# Modulated quasi-static tensile testing for advanced characterization of elastomer mechanics

Mahmoud Hussein, Daniel Mathieu, Gildas l'Hostis, Bernard Durand

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Mahmoud Hussein, Daniel Mathieu, Gildas l'Hostis, Bernard Durand. Modulated quasi-static tensile testing for advanced characterization of elastomer mechanics. Polymer Testing, 2025, 147, pp.108825. 10.1016/j.polymertesting.2025.108825. hal-05405324

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# Modulated quasi-static tensile testing for advanced characterization of elastomer mechanics

Mahmoud Hussein*, Daniel Mathieu, Gildas L'hostis, Bernard Durand

Laboratoire de Physique et Mécanique Textiles, Université de Haute Alsace, ENSISA, 11 rue des Frères Lumières, 68093, Mulhouse, Cedex, France

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

## ARTICLE INFO

**Keywords:**

Modulated tensile testing-Digital image correlation-thermomechanical analysis-viscoelasticity

## ABSTRACT

Polymers exhibit complex mechanical behavior. Rubber or similar materials are known to display both visco-elasticity and nonlinear elasticity. However, quasi-static mechanical tests alone are insufficient for capturing the viscous component of their response. While Dynamic Mechanical Analysis (DMA) provides valuable insights, it is inadequate to characterize the material's behavior under large deformations fully. A hybrid approach combining a quasi-static tensile test with modulation is proposed to address these limitations. During these tests, deformation fields obtained via Digital Image Correlation (DIC) can be complemented by infrared thermography. The experimental setup is based on a modified DY22 universal testing machine. Tests were performed on a rubber band designed for sports applications. The mechanical results, including the load-elongation relationship and field measurements, reveal distinct differences between the quasi-static and modulated components. These differences are particularly evident in the evolution of the elastic modulus and Poisson's ratio as a function of elongation. The measurements provide detailed insights into the material's mechanical response at each loading stage. The results demonstrate that integrating modulated testing techniques with thermomechanical analysis allows for a more comprehensive characterization of materials for engineering applications.

## 1. Introduction

Polymers are essential in modern engineering, as well as in medical and industrial applications. Since their widespread adoption in the twentieth century, scientific interest in these materials has grown significantly. Their advantages over conventional materials, such as lower production costs and enhanced mechanical properties when reinforced with fibers, have been key drivers of this interest.

Understanding and accurately predicting polymer mechanical behavior is crucial for optimizing their performance in real-life applications. Mechanical behavior, which describes how a material responds to external forces, encompasses stress and strain, elasticity, plasticity, toughness, and failure modes. High performance in load-bearing structures is particularly critical in safety-sensitive industries such as aerospace and medical applications.

Tensile, compression, and shear tests are widely used conventional mechanical testing methods for evaluating the fundamental properties of metals, ceramics, and polymers. These tests provide essential data on key mechanical parameters, including yield strength, elastic modulus, and ultimate tensile strength, which defines the maximum stress a

material can withstand before failure. For instance, in a tensile test, the material is subjected to a uniaxial tensile load until fracture, allowing for the assessment of both strength and ductility. An important factor influencing mechanical behavior is the loading rate. Unlike purely elastic materials, polymers exhibit a time-dependent response due to their viscoelastic nature, even in the reversible deformation regime. This dissipative component persists over a broad temperature range, making the loading speed a critical parameter in mechanical characterization. The influence of the material's viscous component can be examined by varying the strain rate. Various experimental approaches exist, ranging from quasi-static tensile testing using universal testing machines to high-strain-rate impact tests under extreme loading conditions [1–6].

Due to their viscoelastic nature, the mechanical response of polymers to stress or strain depends not only on the applied force but also on the loading rate and the duration of the applied force. As a result, polymers exhibit time- and rate-dependent phenomena such as creep, characterized by progressive deformation under constant stress, and stress relaxation, which refers to the gradual decrease in stress under constant strain. These behaviors are not adequately captured by standard static tests, such as tensile and compression tests, which are typically

<footnote>* Corresponding author.
E-mail address: Mahmoud.hussein@uha.fr (M. Hussein).</footnote>
performed under steady, constant strain-rate conditions.

A comprehensive understanding of the mechanical behavior of polymers requires the application of periodically varying loads. Dynamic Mechanical Analysis (DMA) is the most widely employed technique. In this method, oscillatory forces are applied to the material at different frequencies and temperatures, enabling the measurement of both elastic and viscous responses. By adjusting the excitation frequency, the storage modulus, which characterizes the elastic behavior, and the loss modulus, which quantifies viscous dissipation, can be determined. These parameters are essential for evaluating the long-term performance of polymers under cyclic loading conditions [7]. DMA can be conducted across multiple scales, from micrometers to centimeters and beyond. Furthermore, vibrating microprobes have been utilized to investigate the surface properties of polymers at microscopic scales, providing additional insights into their mechanical behavior [8].

The available DMA equipment in our laboratory cannot characterize viscoelastic parameters under large deformations. To address this limitation, a hybrid approach combining quasi-static stress and DMA has been under development for several years [9]. This method involves superimposing a low-amplitude modulation onto the quasi-static stress, allowing for the temporary release of stresses within the material. This approach enables a more comprehensive characterization of the viscoelastic behavior under conditions that better reflect real-world mechanical loading.

Unlike DMA, where the analyzed data are limited to the force generated by the sample and the imposed deformation, the proposed approach aims to characterize the material response under large deformations on sizable samples. This necessitates measuring displacement and deformation fields on the material's surface. These fields are conventionally obtained using Digital Image Correlation (DIC), a technique that has gained widespread adoption due to advancements in digital imaging and data processing technologies [10]. In recent years, DIC has been increasingly utilized for the study of stress-strain behavior [11–13], residual stress analysis [14–16], and fracture propagation [17, 18]. Its ability to provide full-field, non-contact measurements makes it a valuable tool for characterizing complex mechanical responses in polymeric materials.

In the case of homogeneous deformation, the applied force is sufficient to infer the stress state. However, additional measurements are required to estimate the associated stress distribution when inhomogeneous deformation. One effective approach is to measure the temperature field on the specimen, as thermal variations can provide insights into the material's mechanical response [19]. Thermal cameras, also referred to as infrared cameras, are used to capture temperature distributions during mechanical testing, providing complementary data. These thermal images show the effects of mechanical loading on heat generation and dissipation within the material, as thermo-mechanical coupling occurs under specific conditions. Many materials, particularly polymers, undergo localized heating or cooling due to mechanical deformation. This phenomenon is caused by internal friction, viscoelastic dissipation, or plastic deformation, which can influence the material's mechanical properties. Changes in stiffness, strength, and fracture behavior may result from these thermo-mechanical interactions. By monitoring real-time temperature variations, a more detailed analysis of these coupled effects can be conducted, allowing for a deeper understanding of material behavior under various loading conditions.

The combination of these technologies enables a comprehensive understanding of material behavior. For instance, during a mechanical test with modulation, real-time deformation can be captured by an optical camera, while a thermal camera records simultaneous temperature variations. This dual approach provides deeper insight into the material's response under complex loading conditions and environmental influences, allowing for a more thorough characterization of its mechanical and thermal behavior.

This study aims to develop and validate a modulated quasi-static

tensile testing methodology for accurately characterizing the mechanical behavior of polymer materials, with a specific focus on rubber. Traditional mechanical tests, such as standard tensile testing and Dynamic Mechanical Analysis (DMA), provide valuable insights into material properties but often fail to capture nonlinear, time-dependent, and thermomechanical effects under realistic loading conditions. A hybrid approach is proposed to address this limitation, integrating controlled oscillatory loading, Digital Image Correlation (DIC) for full-field strain measurements, and infrared thermography to assess heat generation and dissipation associated with viscoelastic deformation.

This methodology enables a more detailed investigation of the relationship between strain evolution, stress response, and temperature variations. It facilitates the quantification of both dynamic and quasi-static mechanical properties, including Poisson's ratio and elastic modulus. Furthermore, energy dissipation mechanisms are examined by correlating thermal and mechanical data. The ultimate objective is to establish a robust testing framework that enhances understanding of polymer behavior under cyclic and quasi-static loading conditions, providing valuable insights for biomechanics, soft robotics, and industrial engineering applications.

The modified tensile machine used for testing is presented in the following sections. Details of the experimental setup, including the arrangement of the cameras, are described. The data processing methods are then explained. In the section on experimental results, the thermo-mechanical behavior of rubber is examined using advanced techniques such as Digital Image Correlation (DIC). Finally, the conclusions summarize the key findings of this study and discuss future research directions, emphasizing the importance of improved testing methodologies for optimizing polymers and polymer-containing materials.

## 2. Material and methods

### 2.1. The modified universal testing machine

Modifying the tensile testing apparatus began with repurposing an old, rarely used Adamel Lhomargy DY22 dynamometer. A significant upgrade was necessary to meet our requirements. The redesign focused on both mechanical and software aspects. Mechanical modifications aimed to optimize the drive system within the modulation unit, while software improvements ensured precise control and synchronization during experiments.

A modulated tensile testing system was implemented on the old Adamel Lhomargy DY22 universal testing machine by integrating a modulation mechanism into the conventional tensile setup. This device, illustrated in Fig. 1, represents the modulation mechanism, which is purely mechanical. Designed by one of the authors of this work, its components were dimensioned using finite element analysis to ensure that the generated motion remains unaffected by the forces exerted by the specimens during testing. The mechanism is driven by a brushless motor (Fig. 2) with a power of 0.79 kW, a torque of 2.5 Nm, and a nominal speed of 3000 rpm. The motor is controlled by a SERVOSTAR 300 motor drive, which regulates its rotational speed. The speed is adjusted dynamically by sending a reference speed signal from LabVIEW to the motor drive, ensuring precise synchronization with the experimental requirements.

The modulation system, Fig. 1, consists of a circular eccentric cam with an offset of 3 mm, enclosed within an oblong slot of a large control wheel angularly adjustable. This part, angularly positioned outside the modulation unit's casing, converts the cam's rotary motion into linear displacement. As the cam rotates, it alternately pushes and pulls a mobile axis, transmitting the motion to the lower grip that holds the specimen during testing. This setup effectively transforms the motor's continuous rotation into a sinusoidal translational movement with an adjustable amplitude tailored to the experimental requirements.

The adjustment is performed by operating a gear and locked in place
<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic diagram of the Modulation mechanism.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Modulation frequency control electrical panel and the modulation device.</figcaption></figure>

during operation by a double ratchet wheel located outside the mechanism. Each wheel offers fifty possible positions. Three full turns of the gear are required for the internal components to return to their initial configuration. The sinusoidal movement of the clamp pusher varies in amplitude between 0 and 3 mm for a quarter turn of the large control wheel. This results in 38 unique values for the mechanical adjustment of the translation amplitude of the lower clamp. These values follow a sinusoidal progression, as described by the following equation.

$$Amplitude_{selected} = Amplitude_{max} \cdot cos(\theta_{adjusted}) \qquad Eq. 1$$

Where: the adjusted angular position is defined by the position of the external adjustment gear:

$$\theta_{adjusted} = 2\pi \cdot \frac{step}{150} \qquad Eq. 2$$

This formulation ensures precise and repeatable control over the modulation amplitude.

A tracking plate was integrated into the system using a structure attached to the original machine to enable continuous tracking of the sample during elongation. This plate, mounted on slides, moves at half the speed of the moving crosshead, ensuring that the center of the

sample remains in view at all times. This motion is achieved through a pulley and cable system. A height adjustment mechanism is included to align the plate with the fixed clamp, allowing precise targeting of the sample center regardless of the specimen's length. This adjustment is facilitated by a screw-nut mechanism, which maintains the steel cable tension and enables fine positioning of the plate. Before measurements begin, the position can be verified and precisely adjusted using a graduated target observed through the optical camera. This setup provides a cost-effective solution for monitoring areas of interest, avoiding the complexity and expense of fully automated tracking systems. The configuration is specifically designed to capture thermomechanical effects in the samples, simultaneously recording both thermal and displacement fields within the region of interest. A schematic diagram of the system is presented in Fig. 3.

The sample's infrared radiation is first reflected by a semi-transparent mirror coated with a thin layer of gold. A secondary mirror then redirects the reflected light to a thermographic camera. The gold-coated mirror allows visible light transmission with a specific attenuation coefficient, provided that (as shown in Fig. 4) the gold deposition is properly executed. Since the mirror is fabricated in the laboratory using a metallization machine, the parameters of this process
<figure><img src="image_5.png" /><figcaption>Fig. 3. Machine configuration.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Influence of metallization conditions on optical attenuation and infrared reflection.</figcaption></figure>

must be carefully defined. The emission current, the metallization time, and the thickness of the deposited gold layer serve as fundamental parameters for ensuring proper fabrication.

The objective is to achieve a metallic deposition thickness with a reflection coefficient close to 100 % while allowing sufficient visible light transmission. To measure the reflection coefficient at the angle of incidence used and for the wavelength detected by the thermographic camera, a black body is created using a black-painted metallic piece heated to an initial temperature between 50 and 60 °C. This temperature range ensures that the object remains visible to the camera, regardless of the quality of the gold deposition.

A miniature temperature probe, positioned close to the surface observed by the camera, is used to monitor the object's temperature as it cools naturally. Simultaneously, the thermographic camera records the temperature. Since the reflection coefficient is independent of light intensity, waiting for the complete cooling of the metallic body was deemed unnecessary. Thus, the lowest temperatures for these measurements remain at least 10 °C above the laboratory temperature, which is set according to NF EN ISO 139/A1 - September 2011 (20.0 ± 2 °C). The collected data indicate that the metallization duration has little impact on infrared reflection. However, this parameter significantly influences the attenuation of transmitted light, which is critical for optical camera usability. As a result, a deposition duration of 1 min was selected, as it provides a reliable metallization process, ensuring a high reflection coefficient for the thermographic camera while maintaining moderate attenuation of transmitted light. After manufacturing the final semi-reflective mirror, a calibration of the temperature

perceived by the thermographic camera was undertaken.

## 2.2. The measurement devices

An optical camera (Dalsa) with a maximum acquisition frequency of 100Hz was used and integrated into the tensile testing. The camera's resolution is 1400x1024 pixels. Image acquisition is done using the Hiris software from RD Vision, which allows the recording frequency to be adjusted. For this work, this frequency was fixed at 19 Hz, based on the experimental conditions.

A requirement for using the camera images for displacement field measurement was to quantify the distortion that its optical system and its position relative to the specimens could induce. Calibration was performed using the MATLAB Camera Calibrator application. This application facilitates the accurate estimation of intrinsic parameters. Instead of a standard checkerboard, a grid pierced with circular holes was used as the calibration target. This target consisted of 3 columns × 4 rows, pierced with holes 2 mm in diameter and spaced 3 mm apart. The pattern was placed in the camera field of view, and several images were captured at different orientations and positions within the camera field of view to ensure reliable calibration. After analyzing the results, a standard two-parameter model appeared sufficient to obtain an average reprojection error of less than 0.2 for almost all calibration images. The obtained parameters showed the existence of a low negative radial distortion. This resulted in a displacement not exceeding 2 % of the theoretical position on the edges of the image. This procedure also made positioning the camera relative to the specimens on the moving stage
easier. The intrinsic parameters of the optical camera were subsequently considered in the pre-processing of the optical images.

Our setup uses a thermal camera FLIR (SC3000) with a resolution of 320x240 pixels. The capture card used with this camera can record 50 frames per second in full definition. This thermal camera permits, with mechanical tests in modulation, measurement of the variations in temperature on the surface of a sample when it is subjected to loading. Such variations often express dissipative mechanisms of energy related to internal friction, material damage, or fatigue. Thanks to the thermal camera, it becomes possible to observe the temperature distribution on the sample in real-time and thus obtain fundamental information about the thermomechanical phenomena connected with material deformation.

Measurement of the imposed modulated linear displacements requires real-time accuracy. This was done with a Linear Variable Differential Transformer (LVDT) sensor. This device can track the changes accurately and sensitively, as environmental factors such as temperature or electromagnetic interference do not affect its workings.

Both cameras capture the thermal and optical responses during tests and are integrated with a trigger mechanism. This trigger ensures that recording starts simultaneously in all systems, keeping the visual and thermal data temporally aligned with the mechanical parameters. The force sensor, which is connected to the system, measures the applied force during testing. All elements, including the motor drive, LVDT sensor, cameras, and force sensor, are centrally controlled and synchronized using LabVIEW software. LabVIEW will be the integrating hub to which all these components shall be connected, their timing and functionality controlled, and communications and data logging will be made for further analysis.

## 2.3. The material

The specimens tested in this study were obtained from elastic bands composed of 100 % natural rubber latex, initially sold for sports training. Each band has a length of 1.04 m, a width of 13 mm, and a thickness of 4.8 mm. The manufacturer of this commercial product specifies the resistance level as a force in kgf for an additional 200 % elongation relative to the original length. At this stage, no detailed information is available regarding the formulation of the tested rubber, including the type of crosslinking system or whether additives such as fillers are present. To address this uncertainty, thermogravimetric analysis (TGA) was carried out. This analysis was carried out in two stages with a temperature increase rate of 10 °C per minute up to a temperature of 800 °C. The first stage consisted of carbonizing the sample in a nitrogen atmosphere. Then air was introduced for the rest of the temperature increase. This allowed the non-volatile carbon elements to be burned. A third, less defined peak is observed at 478 °C, which can be associated with the late decarbonization of the filling elements or the dehydration of their mineral part. Prior to this final degradation stage, the filler content is estimated at around 25 %, while the residual ash content is close to 20 %, suggesting the presence of mineral fillers or reinforcing agents in the composition. These results indicate that the material is not a purely unfilled rubber but contains a significant amount of fillers. However, further analyses would be necessary to identify the nature of these fillers and to characterize the crosslinking system used. Tensile tests were conducted until failure to gain a better preliminary understanding of the mechanical properties of this material. Fig. 5 presents a typical stress-strain response of this material under uniaxial tension along the length of the band. A Young's modulus of 2.13 MPa was determined using the initial configuration as a reference. The breaking point occurs at 74.5 N, corresponding to an elongation of 440 % relative to the initial length.

The selected samples were chosen for their mechanical behavior, which is often initially characterized as hyper-elastic. However, this material exhibits high thermo-sensitivity and a hysteretic cyclic response, providing notable damping properties in practical

applications. The energy effects (primarily heat generation) associated with elastomer deformation have been a research subject since the industrial introduction of natural rubber, with studies conducted by pioneering researchers. Notably, Joule [34] investigated entropic effects, while Flory [33] and Anthony [32] explored the well-known thermo-elastic inversion phenomenon. This phenomenon causes elastomers to contract when subjected to a temperature increase beyond a certain stress threshold, which is opposite to the behavior of most conventional materials.

This study analyzes the material's mechanical, thermal, and energetic response under modulated quasi-static loading. This energy-based approach will help determine whether the microstructural changes induced by deformation are coupled or independent and whether they exhibit dissipative behavior to varying degrees.

As shown in Fig. 6, a random speckle pattern was applied to the samples to track displacement on the material's surface and accurately calculate deformations. Since the samples are made of rubber, they are not reflective, and only a black graphite-based coating (Leit-und Gleichlack) was used to create this pattern using an airbrush spraying technique, ensuring a fine and evenly distributed random pattern. The graphite-based coating was chosen based on GOM company recommendations, as it provides strong adhesion to rubber surfaces, high contrast under optical imaging, and low reflectivity, all of which enhance Digital Image Correlation (DIC) accuracy.

The initial distance between the clamps was set to 110 mm. During the tests, the optical camera observed a region near the center of the specimen. This image captures a surface area of 45 × 33 mm, corresponding to a portion of the sample that is 400 pixels wide and 1024

<figure><img src="image_7.png" /><figcaption>Fig. 5. Sample stress-strain curve.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Sample coated with a speckle pattern.</figcaption></figure>
pixels high. Within this observation window, a Region of Interest (ROI) was defined for digital image correlation (DIC) to track surface deformation. The ROI was continuously adjusted to remain within the camera's field of view, ensuring accurate strain measurements. The ROI was updated every 20 s to capture deformation progression, allowing for sequential tracking of surface changes over time.

## 2.4. Detailed conditions for modulated quasi-static tensile test

A sinusoidal modulation is superimposed on a base traverse speed in the modulated quasi-static tensile test. This characterizes the material response due to dynamic elongation. The main test parameters are crosshead speed (Speedcrosshead), modulation frequency (f), sample initial length (L0), and modulation amplitude.

For the measurement, the modulation frequency is set to 1 Hz even though our device can be used up to a frequency of 2 Hz without any problem. Based on the quasi-static elongation of the specimen, the modulation amplitude is determined, which can be adjusted concerning material-specific deformation properties. With these parameters, the modulation pattern introduces sinusoidal variations in deformation, thus enabling detailed analyses of time-dependent and amplitude-sensitive responses.

The modulation induced by the device must be adjusted in frequency and amplitude. The objective is to ensure that, in each cycle, the material's relaxation can be observed to study it. This induces a constraint on the speed of the moving crosshead:

$$ \text{Speed}_{\text{crosshead}} \le 2\pi.f.\text{Amplitude}_{\text{max}} \cdot \cos(\theta_{\text{adjusted}}) \quad \text{Eq. 3} $$

By setting a ratio of 1/6 between the speed of the moving crosshead (20 mm/min) and the maximum instantaneous speed resulting from the modulation, we obtain an amplitude of 0.226 mm for the modulation.

## 2.5. Data processing

### 2.5.1. DIC

DIC (Digital Image Correlation) is a non-contact optical measurement technique that uses image analysis to get full-field displacement and strain fields at various scales [20]. Initially introduced in laser speckle metrology as a measure of surface displacement components by Peters and Ranson in the 1980s [21]. DIC has matured and become a valuable method for measuring several characteristics as a result of the rapid rise of digital imaging and data-processing tools [22]. In recent years, it has been increasingly used to study stress-strain deformation behavior [23–25], residual stresses [26–28], and fracture propagation [29,30].

The open-source algorithm NCORR [31] was used to get the displacement and the strain. After loading the reference and sequence images, the area of interest and the calculation parameters must be defined. A radius of 30 pixels was selected for the identification sub-region to encompass enough identifiable spots. A sub-spacing of 5 pixels was chosen. This last value allows a greater calculation speed at the expense of a loss in resolution. Four sub-regions were also defined to accelerate the identification of displacements. Finally, we chose the option of significant displacement analysis with automatic updating of the reference image in the event of a change that is too important to ensure correlation.

### 2.5.2. Analysis of mechanical data

During a modulated quasi-static stress, the distance B(t) between the clamps follows the following theoretical law:

$$ B(t) = B_{\text{init}} + \text{Speed}_{\text{crosshead}} \cdot t + \text{Amplitude}_{\text{max}} \cdot \cos(\theta_{\text{adjusted}}) \cdot \sin(2\pi \cdot f \cdot t) \quad \text{Eq. 4} $$

With: Binit, the initial distance between clamps.

The previous equation shows the sample will undergo longer traction

than relaxation during a cycle. Thus, the sample's elongation can be subdivided into two parts. The first corresponds to stress levels already reached during the previous cycle, while the second subjects the material to elongation levels not yet reached.

This stress analysis shows that the study of the material behavior can be done finely by considering separately each sub-part of a stress cycle. The difficulty of this approach will be the number of values collected during each evolution. Another possibility is to treat each cycle globally as a single entity. This assumes global analysis of the response of the material to the stress, namely, in this example, the force measured by the force sensor of the test machine. Following the imposed elongation, an overall force appears. The following relation can describe it:

$$ F(t) = F_{\text{init}} + g(t) + h(t) \cdot \sin(2\pi \cdot f \cdot t + \varphi(t)) \quad \text{Eq. 5} $$

With: Finit, the pretension force that is applied to the sample before testing, g(t), the quasi-static component of the force as a function of time, h(t) the evolution of the amplitude of the pulsatile evolutions of the overall force and φ(t) the phase shift with the imposed strain.

The function h(t) allows the definition of a complex stiffness function of the sample at different times and, therefore, levels of elongation. This is done like the approach developed during the DMA (Differential Dynamical Analysis) experiments. We define a complex stiffness K* of the material at each elongation as follows:

$$ K^* = \frac{h(t)}{\text{Amplitude}_{\text{max}} \cdot \cos(\theta_{\text{adjusted}})} e^{i \cdot \varphi(t)} \quad \text{Eq. 6} $$

This complex stiffness has a fundamental part denoted K' (storage stiffness) and an imaginary part denoted K'' (loss modulus). The fundamental part represents the purely elastic behavior of the recoverable material during the cycle. The imaginary part represents the viscous component of the material's behavior. It is associated with the energy the material dissipates during a stress cycle. The ratio between the loss modulus and the storage modulus is the loss factor tan(φ(t)). This is why φ(t) is also called the loss angle, which is typically determined from the phase shift between sinusoidal stress and sinusoidal strain. To define the complex stiffness and the phase shift, it is necessary to isolate the quasi-static part of the force signal and demodulate the remainder, and this at each stress level. To obtain these dynamic stiffnesses, it is first necessary to separate the components of the force signal. This is obtained by a moving average filter based on the FFT analysis of the LVDT sensor signal. On one side, we get the quasi-static force's evolution; on the other, we get the modulated part by subtraction. The amplitude at each instant is obtained by integrating the square of the signal over a width of one oscillation. The phase of the modulated force is extracted using a convolution with two analysis windows offset by 90°. These analysis windows correspond to a wavelet for the excitation frequency.

A specific analysis program was developed on Matlab to perform these analyses. This software also identifies mechanical parameters during the traction phase separately from the compression phase. the interface of this program is shown in Fig. 7.

## 3. Experimental results

Rubber bands' mechanical behavior is elastic but inherently nonlinear. As reported in the literature, the elastic response of this elastomer under tensile loading conceals a transition in its thermomechanical behavior. This transition occurs relatively early in the deformation process. Therefore, the behavior of the samples was specifically examined up to a deformation close to 8 % of their strain at failure.

### 3.1. Optical image analysis

Following the test, more than 2000 images must be processed. Due to deformation, approximately 75 % of the initial specimen area remains
<figure><img src="image_9.png" /><figcaption>Fig. 7. Initialization window of the program used to process the data.</figcaption></figure>

visible in the final image, while 25 % of the observed sample is no longer available for characterizing the material behavior. To enhance the accuracy of the analysis, the image sequence is divided into five groups, allowing for a more precise readjustment of the areas of interest during testing. Additionally, this strategy helps to mitigate the accumulation of computational errors in the displacement field identification procedure. The reliability of this process is quantified by monitoring the standard deviation of the deformation field values.

Before analyzing the deformation results, the displacement fields measured by the Ncorr software along the horizontal and vertical directions of the images are examined. Fig. 8 presents a representative example of the displacement field in both directions for a deformation of 6.5 %.

The displacement field reveals two key aspects: the apparent rigid body displacement induced by the experimental setup and, to a lesser extent, the potential heterogeneity in the sample's response. The latter becomes more pronounced when contour lines are displayed, as shown in Fig. 8. Deformation is the primary focus in material properties, as it is a fundamental parameter for characterizing mechanical behavior. The test aims to identify a region where deformation can be considered homogeneous. However, visualizing the deformation field alone is insufficient to confirm this assumption. Displacements are measured in pixels, where U represents displacement along the horizontal direction, and V corresponds to displacement along the vertical direction.

In Fig. 9, the deformation field in the tensile direction is shown at the end of the first sequence of 400 images and the beginning of the subsequent sequence. Although the average deformation level appears

similar, significant differences in heterogeneity distribution are observed. This discrepancy is attributed to the accumulation of minor errors during the displacement identification process. To maintain adequate resolution, the area selected for deformation calculations remains relatively limited, preventing the smoothing of computational inaccuracies.

The plot in Fig. 9(C), which presents the standard deviation of deformation values in both the tensile and transverse directions, confirms that localization errors increase progressively during the analysis of an image sequence. However, at the start of each new sequence, the standard deviation resets to a baseline value before increasing again, generally linearly, as more images are processed. Occasionally, this trend is disrupted when the Ncorr algorithm reinitializes by selecting a new reference image, particularly during large deformations.

In addition, The overall trend is that the standard deviations exhibit periodic variations. A time-frequency analysis indicates that this periodic component has a fundamental frequency corresponding to the modulation of the tensile elongation. The spectral power at this frequency decreases at the beginning of a new sequence. However, due to the presence of associated harmonics and the weak nature of the signal, the extraction of meaningful information related to the mechanical behavior of the specimens remains challenging. Consequently, it is reasonable to assume that the observed area undergoes homogeneous deformation for the subsequent analysis of the results. This assumption enables the analysis to focus exclusively on average deformation values.

After concatenating the values for each image sequence, it becomes evident that the modulation of elongation translates directly into a modulation of deformation in both the tensile and transverse directions. Several insights can be derived from these signals obtained from image processing. The first is the quasi-static behavior, which can be easily extracted using a moving average over the modulation period. The second is the purely periodic component, characterized by its amplitude over time and the phase difference between the two directions. This decomposition is illustrated in Fig. 10.

The initial configuration is used as the reference to plot the graphs in this figure. However, it is well known that for large elongations, as permitted by the material studied here, this approach deviates from what the specimen experiences. It would be more appropriate to consider the true or natural strain, the natural logarithm of the ratio between the specimen's current and initial lengths.

We aim to study the modulated part of the specimen's mechanical behavior. The overestimation of strain in this case (maximum strain of 35 % instead of 30 %) does not affect our analysis from this perspective. Regarding modulated strain, taking the initial length as the reference highlights the stability of the modulated elongation despite the increase in force applied to the specimen. This observation confirms the effectiveness of our modulation mechanism. This engineering strain does not

<figure><img src="image_10.png" /><figcaption>Fig. 8. Displacement field in pixels during the modulated tensile test after 21s (Euler's point of view).</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 9. Deformation field at the end of the first sequence of 400 images (pixel/pixel) (A) and the start of the following sequence (Lagrange point of view). The standard deviation of deformations in the direction of traction and transversely when divided into five analysis sequence.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Average deformations of the tensile direction, the transverse direction, and periodic components of these deformations.</figcaption></figure>

represent the actual strain experienced by the specimen at a given deformation level. In reality, the true strain induced by modulation is inversely proportional to the length reached through quasi-static deformation. However, since this effect is the same for deformations in both observed directions, taking it into account would not affect the phase shift measurements or the dynamic Poisson's ratio coefficient.

The purely periodic component exhibits an amplitude that varies with the deformation. A linear interpolation (Equation (7)) shows that this amplitude remains relatively constant in the direction of the applied load. The increase is limited to +4 % for an average value of 0.2461 %, consistent with the amplitude of the elongation modulation, which will be discussed in the section analyzing the force exerted by the specimen.

The lateral deformation is unconstrained and, therefore, influenced by variations in longitudinal deformation. As elongation increases, the alignment of the elastomer's macromolecules is modified, making it reasonable to assume that transverse deformation becomes more

restricted with increasing longitudinal deformation. A linear interpolation can be applied to describe the evolution of the amplitude of transverse deformation oscillations. This trend indicates that the amplitude decreases by 33 % from its initial value.

$$ \epsilon_{\text{modulated}}(\epsilon_{\text{mean}}) = \text{Amp}(\epsilon_{\text{mean}}) \cdot \sin(\omega \cdot \epsilon_{\text{mean}} + \varphi(\epsilon_{\text{mean}})) $$

$$ \text{Longitudinal directionAmp}(\epsilon_{\text{mean}}) = 3.1305 \cdot 10^{-4} \cdot \epsilon_{\text{mean}} + 2.4086 \cdot 10^{-3} $$

$$ \text{Lateral directionAmp}(\epsilon_{\text{mean}}) = -1.1321 \cdot 10^{-3} \cdot \epsilon_{\text{mean}} + 1.0894 \cdot 10^{-3} $$

When the periodic lateral deformation is plotted against the longitudinal deformation over a few modulation periods, hysteresis loops become apparent. This observation suggests the presence of a slight phase shift between the two deformation modes. The phase difference between lateral and longitudinal deformation was calculated, revealing that the
transverse deformation consistently lags behind the longitudinal deformation by 3.4°, with a standard deviation of 3°. This variability indicates the challenge of accurately measuring this parameter. Occasionally, a temporary reversal of the phase shift is observed (Fig. 11).

This result suggests that longitudinal and lateral deformation coupling is not entirely direct. A damping effect in this coupling could explain this observation. Thus, the classical Poisson's ratio, the negative ratio of lateral deformation to longitudinal deformation, can be supplemented by a complex coefficient characterized by its magnitude and the previous phase angle. Without the elongation modulation system, this aspect of the coupling between deformation directions would not have been identifiable.

Furthermore, contrary to assumptions under low-intensity loading, this coefficient varies with deformation. The ratio of the amplitude values enables the definition of a dynamic Poisson's ratio as a function of the longitudinal deformation level. These values can be compared with those obtained from quasi-static deformations (Fig. 12).

The Poisson's ratio calculated using the average deformation values exhibits a two-phase evolution. During the initial phase, the Poisson's ratio decreases rapidly, indicating a reorganization or alignment of macromolecules in the direction of the applied load at the onset of deformation. This initial behavior ceases beyond 10 % longitudinal deformation, suggesting that the macromolecular network can no longer deform as quickly as it did at the start. This threshold aligns with the onset of thermodynamic inversion typically observed in this type of material.

Beyond this point, the Poisson's ratio exhibits a linear decrease. In contrast, the dynamic Poisson's ratio values display more significant variability due to the inherent sensitivity of deformation measurements to fluctuations. A standard deviation of 0.033 is observed relative to the assumed linear evolution of these values. Despite this variability, elongation modulation facilitates investigating the material's response under faster loading conditions, where reduced time is available for macromolecular reorganization. As the deformation rate increases, the modulation reveals a growing discrepancy between the dynamic Poisson's ratio and the values derived from the quasi-static component of the material's response.

The Poisson's ratio measurements estimate the evolution of the specimen's cross-section as a function of the material's deformation. This analysis assumes that the coupling in the thickness direction is of the same magnitude as that in the lateral direction. This assumption enables the investigation of quasi-static and dynamic stresses using the measured load data.

## 3.2. Mechanical analysis

After separating the modulated component from the quasi-static component (Fig. 13), the complex modulus defined by its magnitude and loss angle can be measured at each quasi-static deformation level. The stresses are calculated while incorporating the corresponding

<figure><img src="image_13.png" /><figcaption>Fig. 11. Phase shift between the lateral and Longitudinal.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Quasi-static and dynamic Poisson's ratio of the sample up to a deformation of 33 %.</figcaption></figure>

Poisson ratios. The variation in cross-section induced by the modulation rapidly becomes negligible compared to that resulting from the quasi-static deformation level.

Introducing the reduction in cross-section leads to a more linear stress-strain curve compared to the assumption of a constant cross-section. The discrete derivative of this evolution provides a quasi-static modulus for each deformation level. These values can be compared to the magnitude of the complex modulus obtained from the periodic component of the force (Fig. 14). The results emphasize the material's sensitivity to the loading rate. For the quasi-static part, the theoretical strain rate, calculated concerning the current configuration, varies from $3.10^{-3}$ to $2.3.10^{-3}$ s$^{-1}$. By making a similar calculation for the modulated part, the amplitude of this strain rate evolves from $1.3.10^{-2}$ to $9.6.10^{-3}$ s$^{-1}$. Therefore, the modulation of the strain results in a loading rate up to four times higher than that induced by quasi-static traction. Classically, excluding thermomechanical effects, higher loading rates yield an increased elastic modulus. This behavior suggests the presence of a damping phenomenon, where the loading rate influences the mechanical response. However, further investigations are required to validate and quantify this explanation.

The previous results were obtained by applying a sliding analysis window and isolating an elongation oscillation and adjacent data points. Consequently, it is impossible to determine whether the material's relaxation induces a response distinct from its tensile behavior. However, this distinction can be achieved by decomposing each oscillation into three phases: the initial extension to a specific deformation, the relaxation phase, and the subsequent extension following relaxation, as illustrated in Fig. 15.

The data is interpolated using third-degree polynomials (Fig. 16) to enhance the visualization of the results. For the specimens analyzed, the behavior during relaxation induced by modulation closely resembles the overall mechanical response. However, the subsequent tensile phase following relaxation results in a more linear decrease in the elastic modulus, reaching its lowest value. The initial extension of the material appears, on average, to exhibit an intermediate behavior, yielding higher moduli than those measured in the quasi-static phase of the response.

It is important to note that identifying the modulus for each sub-sequence is conducted using a limited number of data points compared to a more comprehensive analysis spanning multiple loading cycles. Furthermore, minor irregularities in the mechanical loading system's operation may occasionally occur. This is clearly illustrated in the loss angle graph obtained through windowed FFT processing. Just before the 15 % strain, a brief jolt in the loading system was observed, significantly impacting the identification of this parameter.

Indeed, measuring a minimal loss angle is numerically challenging, especially over portions of a loading cycle. The values obtained are very noisy, as shown in Table 1. As for the deformation results, the average
<figure><img src="image_15.png" /><figcaption>Fig. 13. (A) Quasi-static stress as a function of optically measured strain. (B) Oscillatory component of the stress.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 14. (A) Quasi-static and dynamic modulus of elasticity. (B) Stress loss angle as a function of quasi-static strain.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15. Shows the different phases of each oscillation.</figcaption></figure>

<caption>Fig. 16. Interpolated evolution of the modulus of elasticity at different values of deformation of the sample.</caption>

loss angle is approximately constant with close values, whatever the analysis carried out.

### 3.3. Thermal analysis

Our experimental setup includes a thermographic camera to complement the mechanical analysis. The effective field of view allows measurements over a 60 mm sample length, corresponding to half the initial distance between the clamps. The recorded images capture the speckled surface, enabling the monitoring of displacement fields using the Digital Image Correlation (DIC) method. However, the paint applied to the sample exhibits a higher emissivity than the base material, facilitating the tracking of material point movements through the

thermographic camera. This approach offers the advantage of simplifying displacement compensation when identifying localized heat sources. As a result, there is no need to transfer motion data from the optical camera, a process complicated by factors such as differences in scale and frame rate. Fig. 17 illustrates the feasibility of this measurement. Despite the low resolution of the thermographic images, the modulated component of the deformation remains detectable. However, the precision achievable with optical measurements cannot be attained.

At the applied tensile speed, the maximum potential displacement visible between two consecutive thermographic images is approximately 0.017 pixels. Consequently, the practical benefit of this effect remains limited. However, the difference in emissivity introduces
<table><caption>**Table 1**
Summary of data on the evolution of the complex modulus.</caption><thead><tr><th rowspan="2"></th><th>Modulus variation</th><th colspan="2">Loss angle</th></tr><tr><th>Standard deviation (MPa)</th><th>Average (deg)</th><th>Standard deviation (deg)</th></tr></thead><tbody><tr><td>Extraction using wavelets</td><td>0.031</td><td>2.7</td><td>1.1</td></tr><tr><td>First tensile loading</td><td>0.197</td><td>3.3</td><td>6.3</td></tr><tr><td>Relaxation</td><td>0.063</td><td>2.1</td><td>5.9</td></tr><tr><td>Second tensile loading</td><td>0.261</td><td>2.9</td><td>6.9</td></tr></tbody></table>

This confirms that a 35 % deformation in the loading direction cannot significantly alter the three-dimensional macromolecular network and induce a noticeable change in the phase shift between elongation and measured force.

<figure><img src="image_18.png" /><figcaption>**Fig. 17.** Deformation from thermographic images at the beginning of the tensile test.</figcaption></figure>

heterogeneities in the perceived thermal field due to the presence of paint speckles.

Nonetheless, the average surface temperature of the sample can be calculated to analyze its global thermal behavior, as shown in Fig. 18. The detected temperature fluctuations are close to the sensitivity limit of the thermographic camera used for measurement (20 mK). Additionally, the observed volatility occurs at least three times lower than the modulation frequency of the elongation. Consequently, detecting the influence of modulation on rubber samples using this approach is not feasible.

To improve the differentiation of thermal field measurements. This could be achieved by following two complementary approaches. The first involves modifying the relative position between the camera and the specimen. By forgoing the ability to view the same surface in optical

and thermal imaging, it will be possible to change the camera lens to directly view a more limited sample area, maximizing the number of valuable pixels. The perceived signal will be of better quality because it will be less attenuated by the laboratory atmosphere and reflections from the mirrors. Furthermore, the observed surface may naturally have an emissivity whose variation will depend solely on the quality of the sample and not on an artificially added speckle pattern. These modifications are already being implemented to validate their relevance. However, a pitfall of this approach is that for a specimen of a certain thickness, it will not necessarily be possible to identify with the same acuity an event on the surface opposite to the one directly observed. The second approach, of course, involves increasing the quasi-static and dynamic deformation rate. A higher deformation rate can only amplify the temperature variations at the specimen surface. This approach should prove successful, given the current performance of our test setup. In any case, this is a possibility we want to explore once the modification of the observation conditions is implemented. This will also allow for a broader exploration of the mechanical behavior of this material. By implementing these improvements, it will be possible to maintain the advantage of simultaneous acquisition of stress and thermal fields without requiring complex coordinate transformations between optical and thermal images.

## 4. Conclusion

This study presents the modulated quasi-static tensile testing method applied to an elastomer. This experimental approach improves the characterization of the mechanical behavior of materials in quasi-static uniaxial tension. Indeed, the modulation of the elongation allows the dynamic response of the material to be probed. By integrating digital image correlation (DIC) and infrared thermography, we managed to capture both the evolution of the deformation and the thermal effects in a rubber sample under these conditions. Our samples' behavior was monitored for a deformation limited to 35 % to follow the initial part of the samples. This limit can correspond to a good part of the range of use of this product. In addition, it is generally accepted in the literature that a first thermodynamic transition occurs for this type of material at a lower deformation.

Thus, the evolution of some parameters, such as the quasi-static Poisson's ratio, presents an initial nonlinear behavior testifying to a structural reorganization of the macromolecular network. A remarkable result was that the loading rate dependence is measurable both in the strains and in the force generated by the elongating material. A key finding that requires further investigation is the difference between quasi-static and dynamic Poisson's ratios, suggesting that polymer deformation involves complex interactions between elastic and viscous effects. Furthermore, hysteresis in strain evolution highlights energy

<figure><img src="image_19.png" /><figcaption>**Fig. 18.** Example of a thermal field affected by the emissivity variation due to the speckle and variation of the average temperature as a function of the quasi-static deformation.</figcaption></figure>
dissipation mechanisms, confirming the need for advanced testing techniques. The use of elongation modulation in thermomechanical analysis has not yet provided all the knowledge we expected, namely the possibility of separating the reversible from the irreversible part of the thermographic data.

Future research at several modulation frequencies would be relevant to exploring this testing method further. Other materials will also be interesting to investigate to extend its applicability. In addition, optimizing the sensitivity of thermal imaging and refining image processing algorithms will further improve the accuracy of strain field measurements. By advancing the understanding of elastomer mechanics, this approach paves the way for more reliable material models and improved design strategies in polymer engineering.

## 7. Conclusion

This study introduces a robust and scalable experimental framework that significantly advances the understanding of viscoelastic polymer behavior. The insights gained from this method will contribute to improved material design, reliability, and performance optimization in engineering applications, making it an ideal contribution to Polymer Testing.

### CRediT authorship contribution statement

**Mahmoud Hussein:** Writing – original draft, Software, Methodology, Data curation, Formal analysis. **Daniel Mathieu:** Writing – review & editing, Supervision, Investigation, Conceptualization. **Gildas L'hostis:** Supervision, Resources, Investigation, Writing – review & editing. **Bernard Durand:** Supervision, Resources, Project administration, Conceptualization, Validation, Writing – review & editing.

### Declaration of competing interest

The authors declare that there is no conflict of interest regarding the publication of this paper. All research activities, data collection, and experimental implementations were conducted independently, without any financial, personal, or organizational relationships that could be perceived to influence the work reported in this manuscript.

### Data availability

Data will be made available on request.

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