier, M.-P. Moutrille, J.-B. Le Cam, X. Balandraud, M. Grédiac, Study on the use of motion compensation technique to determine heat sources. application to large deformations on cracked rubber specimens, Exp. Mech. 49 (2009) 561–574.

[18] J. Samaca Martinez, X. Balandraud, E. Toussaint, J.-B. Le Cam, D. Berghezan, Thermomechanical analysis of the crack tip zone in stretched crystallizable natural rubber by using infrared thermography and digital image correlation, Polymer 55 (2014) 6345–6353.

[19] J.R. Samaca Martinez, E. Toussaint, X. Balandraud, J.-B. Le Cam, D. Berghezan, Heat and strain measurements at the crack tip of filled rubber under cyclic loadings using full-field techniques, Mech. Mater. 81 (2015) 62–71.

[20] C. Liu, B. Gu, J. Chen, L. Zhang, Y. Lu, F. Li, Thermo-mechanical coupling analysis of edge-cracked rubber specimen focusing on the crack tip: Experimental observation and numerical simulation, Mater. Today Commun. 31 (2022) 103348.

[21] F. Xiang, K. Schneider, G. Heinrich, New observations regarding fatigue crack paths and their fracture surfaces in natural rubber: Influences of r-ratio and pre-load, Int. J. Fatigue 135 (2020) 105508.

[22] S. Charlès, Contribution of Quantitative Imaging To the Mechanical, Thermal and Calorific Characterization of Rubber (Ph.D. thesis), University of Rennes, 2021.

[23] T.T. Mai, K. Tsunoda, K. Urayama, Crack tip stress intensification in strain-induced crystallized elastomer, J. Mech. Phys. Solids 193 (2024a) 105895.
[24] T.T. Mai, T. Yasui, R. Tanaka, Y. Jotatsu, H. Masunaga, T. Kabe, K. Tsunoda, S. Sakurai, K. Urayama, Unraveling non-uniform strain-induced crystallization near a crack tip in natural rubber, Adv. Sci. 11 (2024b) 2307741.

[25] T.T. Mai, T. Yasui, R. Tanaka, Y. Jotatsu, H. Masunaga, T. Kabe, K. Tsunoda, S. Sakurai, K. Urayama, Mechanically modulated strain-induced crystallization around crack tips in nanofiller-reinforced elastomers, ACS Appl. Mater. & Interfaces 17 (2025) 43690-43701.

[26] J.-B. Le Cam, A. Tayeb, S. Charlès, First evidence and characterization of strain-induced crystallization heterogeneity in natural rubber under homogeneous strain states, Polymer 255 (2022) 125120.

[27] P. Vacher, S. Dumoulin, F. Morestin, S. Mguil-Touchal, Bidimensional strain measurement using digital images, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 213 (1999).

[28] S. Charlès, J.-B. Le Cam, Inverse identification from heat source fields: a local approach applied to hyperelasticity, Strain (2020) e12334, http://dx.doi.org/10.1111/str.12334.

[29] J.-B. Le Cam, Strain-induced crystallization in rubber: A new measurement technique, Strain 54 (2018) e12256.

[30] J.B. Le Cam, P.-A. Albouy, S. Charlès, Comparison between x-ray diffraction and quantitative surface calorimetry based on ir thermography to evaluate strain-induced crystallinity in natural rubber, Rev. Sci. Instrum. 91 (2020) 044902.

[31] J.-B. Le Cam, W. Amoako Kyei-Manu, A. Tayeb, P.-A. Albouy, James J.C. Busfield, Strain-induced crystallisation of reinforced elastomers using surface calorimetry, Polymer Testing 131 (2024) 108341.

[32] X. Balandraud, J.B. Le Cam, Some specific features and consequences of the thermal response of rubber under cyclic mechanical loading, Arch. Appl. Mech. 84 (2014) 773-788.

[33] J.-B. Le Cam, Energy storage due to strain-induced crystallization in natural rubber: the physical origin of the mechanical hysteresis, Polymer 127 (2017) 166-173.

[34] R. Osumi, T.T. Mai, K. Tsunoda, K. Urayama, Pronounced effect of strain biaxiality on high-temperature behavior of strain-crystallizing elastomers, Soft Matter 21 (2025) 5080-5088.

[35] S. Toki, T. Fujimaki, M. Okuyama, Strain-induced crystallization of natural rubber as detected real-time by wide-angle x-ray diffraction technique, Polymer 41, 5423-5429.

[36] S. Trabelsi, P.-A. Albouy, J. Rault, Effective local deformation in stretched filled rubber, Macromolecules 36, 9093-9099.

[37] R.W. Schafer, IEEE Signal Processing Magazine, in: What is a savitzky-golay filter? [lecture notes], vol. 28, IEEE, 2011, pp. 111-117.

[38] H. Zhang, A.K. Scholz, J.de. Crevoisier, F. Vion-Loisel, G. Besnard, A. Hexemer, H.R. Brown, E.J. Kramer, C. Creton, Nanocavitation in carbon black filled styrene-butadiene rubber under tension detected by real time small angle x-ray scattering, Macromolecules 45 (2012) 1529-1543.

[39] K. Bruening, In-Situ Structure Characterization of Elastomers During Deformation and Fracture, Springer International Publishing, 2014.

[40] E. Euchler, K. Schneider, S. Wiessner, R. Bernhardt, G. Heinrich, S.V. Roth, T. Tada, Cavitation damage in tire rubber materials, in: B. Huneau, J.-B. Le Cam, Y. Marco, E. Verron (Eds.), 11th European Conference on Constitutive Models for Rubber, Taylor & Francis Group, 2019, pp. 87-91.

[41] F. Xiang, K. Schneider, M. Schwartzkopf, G. Heinrich, Competition between strain-induced crystallization and cavitation at the crack tip of unfilled and carbon black-filled natural rubber, Macromolecules 55 (2022) 10682-10693.