Investigation on the tensile rheological behavior of pa6 film based on fractional order model

This paper presents the research results on the rheological behavior of Polyamide 6 (PA6) film during uniaxial stretching. The creep and relaxation tests of PA6 films were conducted on dynamic mechanical analysis (DMA) tester under uniaxial tensile conditions. And analyzed the PA6 film creep behavior under different stress and temperature, the stress relaxation behavior under different strain. With the increase of temperature or stress, the creep behavior of PA6 film was enhanced. With the increase of strain, the relaxation of PA6 film showed an enhanced tendency. Fractional-order Maxwell model, fractional-order Kelvin model and fractional-order linear body model were used to fit and analyze the experimental data under different conditions. The results showed that the fractional-order linear body model was better suited for describing the rheological behavior of PA6 film under uniaxial stretching. Then, the fractional-order linear body model was used to fit and analyze the creep and relaxation curves under different conditions, and the rheological properties of PA6 film were discussed according to the fitting results. The results indicated that the rheological elements in the internal microstructure of PA6 film were more easily activated under high stress state, and the macroscopic viscous characteristics were more obvious under high strain state. The above results will provide a theoretical basis to further explore the deformation mechanism of PA6 film during tensile tests.


Introduction
Due to the excellent physical, chemical and mechanical properties, the microstructure of biaxially oriented films can be precisely controlled.Biaxially oriented technology has become one of the main production methods of high performance films [1][2][3].In the actual industrial production, the preparation process of the stretched film involves the complex microstructure evolution and mechanical behavior inside the polymer, so there are some basic scientific problems that have not yet been fully understood [4,5].If a new mathematical constitutive model can be established to describe the stress-strain relationship of stretched films, it will be helpful to further explore the relationship between material microstructure evolution and mechanical properties.Also, classical models based on integer-order calculus (Maxwell, Kelvin and Zener) and various characterization methods have been used to study the deformation behavior of polymers, but there are some limitations in reflecting the real properties and complex mechanical behavior of materials [6][7][8].For example, Maxwell and Kelvin models are only suitable for describing the dynamic mechanical behavior of materials in a narrow frequency range [9].Wang et al [10] compared the effect of the number of Maxwell model units on the fitting accuracy of relaxation time spectrum curve of polyvinyl chloride (PVC) coating films, and found that increasing the quantity of Maxwell units contributes to enhancing the fitting accuracy of the model.However, the above model requires a large number of Maxwell and Kelvin units for practical complex problems, resulting in high computational complexity and not suitable for fast calculation in practical applications [11][12][13].Therefore, it is necessary to introduce new rheological models to accurately describe the complex viscoelastic and mechanical behaviors of polymers.
In recent years, the constitutive equation of fractional order operator has been increasing attention and research [14].Fractional order rheological model with global correlation, by defining the differential and integral item, can use fewer parameters accurately describe a viscoelastic material temperature or in a wider frequency range of complex mechanical behavior [15].It overcomes the disadvantage of inconsistent model fitting with experimental data when using integer-order models to deal with complex physical processes [16][17][18].Li et al [19] conducted a long-term uniaxial compression creep test on polyvinyl alcohol (PVA) fiberreinforced cement-based composite (PVA-ECC), and found that the fractional order Maxwell model can more accurately fit the overall viscoelastic behavior of PVA-ECC polyphase composite than the integer order linear model.Wei et al [20] proposed a temperature-dependent fractional-order constitutive model based on rheological theory to describe the mechanical behavior of different glassy polymers, its effectiveness and predictability were verified through experimental validation.In the work of dynamic viscoelasticity of carbon black filled rubber (CB-Filled rubber) by Yin et al [21], it was found that the fractional mFDZ model can well fit the loss modulus, loss factor and Cole-Cole diagram asymmetry of the actual CB-Filled rubber test [22].Alcoutlabi and Martinez-Vega et al [23] successfully predicted the asymmetric dynamic behavior of Polymethyl methacrylate (PMMA) polymer under real conditions through the correction of the fractional-order Zener model.As a powerful tool, fractional calculus has been proved to play an important role in the dynamic modeling of materials [24].The fractional-order rheological model has been widely applied in the investigation of engineering nonlinear behavior, but describing the constitutive relations of polymers is still a relatively new field and has not been widely explored and applied.Therefore, by studying polymer rheological constitutive models and experimental data, the applicability and effectiveness of these models can be further evaluated, providing theoretical guidance for optimizing materials processing.Due to the existence of multi-scale fractal structures, such as dendrites and spherulites in stretched films [25], the fractional rheological model is very suitable for accurately describing its nonlinear behavior.
In this paper, the creep and relaxation tests of polyamide 6 (PA6) film under tensile mode were carried out by Dynamic Mechanical Analyzer (DMA).Analyzed the PA6 film creep behavior under different stress and temperature, the stress relaxation behavior under different strain.Then using the fractional-order Maxwell model, fractional-order Kelvin model and fractional-order linear body model to fit and analyze the experimental data.The results show that the fractional-order linear body model has better fitting effect.Finally, the fractionalorder linear model was used to fit the experimental data under different conditions.On this basis, the effects of stress and temperature on the creep behavior of PA6 film and the effects of different strains on the relaxation behavior of PA6 film were systematically analyzed.In this study, polyamide 6 (PA6) was used as the research object, and the experimental PA6 raw material was provided by Japan's UBE Co., Ltd, with the brand number 1022B10.PA6 stretch film was prepared by extrusion casting method.The pouring temperature was set at 240 °C-260 °C and the screw speed was 20 r min −1 .By adjusting the extrusion speed and the casting roll speed, the film thickness was controlled at 0.2-0.3mm, and then cooled to 30 °C on the casting roll.The PA6 film was cut into a sample with a size of 10 × 4 × 0.24 mm 3 , and then using the DMA to test the creep and relaxation of the sample under tensile mode.The creep test was carried out at different stresses (1 MPa, 1.5 MPa, 2 MPa) and temperatures (25 °C, 40 °C, 50 °C), while the relaxation test was carried out under different strain(0.02,0.04, 0.06),the loading frequency was 1 Hz and the time was 30 min.The sample size and test conditions are shown in table 1, table 2, and table 3, respectively:

Creep behavior under different stress states
When the temperature is 25 °C and the creep time is 30 min, the creep curves of PA6 film under different stress (1 MPa, 1.5 MPa, 2 MPa) are shown in figure 1.It can be seen that within the set stress range, the creep behavior of the PA6 film under three conditions shows two stages of initial creep and steady creep, and no accelerated creep phenomenon occurs, which may be attributed to the short set time or insufficient stress.From figure 1, it can also be observed that with the increase of stress, the creep length of PA6 film gradually increases, accompanied by a long duration of initial creep and an increase in steady-state creep rate with stress.

Creep behavior at different temperatures
In order to investigate the effect of temperature on the creep behavior of PA6 film, we set the stress to 2 MPa and the creep time to 30 min during testing.The creep test of PA6 film was carried out at different temperature (25 °C, 40 °C, 50 °C), and the creep curves as shown in figure 2. It can be seen that the three sets of creep curves only show two stages of initial creep and steady state creep.At the same time, the length of the creep curve gradually increases with increasing temperature, and a similar trend occurs with the previous set of tests.Indicating that the stress and temperature promote the creep of PA6 film and achieve the same effect.The above law is consistent with the time-temperature equivalence principle in rheological [26].It can also be seen from figure 2 that as the temperature increases, the initial creep time of the PA6 film continues to increase, but the steady-state creep curve is gentler, indicating that the steady-state creep rate decreases.So, we can infer that at higher temperatures, the creep phenomenon of PA6 film has been basically completed in the initial stage.

Relaxation behavior under different strain conditions
In order to further study the relaxation behavior of PA6 film under uniaxial tension, we tested PA6 film under different strain (0.02, 0.04, 0.06) at 25 °C for 30 min in relaxation mode.The relaxation curves under different strains as shown in figure 3. It can be seen that with the increase of strain, the stress relaxation phenomenon is more obvious.Similar to the creep curve, the relaxation curve also shows two stages: the initial stage and the steady-state stage.It can also be seen from figure 3 that the initial relaxation stage becomes longer with the increase of strain.This may be related to the looser internal molecular chains of the polymer under high strain conditions.

Fractional-order rheological model
In order to further analyze the rheological mechanism of PA6 film under uniaxial tension, we used fractional Maxwell model, fractional Kelvin model and fractional linear model to fit the creep and relaxation curves.

Fractional-order maxwell model
The fractional-order Maxwell model consists of a linear spring and a spring pot in series, which is shown in figure 4.  The strain of spring and the spring pot under the action of stress σ is ε 1 and ε 2 , respectively, where: E , e e e = + and the constitutive equation of the fractional Maxwell model is obtained as follows : Examining the static viscoelastic behavior of the fractional-order Maxwell model, assuming a unit step load, the Laplace transform and inverse transform of equation (1) can be obtained as the stress-strain relationship between the creep and relaxation processes: is the Mittag-Leffler function [27].

Fractional-order kelvin model
The fractional Kelvin model is composed of a spring and a spring pot in parallel, as shown in figure 5.
It is assumed that under the action of total stress σ, the stress of the spring and the spring pot is σ 1 and σ 2 respectively, and the strain of the spring and the spring pot is ε.Where: E , To investigate the static viscoelastic behavior of the fractional-order Kelvin model, assuming a unit step load, the Laplace transformation and inverse transformation of equation (4) can be used to obtain the stress-strain relationship in the creep process and relaxation process:   e e e = + Then the constitutive equation of the fractional linear body is obtained: Assuming that a unit step load acts, the Laplace transform and inverse transform of (7) can be used to obtain the stress-strain relational expressions of the creep process and relaxation process.The creep curve of the temperature of 25 °C and the stress of 2 MPa were fitted using three fractional-order models.The fitting curves and the obtained elastic modulus parameters (E 0 , E 1 , E 2 ) are shown in figure 7. The fitting results show that the three fractional-order rheological models mentioned above fit well with the experimental data.At the same time, the correlation coefficient R 2 of the three models were both more than 0.99.Indicating that the fractional-order rheological model has a strong advantage in simulating the rheological behavior of viscoelastic materials.Comparing the fitting results of the three models, it was found that the correlation coefficient R 2 (0.997) of the fractional-order linear body is the highest.And we observed in the fitting process that if the fitting accuracy of the fractional-order Kelvin model and the fractional-order Maxwell model is further improved, the obtained elastic modulus will be significantly different from the experimental values, while the fractional-order linear body model still has a good fitting effect.From this we can concluded that the fractional-order linear body model is more suitable for describing the rheological behavior of PA6 film under uniaxial tension than the other two models.

Creep behavior under different stress states fitting analysis
In order to further investigate the creep behavior under different stress conditions, we used the fractional linear body model to fit the three sets of creep test curves at 25 °C.The fitting results are shown in figure 8 and table 4. It can be seen from table 4 that the elastic modulus parameters (E 0 , E 1 ) and the relaxation time (τ) and the fractional order (α) decrease with increasing stress.Based on the variation of elastic modulus and relaxation time, we can infer that under high-stress conditions, the rheological units of the internal microstructure of PA6 film are more prone to activation, showing shear thinning.According to the definition of fractional calculus, the fractional order (α) ranges from 0 to 1, and as it approaches 0, the macroscopic properties of the material become more like solid.From the variation law of the fractional order (α) obtained by fitting, the fractional order α is basically maintained within the range of 0.5 or less, indicating that the PA6 film exhibits an apparent elastic characteristic under the experimental conditions of this group.The phenomenon is consistent with the previous analysis, which indicates that the creep phase only shows two phases of initial and steady state.

Creep behavior under different temperatures states fitting analysis
The creep curves at different temperatures were analyzed by using a fractional-order linear body model.The results of the fitting are shown in figure 9 and table 5. From the obtained model parameters, it can be found that with the increase of temperature, the elastic modulus E 0 gradually increases, while the elastic modulus E 1 , the relaxation time τ and the fractional order α decrease.According to the deformation law of softening of the polymer with increasing temperature, we can speculate that in the fractional-order body linear model, the elastic pot is mainly used to simulate the viscoelastic deformation characteristics of PA6 film, while the spring in series with the fractional-order Kelvin model is mainly used to characterize the apparent elastic characteristics of a   material.The range of variation of the fractional order α also reflects the deformation of the material in an apparent elastic region.

Creep behavior under different stress states fitting analysis
In order to further analyze the stress relaxation phenomenon of PA6 film, we used the fractional-order linear body model to analyze the relaxation curves under different strain conditions.The fitting results are shown in figure 10 and table 6.It can be observed that as the strain increases, the elastic modulus E 0 and the relaxation time τ decrease gradually, the elastic modulus E 1 does not change significantly, while the fractional order α increases.Combined with the relaxation deformation law under different strains, we can get that in the stress relaxation process of PA6 film, the stress reduction mainly results from the relaxation of spring E 0 .The increase of the fractional order α indicates that the macroscopic viscous characteristics exhibited by the PA6 film are more obvious under high-strain conditions.

Conclusion
In this study, the polyamide 6 (PA6) film was used as the research object.Based on the fractional calculus theory, the rheological behavior of PA6 film under uniaxial tension was studied.The following conclusions were obtained:  (1) Through creep and relaxation tests on PA6 films under uniaxial tension, it was found that the creep phenomenon significantly intensifies with increasing stress and temperature.The result shows that the effects of stress and temperature on the creep behavior of PA6 film are consistent.And there are only two stages of initial creep and steady state creep in the set experimental range.Under different strain conditions, it is found that the relaxation phenomenon of the PA6 film became more obvious as the strain increased.
(2) The fractional-order Maxwell model, fractional-order Kelvin model and fractional-order linear body model are derived based on fractional order theory.The creep curves of the temperature of 25 °C and the stress of 2 MPa were respectively analyzed by three models.It was found that the fractional-order linear body model is more suitable for describing the rheological behavior of PA6 film under uniaxial tension.
(3) Fitting the creep curve and the relaxation curve under different conditions using a fractional-order linear body model.According to the model parameters, it can be found that the rheological unit of the internal microstructure of the PA6 film is easier to activate under high stress conditions; In the fractional-order linear body model, the elastic pot is mainly used to simulate the viscoelastic deformation characteristics of the PA6 film, while the spring in series with the fractional-order Kelvin model is mainly used to characterize the apparent elastic characteristics of the material; in the stress relaxation process of the PA6 film The reduction of stress mainly comes from the relaxation of spring E 0 .Under high strain state, the macroscopic viscosity characteristic of PA6 film is more obvious.

Figure 1 .
Figure 1.Creep curves at different stress with temperatures of 25 °C.

Figure 2 .
Figure 2. Creep curves at different temperatures with a stress of 2 MPa.

Figure 3 .
Figure 3. Relaxation curves at different strain with temperatures of 25 °C.
Then the total strain of the model is: , 1 2

data fitting and analysis 4 . 1 .
Comparison of three fractional-order models

Figure 7 .
Figure 7. Different fractional-order models analyze the creep behavior at 25 °C under a stress of 2 MPa: (a) fractional-order Maxwell model; (b) fractional-order Kelvin model; (c) fractional-order linear body model.

Figure 8 .
Figure 8. Fractional-order linear body model fitting analysis of creep behavior under different stress states at 25 °C.

Figure 9 .
Figure 9. Fractional-order linear body model fitting analysis of creep behavior at different temperatures when the stress is 2 MPa.

Figure 10 .
Figure 10.Fractional-order linear body model fitting analysis of relaxation behavior under different strain states at 25 °C.

Table 1 .
Creep test conditions at 25 °C under different stress conditions.

Table 2 .
Creep test conditions at different temperatures with stress of 2 MPa.

Table 3 .
Relaxation test conditions under different strain conditions at 25 °C.
Fractional-order linear body model is composed of a spring and a fractional differential Kelvin model in series.The schematic diagram is shown in figure6.Assuming that under the action of stress σ, the strains of the spring and the bullet are ε 1 and ε 2 respectively, where E a a are.Then the total strain of the model is: . 1 2

Table 4 .
The model parameters obtained by fitting and analyzing the creep behavior under different stress states at 25 °C using the fractional-order linear body model.

Table 5 .
The model parameters obtained by fitting and analyzing the creep behavior at different temperatures at a stress of 2 MPa using the fractionalorder linear body model.

Table 6 .
The model parameters obtained from the fitting analysis of the relaxation behavior under different strain states at 25 °C.