Programmable ultrasonic modulation of viscoelasticity in polymer-based elastomers: Experiments and constitutive modeling

Abstract

One of the central challenges in soft matter mechanics is to achieve reversible and programmable modulation of viscoelasticity in polymer-based elastomers at small strains, which is crucial for precision engineering and advanced functional devices. Conventional approaches are constrained by irreversibility and lack of dynamic control. In this study, it is demonstrated that ultrasonic vibration (19–22 kHz) enables dynamic, reversible, and tunable modulation of the mechanical response in such materials. Uniaxial compression experiments combined with constitutive and inverse modeling reveal a reversible transition from viscoelastic, dissipative behavior to an elastic-dominated, stable state. The standard linear solid (SLS) model links macroscopic mechanical changes to molecular-level dynamics, such as chain alignment and mobility. Experimentally, ultrasonic vibration suppresses viscoelastic relaxation and energy dissipation, induces negative hysteresis, and enables tunable, reversible hardening, all strongly dependent on vibration frequency and power. Quantitatively, a typical 20% increase in the instantaneous elastic modulus and over 80% reduction in the delayed elastic modulus and viscosity are achieved under ultrasonic vibration. These results clarify the mechanism by which ultrasonic vibration regulates viscoelasticity and provide practical guidance for designing adaptive polymer systems in applications such as ultrasonic-assisted polishing, soft robotics, and flexible electronics.

1. Introduction

Polymer-based elastomers are widely used in advanced applications due to their flexibility and tunable mechanical properties [1], [2], [3], [4], [5]. Their macroscopic behavior is governed by viscoelasticity [6], [7], rooted in complex molecular chain dynamics [8], [9]. Conventional tuning methods, such as chemical modification or filler addition [10], [11], often cause irreversible changes and lack precision. This highlights the need for dynamic and precise control, especially at small strains, which is crucial for precision engineering and adaptive devices.

Low-frequency ultrasonic vibration (20–40 kHz) provides efficient energy transfer and strong field-matter interaction, and is increasingly used to modulate the mechanical properties of soft matter [12]. However, systematic and mechanistic studies on how ultrasonic vibration enables tunable and reversible adjustment of viscoelasticity in polymer-based elastomers remain scarce. Previous studies have either focused on metals [13], [14], [15], or on similar elastomers without considering ultrasonic effects [16], or failed to address unloading behavior and underlying mechanisms [17], [18], leaving the controllable modulation of bulk viscoelasticity in polymer-based elastomers largely unexplored; see also recent reviews and references therein [19], [20].

To address these gaps, this study integrates uniaxial compression experiments with constitutive and inverse modeling [21], [22], [23] to quantitatively elucidate the field-driven regulation of viscoelastic properties in polymer-based elastomers under ultrasonic vibration at small strains.

The key quantitative findings and mechanistic insights are as follows:

• •This work quantitatively demonstrates that ultrasonic vibration at small strains enables dynamic, reversible, and programmable tuning of viscoelasticity in polymer-based elastomers. Typical results show a 20% increase in instantaneous elastic modulus and over 80% reduction in delayed modulus and viscosity.
• •By extracting parameters of the standard linear solid (SLS) model under various ultrasonic conditions, this work establishes a direct quantitative link between field-driven molecular chain alignment and macroscopic mechanical response.
• •The discovery and mechanistic explanation of negative hysteresis and fully reversible hardening under ultrasonic fields reveal a fundamentally distinct, non-destructive adaptation mechanism, which is tunable by ultrasonic frequency and power.

Taken together, these pioneering findings, for the first time, clearly demonstrate the unique and tunable effects of ultrasonic vibration on the viscoelastic behavior of polymer-based elastomers. They also highlight the urgent need for deeper mechanistic understanding at the molecular level to fully exploit these effects. Building on this foundation, the present study systematically elucidates the underlying mechanisms by which ultrasonic vibration modulates viscoelasticity, thereby providing a robust theoretical basis for the rational design of polymer systems with tunable and reversible mechanical properties for advanced applications.

2. Experiments

To systematically investigate the modulation of viscoelasticity in polymer-based elastomers by ultrasonic vibration, uniaxial compression tests were conducted using a custom ultrasonic system integrated with a nano-precision computer numerical control (CNC) machine. This setup enables precise control of vibration parameters and loading conditions. A high-sensitivity force sensor system ensures accurate force measurements at small strains. Force–displacement data were converted to engineering stress and strain for quantitative analysis. The following sections detail the experimental setup, specimen preparation, procedures, and data processing.

2.1. Descriptions

Uniaxial compression tests were conducted using an ultrasonic system integrated with a nano-precision CNC machine, as illustrated in Fig. 1. The system consists of computer-controlled X, Z, B, and C axes, a spindle, and an ultrasonic toolholder equipped with a piezoelectric transducer. The ultrasonic system generates ultrasonic vibrations with a frequency range of 18–24 kHz and an amplitude of 1-$5\mathrm{\mu }\mathrm{m}$ at the tool end. Vibration parameters, including frequency and power, are precisely controlled via a digital ultrasonic generator. When the generator is deactivated, the system operates under conventional conditions for comparison.

Fig. 1.

Schematic of the ultrasonic-assisted compression testing platform with nano-precision control and high-sensitivity force measurement.

The test specimen was a polymer-based elastomer (SatisLoh Company), with its compressive properties summarized in Table 1. A cylindrical specimen (diameter: 19 mm; thickness: 12.5 mm) was affixed to a titanium alloy tool head and brought into contact with a sapphire disk (diameter: 25.4 mm; thickness: 2 mm), which served as the reaction end during compression. The compressive properties of both the polymer-based elastomer and the sapphire are listed in Table 1 for comparison. Owing to its extremely high stiffness and strength [24], the sapphire was considered rigid during the tests.

Table 1.

Compressive properties of the polymer-based elastomer and the sapphire (C-plane, [0001] direction, from [24]).

Material	Compressive elastic modulus (GPa)	Compressive strength (GPa)
Polymer-based elastomer	0.005	0.001
Sapphire (C-plane)	435	1.97

Compression was applied along the $Z$-axis at a constant speed through controlled loading-unloading cycles. The force was measured using a Kistler 9256C2 dynamometer (sensitivity: -26 pC/N) connected to a charge amplifier (Kistler 5080 A), with a force measurement range of −250 N to 250 N and a minimum resolvable force of 0.1 mN. Data acquisition was performed using a National Instruments USB-6281 card at a sampling rate of 150 kHz, ensuring accurate measurement within the ultrasonic frequency range [25]. Due to the limited storage capacity of the data acquisition system during continuous monitoring, the sampling rate was reduced to 5000 Hz, as implemented in Section 2.7. This lower rate was sufficient to capture the main features of the quasi-static loading signal and the envelope of ultrasonic modulation. To minimize the influence of material history and ensure the independence of each measurement, a minimum rest interval of 10 minutes was maintained between consecutive tests. All experiments were conducted at room temperature (approximately 20 $°\mathrm{C}$) to minimize the influence of external thermal fluctuations.

The acquired data were processed and analyzed using MATLAB software. No additional post-processing was applied.

2.2. Stress–strain behavior

The measured force–displacement data were converted into engineering stress–strain curves based on the geometry of the cylindrical specimen. For a specimen with diameter $d=19\mathrm{mm}$ and initial height $h_0=12.5\mathrm{mm}$, the engineering strain $\varepsilon$ and engineering stress $\sigma$ were calculated as follows [26]:

$$A=\frac{\pi d^2}{4},$$ (1)

$$\varepsilon =\frac{\Delta h}{h_0},$$ (2)

$$\sigma =\frac{F}{A},$$ (3)

where $A$ is the cross-sectional area of the specimen, $\Delta h$ is the axial displacement, and $F$ is the applied compressive force.

The corresponding engineering strain rate $\dot{\varepsilon }$ was determined by:

$$\dot{\varepsilon }=\frac{v}{h_0},$$ (4)

where $v$ is the compression velocity.

These stress–strain curves were used to characterize the mechanical response of the material.

2.3. Forward shift

Fig. 2 presents the stress–strain curves and their enclosed hysteresis loop areas of the specimen subjected to strains of 0.004 and 0.008 with and without ultrasonic vibration. The loading curves exhibit a stiff response [18], whereas the unloading curves display different behaviors depending on the presence of ultrasonic vibration. Without ultrasonic vibration, the unloading curves display hysteresis behavior [16], where the stress at a given strain during unloading is lower than during loading, resulting in a positive hysteresis loop area and residual strain.

Fig. 2.

Stress–strain curves and their enclosed hysteresis loop areas of the specimen under uniaxial compression at maximum strains of 0.004 and 0.008. Left: without ultrasonic vibration; right: with ultrasonic vibration (resonant mode, 20745 Hz frequency, 100% power). The inset in the lower right corner magnifies the hysteresis loop areas near $\varepsilon =0.004$, highlighting the differences between loading and unloading curves under ultrasonic vibration.

In contrast, under ultrasonic vibration, both the loading and unloading curves exhibit higher stress at the same strain compared to the non-ultrasonic condition. This indicates that the application of ultrasonic vibration enables more efficient force transmission within the material [27], resulting in a stiffer response. More importantly, the unloading curves under ultrasonic vibration show a pronounced forward shift, meaning that at the same strain, the unloading stress exceeds the loading stress, which is opposite to the conventional hysteresis behavior. To better illustrate the subtle differences between loading and unloading curves at small strains, an inset in the lower right corner provides a magnified view of the hysteresis loop areas near $\varepsilon =0.004$, clearly illustrating the forward shift and phase reversal under ultrasonic vibration, which result in negative hysteresis loop areas.

To quantitatively evaluate these effects, the areas enclosed by the hysteresis loops were calculated for different compression strains, as summarized in Table 2. The results show that the hysteresis loop area, representing the energy dissipated per cycle, is significantly reduced under ultrasonic vibration. Specifically, at a strain of 0.008, the area decreases from $3.5\times 10^{-6}\mathrm{MPa}$ (positive) to $-1.1\times 10^{-6}\mathrm{MPa}$ (negative), and at a strain of 0.004, from $1.3\times 10^{-6}\mathrm{MPa}$ to $-1.7\times 10^{-7}\mathrm{MPa}$. These results demonstrate that ultrasonic vibration fundamentally alters the energy dissipation mechanism [28], making the material response more elastic and less dissipative.

Table 2.

Hysteresis loop areas calculated at different maximum strains with and without ultrasonic vibration. Positive values correspond to conventional hysteresis, while negative values indicate the occurrence of forward shift in the unloading curve under ultrasonic vibration.

${\varepsilon }_{\max }$	Hysteresis loop area (MPa)
	Without ultrasonic vibration	With ultrasonic vibration
0.008	3.5 × 10−6	−1.1 × 10−6
0.004	1.3 × 10−6	−1.7 × 10−7

To further clarify the reversibility and accumulation of these effects, the following section examines the hardening behavior during cyclic loading.

2.4. Hardening

Fig. 3 presents the stress–strain curves of the specimen subjected to a strain of 0.008 with and without ultrasonic vibration. Over four loading-unloading cycles, the curves were stable and repeatable, indicating a consistent mechanical response. Under ultrasonic vibration, the peak stress increased progressively with each cycle. A similar hardening trend was observed at a lower strain of 0.004, as shown in Fig. 4.

Fig. 3.

Stress–strain curves of the specimen under cyclic uniaxial compression at a maximum strain of 0.008. Left: without ultrasonic vibration; right: with ultrasonic vibration (resonant mode, 20745 Hz frequency, 100% power). $N$ denotes cycle number.

Fig. 4.

Stress–strain curves of the specimen under cyclic uniaxial compression at a maximum strain of 0.004. Left: without ultrasonic vibration; right: with ultrasonic vibration (resonant mode, 20745 Hz frequency, 100% power). $N$ denotes cycle number.

The ultrasonic-induced hardening effect was quantitatively evaluated by the increment of peak stress between consecutive cycles, as summarized in Table 3. For both 0.004 and 0.008 strains, the stress increment per cycle remained nearly constant, with mean values of $0.0012\pm 0.0001\mathrm{MPa}$ and $0.0006\pm 0.0001\mathrm{MPa}$, respectively. The low standard deviation (SD) demonstrates the high repeatability and controllability of this effect. In addition, the hardening rate increases with strain, indicating a strain-dependent enhancement of the underlying mechanism.

Table 3.

Increment of peak stress between consecutive unloading cycles ($\Delta I_n$) at different maximum strains under ultrasonic vibration. The three listed values correspond to consecutive cycles. The mean and standard deviation (SD) indicate the repeatability and stability of the hardening effect.

${\varepsilon }_{\max }$	$\Delta I_1$ (MPa)	$\Delta I_2$ (MPa)	$\Delta I_3$ (MPa)	Mean $\pm$ SD (MPa)
0.008	0.0012	0.0013	0.0012	0.0012 ± 0.0001
0.004	0.0005	0.0006	0.0006	0.0006 ± 0.0001

The hardening effect induced by ultrasonic vibration is fundamentally different from conventional strain hardening, which typically occurs at large plastic strains and leads to irreversible microstructural changes [29]. In contrast, the hardening observed here arises at small, reversible strains and is closely linked to the material’s intrinsic elasticity. Under ultrasonic vibration, the peak stress increases progressively and repeatably with each cycle, but once the ultrasonic vibration is removed, the material returns to its original mechanical state. This demonstrates that ultrasonic vibration enables a tunable and repeatable mechanical response, fundamentally distinct from the permanent hardening associated with conventional plastic deformation.

To further elucidate the origins and controllability of this effect, the influence of ultrasonic parameters on the mechanical behavior is systematically investigated in the following section.

2.5. Parameter dependency

As shown in Fig. 5 and Table 4, the peak stress varies with ultrasonic frequency, reaching a maximum at the resonant frequency (20745 Hz) with a mean value of 0.0146 MPa. This represents an increase of approximately 17.7% compared to the non-ultrasonic condition (0 Hz, 0.0124 MPa). At resonance, the coefficient of variation (CV) is also among the lowest (0.0030), indicating highly stable and repeatable mechanical behavior. For frequencies above resonance, the peak stress gradually decreases, while frequencies below resonance lead to greater variability and less consistent responses.

Fig. 5.

Stress-time curves of the specimen under cyclic uniaxial compression at ultrasonic frequencies of 0, 19708, 20745, 21004, 21263, 21782 Hz. A number of holding period of 3 s was applied at strains of 0.008.

Table 4.

Mean, SD, and coefficient of variation (CV) of peak stress at different ultrasonic frequencies. Four peak values are listed in chronological order.

Frequency (Hz)	Peak 1 (MPa)	Peak 2 (MPa)	Peak 3 (MPa)	Peak 4 (MPa)	Mean $\pm$ SD (MPa)	CV
0	0.0124	0.0124	0.0125	0.0124	0.0124 ± 0.0001	0.0081
19708	0.0142	0.0144	0.0147	0.0150	0.0146 ± 0.0003	0.0205
20745	0.0146	0.0146	0.0145	0.0145	0.0146 ± 0.0001	0.0030
21004	0.0136	0.0136	0.0135	0.0137	0.0136 ± 0.0001	0.0074
21263	0.0133	0.0134	0.0133	0.0133	0.0133 ± 0.0001	0.0056
21782	0.0130	0.0130	0.0129	0.0130	0.0130 ± 0.0000	0.0031

Fig. 6 and Table 5 further reveal the influence of ultrasonic power. As the power increases from 25% to 100%, the mean peak stress rises monotonically (from 0.0129 MPa to 0.0171 MPa), indicating enhanced force transmission and material stiffening. However, this improvement is accompanied by a marked increase in variability: the SD and CV grow from 0.0001 and 0.0077 at 25% power to 0.0017 and 0.0994 at 100% power, respectively. These results suggest that, although higher ultrasonic power can further increase peak stress, it may also introduce greater response variability.

Fig. 6.

Stress-time curves of the specimen under cyclic uniaxial compression at ultrasonic powers of 25%, 50%, 75%, and 100%. These power tests were conducted after the frequency tests in Fig. 5 on the same specimen, with holding periods of 3 s applied at strains of 0.008.

Table 5.

Mean, SD, and CV of peak stress at different ultrasonic powers. Four peak values are listed in chronological order.

Power (%)	Peak 1 (MPa)	Peak 2 (MPa)	Peak 3 (MPa)	Peak 4 (MPa)	Mean $\pm$ SD (MPa)	CV
25	0.0128	0.0129	0.0130	0.0130	0.0129 ± 0.0001	0.0077
50	0.0134	0.0139	0.0143	0.0147	0.0141 ± 0.0006	0.0426
75	0.0140	0.0148	0.0157	0.0165	0.0153 ± 0.0011	0.0719
100	0.0151	0.0165	0.0177	0.0189	0.0171 ± 0.0017	0.0994

These results demonstrate that both ultrasonic frequency and power serve as effective and tunable parameters for modulating the mechanical properties of the material. The optimal mechanical performance, characterized by high peak stress and low variability, is achieved near the resonant frequency and at moderate power levels. Excessive power, although beneficial for increasing peak stress, may compromise response uniformity and process stability [30].

As observed in Fig. 5, Fig. 6, the results for the same parameter set obtained from two different experimental groups (Table 4, Table 5) exhibit discrepancies, particularly in the mean values of peak stress. This inconsistency is analyzed and discussed in detail in the following section.

2.6. Relaxation-induced softening

Most experiments in this section, including those on time-dependent effects, were conducted within a single day on the same specimen, minimizing material history effects and ensuring high consistency. In contrast, parameter dependency experiments (i.e., tests with different ultrasonic frequencies and powers) were performed in separate groups on another day, with longer intervals between groups. This led to a pronounced softening phenomenon observed only in the parameter dependency experiments, primarily due to viscoelastic relaxation during unloaded intervals.

As shown in Fig. 5, Fig. 6 and Table 4, Table 5, the four peak stress values within each group remained highly consistent, indicating excellent repeatability under identical conditions. However, the mean peak stress of earlier groups was higher than that of subsequent groups, while within-group variability (SD and CV) remained nearly unchanged. This systematic decrease in peak stress between groups demonstrates that the observed softening effect mainly manifests as a reduction in absolute stress, rather than increased variability.

Despite sufficient rest intervals, some degree of relaxation-induced softening is inevitable due to the inherent viscoelasticity of the material. This highlights the necessity of considering time-dependent effects, especially during unloaded periods, when interpreting the mechanical response of the material under ultrasonic vibration.

To further elucidate the influence of time-dependent effects, the following section systematically investigates the role of strain rate and holding periods.

2.7. Time dependency

Fig. 7 presents the stress–strain curves of the specimen at strain rates of 0.008, 0.004 and 0.0008 ${\mathrm{s}}^{-1}$ with and without ultrasonic vibration. Without ultrasonic vibration, the loading and unloading curves at different strain rates nearly coincide, indicating negligible rate sensitivity and stable viscoelastic behavior. In contrast, under ultrasonic vibration, the peak stress decreases noticeably with increasing strain rate, revealing a distinct rate-dependent softening effect.

Fig. 7.

Stress–strain curves of the specimen under uniaxial compression at strain rates of 0.008, 0.004, and 0.0008 ${\mathrm{s}}^{-1}$. Left: without ultrasonic vibration; right: with ultrasonic vibration (resonant mode, 20745 Hz frequency, 100% power).

To further investigate time-dependent behavior, Fig. 8 shows stress-time curves under sustained uniaxial compression with intermittent holding periods (strain rate: 0.0008 ${\mathrm{s}}^{-1}$, 60 s per hold), while Fig. 9 presents the corresponding stress–strain curves without holding periods for comparison.

Fig. 8.

Stress-time curves of the specimen under sustained uniaxial compression with holding periods of 60 s. Left: without ultrasonic vibration; right: with ultrasonic vibration (resonant mode, 20745 Hz frequency, 25% power).

Fig. 9.

Stress–strain curves of the specimen under sustained uniaxial compression without holding periods of 60 s. Left: without ultrasonic vibration; right: with ultrasonic vibration (resonant mode, 20745 Hz frequency, 25% power).

Without ultrasonic vibration, pronounced stress relaxation is observed during each holding period, as the stress gradually decreases under constant strain; during unloading, a slight stress increase reflects viscoelastic recovery. In contrast, under ultrasonic vibration, the stress remains nearly constant during holding, with negligible relaxation or recovery in both loading and unloading. This indicates that ultrasonic vibration can effectively suppress the time-dependent stress evolution commonly seen in viscoelastic materials. A slight stress increase at low strains during unloading may still be observed, but overall, the time-dependent effects are greatly reduced compared to the non-ultrasonic condition.

Overall, these results demonstrate that ultrasonic vibration not only alters the rate dependence of the material, but also fundamentally suppresses its intrinsic time-dependent relaxation behavior, leading to a more stable and predictable mechanical response under cyclic and sustained loading.

3. Constitutive model

To quantitatively interpret the experimental results and uncover the underlying mechanisms, an inverse modeling approach was used. The measured stress-time data were fitted with analytical solutions of the selected constitutive model (SLS), and model parameters were obtained via nonlinear least-squares optimization. This enabled direct extraction of key physical parameters ($E_1$, $E_2$, $\eta$), linking ultrasonic modulation to changes in viscoelastic properties. The workflow included: (1) model selection; (2) deriving analytical solutions for the loading protocols; (3) fitting to experimental data; and (4) analyzing the physical significance and trends of the fitted parameters under different ultrasonic conditions.

3.1. Model selection

The above experiments demonstrate that, for small-strain deformation, a suitable constitutive model must capture three essential features: (1) tunable and reversible hardening induced by ultrasonic vibration; (2) significant suppression of time-dependent effects, resulting in a more stable mechanical response; and (3) a fundamental shift in energy dissipation from viscoelastic loss to elastic-dominated behavior.

To interpret these phenomena, the standard linear solid (SLS) model [6], [23], [31] was chosen for its clear physical meaning. With parameters $E_1$, $E_2$, and $\eta$, the SLS model quantitatively describes both instantaneous and delayed elastic responses, as well as time-dependent stress relaxation. This allows direct linkage between experimental observations and the underlying physical mechanisms modulated by ultrasonic vibration.

Classical linear elastic and standard viscoelastic models, such as the Maxwell and Kelvin–Voigt models, were also considered but found inadequate. The Maxwell model lacks a long-term elastic component and cannot capture the stable elastic response observed under ultrasonic vibration. The Kelvin–Voigt model fails to describe time-dependent stress relaxation. As a result, neither model can simultaneously account for the reversible hardening and suppression of time-dependent effects induced by ultrasonic vibration. Although empirical models may offer better fitting accuracy, they lack clear physical interpretability and predictive power.

In contrast, the SLS model provides a balanced and physically interpretable framework that captures all key features observed in the experiments. It accurately describes both ramp loading/unloading and stress relaxation during holding periods, using analytical solutions that correspond directly to the experimental protocols. The model parameters are directly linked to the mechanisms modulated by ultrasonic vibration, enabling robust analysis and interpretation of the tunable viscoelastic behavior of polymer-based elastomers.

While more complex viscoelastic models, such as the generalized Maxwell model, can capture multi-stage relaxation if needed, the SLS model offers a practical balance between accuracy and interpretability for this study. However, it should be noted that the SLS model assumes a single relaxation time and linear viscoelasticity, which may limit its applicability to materials with pronounced multi-mode or nonlinear relaxation behavior.

3.2. Descriptions

As shown in Fig. 10, the SLS model describes the viscoelastic response of materials by combining instantaneous and delayed elastic elements with a viscous element. Its differential form in the Maxwell representation is [31]:

$$\sigma \left(t\right)+\frac{\eta }{E_2}\frac{d\sigma \left(t\right)}{dt}=E_1\varepsilon \left(t\right)+\frac{\eta (E_1+E_2)}{E_2}\frac{d\varepsilon \left(t\right)}{dt},$$ (5)

where $\sigma \left(t\right)$ and $\varepsilon \left(t\right)$ are the stress and strain, $E_1$ is the instantaneous elastic modulus, $E_2$ is the delayed elastic modulus, $\eta$ is the viscosity coefficient, and $t$ is time.

Fig. 10.

Schematic of the Maxwell representation of the SLS model, which consists of a Maxwell solid and a spring. $E_1$ reflects the instantaneous elastic modulus, $E_2$ the delayed elastic (viscoelastic) response, and $\eta$ the viscosity coefficient governing time-dependent relaxation and dissipation.

Depending on the loading protocol, the SLS model provides different analytical solutions for stress relaxation. For the more practical case of a linear ramp strain input, i.e., $\varepsilon \left(t\right)=\dot{\varepsilon }t$, the analytical solution is

$$\sigma \left(t\right)=E_1\dot{\varepsilon }t+\eta \dot{\varepsilon }\left[1-exp\left(-\frac{E_2}{\eta }t\right)\right].$$ (6)

Eq. (6) describes the stress response under a constant strain rate as the sum of two contributions: the first term, $E_1\dot{\varepsilon }t$, reflects the instantaneous elastic response, while the second term, $\eta \dot{\varepsilon }\left[1-exp\left(-\frac{E_2}{\eta }t\right)\right]$, captures the time-dependent viscoelastic relaxation, which decays exponentially with a time constant $\tau =\eta /E_2$.

For an ideal step strain input under a constant strain ${\varepsilon }_0$, the analytical solution is given by

$$\sigma \left(t\right)=E_1{\varepsilon }_0+E_2{\varepsilon }_{0}exp\left(-\frac{E_2}{\eta }t\right).$$ (7)

Eq. (7) describes the stress relaxation process, with an initial stress of $(E_1+E_2){\varepsilon }_0$ and a long-term equilibrium value of $E_1{\varepsilon }_0$.

In this work, all SLS parameter fittings are based on the analytical solution that matches the experimental loading protocol, ensuring both physical consistency and fitting accuracy.

To determine the model parameters and assess the agreement between model and experiment, a nonlinear least-squares fitting procedure [22] was applied to the stress-time data for each segment. The quality of the fit was evaluated using two standard metrics: the coefficient of determination ($R^2$) and the root mean square error (RMSE) [32], defined as follows:

$$R^2=1-\frac{\sum _{i=1}^n{(y_i-{\hat{y}}_i)}^2}{\sum _{i=1}^n{(y_i-\overline{y})}^2},$$ (8)

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum _{i=1}^n{(y_i-{\hat{y}}_i)}^2},$$ (9)

where $y_i$ is the experimental value, ${\hat{y}}_i$ is the fitted value, $\overline{y}$ is the mean of the experimental values, and $n$ is the total number of data points. $R^2$ reflects the proportion of variance explained by the model, while RMSE quantifies the average fitting error.

All calculations, including data processing, parameter fitting, and model evaluation, were performed in MATLAB software. Custom scripts were developed to implement the analytical solutions, conduct nonlinear least-squares fitting, and compute the goodness-of-fit metrics for each experimental segment.

The detailed derivations of the SLS model’s analytical solutions for both ramp and step strain inputs are provided in Appendix.

3.3. Fittings

The SLS model fitting results for the loading, holding, and unloading stages at a maximum strain of ${\varepsilon }_{\max }=0.0016$ provide clear quantitative insights into how ultrasonic vibration modifies the viscoelastic response of the polymer-based specimen (see Fig. 11, Fig. 12, Fig. 13). These results are in excellent agreement with the experimental observations described earlier.

Fig. 11.

Stress-time curves fitted by the SLS model for the specimen under uniaxial compression during the loading stage at a strain rate of $\dot{\varepsilon }=0.0008{\mathrm{s}}^{-1}$. Without ultrasonic vibration: $E_1=0.0010\mathrm{MPa}$, $E_2=0.0002\mathrm{MPa}$, $\eta =0.0088\mathrm{MPa}$s, $R^2=0.9941$, RMSE $=4.3\times 10^{-5}\mathrm{MPa}$; with ultrasonic vibration: $E_1=0.0012\mathrm{MPa}$, $E_2=0.0002\mathrm{MPa}$, $\eta =0.0044\mathrm{MPa}$s, $R^2=0.9956$, RMSE $=4.4\times 10^{-5}\mathrm{MPa}$. Left: without ultrasonic vibration; right: with ultrasonic vibration (resonant mode, 20745 Hz frequency, 25% power).

Fig. 12.

Stress-time curves fitted by the SLS model for the specimen under uniaxial compression during the holding stage at a strain of ${\varepsilon }_0=0.0016$. Without ultrasonic vibration: $E_1=1.6267\mathrm{MPa}$, $E_2=0.1396\mathrm{MPa}$, $\eta =2.9170\mathrm{MPa}$s, $R^2=0.0652$, RMSE $=2.2\times 10^{-4}\mathrm{MPa}$; with ultrasonic vibration: $E_1=2.3694\mathrm{MPa}$, $E_2=0.0005\mathrm{MPa}$, $\eta =1.0777\mathrm{MPa}$s, $R^2=-0.0000$, RMSE $=2.1\times 10^{-4}\mathrm{MPa}$. The experimental data (blue dots) were downsampled by a factor of 20 from the original 5000 Hz sampling rate for clarity. Left: without ultrasonic vibration; right: with ultrasonic vibration (resonant mode, 20745 Hz frequency, 25% power).

Fig. 13.

Stress-time curves fitted by the SLS model for the specimen under uniaxial compression during the unloading stage at a strain rate of $\dot{\varepsilon }=0.0008{\mathrm{s}}^{-1}$. Without ultrasonic vibration: $E_1=0.0000\mathrm{MPa}$, $E_2=0.0012\mathrm{MPa}$, $\eta =0.0062\mathrm{MPa}$s, $R^2=0.9940$, RMSE $=3.8\times 10^{-5}\mathrm{MPa}$; with ultrasonic vibration: $E_1=0.0009\mathrm{MPa}$, $E_2=0.0010\mathrm{MPa}$, $\eta =0.0026\mathrm{MPa}$s, $R^2=0.9997$, RMSE $=1.4\times 10^{-5}\mathrm{MPa}$. Left: without ultrasonic vibration; right: with ultrasonic vibration (resonant mode, 20745 Hz frequency, 25% power).

For the loading stage, without ultrasonic vibration, the fitted parameters are $E_1=0.0010\mathrm{MPa}$, $E_2=0.0002\mathrm{MPa}$, and $\eta =0.0088\mathrm{MPa}$s, with $R^2=0.9941$ and RMSE $=4.3\times 10^{-5}\mathrm{MPa}$. With ultrasonic vibration, $E_1$ increases slightly to $0.0012\mathrm{MPa}$ and $\eta$ decreases to $0.0044\mathrm{MPa}$s, while $E_2$ remains unchanged. The high $R^2$ and low RMSE values in both cases indicate excellent model agreement. These results demonstrate that ultrasonic vibration enhances the instantaneous elastic response and suppresses viscous dissipation during loading, resulting in a stiffer and more energy-efficient material response, as reflected by the observed stiffer loading curves and reduced hysteresis.

For the holding stage, without ultrasonic vibration, the fitted parameters are $E_1=1.6267\mathrm{MPa}$, $E_2=0.1396\mathrm{MPa}$, and $\eta =2.9170\mathrm{MPa}$s, with $R^2=0.0652$ and RMSE $=2.2\times 10^{-4}\mathrm{MPa}$. The stress relaxes significantly during the 60 s holding period, as indicated by the relaxation time $\tau =\eta /E_2\approx 20.9$ s. In contrast, with ultrasonic vibration, $E_1$ increases to $2.3694\mathrm{MPa}$, while $E_2$ drops sharply to $0.0005\mathrm{MPa}$ and $\eta$ to $1.0777\mathrm{MPa}$s, with $R^2=-0.0000$ and RMSE $=2.1\times 10^{-4}\mathrm{MPa}$. The relaxation time $\tau$ increases to about $2155$ s, but the relaxation amplitude is negligible due to the extremely small $E_2$, resulting in a nearly flat stress-time curve. This confirms that ultrasonic vibration almost completely suppresses stress relaxation, resulting in a nearly purely elastic and time-independent response during holding, consistent with the experimental observation of negligible stress decay. It should be noted that the $R^2$ values for the holding stage is low (or even negative) under ultrasonic vibration. This is not due to poor fitting, but rather because the stress remains nearly constant, so the variance explained by the model is very small, a known limitation of $R^2$ for nearly flat data, which does not affect the physical validity of the fit.

For the unloading stage, without ultrasonic vibration, the fitted parameters are $E_1=0.0000\mathrm{MPa}$, $E_2=0.0012\mathrm{MPa}$, and $\eta =0.0062\mathrm{MPa}$s, with $R^2=0.9940$ and RMSE $=3.8\times 10^{-5}\mathrm{MPa}$. With ultrasonic vibration, $E_1$ increases to $0.0009\mathrm{MPa}$, $E_2$ decreases slightly to $0.0010\mathrm{MPa}$, and $\eta$ decreases to $0.0026\mathrm{MPa}$s, with $R^2=0.9997$ and RMSE $=1.4\times 10^{-5}\mathrm{MPa}$. Ultrasonic vibration results in a more elastic and less viscous unloading response, as evidenced by the observed forward shift of the unloading curve and reduced hysteresis.

Ultrasonic vibration leads to a typical 20% increase in instantaneous elastic modulus ($E_1$) and over 80% reduction in both delayed elasticity ($E_2$) and viscosity ($\eta$), marking a clear transition from viscoelastic to elastic-dominated behavior with enhanced stiffness and reduced energy dissipation. These quantitative changes in SLS model parameters directly explain the observed shift to a more elastic, stable, and time-independent mechanical response under ultrasonic vibration, providing a robust physical basis for the experimental phenomena.

4. Mechanism analysis

Table 6, Table 7, Table 8, Table 9 present the fitted SLS model parameters ($E_1$, $E_2$, $\eta$) and goodness-of-fit metrics ($R^2$, RMSE) for the specimen, covering both stress relaxation (holding) and loading/unloading stages, with and without ultrasonic vibration. These results quantitatively reveal how ultrasonic vibration modulates the viscoelastic properties of the material.

Table 6.

Fitted SLS parameters and goodness-of-fit metrics for the loading and unloading stages at a strain rate of $\dot{\varepsilon }=0.0008{\mathrm{s}}^{-1}$ without ultrasonic vibration. Each segment corresponds to a loading or unloading window at a specific maximum strain, as indicated by the value of ${\varepsilon }_{\max }$ in the table.

Segment	${\varepsilon }_{\max }$	$E_1$ (MPa)	$E_2$ (MPa)	$\eta$ (MPa$\cdot$s)	$R^2$	RMSE
1	0.0016	0.0010	0.0002	0.0088	0.9941	4.3 × 10−5
2	0.0032	0.0013	0.0005	0.0281	0.9987	3.1 × 10−5
3	0.0048	0.0020	0.0004	0.0054	0.9994	2.7 × 10−5
4	0.0064	0.0024	0.0005	0.0061	0.9997	2.1 × 10−5
5	0.0080	0.0028	0.0005	0.0052	0.9999	1.1 × 10−5
6	0.0080	0.0026	0.0007	0.0033	0.9999	1.8 × 10−5
7	0.0048	0.0015	0.0016	0.0070	0.9999	9.7 × 10−6
8	0.0064	0.0014	0.0013	0.0036	0.9999	1.1 × 10−5
9	0.0032	0.0001	0.0021	0.0084	1.0000	5.8 × 10−6
10	0.0016	0.0000	0.0012	0.0062	0.9940	3.8 × 10−5

Table 7.

Fitted SLS parameters and goodness-of-fit metrics for the loading and unloading stages at a strain rate of $\dot{\varepsilon }=0.0008{\mathrm{s}}^{-1}$ with ultrasonic vibration (resonant mode, 20745 Hz frequency, 25% power). Each segment corresponds to a loading or unloading window at a specific maximum strain, as indicated by the value of ${\varepsilon }_{\max }$ in the table.

Segment	${\varepsilon }_{\max }$	$E_1$ (MPa)	$E_2$ (MPa)	$\eta$ (MPa$\cdot$s)	$R^2$	RMSE
1	0.0016	0.0012	0.0002	0.0044	0.9956	4.4 × 10−5
2	0.0032	0.0017	0.0004	0.0126	0.9993	2.7 × 10−5
3	0.0048	0.0022	0.0004	0.0057	0.9997	2.0 × 10−5
4	0.0064	0.0026	0.0005	0.0060	0.9999	1.5 × 10−5
5	0.0080	0.0029	0.0005	0.0056	1.0000	1.1 × 10−5
6	0.0080	0.0024	0.0012	0.0061	1.0000	6.4 × 10−6
7	0.0048	0.0023	0.0009	0.0058	0.9999	1.7 × 10−5
8	0.0064	0.0020	0.0009	0.0038	0.9998	1.8 × 10−5
9	0.0032	0.0002	0.0022	0.0173	0.9997	1.8 × 10−5
10	0.0016	0.0009	0.0010	0.0026	0.9997	1.4 × 10−5

Table 8.

Fitted SLS parameters and goodness-of-fit metrics for the stress relaxation stages at a strain of ${\varepsilon }_0=0.0016$ without ultrasonic vibration. Each segment corresponds to a constant strain, as indicated by the value of ${\varepsilon }_0$ in the table.

Segment	${\varepsilon }_0$	$E_1$ (MPa)	$E_2$ (MPa)	$\eta$ (MPa$\cdot$s)	$R^2$	RMSE
1	0.0016	1.6267	0.1396	2.9170	0.0652	2.2 × 10−4
2	0.0032	1.6756	0.0541	0.7666	0.0393	2.2 × 10−4
3	0.0048	1.8549	0.0451	0.7179	0.0616	2.2 × 10−4
4	0.0064	2.0441	0.0409	0.6672	0.0864	2.2 × 10−4
5	0.0080	2.2206	0.0378	0.7242	0.1130	2.2 × 10−4
6	0.0064	2.0101	0.0010	1.0323	−0.0001	2.2 × 10−4
7	0.0048	1.7923	0.0009	0.7726	−0.0001	2.2 × 10−4
8	0.0032	1.6012	0.0007	0.5478	−0.0001	2.2 × 10−4
9	0.0016	1.5882	0.0013	0.6102	−0.0001	2.2 × 10−4

Table 9.

Fitted SLS parameters and goodness-of-fit metrics for the stress relaxation stages at a strain of ${\varepsilon }_0=0.0016$ with ultrasonic vibration (resonant mode, 20745 Hz frequency, 25% power). Each segment corresponds to a constant strain, as indicated by the value of ${\varepsilon }_0$ in the table.

Segment	${\varepsilon }_0$	$E_1$ (MPa)	$E_2$ (MPa)	$\eta$ (MPa$\cdot$s)	$R^2$	RMSE
1	0.0016	2.3694	0.0005	1.0777	−0.0000	2.1 × 10−4
2	0.0032	2.2753	0.0010	1.8980	−0.0001	2.1 × 10−4
3	0.0048	2.4034	0.0010	0.4999	−0.0002	2.0 × 10−4
4	0.0064	2.5547	0.0000	0.0046	−0.0000	2.0 × 10−4
5	0.0080	2.6895	0.0218	0.1047	0.0248	2.0 × 10−4
6	0.0064	2.5656	0.0010	2.3440	−0.0001	2.1 × 10−4
7	0.0048	2.4782	0.0009	1.7675	−0.0001	2.1 × 10−4
8	0.0032	2.4899	0.0009	1.1897	−0.0001	2.1 × 10−4
9	0.0016	3.0012	0.0014	1.1430	−0.0001	2.1 × 10−4

The evolution of SLS parameters under ultrasonic vibration directly links molecular-level dynamics to macroscopic mechanical response, explaining key phenomena including hardening, negative hysteresis, and time-dependent effects. Fig. 14 summarizes the proposed molecular mechanisms and energy dissipation pathways. The following analysis integrates experimental data, modeling, and literature to systematically elucidate these mechanisms.

Fig. 14.

Schematic illustration of the proposed mechanism for ultrasonic modulation in polymer-based elastomers. The diagram contrasts the molecular network structure and energy dissipation pathways with and without ultrasonic vibration, highlighting the cooperative roles of hard and soft segments as inferred from experimental results. Hard segments primarily provide elastic energy storage and mechanical strength, while soft segments contribute to flexibility and energy dissipation through chain mobility and entanglement. The color scheme and basic elements are adapted from [33].

• (1)Ultrasonic suppression of viscoelastic relaxation and energy dissipation: In the absence of ultrasonic vibration, the polymer chains are predominantly in a disordered and highly entangled state. In this case, slow relaxation pathways such as chain disentanglement, segmental motion, and local rearrangements are highly active and govern the viscoelastic response. These molecular processes lead to pronounced stress relaxation and significant energy dissipation [34], [35], as reflected by the relatively low instantaneous elastic modulus $E_1$, higher delayed elastic modulus $E_2$, and larger viscosity $\eta$ values in the SLS model. Ultrasonic vibration fundamentally alters the viscoelastic behavior of polymer-based elastomers by promoting molecular chain alignment and suppressing slow relaxation channels [36], [37]. This is reflected in the significant increase of $E_1$ and sharp reduction of $E_2$, leading to a more elastic, time-independent response. The viscosity $\eta$ generally decreases during loading/unloading, indicating reduced energy dissipation, but shows complex, strain-dependent changes during holding due to the interplay of local heating, nonlinear dynamics, and microstructural heterogeneity [38], [39], [40]. These molecular-level changes underpin the observed transition from viscoelastic to elastic-dominated behavior, and lay the foundation for the subsequent phenomena such as negative hysteresis and reversible hardening.
• (2)Ultrasonic-induced transition to rate-dependent viscoelasticity: Ultrasonic vibration induces pronounced rate-dependent softening: as strain rate increases, the stress decreases, in sharp contrast to the nearly rate-insensitive behavior without ultrasonic vibration. This effect arises because ultrasonic vibration enhances molecular mobility and facilitates rapid, reversible rearrangement of weak crosslinks and entanglements [41]. At low strain rates, the network remains highly aligned and elastic; at higher rates, increased chain mobility accelerates stress relaxation [42], reducing stress. This dynamic balance between ordered, load-bearing structures and fast molecular rearrangement is fundamentally distinct from conventional viscoelasticity, and highlights the unique non-equilibrium dynamics enabled by ultrasonic fields.
• (3)Ultrasonic-induced negative hysteresis: signature of energy dissipation pathway reorganization The forward shift of the unloading curve and negative hysteresis under ultrasonic vibration, as captured by slightly reduced $\eta$ and $E_2$ in the SLS model, are fundamentally different from conventional viscoelastic behavior [16]. Mechanistically, this reflects a reorganization of energy dissipation pathways: ultrasonic vibration rapidly mobilizes and aligns molecular chains, suppressing both delayed elastic and viscous responses. As a result, elastic recovery becomes so efficient that unloading stress can exceed loading stress at the same strain, indicating highly reversible energy storage with minimal dissipation. At the molecular level, synchronized chain rearrangement and reduced internal friction [43] enable near-instantaneous stress recovery, making negative hysteresis a clear macroscopic signature of this low-dissipation, non-equilibrium state.
• (4)Ultrasonically tunable and reversible hardening via dynamic molecular alignment: Cyclic loading experiments reveal that ultrasonic vibration induces a repeatable and fully reversible increase in stress, without permanent plastic deformation. The enhancement of $E_1$ confirms that this hardening arises from dynamic molecular alignment and network reorganization, rather than irreversible damage. The process is highly tunable by adjusting ultrasonic parameters (frequency, amplitude, etc.), enabling precise control of mechanical response. Theoretically, this represents a field-driven, non-equilibrium adaptation of the polymer network, where periodic chain alignment transiently increases the density of load-bearing segments [44]. Unlike conventional strain hardening, which is associated with irreversible microstructural changes [45], the ultrasonic-induced hardening is non-destructive and rapidly reversible upon removal of the field. While minor irreversible effects such as localized chain scission may occur at high amplitudes or after many cycles, the dominant mechanism remains reversible, highlighting a dynamic balance between energy input and molecular rearrangement that defines the limits of adaptive mechanical performance.
• (5)Ultrasonic parameter tuning and optimal processing window: Both ultrasonic frequency and power act as effective, tunable parameters for modulating mechanical response. Optimal performance, high peak stress and low variability, is achieved near resonance and at moderate power, while excessive power increases variability. Theoretically, this tunability reflects the strong coupling between external fields and the polymer network: resonance maximizes energy transfer and molecular alignment, but excessive input can trigger nonlinearities, local heating, and microstructural heterogeneity [38], [39], [40], leading to instability. Local heating can enhance molecular mobility and promote chain alignment [46], thereby improving the elastic response. However, excessive local heating may also accelerate relaxation processes; thermal degradation typically occurs only at temperatures much higher than those used in this study [47]. Humidity may affect polymer moisture content and viscoelasticity [48], but for the elastomers studied here, its influence is expected to be limited under typical conditions. Thus, an optimal processing window emerges, governed by the interplay between material viscoelasticity, environmental conditions and ultrasonic parameters, enabling precise and adaptive control of mechanical performance.
• (6)Intrinsic relaxation limits: time-dependent softening and the impact of rest intervals Relaxation-induced softening, observed as a systematic decrease in peak stress between experimental groups, primarily results from viscoelastic relaxation during unloaded intervals [49]. Even with ultrasonic vibration, residual reductions in $E_2$ and $\eta$ persist due to the intrinsic molecular mobility of the polymer. This highlights a fundamental limit: while ultrasonic vibration can markedly suppress time-dependent relaxation during active loading, it cannot fully eliminate entropic recovery and chain segmental motion during rest. Consequently, stored elastic energy is gradually lost, reducing stress upon reloading. The interplay between externally driven alignment and intrinsic relaxation ultimately governs the long-term stability and repeatability of the mechanical response. This is further supported by cyclic and sustained loading experiments, which confirm that ultrasonic modulation of viscoelasticity is stable and reversible within the tested range. The stress–strain response remains consistent across cycles, with no evidence of fatigue, permanent changes, or material degradation, even after prolonged ultrasonic exposure at small strains. These results demonstrate reliable long-term performance under typical conditions, though extremely high amplitudes or prolonged exposure near the thermal degradation threshold may still cause irreversible effects.

Compared to other external field-based methods for tuning polymer viscoelasticity, such as electric, magnetic, or optical fields [42], ultrasonic modulation offers distinct advantages. Ultrasonic vibration penetrates deeply and uniformly into bulk materials, enabling rapid and reversible modulation of molecular mobility without the need for specific chemical groups. In contrast, electric and magnetic fields typically require polar or magnetic groups in the polymer, while optical approaches are limited by light penetration, material transparency, and often require photoresponsive groups. Thus, ultrasonic modulation is broadly applicable and efficient for dynamic control of polymer mechanics.

However, the effectiveness of ultrasonic modulation also depends on key molecular characteristics, such as molecular weight, chain entanglement, and molecular architecture. Elastomers with higher molecular weight and greater entanglement generally show stronger and more tunable responses, while low-molecular-weight or loosely entangled systems may respond less. Crosslinks, block copolymer structures, or bulky side groups can further restrict chain mobility, potentially limiting the modulation effect.

From a practical perspective, ultrasonic modulation is inherently scalable, as transducers and control systems can be adapted for large-scale devices and complex geometries. Implementation will require optimization of transducer placement, coupling efficiency, and field uniformity.

In summary, ultrasonic vibration at small strains induces a transition from viscoelastic to elastic-dominated behavior in polymer-based elastomers. SLS model analysis shows that this transition is governed by field-driven molecular chain alignment and suppression of slow relaxation, resulting in reduced energy dissipation and enhanced stiffness. The observed negative hysteresis and reversible hardening confirm that ultrasonic vibration enables rapid, non-destructive reorganization of the polymer network. These insights provide a mechanistic basis for designing polymer systems with tunable, reversible mechanical properties and offer guidance for practical applications such as ultrasonic-assisted polishing [27], [50], [51], soft robotics [52], [53], and flexible electronics [54], [55].

Looking ahead, in situ small-angle X-ray scattering [56] offers a real-time, non-destructive approach to directly probe microstructural changes, such as chain orientation and entanglement density, induced by ultrasonic fields. This technique would provide critical insights into the molecular mechanisms underlying ultrasonic modulation.

5. Conclusion

This study demonstrates that ultrasonic vibration at small strains enables reversible and programmable modulation of viscoelasticity in polymer-based elastomers. The main findings are as follows:

• •Ultrasonic vibration promotes molecular chain alignment and suppresses slow relaxation processes, enabling dynamic, reversible, and tunable modulation of viscoelasticity. Typical results show a 20% increase in instantaneous elastic modulus and over 80% reduction in delayed modulus and viscosity.
• •A clear transition from viscoelastic to elastic-dominated mechanical response is observed under ultrasonic vibration, characterized by reduced energy dissipation, enhanced stiffness, and suppression of time-dependent relaxation.
• •Ultrasonic vibration induces negative hysteresis and reversible hardening, both of which are strongly dependent on vibration frequency and power. These effects are fundamentally distinct from conventional plastic or strain hardening and are fully reversible upon removal of the ultrasonic field.
• •The SLS model quantitatively links changes in macroscopic mechanical response to underlying molecular dynamics, supporting the understanding and prediction of the tunable behavior of soft polymer systems under ultrasonic fields.
• •Both ultrasonic frequency and power serve as effective, tunable parameters for modulating mechanical properties. Optimal performance is achieved near resonance and at moderate power, while excessive power may increase response variability.

These findings offer practical guidance for the design and optimization of adaptive materials in applications such as ultrasonic-assisted polishing and soft robotics, and highlight the importance of parameter selection for achieving stable and controllable mechanical performance.

CRediT authorship contribution statement

Ying Geng: Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis. Guoyan Sun: Project administration, Funding acquisition, Data curation. Sheng Wang: Writing – review & editing, Project administration, Funding acquisition. Qingliang Zhao: Supervision, Resources, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix. Analytical solutions of the SLS model

The standard linear solid (SLS) model is a classical viscoelastic model that captures both instantaneous and delayed elastic responses as well as viscous effects. Its constitutive equation, in differential form, is given by [31]:

$$\sigma \left(t\right)+\frac{\eta }{E_2}\frac{d\sigma \left(t\right)}{dt}=E_1\varepsilon \left(t\right)+\frac{\eta (E_1+E_2)}{E_2}\frac{d\varepsilon \left(t\right)}{dt},$$ (A.1)

where $\sigma \left(t\right)$ is the stress, $\varepsilon \left(t\right)$ is the strain, $E_1$ is the instantaneous elastic modulus, $E_2$ is the delayed elastic modulus, $\eta$ is the viscosity, and $t$ is time.

(1) Ramp strain input ($\varepsilon \left(t\right)=\dot{\varepsilon }t$)

Consider a constant strain rate input, $\varepsilon \left(t\right)=\dot{\varepsilon }t$, where $\dot{\varepsilon }$ is a constant. Then $\frac{d\varepsilon \left(t\right)}{dt}=\dot{\varepsilon }$.

Substituting into Eq. (A.1):

$$\sigma \left(t\right)+\frac{\eta }{E_2}\frac{d\sigma \left(t\right)}{dt}=E_1\dot{\varepsilon }t+\frac{\eta (E_1+E_2)}{E_2}\dot{\varepsilon }$$ (A.2)

Multiply both sides by $\frac{E_2}{\eta }$ to obtain a standard first-order linear ODE:

$$\frac{d\sigma \left(t\right)}{dt}+\frac{E_2}{\eta }\sigma \left(t\right)=\frac{E_1{E}_2}{\eta }\dot{\varepsilon }t+(E_1+E_2)\dot{\varepsilon }$$ (A.3)

This is a first-order linear nonhomogeneous ODE. The integrating factor is:

$$\mu \left(t\right)=exp\left(\frac{E_2}{\eta }t\right)$$

Multiplying both sides by $\mu \left(t\right)$:

$$exp\left(\frac{E_2}{\eta }t\right)\frac{d\sigma \left(t\right)}{dt}+\frac{E_2}{\eta }exp\left(\frac{E_2}{\eta }t\right)\sigma \left(t\right)$$ (A.4)

$$=\left[\frac{E_1{E}_2}{\eta }\dot{\varepsilon }t+(E_1+E_2)\dot{\varepsilon }\right]exp\left(\frac{E_2}{\eta }t\right)$$ (A.5)

The left side is the derivative of a product:

$$\frac{d}{dt}\left[\sigma \left(t\right)exp\left(\frac{E_2}{\eta }t\right)\right]=\left[\frac{E_1{E}_2}{\eta }\dot{\varepsilon }t+(E_1+E_2)\dot{\varepsilon }\right]exp\left(\frac{E_2}{\eta }t\right)$$

Integrate both sides from $0$ to $t$ (assuming $\sigma \left(0\right)=0$):

$$\sigma \left(t\right)exp\left(\frac{E_2}{\eta }t\right)={\int }_0^t\left[\frac{E_1{E}_2}{\eta }\dot{\varepsilon }\tau +(E_1+E_2)\dot{\varepsilon }\right]exp\left(\frac{E_2}{\eta }\tau \right)d\tau$$ (A.6)

Split the integral:

$$=\frac{E_1{E}_2}{\eta }\dot{\varepsilon }{\int }_0^t\tau exp\left(\frac{E_2}{\eta }\tau \right)d\tau +(E_1+E_2)\dot{\varepsilon }{\int }_0^{t}exp\left(\frac{E_2}{\eta }\tau \right)d\tau$$ (A.7)

Let $a=\frac{E_2}{\eta }$, then use the following integral formulas:

$${\int }_0^t\tau e^{a\tau }d\tau =\frac{t}{a}{e}^{at}-\frac{1}{a^2}{e}^{at}+\frac{1}{a^2}$$

$${\int }_0^t{e}^{a\tau }d\tau =\frac{1}{a}(e^{at}-1)$$

Substitute back:

$$\sigma \left(t\right)exp\left(at\right)=\frac{E_1{E}_2}{\eta }\dot{\varepsilon }\left[\frac{t}{a}{e}^{at}-\frac{1}{a^2}{e}^{at}+\frac{1}{a^2}\right]$$ (A.8)

$$+(E_1+E_2)\dot{\varepsilon }\left[\frac{1}{a}(e^{at}-1)\right]$$ (A.9)

Expand and collect terms:

$$\sigma \left(t\right)exp\left(at\right)=\left[\frac{E_1{E}_2}{\eta a}t{e}^{at}-\frac{E_1{E}_2}{\eta a^2}{e}^{at}+\frac{E_1{E}_2}{\eta a^2}\right]\dot{\varepsilon }$$ (A.10)

$$+\left[\frac{E_1+E_2}{a}{e}^{at}-\frac{E_1+E_2}{a}\right]\dot{\varepsilon }$$ (A.11)

Combine $e^{at}$ terms:

$$\sigma \left(t\right)exp\left(at\right)=\left[\frac{E_1{E}_2}{\eta a}t+\frac{E_1+E_2}{a}-\frac{E_1{E}_2}{\eta a^2}\right]e^{at}\dot{\varepsilon }$$ (A.12)

$$+\left[\frac{E_1{E}_2}{\eta a^2}-\frac{E_1+E_2}{a}\right]\dot{\varepsilon }$$ (A.13)

Now, divide both sides by $exp\left(at\right)$ to solve for $\sigma \left(t\right)$:

$$\sigma \left(t\right)=\left[\frac{E_1{E}_2}{\eta a}t+\frac{E_1+E_2}{a}-\frac{E_1{E}_2}{\eta a^2}\right]\dot{\varepsilon }$$ (A.14)

$$+\left[\frac{E_1{E}_2}{\eta a^2}-\frac{E_1+E_2}{a}\right]exp(-at)\dot{\varepsilon }$$ (A.15)

Substitute $a=\frac{E_2}{\eta }$, so $1/a=\frac{\eta }{E_2}$, $1/a^2=\frac{{\eta }^2}{E_2^2}$:

After simplification, all terms combine to:

$$\sigma \left(t\right)=E_1\dot{\varepsilon }t+\eta \dot{\varepsilon }\left[1-exp\left(-\frac{E_2}{\eta }t\right)\right]$$ (A.16)

Physical meaning: The first term is the purely elastic response, while the second term describes the time-dependent viscoelastic contribution, which decays exponentially with a time constant $\tau =\eta /E_2$.

(2) Step strain input ($\varepsilon \left(t\right)={\varepsilon }_0$)

For a step strain, $\varepsilon \left(t\right)={\varepsilon }_0$, so $\frac{d\varepsilon \left(t\right)}{dt}=0$.

Substitute into Eq. (A.1):

$$\sigma \left(t\right)+\frac{\eta }{E_2}\frac{d\sigma \left(t\right)}{dt}=E_1{\varepsilon }_0$$ (A.17)

Multiply both sides by $\frac{E_2}{\eta }$:

$$\frac{d\sigma \left(t\right)}{dt}+\frac{E_2}{\eta }\sigma \left(t\right)=\frac{E_1{E}_2}{\eta }{\varepsilon }_0$$ (A.18)

This is a first-order linear nonhomogeneous ODE. The integrating factor is again $\mu \left(t\right)=exp\left(\frac{E_2}{\eta }t\right)$.

Multiply both sides by $\mu \left(t\right)$:

$$\frac{d}{dt}\left[\sigma \left(t\right)exp\left(\frac{E_2}{\eta }t\right)\right]=\frac{E_1{E}_2}{\eta }{\varepsilon }_{0}exp\left(\frac{E_2}{\eta }t\right)$$ (A.19)

Integrate both sides from $0$ to $t$:

$$\sigma \left(t\right)exp\left(\frac{E_2}{\eta }t\right)-\sigma \left(0\right)=\frac{E_1{E}_2}{\eta }{\varepsilon }_0{\int }_0^{t}exp\left(\frac{E_2}{\eta }\tau \right)d\tau$$ (A.20)

$$=\frac{E_1{E}_2}{\eta }{\varepsilon }_0\cdot \frac{\eta }{E_2}\left[exp\left(\frac{E_2}{\eta }t\right)-1\right]$$ (A.21)

$$=E_1{\varepsilon }_0\left[exp\left(\frac{E_2}{\eta }t\right)-1\right]$$ (A.22)

So,

$$\sigma \left(t\right)exp\left(\frac{E_2}{\eta }t\right)=\sigma \left(0\right)+E_1{\varepsilon }_0\left[exp\left(\frac{E_2}{\eta }t\right)-1\right]$$ (A.23)

$$\sigma \left(t\right)=E_1{\varepsilon }_0+\sigma \left(0\right)exp\left(-\frac{E_2}{\eta }t\right)-E_1{\varepsilon }_{0}exp\left(-\frac{E_2}{\eta }t\right)$$ (A.24)

$$\sigma \left(t\right)=E_1{\varepsilon }_0+\left[\sigma \left(0\right)-E_1{\varepsilon }_0\right]exp\left(-\frac{E_2}{\eta }t\right)$$ (A.25)

The initial condition is $\sigma \left(0\right)=(E_1+E_2){\varepsilon }_0$, so:

$$\sigma \left(t\right)=E_1{\varepsilon }_0+E_2{\varepsilon }_{0}exp\left(-\frac{E_2}{\eta }t\right)$$

Physical meaning: The initial stress is $(E_1+E_2){\varepsilon }_0$, and it relaxes to $E_1{\varepsilon }_0$ with a time constant $\tau =\eta /E_2$.

(3) Summary

- Ramp strain input ($\varepsilon \left(t\right)=\dot{\varepsilon }t$):

- Step strain input ($\varepsilon \left(t\right)={\varepsilon }_0$):

Data availability

Data will be made available on request.