# Application of full-field measurements and numerical simulations to analyze the thermo-mechanical response of a three-branch rubber specimen

Xavier Balandraud, Evelyne Toussaint, Jean-Benoit Le Cam, Michel Grédiac, Ronny Behnke, Michael Kaliske

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Xavier Balandraud, Evelyne Toussaint, Jean-Benoit Le Cam, Michel Grédiac, Ronny Behnke, et al. Application of full-field measurements and numerical simulations to analyze the thermo-mechanical response of a three-branch rubber specimen. Constitutive Models for Rubber VII, 2011, Dublin, Ireland. hal-05432583

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# Application of full-field measurements and numerical simulations to analyze the thermo-mechanical response of a three-branch rubber specimen

X. Balandraud, E. Toussaint, J.-B. Le Cam & M. Grédiac

Laboratoire de Mécanique et Ingénieries (LaMI), Université Blaise Pascal (UBP),
Institut Francais de Mécanique Avancée (IFMA), BP10448,F-63000 Clermont-Ferrand, France

R. Behnke & M. Kaliske

Institute for Structural Analysis (ISD), Technische Universität Dresden, Germany

**ABSTRACT:** The study deals with the characterization of the thermo-mechanical behavior of rubber. First of all, experiments are performed with a three-branch-shaped rubber specimen previously proposed by Guélon et al. (2009) used for this purpose. This heterogeneous test induces simultaneously the three types of strain states classically employed to identify mechanical properties of rubber (uniaxial and equibiaxial tension, and pure shear), as well as some intermediary states. Recent experimental studies that investigated such heterogeneous tests only considered the deformation field, but not the corresponding thermal field. This aim of the present study is to push forward the idea by measuring both the displacement and thermal fields during this test, and to compare the experimental results with their numerical counterparts predicted by a thermo-mechanical numerical simulation performed at ISD using a novel model for thermo-viscoelasticity – an improved version of the extended-tube model presented by Kaliske & Heinrich (1999) in the case of isothermal hyperelasticity. The comparison between predicted and experimental fields, as well as the correlation between thermal and kinematic fields, will be discussed in the presentation.

## 1 INTRODUCTION

Classically, constitutive relations for rubber are identified and validated from several tests, assumed to be homogeneous, i.e. uniaxial tension, pure shear and equibiaxial tension. In a previous work, (Guélon et al. 2009) have proposed a new method to characterize and to identify the constitutive parameters in the framework of hyperelasticity. For this purpose, the authors used a three-branch sample under tension that induces simultaneously the three homogeneous and intermediate deformation states. In the present study, both thermal and kinematic full field measurements are performed to characterize the thermo-mechanical response of the three-branch-specimen and to compare the experimental results to the predictions of a thermo-viscoelastic model developed at ISD.

## 2 PRESENTATION OF THE THREE-BRANCH TEST

The test was carried out with a MTS 858 Elastomer Test System testing machine. Its loading capacity is

± 15 kN and the loading cell is ± 1 kN. In order to apply biaxial loading, i.e. to generate equibiaxial tension and pure shear at the sample center, a new tensile apparatus has been designed to be adapted to the uniaxial tensile machine (Guélon et al. 2009). In practice, the two vertical branches of the sample were fastened to the grips of the conventional testing machine, whereas the branch perpendicular to the previous ones was fastened in the grip.

The sample geometry, which corresponds to a three-branch sample, was 2 mm thick and 52 mm high, and the branches were 20 mm in width. It is presented in Figure 4. The bottom branch was clamped, the loading was carried out in two steps by applying prescribed displacements along the axis of the two other perpendicular branches. The first step consisted in applying a 12 mm displacement along the y-direction and a 30 mm displacement along the x-direction. During this step, images were stored only with a CCD camera. This first step was necessary to generate fields which were sufficiently heterogeneous on the sample surface. The configuration obtained at the end of the first step is called intermediate config-
uration in the following. The second step consisted in applying a 47 mm displacement along the y-direction. The displacement rate was equal to 15.66 mm/s. At the end of the test, the corresponding global stretch ratios were 2.13 and 1.85, respectively. During this second step, images were stored by both the CCD and IR cameras. Each camera was positioned on each side of the specimen.

## 2.1 Measurement of kinematic fields

During the test, images for kinematic fields were stored using a cooled 12-bit dynamic Sensicam camera. The images correspond to different stretch ratio levels. The charge-coupled device (CCD) of the camera had $1.4 \times 10^6$ connected pixels ($1376 \times 1040$). Uniform lighting at the sample surface was ensured by a set of fixed lamps. Digital Image Correlation (DIC) was used to measure the displacement field on the sample surface. DIC is a full-field measurement technique developed at the beginning of the 1980s (Sutton et al. 1983). This method consists in matching, before and after displacement, the brindled pattern in a physical part of the observed surface of the specimen, called “Region Of Interest” (ROI). The SeptD software was used (Vacher et al. 1999) for this purpose. This optical technique enables us to reach a resolution of 0.03 pixels, corresponding to $2.9 \ \mu\text{m}$ and a spatial resolution (defined as the smallest distance between two independent points) of 10 pixels, corresponding to $975 \ \mu\text{m}$. A set of sub-images is considered to determine the displacement field of a given image with respect to a reference image. This set is referred to as “Zone of Interest” (ZOI). A suitable correlation function was used to calculate the displacement of the center of a given ZOI between two images captured at different stages of an experiment. To improve image contrast, white paint was sprayed on the sample surface before testing. The DIC technique is well-suited for measuring large strains (Sasso et al. 2008, Chevalier et al. 2001).

## 2.2 Measurement of thermal fields

Temperature measurements were carried out with a Cedip Jade III-MWIR infrared (IR) camera, which features a local plane array of $320 \times 240$ pixels and detectors with a wavelength range of 3.5 - 5 µm. The integration time was 1500 µs and the acquisition frequency was set to 150 Hz. The noise equivalent temperature difference (NETD) of the measurement, i.e. the thermal resolution, is equal to about 20 mK for the used temperature range 5 - 40 degrees C. In order to ensure that the internal temperature of the camera was optimal for performing the measurements, it was set up and switched on for one hour before the experiment. The stabilization of the temperature in the camera was necessary to avoid any drift of the measurements during the test.

<figure><img src="image_2.png" /><figcaption>Figure 1. Rheology of the viscoelastic material model, $\bar{F} = \bar{F}^e \bar{F}^i$.</figcaption></figure>

# 3 EXTENDED TUBE MODEL

## 3.1 Isothermal hyperelasticity

The extended tube model, introduced by Kaliske & Heinrich (1999), is a material model which is well suited for the characterization of the mechanical material behavior of particle-reinforced rubber. The isochoric free energy function $\bar{\Psi}_0$ can be derived from statistical mechanics and represents the isochoric material response for the case of isothermal hyperelasticity. A comparison of several material models for hyperelasticity in Marckmann & Verron (2006) highlights the capacity of the extended tube model to fit the experimental data in uniaxial, equibiaxial as well as pure shear strain states. For this reason, the extended tube model is chosen as the starting point for the derivation of a finite visco-elastic material model which is briefly described in the next section.

## 3.2 Isothermal viscoelasticity

For many practical applications, the material response of rubber can be assumed to be purely hyperelastic and often volume changes under deformation are neglected, i.e. the rubber is regarded as incompressible. However, experiments with rubber test samples reveal viscous behavior, time dependent (aging) and load dependent (Mullins effect, damage effects) material characteristics. In addition, dissipative phenomena cannot be taken into account by a purely hyperelastic material model. As a result, the aforementioned hyperelastic extended tube model is further developed to a finite non-linear visco-elastic material model. Its rheology is shown in Figure 1. The rheological Zener element contains an elastic (equilibrium - EQ) and viscous (non-equilibrium - NEQ) branch which is modelled by a Maxwell element. In order to describe finite viscoelasticity, a volumetric-isochoric split of the free energy $\bar{\Psi}_0$ is introduced where the deformation gradient $\bar{F}$ is multiplicatively decomposed into a volumetric $\bar{F}_{vol}$ and isochoric $\bar{F}$ contribution, $\bar{F} = \bar{F} \bar{F}_{vol}$ and $\bar{F} := J^{-\frac{1}{3}} \bar{F}$ with $J := \det \bar{F}$. The isochoric part of the deformation gradient is further split into an elastic and inelastic part, $\bar{F} = \bar{F}^e \bar{F}^i$. The viscosity of the material is assumed to be purely isochoric. The dashpot is governed by the effective creep
rate

$$ \dot{\gamma} := \dot{\gamma}_0 \left[ \frac{\lambda}{\lambda_e} - 1 \right]^c \left( \frac{\tau_v}{\hat{\tau}} \right)^m, \quad \text{with} \quad (1) $$

$$ \lambda := \sqrt{\frac{\bar{I}_1}{3}}, \quad \lambda_e := \sqrt{\frac{\bar{I}_1^e}{3}}, \quad (2) $$

where $\dot{\gamma}_0$ denotes the reference effective creep rate, $\frac{\lambda}{\lambda_e}$ is the inelastic network stretch and $c \le 0$ as well as $m > 0$ are model parameters which control the inelastic network kinematics and the energy activation, respectively. The first invariants can be calculated by $\bar{I}_1 = \text{tr} \bar{C}$ and $\bar{I}_1^e = \text{tr} \bar{C}_e$ from the elastic Cauchy-Green tensor $\bar{C}_e$. The stress term $\tau_v$ can be obtained from the isochoric viscous stress $\tau_{\text{iso}}^v$,

$$ \tau_v := \frac{\sqrt{\tau_{\text{iso}}^v : \tau_{\text{iso}}^v}}{\sqrt{2}}, \quad \tau_{\text{iso}}^v := \mathbb{P} : \bar{\tau}^v. \quad (3) $$

The algorithmic setting is provided in detail in Dal & Kaliske (2009) and Behnke et al. (2011).

### 3.3 Thermo-mechanical coupling

During cyclic loading, elastomeric material shows also a significant heat build up which is due to dissipative phenomena like irreversible network rearrangements, micro cracking or viscous characteristics. All the phenomena of mechanical energy dissipation lead to a critical heating of cyclic loaded elastomeric components which can cause alteration and deterioration. Due to their entropy and high extensibility, elastomers are also characterized by an observable temperature change upon deformation (heating during stretching, cooling during unloading). In the following, mainly these latter effects are studied.

The concept of the thermo-mechanical modelling follows the approach presented in Reese & Govindjee (1998). The kernel of this approach consists of the evaluation of the free energy functions of the elastic and viscous branch as well as the inner energy $e_0$ and the heat capacity $\bar{c}$ at a reference temperature $\Theta_0$. At the Gauss point level, the stresses are always evaluated using the reference potentials $\bar{\Psi}_0^e$ and $\bar{\Psi}_0^v$ which are marked therefore by a subscript 0. The temperature dependency of the material and its properties are then constructed by multiplying the reference energy and heat capacity terms by dimensionless temperature functions $f(\Theta)$, $t(\Theta)$,

$$ \Psi = f_{\text{EQ}}(\Theta) [\bar{\Psi}_0^e(\bar{b}_i) + U_0(J)] + t_1(\Theta) e_{0\text{EQ}} + f_{\text{NEQ}}(\Theta) [\bar{\Psi}_0^v(\bar{b}_{e_i})] + t_2(\Theta) \bar{c}. \quad (4) $$

$\Theta$, $\bar{b}_i$ and $U_0$ are the absolute temperature, the eigenvalues of the left Cauchy-Green tensor $\bar{b}$ and the volumetric free energy function, respectively. For the FE implementation, the governing equations of coupled

thermo-mechanics (balance of linear and angular momentum, the first and second law of thermodynamics in form of the balance of energy and the dissipation inequality as well as a constitutive law for the heat flux) are written in their weak form. For more details, we refer to Reese & Govindjee (1998).

## 4 MODEL PARAMETER IDENTIFICATION

For parameter identification, a cyclic uniaxial extension test is carried out at LaMI. A rubber specimen of cuboidal shape ($l = 22$ mm, $b = 4$ mm, $t = 2$ mm) is placed in a testing machine. The mechanical material response is characterized by the reaction force which is recorded during the test. The test consists of a sawtooth wave motion of the upper grip of the testing machine. At the beginning, 3 cycles with a positive strain amplitude of $\varepsilon = 1.0$ are carried out. They are followed by 3 cycles with a positive strain amplitude of $\varepsilon = 1.75$. The absolute value of the displacement rate during the test is kept constant (500 mm/min). The thermal material response – which is dominated by the thermo-elastic coupling for this loading case – is measured by an IR camera during the experiment. The thermal material properties are physically measurable quantities (heat conductivity, heat capacity at reference temperature) which are taken from Pottier et al. (2009).

The recorded mechanical and thermal data are used in the following for the parameter identification. For this automated task, a parameter identification algorithm has been developed at ISD. It is based on least square minimization of the measured and computed test data. The result is one locally optimized parameter set, specified in Behnke et al. (2011), which is used for the simulation of the three-branch rubber specimen.

## 5 EXPERIMENTAL RESULTS

### 5.1 Displacement distribution measured by DIC

Figure 2 shows the stretch states as functions of the $(X, Y)$ coordinates in the reference configuration using a suitable color code at the end of the first step.

The Equibiaxial Tension (ET), Pure Shear (PS) and Uniaxial Tension (UT) states correspond to the blue, green and red colors, respectively. Intermediate states are defined by a color that is a weighted average of two colors, namely, blue and green for the states between ET and PS, green and red for the states between PS and UT. Figure 2 shows that the central zone of the specimen is mainly in an ET state. The zones corresponding to the three branches are mainly in a UT state. The PS zone is located on a ring-shaped zone located between the ET and UT zones.

A typical view of the displacement fields obtained at the end of the second step is shown in Figure 3 a) and b). The displacement fields during the second
<figure><img src="image_3.png" /><figcaption>Figure 2. Stretch states in the gauge section.</figcaption></figure>

step are necessary to establish the correspondence between the geometry of the coupon at any stage of the loading and the reference geometry. It will then be possible to correctly analyze and process the temperature maps at different loading amplitudes. This is explained in the following section.

## 5.2 Thermal distribution measured by IR thermography

Since the specimen undergoes large deformations, any given material point at its surface clearly moves during the loading cycle, while the zone captured by the detector array of the IR camera remains unchanged. The smaller the distance to a fixed grip, the greater the amplitude of this displacement. Consequently, a given pixel of the IR detector matrix does not correspond to the same material point plotted during the deformation of the specimen surface. The objective is now to explain how to track the material points.

Constructing a suitable motion compensation technique in the context of mechanical testing and temperature measurement using an IR camera has only seldom been addressed in the literature (Pottier et al. 2009). In the present study, the kinematic field measurement is used to track the material point in the present study. A suitable algorithm was developed.

Two configurations must be distinguished: a reference geometry and the current one. For the present application, the reference geometry corresponds to the configuration obtained at the end of the first step of the loading, the so-called intermediate configuration. Considering kinematic and thermal fields at a same time, the suitable compensation technique is built in

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Figure 3. Experimented displacements and temperature variation; a) horizontal displacement; b) vertical displacement; c) θ-temperature variation.</figcaption></figure>

practice with the following procedure:

* kinematic fields are calculated at the center of each ZOI. In the present case, the number of ZOIs is lower than the number of points where temperatures are measured. The first step consists in interpolating the values of the displacements on the thermal grid;
* using these values of displacement, temperatures are then interpolated on a regular grid defined on the reference configuration.

Then temperature variations $\theta$ are obtained by subtracting the temperature field corresponding to the reference configuration. Figure 3 c) gives the temperature variation $\theta$ at the end of the loading procedure. The temperature variation field obtained highlights a high temperature gradient in the two branches between the moving grips. The temperature variation reaches 3 degree C in the zone mainly submitted to uniaxial tension (see Figure 2).

## 6 PREDICTION OF THE THERMO-MECHANICAL RESPONSE

The aim of the simulation is twofold. First, the identified model parameters can be validated by a non-homogeneous test. Second, the simulation capacity of the thermo-mechanical coupled material model and its finite element implementation are demonstrated and used to predict the experimental results. The FE mesh, the prescribed displacements and the boundary conditions are illustrated in Figure 4. The FE mesh consists of 384 isoparametric finite solid and 436 finite surface elements to model the heat conduction through the surface. Only one half of the specimen is modelled with one layer of solid elements in z-direction. The simulation results at the end of the loading are illustrated in Figure 5. They are projected to the intermediate configuration by an in-house post-processing algorithm. The temperature variation field prediction is in good agreement with the temperature variation field measurement.

<figure><img src="image_7.png" /><figcaption>Figure 4. FE mesh and boundary conditions, $\Theta_0 = 292$ K.</figcaption></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Figure 5. Simulated displacements and temperature variation; a) horizontal displacement; b) vertical displacement; c) $\theta$-temperature variation.</figcaption></figure>

## 7 CONCLUSIONS

This study deals with the characterization of the thermo-mechanical behavior of rubber. From an experimental point of view, the heterogeneity induced by the three-branch test has been characterized related to the three classical homogeneous deformation

states. Coupled kinematic and thermal full field measurements allowed us to associate at each point of the sample surface a deformation state and a temperature variation. This experimental test has been simulated using the thermo-viscoelastic model developed at ISD. Numerical simulations satisfactorily predict the thermo-mechanical response measured.
# ACKNOWLEDGEMENTS

Part of the research reported herein is carried out within the DFG research unit 597. The coauthors R. Behnke and M. Kaliske would like to acknowledge gratefully the financial support of the DFG under grant KA 1163/3-2.

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Dal, H. & M. Kaliske (2009). Bergström-Boyce model for non-linear finite rubber viscoelasticity: Theoretical aspects and algorithmic treatment for the FE method. *Computational Mechanics* 44, 809–823.

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