Time-dependent mechanical behavior of tough hydrogels under multiaxial stretching

ABSTRACT Understanding the time-dependent mechanical behavior of tough and viscoelastic hydrogels under complex external loading is crucial. In this study, we utilized tough and viscoelastic hydrogels synthesized through the copolymerization of methacrylic acid and methacrylamide as a model system to investigate their mechanical behavior under multiaxial stretching across a wide range of strain rates. Three stretching modes examined were uniaxial, pure shear, and equal biaxial stretching. Our findings show that under equal biaxial stretching, the hydrogels exhibit higher mechanical properties and energy dissipation compared to uniaxial and pure shear stretching, owing to the greater contribution of hydrogen bonds to energy dissipation in the former stretching mode. Additionally, employing the time-elongation separability method during the stretching process, we observed that the relaxation of dynamic hydrogen bonds in the hydrogels only depends on stretching time, independent of the elongation ratio and stretching modes. We anticipate that this study will yield valuable contributions to the design of durable load-bearing soft materials, particularly in dealing with complex deformation and strain rate responses. GRAPHICAL ABSTRACT

Tough and viscoelastic hydrogels exhibit large deformations and strain rate responses, which led to extensive applications in areas such as healthcare, bioelectronics, and soft robotics [3][4][5].In these applications, hydrogels often involve very complex deformation and strain rate responses, which affect the reliability and stability of material properties [4,5,8,17,[20][21][22][23][24].Understanding the mechanical behavior and damage mechanism of these tough and viscoelastic hydrogels under multi-axial loading modes is indispensable.Currently, there are few studies on biaxial deformation focusing on soft rubber or weak hydrogels [20,21,25].However, little attention has been paid to their mechanical responses of tough and viscoelastic hydrogels under biaxial deformation with various strain-rate loading.
To ensure the accuracy of multiaxial stretching for hydrogels, it is essential to securely and stably clamp the samples to prevent any loosening or sliding.Given the potential risk of stress concentration and damage at the clamping point, it becomes crucial for the sample to possess sufficient mechanical strength.In this study, we investigated the mechanical behavior of a tough and viscoelastic hydrogel, P(MAAm-co-MAA), as a model system under various biaxial tensile modes with different strain-rate loadings.The P(MAAm-co-MAA) hydrogels were synthesized by randomly copolymerizing MAAm and MAA [16,26].These hydrogels, with a water content of approximately 50-70 wt%, exhibit high stiffness and toughness due to the presence of strong inter-and intrahydrogen bonding.Additionally, the hydrogel displays remarkable viscoelastic behavior attributed to the dynamic nature of inter-and intra-hydrogen bonding.The toughness and stiffness exhibited by P(MAAm-co-MAA) hydrogels enable us to investigate their mechanical behavior under biaxial tensile tests.The multiaxial stretching involving three types of stretching modes, including uniaxial extension, pure shear, and equal biaxial tests, was conducted at various strain rates.The mechanical behavior and stress relaxation mechanism of hydrogels were analyzed.

Materials
Methacrylamide (MAAm), methacrylic acid (MAA), 2-ketoglutaric acid were of analytical grade and purchased from Tokyo Chemical Industry Co., Ltd., Shanghai.All the chemicals were used without further purification.

Preparation of P(MAAm-co-MAA) hydrogels
The precursor solution for gel synthesis was prepared, which included the monomers MAAm and MAA, along with the initiator 2-ketoglutaric acid.The total concentration of monomers was set at 5 M, with a molar ratio of MAAm to MAA as 1:9.Additionally, 0.1 mol % of the α-keto was added, relative to the total monomer concentrations.The mixture was thoroughly mixed under light-avoiding conditions.Next, the mixed aqueous solution was injected into the middle of a parallel glass plate with 1 mm thick silicone spacer, and the polymerization was conducted under UV irradiation for 8 h in glove box.The asprepared hydrogels were removed from glass plate and immersed in a large amount of water for at least 7 days to reach the equilibrium state.During this process, water was renewed three times a day and the unreacted monomers were dialyzed.The chemical structure of polymers and the structure illustration is shown in Scheme 1.

Multiaxial tensile measurements
Multiaxial tensile and cyclic test were conducted at 25°C using biaxial film stretching tester KARO IV (Bruker, Germany).The biaxial stretching experiments involved three types of stretching modes, namely uniaxial extension, pure shear, and equal biaxial tests (Scheme 2).The labels 'U,' 'PS,' and 'EB' in the diagram correspond to uniaxial stretching, pure shear stretching, and equal biaxial stretching, respectively.In uniaxial extension, a sample sheet was stretched along the x-direction while keeping free scaling in the y-direction.In pure shear extension, a sample was stretched along x-direction, while the dimension in the y-direction was kept unchanged (λ y = 1).In equal biaxial extension, a sample was stretched equivalently stretched along two directions.In the measurement, the tensile engineering stress, including σ x and σ y , can be recorded as a function of the corresponding elongation ratios λ x and λ y along the x and y directions, respectively.The geometry of the sample sizes for the uniaxial extension, pure shear and equal biaxial test was 75 mm (length) × 60 mm (width) × 1 mm (thickness), 75 mm (length) × 75 mm (width) × 1 mm (thickness), and 75 mm (length) × 75 mm (width) × 1 mm (thickness), respectively.For three types of stretching modes, the samples were stretched to a predetermined elongation ratio 1.6 under various tensile velocities, which corresponds to various engineering strain rates.For the multiaxial cyclic tests, three types of stretching modes, including uniaxial extension, pure shear and equal biaxial test, were also performed.The samples were stretched to a predetermined elongation ratio λ i (=1.6) at an engineering strain rate, and then the clamp returned to its original position at the same strain rate without stopping at the peak stretched elongation ratio λ i (=1.6), completing the tensile cycle.Cyclic tests with various engineering strain rates were performed.In this study, we conducted equal biaxial stretching using five different engineering strain rates.For other deformation modes, seven strain rates were employed.Furthermore, the waiting time after the loading-unloading cycle at a specific strain rate is approximately 15 min.We have confirmed that within this waiting time, the broken hydrogen bonds are fully restored before conducting the second cyclic tensile test.Before the measurements, the biaxial stretching apparatus was operated for 30 min to ensure test reliability.To prevent water evaporation from the samples during the measurement, a silicone oil was coated on the surface of the samples.

Stress-elongation relationships under various stretching modes
We first investigated the mechanical behavior of hydrogels at the engineering strain rate of 0.05 s −1 under different stretching modes, including uniaxial extension, pure shear, and equal biaxial test.P(MAAm-co-MAA) hydrogels with a molar ratio of MAAm to MAA as 1:9, after reaching the equilibrium state in water and having a water content of 70 wt%, exhibit translucency, suggesting macroscopic homogeneity of the sample (Scheme 2).Additionally, previous research has demonstrated that there is no distinct scattering observed in the wide-angle X-ray scattering pattern of the gel in its wet state or in the X-ray diffraction pattern of the gel in its dry state [16].This indicates that the gel lacks a crystalline structure at the nanoscale and exhibits an amorphous nature.Scheme 2 shows the images of the unstretched (λ = 0) and stretched (λ = 1.5) samples under different stretching modes.The hydrogel possesses sufficient strength to ensure that the sample remains undamaged at the clamping point under various biaxial tensile conditions, allowing it to withstand significant deformation and undergo testing.The stress-elongation ratio of samples stretched to λ = 2.0 under various stretching modes are shown in Figure 1.All stress-elongation curves demonstrate yielding behavior at small deformations, resulting in enhanced mechanical properties of the hydrogels.The stress-elongation curves obtained from equal biaxial stretching consistently showed higher values compared to uniaxial and pure shear stretching.Both the x-and y-directions of equal biaxial stretching demonstrated similar stress-strain elongation behavior.Therefore, we presented the stress- elongation curve specifically for the x-direction in the subsequent section.The y-direction of pure shear stretching exhibited the weakest tensile behavior, while the stress-elongation curves for uniaxial stretching and the x-direction of pure shear stretching were nearly identical.The minimum tensile stress is found in the y-direction of pure shear represented as PS-σ y , which limits deformation along the y-axis.Furthermore, the stress in the x-direction of pure shear is approximately twice as high as the stress in the y-direction.It was found that the Young's modulus (E) of the hydrogel was measured at the engineering strain rate 0.05 s −1 as 903 kPa for equal biaxial stretching, 511 kPa for the x-direction of pure shear stretching and uniaxial stretching, and 263 kPa for the y-direction of pure shear stretching.According to linear elastic theory, the E values for equal biaxial stretching, x-direction of pure shear stretching, uniaxial stretching, and y-direction of pure shear stretching can be calculated using the formulas 2G 0 (1+μ)/(1-μ), 2G 0 /(1-μ), and 2G 0 μ/(1-μ), respectively.Here, μ represents the Poisson ratio and G 0 represents the shear modulus of the soft materials.Assuming the materials are incompressible (with μ set as 0.5), the predicted E ratio for different stretching modes is 3:2:1, which closely matched the measured ratio.It means that P(MAAm-co-MAA) hydrogels shows the rubber-like behavior at the engineering strain rate 0.05 s −1 , and the mechanical behavior of hydrogels at small deformation can be described by the linear elastic theory.

Time-dependent mechanical behavior under various stretching modes
P(MAAm-co-MAA) hydrogels are formed through the associations of hydrogen bonds between polymer chains.As a result of the dynamic breaking and re-forming nature of these hydrogen bonds, these hydrogels exhibit viscoelastic behavior, making their mechanical properties significantly influenced by the applied strain rate.Now we systematically investigated the dependence of mechanical behaviors on various strain rate _ ε (from 0.007 s −1 to 0.2 s −1 ) at a fixed peak elongation ratio λ = 1.6 under various stretching modes, which is below the elongation at break of the hydrogels.All stress-elongation curves under various stretching modes strongly depend on the applied strain rate, which are shown in Figure 2. The hydrogels measured at a higher strain rate show the higher mechanical properties.The variations of Young's modulus E with the strain rate under various stretching modes are summarized in Figure 3.The results demonstrate a significant dependence of E on the strain rate, where E increases as the strain rate increases.In the tested range of strain rates, regardless of the deformation mode, the Young's modulus E in biaxial stretching is greater than that in pure shear stretching along the X-axis.The E value for uniaxial stretching is close to that of pure shear, while the minimum E value is observed in pure shear along the Y-axis.At higher strain rates, a larger number of hydrogen bonds do not have sufficient time to relax.As a result, they act as 'chemical crosslinkers' and enhance the mechanical properties of hydrogels.In uniaxial stretching, a higher load is required in the absence of constraints along the Y-axis to achieve faster deformation through internal alignment and hydrogen bond dissociation of polymers.In the process of pure shear deformation, although the X-axis still reaches the same level of deformation, the constraint along the Y-axis can be considered as loading the material in that direction, guiding the orientation of polymer chains in the Y-axis direction, thereby reducing the stress required in the X-axis direction and affecting the original elastic modulus along the X-axis.In the case of biaxial stretching, where both directions are simultaneously loaded, a large number of hydrogen bonds need to be broken in order to stretch and separate the polymer chains in different directions.As a result, the highest mechanical properties are observed.

The loading-unloading behavior
In order to investigate energy dissipation during stretching, we performed loadingunloading measurements using biaxial film stretching tester under different stretching modes.The loading-unloading curves at various strain rates are presented in Figure 4.It is evident from the curves that a distinct hysteresis loop is observed, indicating the occurrence of energy dissipation during the loading-unloading process.For the high water content of hydrogels, some of the inter-or intra-chain hydrogen bonds may be replaced by hydrogen bonds formed between water and polymer chains.This can result in low hysteresis during the loading-unloading test.In the case of P(MAAm-co-MAA) hydrogels, the presence of hydrophobic motifs (methyl group) on the polymer chain helps stabilize the hydrogel by shielding the hydrogen bonds from water molecules.As a result, the   hydrogels exhibit a large hysteresis loop due to the rupture of hydrophobic interactions and hydrogen bonds.Moreover, the area of the hysteresis loops with higher strain rates across different stretching modes.In order to examine the correlation between energy dissipation and strain rate more comprehensively, we compiled the energy dissipation data at various strain rates for different stretching modes.The energy dissipation (D) was determined by integrating the areas enclosed by the original loading and unloading curves (Figure 5(a)).It is important to note that biaxial stretching involves two stretching directions.The calculation of D is as follows: Where W 0 and W i referred to work of stretching in the first loading and reloading, respectively.W i could be calculated according to the following equation: The unloading process can be considered as the reloading process.The energy dissipation factor, Δ ¼ D=W 0 , and D as a function of strain rate are illustrated in Figure 5. D displayed clear dependence on the strain rate (Figure 5(a)), with greater energy dissipation observed under equal biaxial stretching compared to the other two stretching modes.This can be attributed to the total deformation occurring in two stretching directions, leading to more fracture of dynamic hydrogen bonds.Compared to uniaxial stretching, the deformation of the sample is constrained in the Y-axis direction during pure shear cyclic stretching, resulting in a larger number of dynamic bonds being broken.This leads to enhanced energy dissipation.D and Δ approximately followed an exponential relationship, D _ ε 2 .Furthermore, D dependence on Δ was not influenced by the stretching modes.Similarly, Δ exhibited similar dependence on the strain rate across different stretching modes (Figure 5(b)).Both D and Δ gradually increased with higher strain rates, implying that more hydrogen bonds contribute to the energy dissipation at high strain rate.
Since different stretching modes lead to varying levels of energy dissipation, it is essential to comprehend the dynamic breakage of hydrogen bonds during each specific stretching mode.For some soft materials from the associations of dynamic bonds, it has been demonstrated that the elongation amplitude can be decoupled from the time dependency [22,27].This leads to a modulus that decreases with deformation time when the breakage of dynamic bonds is governed by the thermal activation process.The time-dependent function of reduced stress σ red (shear modulus) can be expressed as follows: σ red ¼ σ f λ ð Þ , where f λ ð Þ is related to the deformation function, and σ is the engineering stress.To confirm whether or not such behavior is applicable for current systems, the reduced stress σ red of hydrogels as a function of deformation time under various stretching modes from Figure 2 is determined.Assuming that the deformation of hydrogels follows a Neo-Hookean model, the deformation correlation term f λ ð Þ can be described for various stretching modes as follows [21,25] Figure 6 presents the plots depicting the dependence of reduced stress σ red t ð Þ on time at various strain rates under different stretching modes.In all stretching modes, the reduced stress σ red t ð Þ gradually decreases as the stretching time increases.This stress-softening behavior can be attributed to the continuous fracture of hydrogen bonds during the stretching process.Notably, the curves for different strain rates almost overlap, forming a master curve that encompasses a wide range of stretching times, except for some data at the extremely high strain rate (=0.2 s −1 ).It means that the relaxation of dynamic bonds in the hydrogels was only related to the stretching time, independent of the elongation ratio and stretching modes.For the relaxation behavior of dynamic materials, Long proposed a model that reduced stress σ red t ð Þ is related to the relaxation of dynamic bonds during the deformation.The derived reduced stress σ red t ð Þ can be described as [27].
Where, t R is the breaking time of the dynamic bonds, G 0 is the shear modulus of hydrogel at the unrelaxed state, and α is the positive dimensionless material constant.Furthermore, the fraction of 'chemical crosslinkers' in the total chains ρ can be determined from the ratio of modulus at the unrelaxed state to that at the fully relaxation state.We applied this theory to our system and successfully fitted the experimental data using equation 7, as depicted in Figure 6.The relaxation curves for all stretching modes exhibited excellent agreement with the fitted parameters.These parameters, including G 0 (~250 kPa), α (~2.3), t R (~2 s) and ρ (~30%), are summarized in Table 1.Interestingly, we observed that these parameters had nearly identical values regardless of the stretching mode employed.This indicates that the amount of broken hydrogen bonds may vary under different stretching modes, but the breakage of dynamic bonds primarily depends on the loading time, rather than the specific stretching mode.It was also found that these parameters remain consistent regardless of the stretching mode used in dual-cross-link poly(vinyl alcohol) hydrogels [21].This suggests that the parameters obtained through this method are universal and can be utilized to describe the mechanical properties of other types of tensile viscoelastic hydrogels.Furthermore, the value of ρ is approximately 30%, indicating that around 70% of dynamic bonds undergo relaxation during the different stretching modes.The characteristic bond breaking time is roughly 2 s.The mechanical behavior of dynamic hydrogels is strongly influenced by the relationship between the bond breaking time (t R ) and the applied strain rate (_ ε) [10,12,28].When _ ε � t R ~1 is achieved, the bonds enter a mobile state, resulting in time-dependent behavior of the hydrogel.At strain rates significantly lower than the inverse of the bond lifetime (_ ε � t R <1), the complete relaxation of bonds leads to soft elastic  behavior in the hydrogel.In our study, the investigated strain rates _ ε ranged from 0.007 to 0.2 s −1 , indicating that the hydrogel exhibits viscoelastic properties at high strain rates and a quasi-static state at very low strain rates.In this investigated strain rates, the relaxation of dynamic bonds in the hydrogels was only related to the stretching time, independent of the elongation ratio and stretching modes.

Conclusions
Tough and physical P(MAAm-co-MAA) hydrogels from the associations of hydrogen bonds were prepared by using random radical copolymerization.The mechanical behavior under complex external loading (uniaxial stretching, pure shear stretching and equal biaxial stretching) was systematically investigated over a wide range of strain rate.The hydrogels measured at a higher strain rate show the higher mechanical properties.The stress-elongation curves from equal biaxial stretching consistently exhibited higher values than those from uniaxial and pure shear stretching.The stress-elongation behavior was similar in both the x-and y-directions of equal biaxial stretching.The y-direction of pure shear stretching displayed the weakest tensile behavior, while the stress-elongation curves for uniaxial stretching and the x-direction of pure shear stretching were nearly indistinguishable over a wide range of strain rate.Biaxial stretching causes the most damage (energy dissipation) to the sample, followed by pure shear stretching, while uniaxial stretching causes the least amount of damage to the sample.Moreover, by employing the time-elongation separability method, it has been observed that the damage or relaxation of hydrogen bonds in hydrogels is solely determined by the stretching time, irrespective of the elongation ratio and stretching modes.

Figure 1 .
Figure 1.The typical engineering stress-elongation curves of hydrogels involving three stretching modes at the strain rate of 0.05 s −1 .U, PS-X, PS-Y and EB correspond to the stress-elongation curves under the uniaxial stretching, pure shear stretching along X direction, pure shear stretching along Y direction, and equal biaxial stretching, respectively.

Figure 3 .
Figure 3.The strain rate dependent of modulus under various stretching modes: uniaxial stretching (U), x-direction of equal biaxial stretching (EB-X), y-direction of equal biaxial stretching (EB-Y), x-direction (PS-X) and y-direction (PS-Y) of pure shear stretching.

Figure 5 .
Figure 5. Energy dissipation (a) and energy dissipation factor (b) as a function of strain rate at different stretching modes.

Table 1 .
Fitted structural parameters of hydrogels under various stretching modes.