Requisites on material viscoelasticity for exceptional points in passive dynamical systems

Recent progress in non-Hermitian physics and the notion of exceptional point (EP) degeneracies in elastodynamics have led to the development of novel metamaterials for the control of elastic wave propagation, hypersensitive sensors, and actuators. The emergence of EPs in a parity-time symmetric system relies on judiciously engineered balanced gain and loss mechanisms. Creating gain requires complex circuits and amplification mechanisms, making engineering applications challenging. Here, we report strategies to achieve EPs in passive non-Hermitian elastodynamic systems with differential loss derived from viscoelastic materials. We compare different viscoelastic material models and show that the EP emerges only when the frequency-dependent loss-tangent of the viscoelastic material remains nearly constant in the frequency range of operation. This type of loss tangent occurs in materials that undergo stress-relaxation over a broad spectrum of relaxation times, for example, materials that follow the Kelvin–Voigt fractional derivative (KVFD) model. Using dynamic mechanical analysis, we show that a few common viscoelastic elastomers, such as polydimethylsiloxane and polyurethane rubber, follow the KVFD behavior such that the loss tangent becomes almost constant after a particular frequency. The material models we present and the demonstration of the potential of a widely available material system in creating EPs pave the way for developing non-Hermitian metamaterials with hypersensitivity to perturbations or enhanced emissivity.

While a plethora of research has studied EPs and their applications in PT-symmetric systems with balanced gain and loss, entirely passive (with no gain) systems with differential-loss have also been shown to exhibit EPs [1,8,15,22,23]. Loss or dissipation is often found to be an intrinsic property of materials, whereas gain is induced often by pumping energy into the system from external sources via piezoelectric [24,25], electroacoustic [7,11], piezoacoustic [26,27], non-Foster circuits [28], and electromagnets [29]. Creating gain needs sophisticated, active, positive feedback control circuits, making systems energy expensive, bulkier, and difficult to control each active element [25,[28][29][30]. Realizing EPs in passive systems can make implementation more straightforward and integration into devices and structures seamless for engineering applications. EPs are branch point singularities in the parameter space of a physical system where the eigenvalues and corresponding eigenvectors coalesce and become degenerate [31,32]. In contrast to degeneracy points in Hermitian Hamiltonians (diabolic points), where the bifurcation is linear, EPs exhibit higher-order bifurcations [33,34] (figure 1). Any perturbation in the vicinity of an EP results in a bifurcation of degenerate eigenvalues in the orthogonal direction (orthogonal bifurcation), making EPs very sensitive to external interference. The sensitivity of EPs has been exploited in developing hypersensitive gyroscopes [35,36], nanoparticle sensors [4,33,34,37], and accelerometers [14,38]. However, achieving a similar level of sensitivity in passive non-Hermitian systems is challenging regardless of their other utility, such as in enhancing emissivity in proximity to the EPs [15]. Unlike the sharp orthogonal EP phase transition in PT-symmetric systems, bifurcations in passive non-Hermitian systems exhibit an approximate transition with an avoided crossing of eigenmodes [1,39,40].
In this article, we report material design considerations for realizing EPs with sharp bifurcation in a passive non-Hermitian metamaterial in an elastodynamic framework. Such a metamaterial can be realized by constituent elements containing a coupled mechanical oscillator (dimer) having differential loss. Incorporating a resonant element with a viscoelastic material for the dissipative component, while the other resonator being an elastic conservative element will allow introducing non-Hermiticity to the metamaterial. Until now, theoretical investigations to realize EPs in coupled mechanical oscillators with passive differential damping have predominantly been limited to employing springs and viscous dashpots as a means to introduce non-Hermiticity [39,41,42]. Nevertheless, the springs and dashpots inadequately capture the physical characteristics exhibited by the majority of real viscoelastic materials, such as common rubbers and polymeric foams. While numerous plastics, rubbers, and foams exhibit viscoelasticity that can be described by different viscoelastic models, not every viscoelastic material will lead to a non-Hermitian metamaterial with a clear EP. The inherent-coupling between the storage and loss moduli (i.e. the real and imaginary parts of the complex dynamic modulus) by the Kramers-Kronig relation [43,44] makes the experimental realization a significant challenge and requires a judicial design [15]. We investigate different viscoelastic material models in comparison to a coupled oscillator model consisting of differential structural damping and formulate requisites on the frequency-dependent loss tangent to achieve a sharp EP bifurcation that is critical for exploiting its hypersensitivity to system parameters. Our findings will guide the experiments [15] on achieving sharp EPs in systems containing viscoelastic materials to create non-Hermiticity. Because the governing non-Hermitian Hamiltonian of our system is mathematically similar to all other bimodal physical systems, our results can be applied broadly to other fields of physics as well.
This article has the following structure: we first discuss general rate independent frictional (structural) damping and compare the occurrence of EPs in a coupled oscillator dimer for the PT-symmetric case with balanced gain and loss and for the passive non-Hermitian case with differential loss. In the following sections, we revisit the basics of viscoelastic materials to test the feasibility of using them in creating non-Hermiticity. We individually analyze three different viscoelastic material models to identify the best suited viscoelastic material to create a passive elastodynamic metamaterial that supports the formation of an EP. Finally, we report on experimentally measured dynamic properties of viscoelastic elastomers-polydimethylsiloxane (PDMS), polyurethane rubber, natural rubber, and polyurethane foam and demonstrate a concept experimental design to achieve EPs.

Coupled oscillators with differential frictional damping
Discrete coupled oscillators exhibit two individual vibration modes, one in-phase and one out-of-phase. For Hermitian systems, the resonant frequencies of these modes are real and distinct. However, these modes begin interacting with each other when appropriate damping is introduced, resulting in an EP degeneracy.
The most common source of damping in structures is Coulomb friction, which opposes the relative motion between surfaces. A coupled oscillator system with frictional damping is shown in figure 2(a). Two oscillators of masses m 1 and m 2 and Hookean springs of stiffnesses k 1 and k 2 , respectively, are mounted on rigid walls and coupled together by another Hookean spring of stiffness κ. Frictional forces of magnitudes γ 1 N 1 and γ 2 N 2 are resisting the motion of m 1 and m 2 , respectively, where γ 1 , γ 2 are coefficients of friction and N 1 , N 2 are the contact forces. The governing equations of motion for the oscillators can be written as where x 1 (t) and x 2 (t) are the instantaneous positions of m 1 and m 2 oscillators respectively, and sgn is the signum function, e.g. sgn (ẋ 1 ) =˙x 1 |ẋ1| . In a system with Coulomb damping, free vibrations undergo a gradual linear decay in oscillations, but it is not possible to arrive at a steady-state solution. Thus, we consider structural damping by assuming N 1 = k 1 x 1 and N 2 = k 2 x 2 , which means that the friction force is due to the structural (or hysteretic) damping property of the spring's constituent material. Considering steady state solutions for x 1 and x 2 , i.e. x 1 (t) = X 1 e iωt and x 2 (t) = X 2 e iωt , which gives sgn (ẋ 1 ) = iωX1 where X 1 and X 2 are the complex amplitudes of the vibrations of the two individual oscillators. Unlike viscous damping, structural damping is rate-independent. It is usually found in metals, where it arises due to crystal defects and internal friction from grain boundary sliding.
Assuming m 1 = m 2 = m and k 1 = k 2 = k for symmetry, and ω 0 = √ k m , Ω = ω ω0 , andκ = κ k for making the parameters dimensionless, the final equations of motion in matrix form can be written as For non-trivial solutions of eigenvalues (Ω) and eigenvectors {ψ}, substituting the determinant of [H] equal to 0 yields, Solving the above equation yields the two complex natural frequencies of the system, Ω 1 and Ω 2 . In figure 2, we plot the real and the imaginary parts of Ω 1 and Ω 2 as functions of normalized coupling (κ) for both PT-symmetric (figure 2(b)) and passive non-Hermitian (differential loss only) cases (figure 2(c)). To balance the gain and the loss in the PT-symmetric case, we consider γ 1 = −γ and γ 2 = γ = 0.25. As the coupling increases, an EP emerges atκ ep = γ (refer to SI for derivation). In the passive non-Hermitian case with no gain, we consider the simplest differential-loss condition (zero loss-loss) by assuming γ 1 = 0 and γ 2 = γ = 0.5 (refer to figure S1 in SI for other cases of differential loss). In such a system, the EP appears at κ ep = γ 2 (refer to SI for derivation). Noteworthy is the sharp and orthogonal bifurcation at the EP. Here, sharpness and orthogonality are characterized by a zero gap between the complex natural frequencies i.e. |Ω 1 − Ω 2 | = 0 and a slope of tan −1 ( ∂Ω ∂κ ) ≈ π/2 at the EP-which implies that an infinitesimal variation inκ will result in a large splitting of frequencies (i.e. Ω 2 − Ω 1 ), suitable for highly sensitive EP-based sensors. It is worth highlighting that, unlike the PT-symmetric case (figure 2(b)), where eigenfrequencies are real in the exact-phase (κ >κ ep ) and complex in the broken-phase (κ <κ ep ), eigenfrequencies in the passive non-Hermitian case are always complex. Thus, the Hamiltonian for the passive non-Hermitian case does not need to satisfy the pseudo-Hermitian condition to exhibit an exceptional point [45]. We also observe a similar bifurcation in corresponding eigenvectors ψ = [X 1 /X 2 , 1] (figure 3) for the passive non-Hermitian  case, a hallmark feature of EPs. At the EP, the magnitude of eigenvectors | X1 X2 | ≈ 1 and phase difference arg( X1 X2 ) ≈ π /2 for both the modes, which suggest that the modes are degenerate. Conclusively, we have shown a viable pathway to realizing an EP degeneracy in a passive (with no gain) physical system with differential loss. Despite the simplicity, the above system with differential structural damping exhibits a sharp EP degeneracy in a 2 × 2 Hamiltonian system similar to coupled optical waveguides [46], LCR circuits [47], acoustic cavities [48], microwave cavities [49], and quantum harmonic oscillators [50].

Viscoelastic solids as dissipative elements
While structural damping is the simplest model for material damping, it is not the most common type of damping present in engineering materials that are used in applications requiring damping. Usually, engineering materials exhibit viscous damping, which is intrinsically connected to materials' elastic properties, making the materials viscoelastic [43,44]. In contrast to structural damping, where the stiffness and damping parameters (k 1 , k 2 , γ 1 , γ 2 ) are constants, viscoelastic materials are rate dependent. Viscoelastic materials distinctively exhibit a time decaying relaxation modulus (E(t) = E ∞ + E t (t)) whose Fourier transform gives a complex dynamic modulus with frequency dependent real and imaginary parts is called the storage modulus, which is associated with the amount of elastic energy that the material stores under deformation. The imaginary part E ′ ′ (ω) is called the loss modulus, which is associated with the energy that the material dissipates under cyclic loading. The ratio , known as the loss-tangent, is a quantitative measure of the damping capacity of a viscoelastic material.
To realize the non-Hermitian metamaterial, we replace the structurally damped spring of the coupled oscillator model with a viscoelastic material of dynamic modulus E d (ω), cross-sectional area A, and length L (figure 6(a)). The resultant dynamic stiffness of the material is given by The modified governing equations for the non-Hermitian metamaterial with viscoelastic damping elements in frequency domain can be written as, where X 1 and X 2 are complex amplitudes,κ = κ k , Ω = ω ω0 , ω 0 = √ k m , and k is the stiffness of Hookean spring (non-lossy) as shown in figure 6(a).
We investigate three different models of viscoelastic materials in comparison: the Kelvin-Voigt (KV) model, the standard linear solid (SLS), and the KV fractional derivative (KVFD) model. The relaxation moduli and the corresponding frequency-dependent dynamic moduli of the three viscoelastic materials are listed in table 1.

KV materials
KV is the simplest model of a viscoelastic solid. It consists of a spring and a Newtonian dashpot in parallel combination ( figure 6(b)). Under sinusoidal loading, the complex dynamic modulus of the KV solid is given by where E ∞ is the modulus of the spring, η is the viscous damping coefficient, and ω is the frequency of the dynamic load. Notably, KV materials exhibit a frequency-independent storage modulus (E ∞ ) and a loss modulus (ωη) that is linearly dependent on frequency. Assuming k = E ∞ A L for symmetry in real part of stiffness and A L η mω0 = γ as the dimensionless damping coefficient, the governing equations can be written in matrix form as follows Here, we first consider the PT-symmetric case (γ 1 = −γ, γ 2 = γ). To balance with loss, the gain must also be linearly dependent on frequency. A physical example of such gain is the self-excitation of spring-block on a moving conveyor due to a velocity-dependent friction coefficient [51]. On putting the determinant of the Hamiltonian matrix in equation (9) equal to zero and solving for Ω, we observed a sharp EP at κ ep = 1 2 (γ 2 + 2γ) (figures 4(a) and (b)) [39], with eigenfrequencies very similar to the PT-symmetric case in structural damping ( figure 2(b)). For the passive non-Hermitian case (γ 1 = 0, γ 2 = γ), the real and imaginary parts of the complex natural frequencies Ω 1 and Ω 2 are plotted as functions ofκ in figure 6(b). Contrary to the structural damping case, the passive non-Hermitian metamaterial made of a KV material does not exhibit a sharp EP bifurcation. Instead, we observe a region of approximate transition [39,40] where the modes gradually bifurcate. In this region, we identify that for a particular coupling of κ c ≈ 1 4 ( 2 5 γ 2 + 2γ ) (figures 4(c) and (d)), the eigenfrequencies make the closest approach without merging such that the gap between them i.e. |Ω 1 − Ω 2 | is minimum ( figure 4(c)). We hypothesize that the indistinct EP transition observed here ( figure 6(b)) is due to a strong linear dependence of tan(δ) on the frequency, i.e. to achieve a sharp EP bifurcation, tan(δ) should converge to a constant value as a function of frequency  (figure 5). The strong dependence of tan(δ) on frequency in KV solid is associated with Dirac delta function (δ(t)) in the transient part of the relaxation modulus (table 1), which indicates that the material relaxes abruptly at t = 0 and does not relax further afterwards (δ(t > 0) = 0). To test our hypothesis of the effect of frequency dependent tan(δ), we next investigate SLS and KVFD viscoelastic solids, which undergo stress relaxation for longer durations.

SLS materials
The SLS is a viscoelastic material model consisting of a spring connected in parallel to a Maxwell element (a spring and a dashpot in series) as shown in figure 6(c). The relaxation modulus of SLS decays exponentially with time, and the corresponding storage and loss moduli are both frequency-dependent (table 1). Assuming E ∞ and E t as the moduli of springs, η as the damping coefficient of the dashpot and τ = η Et as the relaxation time, the dynamic modulus can be written as, Assuming , Ω = ω ω0 , ω 0 = √ k m , and T = ω 0 τ (dimensionless relaxation time), the normalized dynamic stiffness can be written as, .
Since it is physically not possible to create gain and maintain symmetry in material systems with the real part of modulus also frequency dependent, from here onwards, we only investigate passive non-Hermitian systems with differential loss (noLoss-Loss). By substituting equation (11) in equations (6) and (7) and solving for Ω, we obtain two resonant frequencies Ω 1 , Ω 2 . The real and imaginary parts of Ω 1 and Ω 2 are plotted in figure 6(c). In this case, we did not observe any EP formation for all reasonable values of material parameters. An EP only starts to appear when T is very small (T < 1), which is physically not possible. In figure 5, comparison of tan(δ) for T = 20π and T = 0.74 indicates that, for shorter relaxation times, the tan(δ) has a plateau like regime and is almost constant for Ω ≈ 1 resulting in the formation of an exceptional point ( figure 10). However, for longer relaxation times, the tan(δ) quickly decays to almost zero, making the imaginary part of the dynamic modulus too small to show any effect. Supposedly, a material model representing a weighted sum of multiple relaxation times-both short and long-will resolve this issue. Usually, experimental measurements of the relaxation modulus of viscoelastic materials are best described by a Prony series, a sum of exponentials physically represented by the generalized Maxwell (GM) model [52][53][54]. The GM model is similar to the SLS model with multiple Maxwell elements connected in parallel, resulting in a discrete spectrum of relaxation times [43,44]. The relaxation modulus E(t) and dynamic modulus E d (ω) for the GM model can be written as, . (13) However, fitting these many terms to experimental data is neither practical nor desirable. So, a power law relaxation function (E(t) = E ∞ + E α t −α ) is generally used for materials that relax over a large spectrum of relaxation times [55][56][57][58]. For a sufficient number of terms, the Prony series (12) is approximately equivalent to a shifted power law relaxation function E(t) = E ∞ + E α (t + t l ) −α (t l ⩾ 0, 0 < α < 1) as follows (more details in appendix B), where Γ is the gamma function. Equation (14) can be converted into an approximate discrete sum using trapezoidal rule as follows where τ m = τ 1 , τ 2 , τ 3 , . . . , τ M are relaxation times and E m are corresponding coefficients of a discrete relaxation spectrumĤ(τ ) given by [43]Ĥ Equations (15) and (16) suggest that instead of using the Prony series, the simpler power-law relaxation modulus can be used to model realistic viscoelastic materials [57,[59][60][61]. The physical representation of the power-law relaxation is the KVFD model. Thus, in the next section, we investigate the possibility of achieving an EP using a KVFD solid as the non-Hermitian element.

KVFD materials
The KVFD solid is similar to the KV solid with a springpot replacing the dashpot. The springpot represents an interpolation between a spring and a dashpot. For an applied strain ϵ, the stress response of a springpot will be σ = η d α ϵ dt α , where , 0 ⩽ α ⩽ 1, with α = 1 indicating a dashpot and α = 0 indicating a spring. The KVFD model we have considered here consists of a springpot connected in parallel to a spring (figure 6(d)). Usually, one or two springpots in parallel are sufficient to accurately model both transient and dynamic behaviors of a viscoelastic material [59].
The relaxation modulus of the KVFD solid obeys a power-law decay as follows, Sometimes, a more general function-shifted power law (E(t) = E ∞ + E α (t + t l ) −α ) is used to avoid the divergence of the transient term at t = 0. From equation (14), we can extract the relaxation spectrum H(τ ) as follows In contrast to GM model (17), relaxation spectrum for power law model (19) is continuous. In figure 7(a), the normalized relaxation spectrum of the power law (H(τ )/E α ) is plotted as a function of relaxation time for different combinations of α and t l . For α = 1 and t l = 0 (KV solid), shorter relaxation times are more dominant as compared to longer, which is the reason why KV solid undergoes an abrupt relaxation at t = 0. For t l > 0 (shifted power law), the relaxation spectrum peaks at τ = t l /α, but decays in both directions from the peak. For t l = 0 and α = 0.25 (KVFD), both shorter and longer relaxation times  dominate. The effect of such relaxation behavior can be observed in the loss-tangent (figure 5), which stays almost constant for higher frequencies, making KVFD materials a potential choice for achieving EPs.
The normalized dynamic stiffness (table 1) for the KVFD solid shown in figure 6(d) can be written as, where Substituting equation (20) in equations (6), (7) and solving for Ω gives two converging solutions, Ω 1 and Ω 2 . Figure 6(d) shows the real and imaginary parts of both natural frequencies undergoing a sharp EP bifurcation. The characteristics of bifurcation near the EP for KVFD material are similar to the structural damping case described earlier (refer to figure S2 in SI for a comparison of eigenfrequencies). Analogous to structural damping, for KVFD, tan(δ) is almost constant and coverges to tan as Ω → ∞. This verifies our hypothesis that for a sharp EP bifurcation to occur, the tan(δ) should not diverge with frequency and should stay almost constant in the frequency range of interest. It is important to note that although EPs are single-frequency phenomena, the occurrence of sharp bifurcation relies on the behavior of tan(δ) in the surrounding frequency range. Remarkably, compared to structural damping, which is rarely observed in engineering materials, many soft polymers and biological materials exhibit viscoelastic behavior that is best described by fractional derivative models. Figure 7(b) shows the storage and loss modulus of PDMS (Sylgard 184, 1:20 ratio of curing agent to liquid elastomer) that we measured experimentally by performing the dynamic mechanical analysis (DMA) on our custom-built dynamic mechanical analyzer ( figure 8(a)). It strongly follows the KVFD model with parameters α = 0.25, E ∞ = 1.36, η = 0.37 (refer to appendix C for details on DMA). In figure 8(b), we compare the frequency-dependent loss tangent of PDMS, polyurethane rubber, natural rubber, and polyurethane foam that we measured using DMA. PDMS and polyurethane rubber display a gradual dependence on the frequency, while polyurethane foam exhibits no dependence of tan(δ) on frequency. Both of these characteristics are suitable for creating a sharp exceptional point. However, natural rubber demonstrates a strong dependence on frequency, making it unsuitable for the intended purpose. Therefore, the utilization of elastomers, such as PDMS and polyurethane rubber, as well as foams like polyurethane foam, in passive non-Hermitian systems enables the realization of sharp EPs experimentally. However, it should be noted that the low elastic modulus of foams leads to a lower ratio of elastic stiffness to the mass of the oscillator. Consequently, the usage of foams in non-Hermitian systems will be limited to very low-frequency ranges. Furthermore, the densification of the foam under significant compression strains causes a steep increase in modulus due to nonlinearity. This situation presents challenges in preserving the parity symmetry within the stiffness matrix. In our previous work, we experimentally demonstrated an EP in a passive dynamical system with PDMS serving as the damping component [15]. The occurrence of EP in experiments is highly sensitive to even the slightest variations in the symmetry of the real part of the stiffness. Therefore, it is essential to precisely characterize the frequency-dependent storage and loss modulus of the viscoelastic material. To achieve this, we conducted precise measurements of the PDMS's frequency-dependent storage and loss modulus using our custom-built dynamic mechanical analyzer (see figure 8(a) and appendix C). To conduct DMA experiments, we initially apply a static precompression stress to secure the sample and carry out local sinusoidal small-strain dynamics to measure the dynamic mechanical properties. It is noteworthy that, due to potential nonlinearities, the storage and loss modulus can vary based on the initial precompression. Therefore, when experimentally characterizing the EP, the precompression stress applied to the system must be identical to the stress applied to the viscoelastic material (PDMS in our case) for measuring the frequency-dependent storage and loss modulus. In contrast to prior studies focusing on enhanced sensitivity, in this study, we demonstrated that the actuation forces (emissivity) can be amplified in proximity to the EPs [15]. Notably, a twofold enhancement of the applied force is achieved using an entirely passive dissipative non-Hermitian system, when it operates near the EP compared to far away from the EP.
In experiments, the real and imaginary parts of eigenfrequencies can be characterized by measuring the frequency response of the system under sinusoidal excitation. Figure 9(a) shows our model system under forced vibration, with the left boundary being driven by applying a sinusoidal displacement u(t) = Ue iωt .
The two oscillators are coupled together by a tunable spring whose stiffness can be adjusted to change the coupling. The required differential-damping is achieved by a Hookean spring of stiffness k (no-Loss) and a PDMS cylinder of area A, length L, and dynamic modulus E d (Loss) ( figure 7(b)).
We assume, m = 4.635 g, A = 28.27 mm 2 (6 mm diameter) and L = 1.95 mm. For the known dynamic stiffness of PDMS (k d = E d A L ), we estimated the corresponding required optimal stiffness of the Hookean spring (k) by minimizing the gap between the resonant frequencies (|ω 1 − ω 2 |) such that the bifurcation at EP is sharpest ( figure 9(b)). Theoretically calculated, real (ω r ) and imaginary parts (ω i ) of the resonant frequencies are plotted in figures 9(c) and (d) in the parameter space of κ and k. For k = 79.12 kN m −1 , the gap between the two natural frequencies approaches almost zero at the EP, which is an indication of a sharp bifurcation. Figure 9(e) shows the theoretical frequency response of the second oscillator, a sharp bifurcation of a single narrow resonant peak at low couplings (κ < κ ep ) into two broad peaks of the same linewidth at higher couplings (κ > κ ep ). From the frequency response, the real part of the resonant frequencies can be estimated from the position of the resonant peaks and the imaginary part by measuring the half-width of the peak at 1/ √ 2 of peak height (figure 9(e)).

Conclusion
In this work, we investigated the viability of achieving sharp orthogonal EP branch-point singularities in passive non-Hermitian systems with viscoelastic materials serving as the dissipative component. Using a coupled oscillator as the model system, we compared different viscoelastic solids with the general rate-independent hysteretic damping to identify the requisites on frequency-dependent dynamic properties for the creation of an elastodynamic metamaterial that support the formation of an EP. We show that for an EP to form, the loss-tangent of viscoelastic material should stay almost constant in the frequency range of operation-a characteristic encoded in the relaxation spectrum of the material. Our work provides a critical framework to design passive (without gain) non-Hermitian systems by using viscoelastic materials for realizing non-Hermiticity. This paves the way for the development of passive non-Hermitian devices for sensing, actuation, and energy-harvesting.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
Appendix A. Effect of relaxation time in SLS model on eigen frequencies Changing variables s → t and c → −t l (t l > 0) Multiplying E α and adding E ∞ on both sides Changing variable x → 1

Appendix C. Dynamic mechanical analysis
We perform dynamic mechanical analysis (DMA) using our custom-built dynamic mechanical analyzer, which consists of a preloaded piezoelectric actuator (Physik Instrumente P841.10) and a dynamic force sensor (PCB 208C01) ( figure 8(a)). To conduct DMA experiments, we first attach the sample between the actuator and the force sensor and apply a static precompression strain to hold the sample in place. Using a MATLAB script and an NI data acquisition board, we supply a sinusoidal input voltage with a specific frequency and amplitude to the actuator. The resulting vibration of the actuator's tip causes a sinusoidal compression and decompression of the sample. The dynamic force sensor measures the sinusoidal force response of the sample. The tangent of the phase shift between the transient force and displacement signals represents the loss tangent of the material. We perform a frequency sweep from 0.1 Hz to 1200 Hz to calculate the loss tangent as a function of excitation frequency.