This study discusses the feasibility of using a combined torsional-axial rheometer to indirectly measure the complex Poisson’s ratio based on shear and Young’s modulus. For this purpose, isothermal frequency sweeps in torsion and extension are performed sequentially on the same cylindrical specimen and under the same environmental conditions. The method is tested on two amorphous polymers, a semicrystalline polymer, a polymer blend, and a copolymer. The article includes an extensive literature review and an uncertainty assessment of the method to provide a basis for subsequent data comparison with existing research. The experimental data show a monotonic increase in the complex Poisson’s ratio up to 0.5 as the temperature approaches α-relaxation for all samples, except for the amorphous polymer. The latter shows a local minimum in the complex Poisson’s ratio observed near α-relaxation, which disappears after thermal annealing of the sample above the α-relaxation temperature. The real and imaginary parts of the complex Poisson’s ratio are additionally determined by evaluating both phase shift angles from torsional and extensional measurements. All polymers show a certain offset between the torsional and extensional phase shift angles in the glassy state, which gradually decreases as the temperature approaches α-relaxation. The complex Poisson’s ratio results are in good agreement with the literature data obtained by existing methods. This confirms that the method is applicable to polymers up to α-relaxation temperatures with significant time savings due to the nondestructive approach. This is of particular interest, given the limited availability of data in the literature.

## I. INTRODUCTION

In isotropic and homogeneous samples, Poisson’s ratio, $\nu $, defines the transverse deformation experienced by the material when a longitudinal deformation is applied [1–3]. In classical linear elasticity, Poisson’s ratio is a well-defined material property [4] that characterizes changes in the size and shape of a material when it is subjected to a strain [3]. Thus, Poisson’s ratio is a measure of the compressibility of the material. For isotropic materials, Poisson’s ratio can also be defined in terms of bulk modulus *B* and shear modulus *G* as $\nu =[ ( 3 ( B / G ) \u2212 2 ) / ( 6 ( B / G ) + 2 )]$ [3]. This definition helps us to delimit the physical range in which Poisson’s ratio can be found $( \u2212 1 \u2264 \nu < 0.5)$ in order to satisfy that $ 0 \u2264 B / G < \u221e$. Thus, in isotropic materials, a Poisson’s ratio approaching 0.5 means approaching the incompressibility limit as observed in rubbers and liquids, while a Poisson’s ratio close to 0 refers to a compressible material behavior as observed for glasses or minerals. A Poisson’s ratio equal to 0 can be found in gases, while a Poisson’s ratio smaller than 0 is found in auxetic materials with network structures as reported for some foams [3]. Within linear elasticity, the relevance and physical significance of Poisson’s ratio as a material property when characterizing mechanical properties is undoubted (see, for instance, the review by Greaves *et al.* [3]).

In purely elastic materials, Poisson’s ratio is a constant that probably varies only with temperature [4]. For viscoelastic materials, Poisson’s ratio is time (or frequency)-dependent [1–10]. In addition, several studies analytically show that Poisson’s ratio in viscoelastic materials may be a function of the stress or the strain history (see, for example, [4] or [9]). This strain-history dependence has also been experimentally observed on amorphous thermoplastics [11], semicrystalline polymers [12], and thermosets [13]. According to these observations, for a correct characterization of the viscoelastic Poisson’s ratio, not only the dependence on temperature, but also its dependence on time (or frequency) is required. In other words, a time- (or frequency) independent Poisson’s ratio is insufficient to describe the behavior of viscoelastic materials [8]. However, the time (or frequency) and strain-history dependency radically complicate the experimental characterization of the viscoelastic Poisson’s ratio with current experimental techniques [1,2,8].

Despite these challenges, Poisson’s ratio remains an essential element for the structural mechanical design of viscoelastic materials [14]. It is of special importance at the interface between different polymers since there the stresses are highly sensitive to Poisson’s ratio [1,8]. Therefore, Poisson’s ratio is a typical input for solving three-dimensional mechanical deformation in finite element simulations [14–16]. In these simulations, however, it is common to assume a constant value in the elastic or plastic regime [3], ignoring the variability of the viscoelastic Poisson’s ratio and therefore inducing errors in mechanical behavior predictions. In addition, time-dependent data of the viscoelastic Poisson’s ratio are required for predicting the viscoelastic relaxation of composite materials [17] and elastomers [18]. A proper prediction of the viscoelastic Poisson’s ratio might help in reducing the number of experimental parameters to fit analytical models [13]. Finally, the viscoelastic Poisson’s ratio links shear modulus, Young’s modulus, and bulk modulus [3] for isotropic materials [19]. This has a practical significance in that measuring the time-dependent bulk modulus is an even more complicated task than measuring the viscoelastic Poisson’s ratio [8]. Despite the crucial role of Poisson’s ratio in viscoelastic material characterization, there is a scarcity of data in the literature, with only a few studies reporting reliable time-dependent (or frequency-dependent) data [1].

The possibility of inferring the viscoelastic Poisson’s ratio from Young’s modulus and shear modulus measurements obtained with a combined torsional-axial rheometer is discussed in this article. The moduli are measured on the same specimen and in the same environment according to the standard protocol introduced by Tschoegl *et al.* [1]. The viscoelastic Poisson’s ratio, defined by [1], is obtained as a function of frequency and temperature.

The article is organized as follows: Section II provides a comprehensive review of the literature on the measurement of the viscoelastic Poisson’s ratio. Section III details the experimental setup, including the method based on a rheometer for indirect measurement, following the standard protocol. Section IV describes the complex Poisson’s ratio and viscoelastic quantities measured by the rheometer. Section V presents considerations of geometry effects on the measurement of the complex Poisson’s ratio with the rheometer. The experimental results show dynamic mechanical data for various materials such as polymethylmethacrylate (PMMA), polycarbonate (PC), polypropylene (PP), PC + acrylonitrile butadiene styrene (ABS), and thermoplastic Polyurethane (TPU). An experimental uncertainty assessment and a comparison of the data with the literature are presented in the discussion.

## II. LITERATURE REVIEW

Depending on the operating mode, the classical methods for measuring the viscoelastic Poisson’s ratio can be divided into direct or indirect methods.

### A. Direct methods

Direct methods are based on the simultaneous measurement of transverse and axial deformations by strain gauges [20–27], contact extensometers [12,13,28–30], or noncontact optical approaches. Strain gauges or extensometers are usually glued to the sample while the measurement is performed. For stiff materials, such as sandstone samples or polymers in their glassy state, bonded strain gauges or extensometers have provided good results (see, for instance, [12,13,25–27]). However, as the polymer softens at temperatures near and above its glass transition, there are limitations to the use of these contact methods [16,18,31]. The main disadvantage is that both the strain gauge and the extensometer can compromise the homogeneity of the specimen, produce stress concentrations, or restrict the free deformation of the specimen [18]. With soft materials, these disadvantages become more significant as the contact can affect the measurement due to additional weight and stiffness [16,31]. It was also observed that as the temperature increased, changes in the epoxy adhesive used to glue the gauge and the polyimide gauge itself can affect the time-dependent behavior of the sample [28]. Therefore, correction factors may be required to account for the reinforcement of the sample [17,28]. In the case of soft materials such as elastomers in their rubber-elastic plateau, techniques based on noncontact optical measurements appear to be more appropriate [16,18]. The group of optical methods includes electro-optical extensometers [32], Moiré interferometry [17,33], or videoextensometry, based on camera acquisition and image postprocessing [14,34]. The main advantage of noncontact methods is that they allow strain determination without interfering with sample deformation during the measurement. Noncontact methods based on image processing, however, are less effective for small strains [18]. Within this group, postprocessing based on digital image correlation (DIC) has gained attention in the last decade in order to reduce strain uncertainty [16,18,35–40]. DIC compares digital images in different stages of deformation and recognizes patterns with different light intensities in the selected region of interest [16,18]. The accuracy of DIC, however, can be affected by out-of-plane translation and rotation of the specimen, as well as lens distortion [16]. Other sources of errors are related to material structural defects, nonuniformity of the specimens, and the speckle pattern painted over the sample. In general terms and similar to other techniques based on imaging, the accuracy and effectiveness drop for small strains [16]. These techniques, however, have provided good results on soft specimens when the total deformation is relatively high as in uniaxial quasistatic tests performed on TPU [16] or in creep tests performed on different commercial elastomers [18]. Xu and Juang recently stated that errors in Poisson’s ratio, of about 15%, are to be expected in uncorrected two-dimensional DIC [16]. Smaller standard deviations in the range between 1.5–5.1 and 2.2–6.5 mm/m for axial and transverse creep strains are documented in a later study presented by Sotomayor del Moral *et al.* [18]. Another inherent limitation of optical methods is that the sample must be optically accessible, which is not always a viable option when the experiments require a closed chamber setup for temperature and, e.g., humidity control. After a comprehensive review of the state-of-the-art techniques using direct methods, it becomes apparent that obtaining an accurate and precise viscoelastic Poisson’s ratio measurement can be challenging when the sample undergoes significant stiffness changes during the measurement, such as in the case of amorphous polymers measured below and above their glass transition temperature.

Given the strain or stress-history dependence of the viscoelastic Poisson’s ratio, it is also important to distinguish between different test modalities when characterizing it [8]. Following the considerations of Tschoegl *et al.* [1] and other later studies (e.g., [13] and [17]), the literature usually differentiates between the viscoelastic Poisson’s ratio, when obtained from relaxations tests, and lateral contraction ratio, when obtained from quasistatic tensile or creep tests [1,13,17]. Using direct methods, several experiments were performed in quasistatic tensile tests presetting the strain rate [11–16,21,41]. Most of these experiments indicate that the lateral contraction ratio decreases with the strain rate as shown by Frank for PMMA, polyvinyl chloride (PVC), and polystyrene (PS) [11,15] and by Pandini and Pegoretti for semicrystalline polybutylene terephthalate (PBT) [12] and epoxy resins [13]. Experimental data also show that the lateral contraction ratio increases with the axial strain. Frank, for instance, reported an increase in the lateral contraction ratio in PMMA with the nominal strain from values of about 3% at temperatures measured between 23 °C and 80 °C [11,14,16]. Similar increases in the lateral contraction ratio with strain were reported by Frank in PVC and PS [11]. Perhaps, one of the most complete databases found in the literature for amorphous and semicrystalline polymers is the chapter “Quasi-static tensile test – Poisson ratio of thermoplastic materials – data” [41] in the book “Polymer Solids and Polymer Melts–Mechanical and Thermomechanical Properties of Polymers.” The chapter summarizes many data obtained from other sources such as [14,42 and 43], which in turn, are based on previous experimental works from [11 and 44]. The results shown therein indicate that for all polymers reported, the lateral contraction ratio increases monotonically with temperature toward values close to 0.5, as the temperature approaches the glass transition temperature. The temperature dependence of Poisson’s ratio from Fig. 26 in [41]. is partially consistent with the data obtained by Schenkel [44]. These results were inferred indirectly from independent measurements of the bulk modulus as obtained from [45] and the torsional modulus as provided by the manufacturer. On the other hand, the data from Fig. 4.27 in [41] show results obtained by Frank [11] in quasistatic extensional tests with a presetting strain of less than 1%. In this case, Frank measured the lateral contraction ratio directly by applying markers and measuring transverse and axial deformations with a biaxial optical technique consisting of up to four cameras [11].

To study the time-dependency of the viscoelastic Poisson’s ratio, some experimental data based on relaxation tests are also found in the literature [11–13,18,26,29,32]. It should be noted that in relaxation tests, the sample shows a relaxation behavior in the axial direction but a retardation behavior in the transverse direction [1,2]. Therefore, the term delay time is typically used in literature to characterize the characteristic time of the viscoelastic Poisson’s ratio. As for quasistatic tests, Pandini and Pegoretti showed that the isochronous viscoelastic Poisson’s ratio in two different epoxy resins increased with the applied extensional strain [13]. To a lesser extent, creep tests are found in the literature to study the time-dependent lateral contraction ratio [11,17,18,34]. In the time domain, most of the experimental data, independent of the selected test (relaxation or creep test) and the polymer studied (amorphous, semicrystalline, or cross-linked), show a monotonous increase of the viscoelastic Poisson’s ratio (or the lateral contraction ratio) with time. See, for instance, [33] for the case of PMMA at temperatures ranging between 22 and 125 °C; [15,42] for the case of PVC or PS at room temperature; [12] for the case of semicrystalline PBT at temperatures ranging between 0 and 60 °C; [18] for the case of different elastomers at temperatures of 25, 50, and 80 °C; or [13,17,26] for the case of epoxy resins at different temperatures. In fact, Tschoegl *et al*. suggested that the viscoelastic Poisson’s ratio must be a monotonic function with time, according to the theory of viscoelasticity [1]. Later, Lake and Wineman theoretically proved that the viscoelastic Poisson’s ratio may actually be a nonmonotonic increasing function depending on the material system [8]. The authors theoretically and exemplarily showed that the viscoelastic Poisson’s ratio may decrease with time in different lattice structures as presented for honeycombs structures, foams, or thermoelastic damping [8]. Later, Grassia *et al*. also showed theoretically and based on previous literature data, that for linear amorphous polymers in the vicinity of α-relaxation, a nonmonotonic Poisson’s ratio behavior may be expected [2].

Different experimental data on the viscoelastic Poisson’s ratio obtained by direct methods can also be found in the frequency domain [10,11,24,25,27,28,30,46]. According to [1], a (complex) frequency-dependent Poisson’s ratio can be defined as the lateral contraction ratio measured in an infinitesimally small uniaxial deformation of a viscoelastic material in response to a steady-state sinusoidally oscillating strain. Literature data for polymers in the low-frequency domain, say, between 0.01 and 100 Hz, are scarce. In general, a decrease in the complex Poisson’s ratio is observed with increasing frequency, as shown for PMMA [24,27,30,46]. Data on the dependence of the complex Poisson’s ratio with temperature are even more scarce, probably due to technical limitations. Kästner and Pohl measured the real part of the complex Poisson’s ratio in a frequency range between 5 × 10^{−4} and 10^{−1 }Hz, and at temperatures between 90.5 and 106 °C [46]. For that purpose, they measured the changes in dimensions of the rectangular cross section of a cantilevered beam [1,46]. Yee and Takamori attached an extensometer to annealed and quenched PMMA specimens and clamped them in a servohydraulic tester for oscillating extensional measurements [30]. They were able to directly measure the complex Poisson’s ratio and the complex Young’s modulus in the samples at a frequency between 0.01 and 10 Hz and at temperatures between −80 and 100 °C [30]. Using a stress-control experimental setup, consisting of a hydraulic load frame and strain gauges for measuring the strain (after correcting the gauge reinforcement on the sample), Arzoumanidis and Liechti measured the frequency dependency of a urethane adhesive at temperatures ranging between −40 and 115 °C in a frequency range between approximately 0.01 and 10 Hz [28]. This comprehensive literature review revealed only these three contributions that utilized direct methods to obtain the complex Poisson’s ratio in viscoelastic materials with temperature dependencies for frequencies below 10 Hz.

### B. Indirect methods

Another possibility discussed in detail by Tschoegl *et al.* [1] is to infer the viscoelastic Poisson’s ratio from the measurement of two independent moduli, e.g., Young’s modulus and shear modulus or shear modulus and bulk modulus [8]. This type of method, valid for isotropic samples, was termed as the indirect method. From a theoretical point of view, the determination of bulk modulus and shear modulus may be more appropriate as these directly represent changes in size and shape, respectively [3]. However, from an experimental point of view, the bulk modulus is very difficult to measure [1], which makes both the shear modulus and Young’s modulus favored candidates to be measured for indirect calculation. One of the first documented tests inferring Poisson’s ratio from indirect measurements was performed by Koppelmann in 1958 [47]. Using a rolling weight, which is driven by a rotary actuator that induced a sinusoidal back-and-forth motion, Koppelmann measured the dynamic extensional and torsional shear modulus in rectangular specimens of PVC and PMMA. The author measured the complex Poisson’s ration in a broad range of frequencies at room temperature [47]. For PVC, he obtained values of the complex Poisson’s ratio equal to 0.32 and independent of frequency. For PMMA, he obtained a nonmonotonic increase starting from values of about 0.30 with a local minimum of about 0.12 at a frequency of about 10 Hz [47], with a relative error of ±2% on Young’s modulus and shear modulus. In the time domain, and performing relaxation tests, Tsou *et al*. measured the viscoelastic Poisson’s ratio of different polymer films of cellulose acetate and PC from independent tests in bending and tension [48]. For PC, the authors obtained a nonmonotonic function of the viscoelastic Poisson’s ratio with a starting value of about 0.40 at short times, a local maximum of about 0.45, and a decrease to about 0.34 at long times [48]. This trend, which seems to contradict the data obtained by direct methods, has been attributed by other authors to the inherent error of the indirect method when the value is derived from two different instruments [1,2,17]. Regarding the determination of the viscoelastic bulk modulus as one of the required two independent moduli, dilatometric [49], acoustic [19], and piezo-electric methods [50] were reported. Their practical relevance is limited to date, as none of them is standardized and the corresponding devices are either self-built or rarely available. As a consequence, only a few attempts were made to infer Poisson’s ratio based on the experimental bulk modulus data [2,51,52] or vice versa [33,38,53].

Experimental inferences of Poisson’s ratio from independent moduli, indeed, require high accuracy in the estimation of each of the moduli as well as special attention in the design of the experiment [1,2,8]. In addition, ideal sample isotropy is a prerequisite.

There are two main challenges in the case of the indirect method. First, in most viscoelastic samples such as polymers, the modulus depends not only on temperature and time (or frequency) but also on other factors such as the aging time after preparation, humidity, or specimen preparation [1,2,8]. Second, since the viscoelastic Poisson’s ratio depends on the strain history, other factors such as the loading path [4,9] must be exactly the same when the specimen is measured independently. These limitations make it challenging to obtain a reliable uncertainty for the viscoelastic Poisson’s ratio when using the indirect method with two independent measurements on different specimens. A much higher uncertainty of the result must be expected if the measurement is obtained on two different instruments as performed by [48] and is even more accentuated when temperature control is also required. It is important to note that even a small temperature variation between the two instruments can significantly increase the measurement uncertainty, thereby limiting the practical usability of the viscoelastic Poisson’s ratio [1,54]. This is particularly critical at temperatures close to the glass transition.

### C. The standard protocol

For this reason, Tschoegl *et al*. [1] presented a protocol stating that for measuring the viscoelastic Poisson’s ratio indirectly, it is required to use the same specimen, measured in the same environment, at the same time (or frequency), and with high accuracy and precision. Only this procedure, termed the standard protocol (see Sec. I C of [1]), can satisfy identical initial and boundary conditions during the measurement. To date, it seems that indirect measurements of two moduli by means of wave transmission in longitudinal and transverse directions have been the only alternative for satisfying this protocol [53,55–60]. This method seems, however, to be not applicable in the low-frequency range of usual mechanical excitation, say between 0.01 and 100 Hz [1,60]. Using these wave-propagation techniques, Waterman obtained real and imaginary parts of the complex Poisson’s ratio in isotactic PP and high-density polyethylene (HDPE) at 5 MHz at temperatures ranging between −100 and 150 °C [57]. They found local plateaus in the real part and local maxima in the imaginary part of the complex Poisson’s ratio close to the relaxation transitions as expected in the classical viscoelastic theory [57,61]. The values of the imaginary part were found to be negative and between 10 and 100 times smaller than the real part over the entire temperature range [57,61]. As we will discuss later in Secs. IV and VII B, the sign of the imaginary part found in the literature depends on the convention adopted. Wada *et al*. obtained similar orders of magnitude in the imaginary part of the complex Poisson’s ratio for styrene-butadiene rubber (SBR) measured at 20 °C [56]. At that temperature, they also found a maximum at a frequency of about 1 MHz. Kono obtained a temperature-dependent complex Poisson’s ratio for PS and PMMA at 1 MHz in a temperature ranging between 20 and 190 °C [54]. Assuming a negligible imaginary part of the bulk and shear moduli, he found a local minimum of the complex Poisson’s ratio at a temperature close to α-relaxation for the PMMA sample (in agreement with the later theoretical statements introduced by [8] and [2]). To the best of the authors’ knowledge, no experiment with mechanical excitation has satisfied the standard protocol introduced by Tschoegl *et al*. for the determination of the complex Poisson’s ratio and its dependence on temperature and frequency. In a frequency range below about 1 MHz, there is no experimental data that include temperature dependence of the complex Poisson’s ratio. Tschoegl *et al*. indeed pointed out that a novel instrument would be required to conduct this type of measurement [1].

In recent years, however, commercial rotational rheometers allow the integration of an additional linear drive for measurements in the axial direction. This possibility opens new horizons for material characterization in two different loading directions, torsion and extension, see, for example [62],^{1}

^{1}Agudo, J. R., G. Arnold, A. Braun, F. Barth, and M. Schäffler, “Challenges and solutions in the rheometry of soft materials: special emphasis on Rheo-Optics,” Webinar, May 2019. See https://youtu.be/OUmhZGXkuKI.

*et al.*[1]. Although this allows multiple moduli to be measured independently on the same specimen and under the same conditions, the accuracy and precision of each modulus need to be sufficient to estimate the Poisson’s ratio with acceptable uncertainties. In the case of DMA, this precision and accuracy must be maintained over the entire temperature and frequency range. At low temperatures, when the sample is stiff, it must be ensured that axial and torsional compliances of the rheometer do not distort the measurement. At high temperatures, on the other hand, it must be ensured that the resultant force and torque for an infinitesimal strain is sufficient to avoid noise in the measurement and that the specimen creep does not distort the geometry of the sample. Furthermore, at frequencies above about 10 Hz, inertial effects must not interfere. Finally, boundary effects induced by, e.g., clamping [64] or the specimen geometry should not impose a considerable effect. The latter factor is well known for torsional measurements on rectangular specimens as a consequence of effects related to torsional warping (see [64–66]). If these assumptions are met, and both the specimen geometry and the temperature and frequency range are selected in a way that measurement artifacts are minimized, the indirect method based on this type of rheometer appears as a very promising technique for measuring the viscoelastic Poisson’s ratio within a broad range of temperature and time (or frequency).

The main advantages of the method are that the measurement is accurate at low strains over the entire temperature and frequency range studied, and not altered by contact methods such as gauge strains at high temperatures. In addition, no optical accessibility is needed and the data obtained represent the sample bulk property at the macroscopic level in favor of spatially limited information obtained from strain gauges [67]. Finally, the dependence of the Poisson’s ratio on time (or frequency) and temperature is carried out in a single measurement, which results in substantial time savings. The feasibility of measuring the complex Poisson’s ratio and its dependence on temperature and frequency, using a combined torsional-axial rheometer, is tested and discussed in detail in this contribution.

## III. EXPERIMENTAL SETUP

A combined torsional-axial modular compact rheometer (MCR) 702e MultiDrive from Anton Paar was used to experimentally measure the complex Poisson’s ratio of five different polymers. A CTD 600 convection oven connected to a GCU 20 cooling unit was used for temperature control during the measurement. The oven has optical access perpendicular to the sample, which was used to connect a high-resolution digiEye600 camera for optical monitoring during the experiments. Measurements were performed from low to high temperatures, and the range was set individually for each polymer to include the glassy state and α-relaxation. Within this range and in steps of 5 °C, a frequency sweep between 30 and 0.1 Hz was performed in torsional shear, immediately followed by the same frequency sweep in extension. For this purpose, a linear motor was coupled to the lower part of the instrument. Both torsional measurements performed with the upper motor and extensional measurements performed with the lower motor were performed with the instrument operating in the CMT (combined motor transducer) mode. This means that the torque and force were measured on the moving magnetic motor [68,69]. Historically, this mode is also known as stress controlled. Measurements combining torsional and axial motors were performed on rectangular and cylindrical specimens using a solid rectangular fixture and a solid circular fixture as the measuring system, respectively. Other experiments on disk-shaped specimens were performed with a plate-plate configuration. In that case, the lower linear motor was removed, and a second rotational motor was installed on the bottom of the MCR 702e MultiDrive. The device then operated in SMT (separated motor transducer) or historically strain-controlled. This means that the lower motor drives the lower geometry and the torque in the upper motor was measured as the torque required to keep the upper geometry nearly fixed.

Amplitude of up to 0.01% and within the linear viscoelastic range was selected for each frequency sweep in torsion and extension. For the extensional case, a static prestretch is required to ensure that the amplitude is always positive during the experiment. This prestretch was applied in the experiment by superimposing a static force equal to 1.2 times the dynamic force required for the desired strain amplitude. A monitoring time of 300 s within the set temperature was chosen to ensure that the sample had reached thermal equilibrium prior to measurement. This type of measurement protocol is commonly referred to as a TTS (time-temperature superposition) experiment. The novelty of the method described here is that it is performed in two loading directions, torsional and extensional. Therefore, the complex shear modulus, $ G \u2217(\omega )$, and the complex elastic modulus, $ E \u2217(\omega )$, as well as the phase shift angle for torsion, $ \delta G$, and extension, $ \delta E$, were obtained on the same sample, under the same conditions and in a single experiment.

Two amorphous polymers, PMMA and PC, a semicrystalline PP homopolymer, a blend of PC and ABS, and a block copolymer TPU were considered in this paper. The cast acrylic PMMA samples were provided by CNC Fertigungstechnik GmbH. The commercial Makrolon^{®} PC and Bayblend^{®} (PC + ABS) samples were provided by Covestro AG. The PP sample used was a Purell HP371P from LyondellBasell Industries N.V. Finally, the ester TPU from the IROGRAN series was provided by Huntsman International LLC. The PMMA, PC, and PC + ABS samples were provided as rectangular specimens. The PP sample was provided in the form of pellets, while the TPU samples were provided in the form of 2 and 3 mm thick sheets. For TPU, rectangular specimens of 30 mm length × 10 mm width × 2 mm thickness and disk-shaped specimens of 12 mm diameter × 2 mm thickness were punched from the 2 mm sheets.

Cylindrical specimens 45 mm in length and 4.8 mm in diameter were heat processed for all polymers. Extreme care was taken to produce samples free of imperfections, inclusions, or voids. Two different preparation methods were used. Samples supplied as rectangular specimens were thermoformed in a cylindrical silicone mold. The mold was constructed from a silicone casting kit provided by TFC Troll Factory^{®} using a 4.80 mm diameter steel cylinder as a negative. Rectangular samples of PMMA, PC, and PC + ABS with a size of 4 mm × 3 mm were placed in the mold and heated in a Memmert UN30 oven at a temperature 20 °C above the glass transition temperature for a period of 2 h. For the TPU sample, the 4 × 3 mm^{2} specimen was carefully cut from the 3 mm sheet with a scalpel, placed in the mold, and introduced into the oven at a temperature of 155 °C for two hours. After two hours, the oven was turned off and the mold was allowed to cool slowly for 24 h. The mold was then removed and the samples were carefully separated from the mold.

In the case of PP, the cylindrical samples were processed directly from pellets using a vacuum compression molding (VCM) tool provided by MeltPrep^{®}. The VCM tool consisted of a sample holder connected to a vacuum source, a piston, and a lid. The interior of the tool contained polytetrafluoroethylene-coated films to prevent sample adhesion. After filling the tool, a pressure of approximately 3.2 bar was applied. The tool was then placed on a preheated plate at a temperature of 190 °C for 15 min. During this process, the piston pressed the sample into a circular tube containing a 4.8 mm diameter mold. Finally, the tool was placed on a cooling plate connected to compressed air for rapid cooling of the PP sample. After 5 min, the sample was removed from the chamber.

The sample diameter was measured at different vertical positions using a digital caliper with a resolution of 0.01 mm. Only cylindrical samples with a standard deviation of less than 1% in diameter were used for the measurements. In addition, the specimens were continuously monitored with the camera during the measurement to verify that no shape distortion occurred.

To avoid significant physical aging effects on amorphous polymers induced by the manufacturing process, the amorphous samples were further subjected to an annealing process. For this purpose, the samples were slowly heated in the CTD 600 at 2 K/min to a temperature slightly above their glass transition. The samples were then held at this temperature for 90 min to allow for thermodynamic equilibration. Finally, the samples were cooled at 5 K/min to the temperature set in the TTS experiment. During the annealing process, a static vertical force of 0 N was applied to avoid sample deformation.

## IV. DEFINITION OF POISSON’S RATIO IN THE FREQUENCY DOMAIN

*M*, and the deflection angle, $\phi $. For extensional measurements, the primary measured variables correspond to the force,

*F*, and the axial displacement,

*s*. To transform primary variables into viscoelastic quantities, we use conversion factors that are dependent on the deformation mode, i.e., torsional or extensional and on the sample geometry. For torsional measurements, the shear stress and shear strain constants are defined as $ C S S= \tau A / M$ and $ C S D= \gamma A / \phi $, where $ \tau A$ represents the shear stress amplitude and $ \gamma A$ is the shear strain amplitude in the oscillatory experiment. Similarly, for extensional measurements, the extensional stress and extensional strain constants are defined as $ C N S= \sigma A / F$ and $ C T S= \epsilon A / s$, where $ \sigma A$ represents the extensional stress amplitude and $ \epsilon A$ is the extensional strain in the oscillatory experiment. Since $| G \u2217 ( \omega )|= \tau A / \gamma A$ and $| E \u2217 ( \omega )|= \sigma A / \epsilon A$, Eqs. (14) and (15) can be rewritten as

*t*and

*w*are the thickness and the width of the sample, respectively, and

*u*represents the width-to-thickness ratio, i.e., $u= w / t$. For the case of the RheoCompass

^{TM}software (Anton Paar) used for the measurements, an approximation of $ J T$ is based on the following polynomial approximation:

*u*[66]. Accordingly, the factor $ C S S / C S D$ reads

*u*= 1) and increase with

*u*, showing a maximum deformation of 10% when

*u*> > 1 [66]. The fact that the sample must be clamped during the measurement violates the free warping assumption in the real measurement [64–66]. This phenomenon, known as the warping torsion effect, leads to a deviation in the absolute shear modulus $| G \u2217 ( \omega )|$ as obtained from torsional experiments. Warping torsion effects have been well documented in the literature, for example, by comparison of torsional and shear measurements on SBR samples (see, for example, [66] and the literature cited therein). The restrained warping contribution in torsion becomes larger with increasing

*u*and reducing the length-to-width ratio, $p= l / w$. The lower the

*p*, the greater the clamping effect contribution and, therefore, the greater the presence of compression superimposed to the primary torsion in the sample. The contribution of the restrained warping contribution in torque is not considered in Eqs. (24) and (25). There are several correction proposals in literature to consider the effect of restrained warping contribution. The authors recommend the article by Dessi

*et al.*[66] as an excellent reference for the use of different correction models, depending on the geometry of the sample. For rectangular samples with a

*p*> 1, which is usually strongly recommended by rheometer manufacturers, the original corrections presented by Szabo [70] based on Timoschenko’s approach seem to be a good approximation [65,66]. Indeed, starting from 2020, the commercial software used for the measurements allows the selection of an improved version of the geometric conversion factors for rectangular specimens, which directly solves the primary torsional inertia moment as shown in Eq. (20) up to

*n*= 2. The improved version also uses a modified Szabo correction in order to recalculate the stress/strain ratio for torsional measurements. Szabo’s approach corrects the effective length of the specimen by assuming that this new length is subjected only to a primary torsion as described by the classical Saint Venant solution. In RheoCompass

^{TM}, the corrected length is given by

*et al.*presented a Szabo correction with up to four terms for contiguous domains depending on the length and aspect ratio of the sample. In the software, it was decided to use the expression with four terms to avoid possible mathematical discontinuities. The dimensionless factor $K( u , \nu )$depends not only on

*u*but also on Poisson’s ratio $\nu $ [70]

The primary torsional inertia moment, as shown in Eq. (20), is only solved up to *n* = 2 in the software. Note that there is no discernible benefit to expanding the equation further, especially when other sources of uncertainty are considered.

The free length of the specimen now appears in the new geometrical conversion factor. First, this adds one more factor to consider when estimating the uncertainty. Second, this increases the difficulty of the measurement since the absolute sample length may change with temperature. Most importantly, an estimate of the complex Poisson’s ratio is necessary for the correction of the complex shear modulus. Although it appears that $K( u , \upsilon )$ may not change strongly with Poisson’s ratio [66], the use of Eqs. (30) and (31) imply an iterative process in order to obtain the complex Poisson’s ratio as a function of the frequency and temperature. These circumstances complicate the indirect measurement of $ \nu \u2032(\omega ), \nu \u2032 \u2032(\omega )$, and therefore, $| \nu \u2217 ( \omega )|$ for the case of rectangular samples.

For cylindrical samples, there are no warping deformations due to torsion. Thus, there is no torque contribution due to warping, and therefore, no correction of the complex shear modulus is necessary. In addition, for oscillatory measurements on cylindrical samples such as those considered here, a strain of 0.01% on specimens of 30 mm free length and 4.8 mm diameter corresponds to deflection angles of the order of 1.6 mrad. Therefore, the specimen is assumed to be subjected to pure torsion under these conditions. The deflection angle is so small during the experiment that Rivlin normal forces required to maintain pure torsion can be neglected over the entire range of temperatures and frequencies evaluated in this article [71,72].

*D*is the diameter of the sample.

## V. EFFECTS OF DEFORMATION MODES, SAMPLE GEOMETRY, AND FREQUENCY

Figure 1 shows the absolute complex shear modulus and Young’s modulus as a function of temperature obtained for TPU using different deformation modes such as extension, torsion, and shear. The curves show results at a frequency of 1 Hz applying a strain of 0.01% within the linear viscoelastic range during the entire temperature range. Three different specimen geometries were compared in the experiments: rectangular samples with a free length of 30 mm, a width of 10 mm, and a thickness of 2 mm [see Fig. 1(b)]; cylindrical samples with a free length of 30 mm and a diameter of 4.8 mm [see Fig. 1(c)]; and disk-shaped samples with a thickness of 2 mm and a diameter of 12 mm [see Fig. 1(d)]. Extensional and torsional experiments were performed using a configuration consisting of the upper rotational drive and the lower linear drive, as described in the experimental setup. Measurements in shear with a plate-plate measuring system were performed removing the lower linear drive and installing a second rotational motor in a separate motor transducer (SMT) configuration.

Closed black squares depict the results of $| E \u2217|$ for extensional measurements on rectangular samples [A1 in Fig. 1(b)]. Closed blue circles show the results of $| G \u2217|$ for torsional measurements on rectangular samples [B1 in Fig. 1(b)]. The classical Saint-Venant’s solution, as depicted in Eq. (20), was selected in the software for obtaining the absolute shear modulus. All experiments were performed according to the standard protocol. This means that the curves of $| E \u2217|$ and $| G \u2217|$ were obtained in the same experimental run. Open black squares and open blue circles depict the results of $| E \u2217|$ for extensional measurements and $| G \u2217|$ for torsional measurements, respectively, in cylindrical samples [A2 and B2 in Fig. 1(c)]. Finally, open red diamonds depict the results of $| G \u2217|$ obtained from shear experiments performed in a SMT configuration with a plate-plate geometry of 12 mm diameter.

Independent of the sample geometry, experiments performed in extension show good agreement in $| E \u2217|$ over the entire measured temperature range. However, the values of $| G \u2217|$ as obtained from rectangular samples overestimate the data obtained from cylindrical and disk-shaped samples [compare closed blue (B1) with open blue symbols (B2) and open red symbols (C) in Fig. 1]. In the glassy state, the overestimation is about 40%. In a temperature range between 25 and 85 °C, the data in rectangular samples are overestimated by about 25%. This overestimation in $| G \u2217|$ as obtained using Saint-Venant’s classical solution and due to the constrained warping effect is to be expected in rectangular samples [64–66,70]. The green dashed line represents the values of $| G \u2217|$ obtained from torsional experiments in rectangular samples, now applying the Szabo correction as defined in Eq. (29). A constant Poisson’s ratio of $ \nu 0=0.49$ was used in Eq. (27) to obtain $ L c o r r( u , \nu )$ in Eq. (26). After correction, $| G \u2217|$ is shifted downward by about 17%, bringing the data closer to the results obtained on cylindrical samples (compare green dashed lines with open blue circles). In the glassy region, the corrected data still overestimates the values in cylindrical samples by a factor of 10%–20%. Data obtained on disk-shaped samples show good agreement with the cylindrical data, except in a temperature range between approximately −5 and 20 °C, where the values measured in shear were found to be lower.

Figure 2 shows the dependence of the complex Poisson’s ratio on temperature in the TPU sample at a frequency of 1 Hz. The closed gray triangles depict the results on rectangular specimens without applying the Szabo correction. Open black squares and closed blue circles depict the results in the same rectangular sample after applying the Szabo correction using $ \nu 0=0.49$ and $ \nu 0=0.35$, respectively. Finally, open and closed red diamonds represent two different experiments performed on cylindrical specimens. Therefore, no correction was implemented. The complex Poisson’s ratio obtained from the rectangular sample without applying any correction remains about 0.2 in the rubber-elastic plateau. This value remains well below the expected value of about 0.48 (see, for instance, [16]), which is indeed expected for a material with rather elastomeric properties. Since the results in $| G \u2217|$ are overestimated, a lower value in Poisson’s ratio is also expected in rectangular specimens, if no correction is applied.

In the temperature range between −10 and 40 °C, both independent experiments performed on cylindrical samples show a complex Poisson’s ratio of 0.49 ± 0.01 in very good agreement with the measurements performed on TPU at room temperature, using digital imaging correlation techniques [16]. The data on cylindrical samples show an increase in $| \nu \u2217|$, from about 0.35 to 0.49, in the region including the glassy state and the glass transition. There is no literature on TPU to compare within this temperature range, but a monotonic growth up to a value of 0.5 as observed for TPU is to be expected in thermoplastic polymers. Above approximately 40 °C, a slight decrease in the complex Poisson’s ratio from 0.49 up to values of about 0.46 is observed. An explanation could be the occurrence of a glass transition in the hard segments of TPU at temperatures similar to those investigated [73].

The data obtained on rectangular specimens after using Eq. (27) with $ \nu 0=0.49$ for the length correction show a complex Poisson’s ratio of about 0.42 in the rubber-elastic plateau (compare diamonds and open black squares). The corrected data still give a deviation of about 8% from the data obtained on cylindrical samples or from other literature data for the rubber-elastic plateau [16]. When using Eq. (27) with a lower Poisson’s ratio in the correction, $ \nu 0=0.35$, the data drop by approximately 3%–4% in the entire range (see closed blue circles). This fact also leads us to suspect that for a sufficient correction of the constrained warping torsion over a wide temperature range, a variable and temperature-dependent Poisson’s ratio value must be properly introduced in Eq. (27). In what follows, only cylindrical samples were considered for the indirect measurement of the complex Poisson’s ratio.

Figure 3 shows the results obtained from frequency sweeps in PMMA at room temperature. The frequency was varied between 30 and 0.1 Hz. The experiment was repeated three times using the same sample. Figure 3(a) shows the results of $| E \u2217|$ and $| G \u2217|$. Considering the three experiments, a standard deviation ranging between 0.2% and 1% in $| E \u2217|$, and between 0.4% and 0.8% in $| G \u2217|$ was measured, depending on the frequency. Since the same sample was used in the experiments, the standard deviation represents only possible random errors related to the primary variables that are measured in the instrument. Note that possible variations in sample diameter or clamping effects are not considered in this kind of experiment.

In the frequency range studied, $| E \u2217|$ increases from about 2700 MPa at 0.1 Hz to 3470 MPa at 15 Hz. A decrease in $| E \u2217|$ is observed at frequencies higher than approximately 15 Hz. This reversal is interpreted as an artifact of the measurement at these frequency levels. The sample manufacturer provides an elastic modulus of 3400 MPa calculated according to ISO 527-1/-2, DIN 53455, and ASTM D 638. $| E \u2217|$ obtained from the dynamic experiment is in good agreement with the data provided by the manufacturer. $| G \u2217|$ ranges from approximately 1000 MPa at 0.1 Hz to 1500 MPa at 30 Hz. In this case, the fit performed according to a power law agrees with the data over the entire measured frequency range. Apparently, no artifacts are observed in torsional measurements, in this frequency range. No values for torsional measurements are provided by the manufacturer. In a frequency range up to 11.6 Hz, the data from Fig. 3(b) reveal a measured complex Poisson’s ratio barely dependent on frequency and equal to 0.30 ± 0.01, considering the three different experiments. The value of the measured complex Poisson’s ratio is in good agreement with other literature data obtained with different methods on PMMA [41]. Since the measured Poisson’s ratio matches the expected literature value, we assume that $| G \u2217|$ was measured with sufficient accuracy. The decrease in Poisson’s ratio at frequencies higher than 15 Hz is mainly attributed to the measurement artifact in the extensional measurement, which seems to be a systematic error. Note that the decrease is small (from 3470 MPa to about 3380 MPa) but has a large effect on the complex Poisson’s ratio. Similar trends were observed in other polymeric samples in the frequency range between 10 and 30 Hz. To mitigate potential uncertainties associated with high frequencies, such as those due to instrument inertial effects, a conservative approach was taken to exclude frequencies above 10 Hz. This is discussed further in Sec. VII A.

## VI. EXPERIMENTAL RESULTS ON COMPLEX POISSON’S RATIO

### A. Amorphous polymers

Figure 4 shows the PMMA results of a complete TTS measurement consisting of isothermal frequency sweeps measured between −20 °C and 150 °C in both extension and torsion. The results of $| E \u2217|$ (closed black squares) and $| G \u2217|$ (closed blue circles) are shown in Fig. 4(a). As expected, $| E \u2217|$ is two to three times higher than $| G \u2217|$ over the entire temperature range measured. A significant softening of the material is observed with a decrease of both moduli when heated above about 105 °C. This softening is associated with the α-relaxation process of the material (main glass transition) due to main chain motions. Figure 4(b) depicts the temperature dependence of both loss factors. The α-relaxation process can also be clearly seen as a maximum in the loss factor at a temperature of about 125 °C. In the temperature range corresponding to the gradual α-relaxation process (between about 80 and 150 °C), both loss factors are equal and independent of the deformation mode (torsion or extension). However, in the temperature range corresponding to the glassy state of the material (temperatures below about 80 °C), the loss factor in extension remains higher than that measured in torsion. From Fig. 4(b), a second thermal event is also observed at a temperature of about 20 °C. This secondary relaxation or β-relaxation process is related to the rotation of the methacrylate side group [74] [see the inset of Fig. 4(b)]. As expected, the relaxation processes show a clear frequency dependence. In Fig. 4(c), which zooms in on the dependence of $ E \u2032 \u2032$ and $ G \u2032 \u2032$ as a function of temperature, this frequency dependence can be clearly discerned. The maximum in $ E \u2032 \u2032$ or $ G \u2032 \u2032$ is shifted to higher temperatures with increasing frequency [see open black diamonds and open blue squares in Fig. 4(c)]. Interestingly, the frequency dependence of the α-relaxation process remains independent of the deformation mode.

Figure 5(a) plots the complex Poisson’s ratio as obtained from Eq. (16) for PMMA as a function of frequency at various temperatures. At temperatures corresponding to the glassy state of the sample, the complex Poisson’s ratio shows little dependence on frequency. However, as the temperature approaches the α-relaxation process, the dependence of the complex Poisson’s ratio on frequency increases. In good agreement with the literature, Poisson’s ratio decreases with increasing frequency [24,27,30,46]. Figure 5(b) shows the complex Poisson’s ratio as a function of temperature at different frequencies. In good agreement with the literature, Poisson’s ratio increases with temperature toward 0.5, when approaching the main glass transition. However, a local minimum is clearly observed at a temperature close to α-relaxation at approximately 110 °C. Another change in the trend of the curve is observed at a temperature around −10 °C, probably due to the β-relaxation of the material.

Other authors have previously observed a local minimum in the complex Poisson’s ratio of PMMA with temperature [55]. Some authors have also demonstrated the feasibility of nonmonotonic growth of the Poisson’s ratio with temperature in amorphous polymers [2,8]. In view of the experimental results, we hypothesize that in our case, the local minimum in the complex Poisson’s ratio could be due to the physical aging of the sample. From Fig. 6(c), in which the relative change in the sample length, $ \Delta L / L$, is plotted as a function of temperature, it can be seen that there is a shrinkage of the sample at a temperature close to the glass transition. $ \Delta L$ is directly measured from the device encoder. The shrinkage, indicated as a local minimum in $ \Delta L / L$, coincides with the local minimum in the complex Poisson’s ratio at a temperature of about 110 °C [compare red circles in Figs. 6(a) and 6(c)]. Visual evidence from images taken during the experiment shows that no other distortion effects such as sagging or Poynting occur at this temperature. Figure 7 shows that the cylindrical specimens do not distort throughout the measurement, reinforcing the hypothesis that the change in $ \Delta L / L$ is primarily due to local shrinkage of the free length. In addition, the shrinkage can be qualitatively confirmed by evaluating the sample length in the images, where it can be seen that the sample size decreases from about 385 pixels [see blue contour in Fig. 7(a)] to 377 pixels [see Fig. 7(f)] between 100 and 125 °C within the specified temperature range.

During the manufacturing process of the specimen, an orientation of polymer chains may occur [75]. The polymer orientation at a local level would relax at a temperature close to the glass transition. This would generate the shrinkage that is observed in Fig. 6(c) or in Fig. 7. Interestingly, the indirect measurement of the complex Poisson’s ratio from $| E \u2217|$ and $| G \u2217|$ seems to be sensitive enough to detect this effect.

The PMMA sample was thermally annealed above the glass transition and the measurements were repeated between 80 and 120 °C (blue squares in Fig. 6). The measurement data confirm that the minimum of the Poisson’s ratio disappeared after this thermal relaxation step. In the temperature range between 80 and 100 °C, the values of the complex Poisson’s ratio practically overlap. Figure 6(b) shows that the frequency dependence in this temperature range is also the same for the sample with and without annealing. No macroscopic shrinkage was observed in this thermally relaxed sample [Fig. 6(c)]. Therefore, physical aging is not expected to be as pronounced as in the first measurement. Physical aging, which tends to be more pronounced in amorphous polymers [75], seems to be the reason for the local minimum in the complex Poisson’s ratio. Consequently, measurements of complex Poisson’s ratio on other amorphous polymers were performed on samples previously annealed, to minimize physical aging phenomena.

From Eqs. (30) and (31), the real $ \nu \u2032$ and imaginary $ v \u2032 \u2032$ parts of the complex Poisson’s ratio can also be determined. Figure 8 compares $ \nu \u2032$ and $ \nu \u2032 \u2032$ in the PMMA sample with and without annealing. In both cases, the real part remains about 10 times larger than the imaginary. The absolute value of the imaginary part varies only between 0 and approximately 0.1, which is in good agreement with literature data [30,57,61]. The imaginary part of the complex Poisson’s ratio different from 0 physically represents a time delay between axial and transverse deformations. Therefore, Fig. 8 shows that the time delay gradually increases with frequency and decreases with temperature to values very close to 0 near the glass transition temperature. This holds for both annealed and unannealed conditions. The imaginary part becomes zero when the real part approaches the theoretical limit value of 0.5, indicating a theoretical incompressibility in the sample. In this case, our experimental results show practically no delay between transverse and axial deformations. It should be noted that at temperatures above the glass transition, a few data points of the Poisson’s loss factor drop slightly below 0. We believe that this is due to the uncertainty of the measurement when the temperature exceeds the glass transition. A detailed analysis of the measurement uncertainty is given in the discussion.

Figure 9 shows the results of the dynamic mechanical measurements for PC. Both complex moduli show a plateau in the glassy state and a softening of about two orders of magnitude at a temperature of around 150 °C [see Fig. 9(a)]. This temperature corresponds to the main glass transition or α-relaxation of the sample. The main glass transition is also clearly seen as a maximum in both loss factors [see Fig. 9(b)]. As in the case of PMMA, the loss factor obtained in extension is higher than that obtained in torsion in the glassy state. As we approach the glass transition, however, both loss factors become similar. Figure 9(c) shows the loss modulus in both deformation modes over a temperature range that zooms in the main glass transition. The maximum in the loss modulus, which is also indicative of the main glass transition, is independent of the deformation mode [see Figs. 9(c) and 9(d)]. The relaxation process can be fitted with an Arrhenius function of the type $f=A exp( E a / R T)$ with a similar frequency dependence, regardless of the deformation mode [see black solid and blue dashed lines in Fig. 9(d) for the fits to extensional and torsional experiments, respectively]. From the Arrhenius fit, with a regression coefficient above 0.99 in both cases, and assuming the universal gas constant *R* = 8.314 J/mol K, a similar activation energy *E _{a}* = 990 KJ/mol can be obtained from extensional and torsional experiments.

The complex Poisson’s ratio results for PC are shown in Fig. 10. Two replicate measurements are shown for two different samples. In both cases, the samples were thermally annealed. The samples were heated from room temperature to 160 °C at a heating rate of 2 K/min. The temperature was held for 90 min, and the sample was cooled down at 5 K/min to 25 (red closed circles) and 80 °C (black closed squares) subsequently. TTS measurements with heating rates of 2 K/min were started at these temperatures. The obtained complex Poisson ratios practically overlap in both measurements [compare red and black curves in Figs. 10(a) and 10(b)]. A slow gradual increase is observed from room temperature up to a temperature of about 120 °C. Then, a more pronounced increase of the complex Poisson’s ratio up to values around 0.5 is observed as the main glass transition is approached [see Fig. 10(a)]. Above 150 °C, a complex Poisson’s ratio slightly above 0.5 is observed, especially at low frequencies. This is due to uncertainties in the complex Poisson’s ratio at these temperature and frequency levels.

In the glassy state, the complex Poisson’s ratio is barely frequency-dependent. A decrease in the complex Poisson’s ratio with frequency is observed at temperatures higher than about 100 °C [see Fig. 10(b)]. As the temperature approaches the glass transition, the decrease in the complex Poisson ratio with frequency becomes more pronounced. Figure 10(c) shows the relative change in the sample length, $ \Delta L / L$ as a function of the temperature. At a temperature of approximately 150 °C, the slope clearly increases, coinciding with the main glass transition. At this temperature, no local macroscopic shrinkage is observed in the sample. This is consistent with a less pronounced physical aging effect in annealed samples. Figure 10(d) plots $ \nu \u2032$ and $ \nu \u2032 \u2032$ for PC. Very similar to the PMMA sample, the real part remains almost an order of magnitude larger than the imaginary part. In the glassy state, the imaginary part is different from 0. At temperatures around 140 °C, the imaginary part decreases to 0, as the temperature approaches the glass transition and the sample approaches incompressibility.

### B. Semicrystalline polymers

Figure 11 shows the dynamic mechanical results for the semicrystalline PP. $| G \u2217|$ and $| E \u2217|$ show a plateau up to a temperature of about −30 °C, corresponding to the glassy state [see Fig. 11(a)]. At higher temperatures, a gradual softening is observed due to the segmental mobility of the amorphous regions of PP confined within the crystalline structure. This temperature corresponds to the glass transition of PP [76,77] and is referred to as β-relaxation in the literature [76,77]. The softening in $| G \u2217|$ and $| E \u2217|$ is not as pronounced as in the case of amorphous polymers due to the crystalline region of the polymer. At about 30 °C, another small plateau is observed in both moduli, followed by another gradual decrease. This second thermal event may be related to the α-relaxation process and is associated with molecular motions within the crystalline regions of PP [76,77]. Both relaxation processes can be clearly seen as maxima in both loss factors [see Fig. 11(b)] or in the loss moduli [see Fig. 11(c)]. Unlike the case of amorphous polymers, in the glass transition of PP, both loss factors are different, depending on the deformation mode. A certain shift of the maxima is even observed depending on the deformation mode. This effect is more evident in the maximum obtained from the loss moduli [see Fig. 11(c)]. Figure 11(d) shows that β-relaxation in the axial direction occurs at significantly lower temperatures (about 2.5 °C) compared to torsion. Interestingly, from the Arrhenius fits to the extensional (solid black lines) and torsional (dashed blue lines) data, two different activation energies of about 360 and 300 KJ/mol can be obtained. α-relaxation shows hardly any frequency dependence, as expected for a thermal event due to the crystalline region [compare open and closed symbols in Fig. 11(d)]. In addition, α-relaxation is independent of the deformation mode. Figure 11(b) shows that as the temperature increases, the two loss factors become similar.

Figure 12 shows the complex Poisson’s ratio as a function of frequency (a) and temperature (b). The dependence of the Poisson’s ratio on frequency is small over the entire measured temperature range. This holds true even well above the main glass transition, at a temperature of 105 °C. This frequency behavior differs from the one observed in amorphous polymers and could, therefore, be related to the presence of a crystalline region in the polymer. Figure 12(b) shows a monotonic increase of the real part of the complex Poisson’s ratio from values of about 0.2 at −50 °C to values of about 0.4 at a temperature of 100 °C. Two slope changes in the curve are seen coinciding with the two thermal events. At −50 °C, the imaginary part is maximum at about 0.04. Above the main glass transition at a temperature of about 0 °C, the imaginary part decreases to about 0.02. This is in contrast to amorphous polymers (e.g., PMMA), where the imaginary part approaches zero at the main glass transition. At higher temperatures, a gradual decrease in $ \nu \u2032 \u2032$ is observed and at about 100 °C, $ \nu \u2032 \u2032$ is very close to zero. Similar to the amorphous case, the imaginary part remains much smaller, compared to the real part of the complex Poisson’s ratio. At 100 °C, the complex Poisson’s ratio is represented by the real part only.

### C. Polymer blends and copolymers

The results for the PC + ABS blend are shown in Fig. 13. The moduli $| E \u2217|$ and $| G \u2217|$ remain slightly below those obtained for the pure PC sample. At a temperature of about 100 °C, softening of both moduli can be observed as the temperatures approach the main glass transition [see Fig. 13(a)]. A maximum in both loss factors is observed at a temperature of about 110 °C [see Fig. 13(b)]. This indicates that the main glass transition temperature is lower than that of the pure PC sample. In the glassy state, the extensional loss factor remains higher than the torsional loss factor. Between −20 and 0 °C, a slight increase in the complex Poisson’s ratio is observed from about 0.34 to 0.39. This could be due to the presence of ABS in the sample. The value remains constant between 0 °C and 60 °C and up to this temperature, the complex Poisson’s ratio shows very little dependence on frequency. From 60 °C, a gradual increase in the complex Poisson’s ratio is observed until it reaches a value close to 0.5 at a temperature of 110 °C, which coincides with the glass transition of the blend. From 60 °C, a clear decrease of the complex Poisson’s ratio with frequency is observed from the experimental results [see Fig. 13(c)].

Finally, Fig. 14 shows the experimental results obtained on the TPU sample. Figure 14(a) shows a gradual decrease in both moduli starting at −40 °C. This temperature corresponds to the glass transition of the soft segment of the polymer. The glass transition of the soft segment is clearly seen as a maximum in both loss factors at a temperature of about −20 °C [see Fig. 14(b)]. In the glassy region, a loss factor in extension is found to be higher than that measured in torsion, following the same trend as for all previous samples. Both loss factors approach each other as the temperature approaches the main glass transition. From a temperature of about −30 °C, both loss factors are equal and completely independent of the deformation mode. This is true for the entire region corresponding to the rubber-elastic plateau. As for amorphous polymers and the polymer blend, a clear frequency dependence of the complex Poisson’s ratio is noticed as the temperature approaches the glass transition [see Fig. 14(c)]. A decrease in the complex Poisson’s ratio with frequency is observed up to a temperature of about −20 °C. From this temperature and throughout the rubber-elastic plateau, a complex Poisson’s ratio very close to 0.5 and practically independent of frequency is noticed. In the rubber-elastic plateau, a slight decrease in the complex Poisson’s ratio is observed from a frequency of about 6 Hz. The complex Poisson’s ratio increases from about 0.32 at −50 °C to 0.5 at −30 °C. A slight decrease in Poisson’s ratio with temperature from about 30 °C can be seen in Fig. 14(d). This is in good agreement with the previous findings as shown in Fig. 2 and could be related to the glass transition in the hard segments of TPU. Throughout the rubber-elastic plateau, with a value of Poisson’s ratio very close to 0.5, which is close to the incompressibility limit, the imaginary value of Poisson’s ratio is zero [see Fig. 14(d)].

## VII. DISCUSSION

The main purpose of this paper, after a thorough review of the literature to date, is to evaluate the feasibility of indirect measurement of the complex Poisson’s ratio in a commercial rheometer. By applying the appropriate geometrical conversion factors, the complex Poisson’s ratio can be calculated as an additional rheological parameter when a torsional and an axial motor are combined in a single measurement [see, e.g., Eqs. (34) and (35)]. In order to comply with the Tschoegl protocol [1], it is important to ensure that both torsional and extensional measurements are performed on the same specimen and under exactly the same experimental conditions. This is *a priori* possible with the measurement protocol presented in this paper. One important finding from the experimental results in Fig. 1 or 2 is that, even under these conditions, cylindrical specimens are highly favored. This is mainly due to the fact that no constrained warping torsion correction is required for cylindrical specimens [64–66,70]. Although several corrections for rectangular samples are presented in the literature [66], these procedures require Poisson’s ratio as an input parameter, and thus, an iterative approach. For example, in Fig. 2, different values of Poisson’s ratio are used as input to correct the torsional measurements in TPU, resulting in a shift of approximately 3-4% in the complex Poisson’s ratio. In this case, a modified Szabo correction was used to recalculate the stress/strain ratio for the torsional measurements. Indeed, this 7% value may be also influenced by the method of correction for the torsional modulus, as described by Dessi *et al.* [66]. It is possible that for the isothermal case, corrections in rectangular samples may be sufficient to experimentally estimate the complex Poisson’s ratio. However, for the sake of simplicity, this study was focused on cylindrical samples.

### A. Experimental uncertainty assessment

To obtain the complex Poisson's ratio from Eq. (34) or (35), for example, we use a constant diameter measured at room temperature before clamping the sample. To estimate the total uncertainty in the diameter, however, we need to consider its changes with temperature during the experiment. The change in diameter may be due to thermal expansion and extensional creep. The second effect tends to occur in amorphous polymers at temperatures above the glass transition, especially during extensional measurement. Note that some static force is always applied to ensure a proper extensional amplitude during the experiment. Sample creep causes not only a change in diameter but also a distortion of the cylindrical shape and is, therefore, highly undesirable during the measurement. From Fig. 6(c) or 10(c), it can be seen that the maximum change in length of the amorphous polymers measured directly on the rheometer is not more than 3.5% in the experiments. Therefore, it can be assumed that the creep of the approximately 5 mm diameter specimens does not significantly deform the specimen at these levels. The optical monitoring of the experiment with the camera allowed us to confirm this statement (see Fig. 7). Figures 6(c) and 10(c) show a gradual increase in $ \Delta L / L$ with temperature, mainly due to the thermal expansion of the sample. A change in the slope of $ \Delta L / L$ with temperature is indeed observed as the temperature approaches the main glass transition. Since we assume isotropic sample properties, it is reasonable to assume that the axial deformation due to thermal expansion is also maintained in the radial direction. However, we understand that the rheometer is not capable of directly measuring changes in diameter. For this reason, the change in diameter with temperature, as estimated from $ \Delta L / L$, is considered as measurement uncertainty. Thus, at temperatures slightly above the glass transition, such as those considered in this paper, maximum relative uncertainties of up to about 3.5% are assumed for $ \delta D / D$ in Eqs. (36) and (37). In future work, quantitative image analysis could be used to assess the sample diameter to further decrease the uncertainty of the Poisson’s ratio measurement.

For axial and angular displacements, $\delta s o=0.01 \mu m$ and $\delta \phi =10 \mu rad$ were assumed. For force and torque, $\delta F=3 mN$ and $\delta M=0.1 \mu N m$ were assumed, respectively. These values represent realistic minima, below which other physical effects are likely to distort the measurement signal in a typical lab environment [79–82]. For further information on rheometer-related disturbances, we recommend Chap. 5 of [79].

Other sources of uncertainty may include the consideration of instrumental inertial effects. These can significantly affect rheometric results in the CMT mode at high frequencies, especially for samples with low moduli or weak gel network structures [79,80,83–86]. In our experiments, this would be particularly critical at higher temperatures. To avoid measurement artifacts, it is critical to distinguish between the torque or force required to deform the sample and the torque or force required to accelerate the instrument. To determine the influence of instrument inertia, a comparison was performed between the total torque and the sample torque, as well a comparison between the total force and the sample force [68,79,83]. A general guideline is to be cautious of rheological results when the total torque exceeds the torque required to deform the specimen by approximately two orders of magnitude [79]. Translating this criterion into a threshold for the raw motor phase [84] would mean that the raw phase should not be greater than approximately 171°. We deliberately avoided introducing additional potential sources of uncertainty related to instrumental inertial effects into the complex Poisson ratio estimation. Therefore, as a conservative approach, we excluded all measurements above 10 Hz. In that frequency range, and for the geometries used in this study, the total force and total torque always remain in the same order of magnitude as the force and torque required to deform the sample. This is true, independent of the temperature and frequency of the experiments. Thus, in the selected frequency range, we do not include an additional source of uncertainty related to inertial effects for Eqs. (36) and (37).

In the glassy state, when the moduli are at their maximum value, effects such as torsional or axial compliance must also be considered [79]. To avoid compliance issues, it must be ensured that $| G \u2217|< C S S C S D 1 Y R$ and $| E \u2217|< C N S C T S 1 Y E$, where $ Y R$ and $ Y E$ are the torsional compliance and axial compliance, respectively. The torsional measurements are within a fairly safe window over the entire temperature and frequency range. For the extensional case, however, we are at the limit of the safety window at temperatures corresponding to the glassy state when using 5 mm diameter samples. This is the necessary trade-off to ensure that the force and torque values remain in a safe interval as the specimen softens. Recall that performing the measurement with a small strain (maximum 0.01%) and within the linear viscoelastic range is also a fundamental criterion [1]. At the same time, the sample diameter is large enough to minimize creep when the glass transition temperature is reached. When using 5 mm specimens, axial calibration is, therefore, of utmost importance, and an additional uncertainty associated with measuring axial compliance in the instrument must be assumed. The corrected value of the axial displacement is given by $s= s o\u2212F Y E$, where $ s o$ is the constant component of the displacement as provided by the axial encoder. Therefore, the uncertainty in *s* can be calculated as $\delta s= ( \u2202 s \u2202 s o ) 2 \delta s o 2 + ( \u2202 s \u2202 F ) 2 \delta F 2 + ( \u2202 s \u2202 Y E ) 2 \delta Y E 2$, where $\delta Y E$ represents the total uncertainty in the axial compliance of the device. Axial compliance was measured in the instrument using a 7 mm diameter steel bar according to the instrument calibration protocol. A value of 0.580 ± 0.001 μm/N was measured after repeating the calibration protocol several times. Thus, a $\delta Y E=0.001 \mu m / N$ was used to estimate the total uncertainty on $\delta s$.

The uncertainty in the phase shift angle obtained from torsional and extensional measurements must be also considered. Considering several replicate measurements performed on PMMA between 10 and 0.1 Hz, we observed a standard deviation in the order of 0.001 rad in both phase shift angles. Therefore, $\delta \delta E=\delta \delta G=0.001 rad$ was set for uncertainty analysis. We note that this value again represents a realistic achievable value that includes external factors such as clamping, to cite an example, and is beyond device performance. Similar standard deviations were observed in the measurements performed on other polymers.

Figure 15(a) shows the relative uncertainty $ \delta \nu \u2032 / \nu \u2032$ of the unannealed PMMA sample as a function of temperature and frequency, after substituting the primary variables and their respective uncertainties in Eq. (36). The relative uncertainty in the real part of the complex Poisson’s ratio is about 9% at the lowest temperature and remains of the order of 5% over a temperature range between room temperature and 115 °C, independent from frequency. At 115 °C, approximately the glass transition temperature, uncertainty analysis shows that $ \delta \nu \u2032 / \nu \u2032$ ranges from 7% at 10 Hz to 22% at 0.1 Hz. The significant increase in the measurement uncertainty after reaching the glass transition is mainly due to two factors. First, the diameter uncertainty is at its maximum at these temperatures. The main factor is that both the force and torque required to maintain the set strain are strongly reduced as the sample softens considerably. If the specimen becomes too soft, the uncertainty in the measurement will be too high. These effects are particularly strong in the case of amorphous polymers, where softening is more pronounced [see Fig. 4(a) for PMMA or Fig. 9(a) for PC]. Indeed, at temperatures higher than 120 °C, Fig. 15(a) shows that the measurement uncertainty exceeds 10%, independent of the frequency. The total uncertainty in $\delta \nu \u2032$ and $\delta \nu \u2032 \u2032$can be used to estimate the total uncertainty in $| \nu \u2217|$, $\delta | \nu \u2217|$, which remains very close to that obtained for $\delta \nu \u2032$ [compare Figs. 15(a) and 15(d)]. Figure 15(b) depicts $ \delta \nu \u2032 \u2032 / \nu \u2032 \u2032$ as a function of temperature and frequency after substituting the primary variables and their respective uncertainties in Eq. (37). While the contribution of $ \delta E$ and $ \delta G$ is only in the order of 0.2% for the total uncertainty in $ \nu \u2032$, they provide the largest contribution (of the order of 30%) to the total uncertainty in $ \nu \u2032 \u2032$. The relative uncertainty in $ \nu \u2032 \u2032$ is considerably larger than in the case of $| \nu \u2217|$ or $ \nu \u2032$ and becomes maximum at around 100 °C when $ \delta E= \delta G$ .

Figure 15(c) shows $| \nu \u2217|$ and $\u2212 \nu \u2032 \u2032$ as a function of frequency and temperature with a colored area reflecting the absolute uncertainty in the measurement. Values in the complex Poisson’s ratio above 0.5 at temperatures above the glass transition are probably due to measurement uncertainty at these levels of force and torque. Interestingly, for the unannealed PMMA specimen, the minimum in the complex Poisson’s ratio is still within an uncertainty of about 7%.

### B. Phase shift angle offset between torsion and extension

The method presented here allows to obtain an imaginary part in the complex Poisson’s ratio as derived from the theory of viscoelasticity [1]. According to this, an imaginary part should appear in the complex Poisson’s ratio as long as $ \delta E\u2260 \delta G$ [see Eq. (35)]. For illustrative purposes, Fig. 15(c) shows $ \nu \u2032 \u2032$ for the unannealed PMMA sample as a function of temperature and frequency, with a colored area indicating the measurement uncertainty. The trend in the measurement showing $\u2212 \nu \u2032 \u2032$ decreasing with temperature appears to be valid, despite the measurement uncertainty. $ \nu \u2032 \u2032$ remains below 0 over the entire frequency and temperature range. The same trend is noticed for all polymers studied [see, for instance, Figs. 10(d), 12(b), 13(d), and 14(d)]. Only at temperatures near the glass transition temperature, some values of $ \nu \u2032 \u2032$ are slightly above zero as observed for PMMA [see Fig. 8(a)], PC [see Fig. 10(d)], or PC + ABS [see Fig. 13(d)]. We believe that this is due to the uncertainty of the measurement at these temperatures [see Fig. 15(b)]. We recall that $ \nu \u2032 \u2032\u22600$ represents a certain time delay between axial and transverse deformations due to material damping. Independent of the convention, one could hypothesize that for nonauxetic materials, a sign change in $ \nu \u2032 \u2032$ during the measurement loses any pragmatic physical sense since this would imply a positive time offset between axial and transverse deformations when the axial strain is superimposed or vice versa.

The results can be interpreted in terms of the loss factor, which correlates with the energy consumed by the sample in the form of internal dissipation. The imaginary part of the Poisson’s ratio can only be nonzero if the phase shift angles in torsion and extension deviate [see Eq. (15) or (35)]. In other words, only if energy dissipation in the internal structure of the specimen is somewhat different depending on the deformation mode, there is a delay between axial and transverse deformations. $tan \delta E>tan \delta G$ was always observed in the glassy state of our experiments, regardless of the polymer studied [see Fig. 4(b), 9(b), 11(b), 13(b), or 14(b)]. At these temperatures, the complex Poisson’s ratio was found to be different from 0.5, indicating a certain compressibility in the material. We can, therefore, try to relate the compressibility of the material to the imaginary part of Poisson’s ratio. The more compressible the material, the greater the relative volumetric change that can be expected in the extensional measurement. Note that the relative change in volume resulting from uniaxial extension can be estimated as $ \Delta V / V\u2248( 1 \u2212 2 | \nu \u2217 |) \epsilon A$, when neglecting the higher strain terms [1]. Even with a small extensional strain, some volume change is expected in extensional measurements when $| \nu \u2217|\u22600.5$. However, no volumetric change is expected when the sample is torsionally deformed. We hypothesize that this is the reason because the sample experiences a different, though slightly different, way of dissipating energy in the glassy state when $| \nu \u2217|\u22600.5$. A different way of energy dissipation as characterized by $tan \delta E>tan \delta G$ can be read from all our measurements in the glassy state. To the best of the authors’ knowledge, this finding is new in the literature for the case of dynamic mechanical analysis. It should be noted that, until now, it has been difficult to obtain $tan \delta E$ and $tan \delta G$ with mechanical excitation on the same sample and under the same conditions with sufficient level of confidence as required by the Tschoegl protocol [1].

It remains an open question to find a physical explanation for a maximum in $\u2212 \nu \u2032 \u2032$ as observed in the unannealed PMMA measurement [see Figs. 8 and 15(c)]. Other authors have found a maximum in the imaginary Poisson’s ratio for PP and HDPE [57,61]. In our case, however, this maximum was found only in unannealed amorphous polymers. Here, it seems that a volumetric change related to the reorganization of the polymer chains at a temperature higher than the glass transition [see the macroscopic shrinkage as shown in $ \Delta L / L$ in Fig. 6(c)] could explain the phenomenon. However, at these temperatures, it appears that the uncertainty in $ \nu \u2032 \u2032$ is also large in our measurements. Due to this limitation, more experiments are needed for a clear assessment.

In all experiments, except for the PP sample, the main glass transition temperature obtained from a maximum of the loss moduli or loss factor, was the same regardless of the deformation mode [see Figs. 3(b) and 3(c), 9(b)–9(d), 13(b), or 14(b)]. Interestingly, for the case of homopolymer PP, the β-transition of the amorphous fractions of PP shows a significant dependence on the deformation mode [see Figs. 11(b)–11(d)]. In the case of PP, the transition of the amorphous region occurs when it is confined within the crystallinity region [87]. The β-transition occurs when the complex Poisson’s ratio is of the order of 0.3 and $tan \delta G andtan \delta E$ are still significantly different. The β-relaxation, in this case, seems to happen at lower temperatures in extension than in torsion [see Fig. 11(d)]. Interestingly, a delay in the relaxation process is consistent with a nonzero imaginary Poisson’s ratio (since at these temperatures $ \delta E\u2260 \delta G$). The crystalline region seems to play a very important role here, making the interpretation of this relaxation shift much more complex than in the amorphous case. The authors hypothesize that obtaining information about the loss factor in two loading directions in the same sample and under the same conditions may help to better interpret the complex relaxation processes in semicrystalline polymers. However, this is beyond the scope of this paper and the connections between bulk rheological properties and microscale structural features remain to be explained in future studies.

In the vicinity of α-relaxation, as the material approaches incompressibility, and for isotropic materials such as those examined in this study, the mode of energy dissipation appears to be similar in both torsion and extension, based on the dynamic mechanical analysis method. This is true for all the specimens examined. However, the authors also hypothesize that this should not necessarily be the case for anisotropic specimens. Relaxation processes that are slightly shifted in temperature could be measured on the same specimen depending on the selected deformation mode, i.e., torsion or extension. It would, therefore, be interesting to test this hypothesis in the case of anisotropic materials. Again, we leave this task for future work.

### C. Comparison with literature data

Figure 16 compares the complex Poisson’s ratio values obtained in this study with the literature data. The error bars represent the uncertainty of the measurement as obtained from the primary variables. For PMMA [Fig. 16(a)], Yee and Takemori’s results from oscillatory measurements at 1 Hz [30] and Frank’s results obtained from quasistationary experiments [11] overestimate our data in a temperature range from about −30 to 70 °C. Above about 70 °C, their measurements are in good agreement with ours. The data obtained by Schenkel [44] show excellent agreement with our results for the annealed sample over the entire temperature range. On the other hand, Kono’s results obtained by wave-propagation techniques [55] show a logical shift of the results toward higher temperatures, since the measurement was performed at 1 MHz. It should be noted that it is not entirely appropriate to compare data obtained by different methods. In addition, different PMMA samples may have different Poisson’s ratio values depending on, for example, the molecular weight, differences in tactility, or the manufacturing process [30]. At room temperature, for example, Poisson’s ratio data between 0.31 and 0.43 are found in the literature [41]. Nevertheless, our experiments yield reasonable Poisson’s ratio data within the measured temperature range when compared with the literature.

The complex Poisson’s ratio of the unannealed PMMA specimen shows a local minimum at a temperature close to α-relaxation. Its appearance has been hypothesized to be related to physical aging effects enhanced by previous quenching during fabrication [77] [compare red circles and blue squares in Figs. 6(a) and 6(c)]. This explains why the minimum, also observed by Kono [55], could be eliminated by performing an annealing step prior to testing [compare red circles and blue diamonds in Fig. 16(a)]. It should be noted that at around 100 °C, the difference in the complex Poisson’s ratio between the annealed and unannealed samples is significant according to the uncertainty study. A difference in the complex Poisson’s ratio of quenched and annealed PMMA samples was also reported by Yee and Takamori [30]. However, their data are limited to a maximum temperature of 100 °C, which is below the range of the local minimum observed here. Apart from aging-related issues, another explanation for the local minimum can be given from a material modeling perspective. Grassia *et al*. proposed to use Kohlrausch–Williams–Watts (KWW) functions to describe the relaxation functions of both the shear modulus and the bulk modulus [2]. Based on this, they derived time-dependent curves for the complex Poisson’s ratio passing through α-relaxation. They emphasize that the relaxation times in shear $ \lambda G$ and bulk $ \lambda K$, which are needed to fit the KWW functions, may differ from each other. This behavior was demonstrated for several linear polymers. For these types of materials, the viscoelastic shear response occurs at longer time scales than the bulk response, implying that $( \lambda G/ \lambda K)\u226b1$. Note that this is also consistent with a bulk relaxation motion of intramolecular nature rather than intermolecular nature [30]. Interestingly, this condition coincides with the appearance of a local minimum in the temperature-dependent complex Poisson’s ratio [2]. Consequently, if we assume that the local minimum is caused by a combination of manufacturing-induced molecular orientation and physical aging, it follows that the viscoelastic response in shear and bulk modes is differently affected in dependence on the thermal and mechanical history of the polymer. As an example, this would imply that either the viscoelastic bulk relaxation response is shifted to shorter times or the viscoelastic shear relaxation response is shifted to longer times when the polymer is in a quenched state relative to the annealed state. Further experiments are required to test this hypothesis, which we leave for future studies.

Figure 16(b) shows the data for PC and polymer blend PC + ABS. At room temperature, Poisson’s ratio values for PC between 0.35 and 0.42 are found in the literature for quasistatic tests [41]. Our values vary between 0.38 ± 0.01 and 0.41 ± 0.02 depending on the measured frequency. Therefore, the data are within the expected range. For PC + ABS, values between 0.35 and 0.37 are reported [41]. Our experiments show a complex Poisson’s ratio of 0.38 ± 0.01, which is also within the expected values. Two replicate measurements performed on two different PC samples collapse completely in the diagram showing the reproducibility of the experiment [compare red circles and blue diamonds in Fig. 16(b)]. PC and PC + ABS show very similar values for the complex Poisson’s ratio up to temperatures of approximately 95 °C. At this temperature, close to the main glass transition of the polymer blend, PC + ABS shows values around 0.5, diverging from PC. The temperature dependence of Poisson’s ratio for PC or PC + ABS was difficult to find in literature. In Chap. 1.1 of [42], a curve for PC can be found, with the literature cited there corresponding to [11]. However, we could not find any PC measurement in this temperature range in [11]. We cannot report how the measurement was performed, but the agreement of our experimental data with the data reported there is very good [compare the dashed red line with red circles in Fig. 16(b)].

Figure 16(c) compares the results for homopolymer PP with the data obtained by Frank in quasistatic tests [11] and with the data obtained by Waterman using wave-propagation methods at 5 MHz [61]. Data from [41] are also shown [see dotted red line in Fig. 16(c)]. Unfortunately, we could not find out by which method these data were obtained. Up to about 10 °C, the literature data overestimate our results. Between 10 and 50 °C, our results are in full agreement with the literature within the range of uncertainty. Above 50 °C, our trend shows slightly lower values than the literature data. For the semicrystalline case, the comparison between different PP samples or different measurement methods is even more critical than for the amorphous case. The history of the polymer during its manufacturing process may have an effect on the crystallinity of the polymer and, most likely, on the subsequent trend of the complex Poisson’s ratio with temperature. Nevertheless, the comparison with the literature suggests again that our data are within the expected values, at least in a temperature range starting from 10 °C. Moreover, replicate measurements over a wide temperature range, as shown in Fig. 2 for TPU, or over a wide temperature and frequency range as in the case of PMMA or PC [see Fig. 16(a) or 16(b)], show standard deviations in the complex Poisson’s ratio of less than 7%. Thus, these standard deviations are in good agreement with the estimated uncertainty values from the uncertainty analysis calculated in Sec. VII A.

Finally, Fig. 17 shows the results of a TTS master curve on the annealed (closed symbols) and unannealed (open symbols) PMMA samples as a function of the reduced frequency, $f a T$. The curves were shifted at a reference temperature of 100 °C, a value close to the main glass transition of the material. RheoCompass^{TM} software was used to horizontally shift the curves based on isothermal frequency sweeps obtained at every 5 °C in both torsion (blue diamonds) and extension (black squares). RheoCompass^{TM} uses a numerical algorithm that automatically finds the horizontal shift parameters based on the minimization of the root mean square error. Exactly the same settings in the software were used for the shift of torsional and extensional data. For the unannealed sample, a temperature range between −20 and 150 °C was considered. For the annealed sample, only a temperature range between 80 and 120 °C was covered in the experiments. Frequency values above 10 Hz were discarded for curve shifting.

The horizontal shift factors for the annealed (open symbols) and unannealed (closed symbols) PMMA samples are shown in Fig. 17(b) for the torsional (blue diamonds) and extensional (black circles) cases.

In the frequency range between approximately 10 and 10^{4} Hz, the moduli obtained from the two master curves overlap with a confidence interval of 95% [see Fig. 17(a)]. Therefore, no significant difference in the moduli is noticed in this frequency domain. Below about 10 Hz, the two curves diverge. At frequencies below 10^{−2} Hz, corresponding to high temperatures, the annealed sample shows $| E \u2217|$ and $| G \u2217|$ values up to 30% higher than the unannealed sample. Figure 17(b) also shows some discrepancy in the horizontal shift factors between the annealed and unannealed samples from a temperature of about 110 °C (compare open and closed symbols). Perhaps, it is worth noting that in the case of the annealed sample, both shift factors are practically independent of the deformation mode. This would imply that for the annealed PMMA, the viscoelastic shear response occurs at similar timescales as the extensional response near α-relaxation. This is consistent with the main glass transitions obtained at the same temperature values, independent of the deformation mode, as shown for PMMA [see Fig. 4(b) or 4(c)]. In the case of the unannealed sample, there appears to be a difference between the two shift factors at temperatures between −20 °C and about 60 °C. This would be consistent with $tan \delta G\u2260tan \delta E$ in a temperature range corresponding to the glassy state. Differences in the shift factors may imply that, in that range covering the β-relaxation, the viscoelastic shear response and the extensional response occur at slightly different time scales.

The complex Poisson ratio obtained from the data generated on the master curve shows good agreement between annealed and unannealed samples up to about 100 Hz. Above this frequency, the data obtained from the annealed specimen show slightly higher values, with what appears to be a plateau at a complex Poisson’s ratio between 0.37 and 0.40. On the other hand, the unannealed sample, measured over a much wider temperature range, shows a monotonic decrease followed by a plateau at values around 0.22, at a frequency of 10^{12} Hz. It should be noted that the master curve of the unannealed sample also includes the β-relaxation of the PMMA sample. For illustrative purposes, a fit using the Williams, Landel, and Ferry (WLF) equation is plotted in Fig. 17(b) according to $ a T=a( T \u2212 100)/b+( T \u2212 100)$. The fit is limited to temperatures near α-relaxation (between 80 and 120 °C). The fit is adjusted for the unannealed torsional (blue dashed line) and extensional (black dashed-dotted line) cases, as well as for the annealed torsional (red solid line) and extensional (green dotted line) cases. In this temperature range, all fit curves appear to be quite close, regardless of the deformation mode, although it appears that the fit curves for the annealed case have slightly lower absolute values of the coefficients *a* and *b* than in the unannealed case [all fit coefficients of the WLF model are summarized in the legend of Fig. 17(b)].

The experimental data are compared with the master curve obtained by Kästner and Pohl at a reference temperature of 101 °C [46]. The dashed red line in Fig. 17(a) is reproduced from [1] according to [46]. Our curve obtained on PMMA after annealing fully agrees with Kästner and Pohl's data from frequencies around 0.1 Hz. Note that although Kästner and Pohl reported a master curve on $ \nu \u2032$ instead of $| \nu \u2217|$, the comparison is still valid since $ \nu \u2032 \u2032$ remains much smaller than $ \nu \u2032$. The good agreement with direct measurements as obtained by [46] again supports the validity of the indirect protocol presented here.

## VIII. CONCLUSIONS

A novel combined torsional-extensional rheometer proved effective in indirectly measuring the complex Poisson’s ratio as a function of the frequency and temperature of different polymers. The use of cylindrical specimens and careful sample fabrication minimized the relative uncertainty in measurements. The experimental uncertainty assessment shows that the complex Poisson’s ratio can be obtained with an uncertainty of about 4%–7% in the polymer samples with a diameter of 5 mm, until the sample reaches a complex Poisson’s ratio close to 0.5 near alpha relaxation. Above this temperature, the uncertainty increases to over 10%, mainly due to sample softening. This assessment does not consider artifacts related to inertial effects because the maximum frequency to consider is cut off at 10 Hz. In addition, this assessment assumes that any change in diameter due to thermal expansion is a measurement uncertainty. For most samples, the complex Poisson’s ratio increased monotonically toward 0.5 as the temperature approached α-relaxation, except for amorphous PMMA, which showed a local minimum that disappeared after thermal annealing of the sample. The complex Poisson’s ratio decreased with increasing frequency, especially near α-relaxation.

A significant offset in the phase shift angles in the glassy state of all polymers was found, indicating a nonzero imaginary part of the complex Poisson’s ratio, and implying a distinct mode of energy dissipation when $| \nu \u2217|\u22600.5$. As the temperature approached the main glass transition and $| \nu \u2217|=0.5$, the offset approached zero. Only semicrystalline PP showed a difference in the phase shift angles after the relaxation of the amorphous region. For PP, a slightly shifted β-transition temperature was obtained, depending on the deformation mode used.

Our results were compared with the existing literature, showing good agreement and confirming the applicability of the method up to α-relaxation temperatures. This is of special relevance due to the limited availability of reliable Poisson’s ratio data in the literature. The indirect approach presented here is unaffected by invasive contact-based devices such as strain gauges, and optical access is not required for the measurement. In addition, the dependence of Poisson’s ratio on frequency and temperature is obtained in a single measurement, resulting in significant time savings, compared to other methods.

The present work has been limited to thermoplastic polymers. Although we have used TPU, which can be considered a thermoplastic elastomer, it remains for future contributions to carry out systematic tests on rubbers. In future work, we would also like to carry out measurements on thermosets, in particular, to test the feasibility of the method for measuring the complex Poisson’s ratio as a function of the degree of cross-linking.

The method also seems to be of great interest for measurements on anisotropic samples such as composites. In this case, it may not be possible to use equations such as those in Eq. (4) to derive the complex Poisson’s ratio. However, performing torsional and extensional measurements on the same sample under the same conditions can provide further information about the anisotropy of the sample. It would be interesting to measure how the $ E \u2217 / G \u2217$ ratio varies with frequency and temperature. Furthermore, it is very likely that the phase shift angles in the two loading directions are completely different, as a consequence of different ways of energy dissipation depending on the anisotropy of the sample. The ratio between the two phase shift angles is expected to represent a novel fingerprint parameter for the characterization of viscoelastic anisotropic materials.

It remains for future contributions to perform indirect measurements of the viscoelastic Poisson’s ratio in the time domain, for example, by performing relaxation tests in both directions. Finally, it also is left for future contributions to consider measurements outside the linear viscoelastic regime to study possible nonlinear effects on the behavior of the complex Poisson’s ratio. From a rheometric point of view, this is of course challenging since sequential measurements on the same specimen are probably not possible, so the simultaneous application of mechanical excitation in torsion and tension by orthogonal superposition appears as a promising alternative.

**ACKNOWLEDGMENTS**

The authors would like to thank Dominik Fauser and Professor Holger Steeb of the Institute of Applied Mechanics at the University of Stuttgart and Dr. Heiko Stettin of Anton Paar GmbH for the many hours of discussion during the preparation of the manuscript. The authors are also grateful to Dr. Daniel Treffer, founder of MeltPrep, for the loan of the necessary Vacuum Compression Molding (VCM) tool used to produce some of the samples.

## AUTHOR DECLARATIONS

### Conflict of Interest

At the time of submission, José Alberto Rodríguez Agudo, Jan Haeberle, Abhishek Shetty, and Christopher Giehl were employees of Anton Paar GmbH, the company that commercially markets the rheometer used in this study. The other authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

## REFERENCES

*Die Querkontraktionszahl von Kunststoffen, Dargestellt am Beispiel Amorpher Thermoplaste*

*KunststoffXtra*

*KunststoffXtra*

*Polymer Solids and Polymer Melts–Mechanical and Thermomechanical Properties of Polymers*

*Mit Kunststoffen Konstruieren: Eine Einführung*

*Saechtling-Kunststoff-Taschenbuch*

*Das Dreidimensionale Viskoelastische Stoffverhalten im Großen Temperatur- und Zeitbereich am Beispiel Eines in der Automobilen Aufbau- und Verbindungstechnik Verwendeten Epoxidharzklebstoffs*

*The Determination of Dynamic Properties of Polymers and Composites*

*Technomer 2019 26. Fachtagung über Verarbeitung und Anwendung von Polymeren*(Institut für Fördertechnik und Kunststoffe, Technische Universität Chemnitz, Chemnitz, 2019).

*Kautsch. Gummi Kunstst.*

**66**(4), 48–54 (2013).

*Klassiker der Technik*

**6**(3),131–136 (2009).

*Complex Fluids in Biological Systems: Experiment, Theory, and Computation*

*The Rheology Handbook: For Users of Rotational and Oscillatory Rheometers*

*Instrument Inertia Correction During Dynamic Mechanical Testing*(TA Instruments, New Castle, DE, 2005).