Characterizing the stress relaxation behavior of unidirectional prepreg through a parallel fractional-order viscoelastic model

In the hot-stamp molding and hot diaphragm forming processes of composites, pressure significantly influences shaping quality. This study establishes a novel parallel fractional-order viscoelastic (PFOV) model with two Scott-Blair elements, achieving remarkable accuracy (R2 = 0.99) with fewer parameters. Unlike traditional models, it incorporates the force history of prepreg, providing a more precise representation of its mechanical response. Comparative analysis against established models underscores its superior ability to capture intricate stress relaxation behaviors. Notably, the model’s reduced parameters enhance its physical interpretability, offering a significant advantage in simulating and predicting prepreg material compression behavior for diverse manufacturing processes.


Introduction
Carbon fiber-reinforced plastic play a pivotal role in various manufacturing processes, including aerospace, automotive, and sports industries, due to their superior strength-to-weight ratio and tailored properties [1][2][3].In essence, the formation of composite materials entails the application of external forces to shape carbon fibers and resin into specific configurations.Diverse forming techniques, including automated lay-up, hot press molding, hot diaphragm molding, and liquid composite molding, are employed for this purpose [4][5][6][7].Within the aircraft manufacturing sector, the widespread adoption of prepregs is attributed to their stable quality and high molding efficiency.For complex components such as frames or stringer, forming techniques like hand layup or automated lay-up may fall short due to subpar molding quality or low efficiency.Consequently, researchers have turned to preforming techniques for fabricating such components.Notable preforming methods include hot-stamp forming and hot diaphragm forming, wherein prepregs are initially shaped under external pressure and subsequently cured in the autoclave.In contrast to hot diaphragm forming, hot-stamp molding is more suitable for the preforming process of complex structures due to its higher applied pressure and the capability to control pressure in different zones [8].
During the preforming process, inappropriate molding parameters can result in defects such as wrinkles and fiber bending, leading to a deterioration in the mechanical performance and service life of the product [9][10][11].Among the molding parameters, pressure holds significant importance.Pressure serves not only as the driving force for the deformation of prepregs during preforming but also impacts the inter-layer slip resistance of prepregs.Elevated interlayer slip resistance can contribute to the occurrence of defects in the final product.Lee et al have investigated the influence of pressure on the preforming process of fabrics, revealing that appropriate pressure can mitigate the unevenness of preforming bodies, reducing in-plane bending and wrinkles in fabrics [12].Yu et al have researched the effects of pressure and inter-layer slip resistance on molding quality demonstrated that pressure induces in-ply tension, averting irregular deformations and wrinkles [13].Brilliant have researched that uneven pressure occurs when the thickness of prepreg sheets is substantial, leading to the manifestation of wrinkles in the final product [14].Therefore, understanding the mechanical behavior of prepregs during compression is crucial for optimizing manufacturing processes and ensuring the structural integrity of the final products.
Unidirectional prepreg materials, composed of resin and fibers, exhibit intricate stress responses under compression, characterized by significant stress relaxation behaviors.The Generalized Maxwell viscoelastic model is widely employed by researchers to elucidate the stress relaxation dynamics of prepreg materials.Somashekar et al investigated stress relaxation in woven fabrics using the five-component Maxwell elementbased model, considering factors like speed and layer count.They found the model reliably predicted stress relaxation across different fiber volume fractions when other test parameters remained constant [15].Lukaszewicz et al utilized power-law and five-order Generalized Maxwell models to analyze prepreg material compression and stress relaxation during automated lay-up.Their study demonstrated that a simple plastic material law with strain hardening accurately models the material response in this process, with good agreement between finite-element modeling and experimental results [16].While the Maxwell model effectively captures stress relaxation in prepreg materials, its limitation lies in introducing numerous parameters, leading to computational complexity and ambiguous physical interpretations.To address this, some researchers have turned to fractional derivatives [17,18].Sourki et al explored stress relaxation in fabric prepreg forming, utilizing fractional derivatives to describe the viscoelastic behavior of prepreg materials and they highlighted the higher accuracy and superior predictive capability of fractional-order models, particularly in the early stages of stress relaxation [18].Faal et al utilized a Zener fractional-order model to study the viscoelastic behavior of glass fiber/polypropylene fabric prepreg, showcasing the model's improved descriptive capability.Their findings demonstrated the superiority of fractional derivatives over conventional approaches in accurately characterizing material behavior, particularly at lower temperature regimes [19].While the Maxwell model is proficient in modeling stress relaxation in prepreg materials, its drawback lies in the complexity introduced by numerous parameters, which obscure physical interpretations.Fractional-order viscoelastic models have been primarily applied to fabric materials, neglecting unidirectional prepreg materials.Unlike fabric prepregs, unidirectional prepreg materials lack fabric weave points, resulting in different stress relaxation behaviors under compression.Therefore, a novel fractional-order viscoelastic model tailored to unidirectional prepregs are imperative for accurately characterizing their stress relaxation behavior, especially considering the extensive use of unidirectional prepreg materials in the aerospace industry.
In this paper, the introduction of the Parallel Fractional-Order Viscoelastic (PFOV) model have been used to capturing the stress relaxation behavior of unidirectional prepreg during compression.Furthermore, stress relaxation experiments were performed on unidirectional prepreg materials, and the results were compared using the PFOV model and the Maxwell model.The comparison confirmed that the model developed in this study can effectively describe the stress relaxation behavior of unidirectional prepreg materials with a reduced number of parameters.

Experimental
The material utilized in this study is M21C unidirectional prepreg with a single-layer thickness of 0.187 mm.
Viscosity is a critical parameter during the molding process of the prepreg.In this study, the viscosity of the prepreg was investigated using a rotational rheometer (HR-2, TA Instruments) [20,21].The prepreg was cut into a circle with a diameter of 25 mm and stacked into a wafer of five plies.The sample was then fixed by the ring on the fixture (figure S1).The viscosity-temperature performance and viscosity-time performance were tested, with a strain rate of 0.01%, frequency of 1Hz, and axial force of 5N.The viscosity-temperature curves from 40 to 180 °C were tested, as well as the viscosity-time curve at the material preforming temperature (80 °C), with a testing duration of 3600 s.
The prepreg was trimmed and stacked into specimens measuring 150 * 100 mm, with the layer count determined by experimental conditions.To ensure the accuracy of the test results, the samples were subjected to pressing under vacuum pressure (Pressure >75 kPa) for 1 h at 23 °C before the commencement of the test.This process effectively removed air from the sample, minimizing potential sources of error.The testing apparatus mirrored that of Hubert et al (figure 1(A)), comprising upper and lower pressure plates affixed to a universal testing machine (ETM204C, Shenzhen Wance Testing Machine Co., Ltd) [22].During fixture installation, a gap gauge was employed to measure the distance between the upper and lower plates, ensuring consistent spacing at different locations to prevent localized excessive pressure during compression.Before testing, the fixture was heated to the experimental temperature.Ahead of each experiment, the samples were preheated for 15 min, and a thermocouple was used for temperature measurement, ensuring the sample temperature reached the experimental level.Once the experiment commenced, the fixture moved downward, applying pressure to the sample.When the fixture reached the predetermined compression distance, the position remained unchanged, conducting a test on the stress relaxation behavior of the specimen, with a relaxation time of 3600 s.
Throughout the experiment, the force applied to the prepreg is illustrated in figure 1(B).In the figure, e 0 represents the strain of the prepreg when compressed to the specified position during the compression experiment, and t 0 represents the time when reaching the specified position during compression.Considering the practical conditions of the hot-stamp molding process, the compression distance in this study is 0.2 mm.

Model approach
The stress relaxation behavior of materials is commonly described using the Generalized Maxwell model.This model comprises an elastic unit and multiple Maxwell units connected in parallel.The expression for the Generalized Maxwell unit can be articulated using a Prony series: Where, n is the order of the Prony series; E 0 is the equilibrium relaxation modulus; e 0 is the strain of the viscoelastic material; E i is the elastic modulus of the Maxwell unit; t i is the relaxation time of the Maxwell unit.
Considering the time t o used in the compression process, equation (1) can be rewritten as: Nonetheless, the generalized Maxwell model comprises a plethora of parameters, with ambiguous physical interpretations for each, thereby introducing substantial complications in data processing and modeling.Therefore, the fractional viscoelastic model, namely, the Scott-Blair model, has been employed to elucidate the relaxation behavior [23], as expressed below: Where Where Γ(•) is the Euler gamma function.As depicted in figure 1(B), in the experiment, the prepreg initially undergoes compression, followed by stress relaxation behavior.Consequently, its strain evolution is: By substituting equation (5) into equation (6), the stress expression for the Scott-Blair model during the stress relaxation phase in this experiment can be derived as follows [17]: Hence, the formulation for the PFOV model is:

Results and discussion
Figure 3 presents the viscosity-temperature curve of the prepreg spanning the temperature range from 40 to 180 °C, alongside the viscosity-time curve at 80 °C.As discerned from figure 3(A), within the 40 °C-80 °C range, the viscosity of the prepreg exhibits a declining trend with increasing temperature.The nadir of viscosity occurs between 80 °C-110 °C, followed by a slight increase within the 110 °C-160 °C.As temperature continues to ascend, the viscosity of prepreg experiences a rapid surge owing to the initiation of curing reactions.Lower viscosity in the prepreg is indicative of reduced interlayer sliding resistance and improved processing performance.Thus, considering the processing performance and practical molding equipment conditions, an optimal preforming temperature of 80 °C is selected.Figure 3(B) investigates the temporal evolution of prepreg viscosity at 80 °C.It can be discerned from the graph that there is no significant variation in prepreg's viscosity over 3600 s, suggesting that resin curing did not occur during the experimental procedure and not impact the experimental outcomes of compression and relaxation.Figure 4 illustrates the fitting results for the stress relaxation behavior of a typical prepreg in the experiment using the Scott-Blair model, the PFOV model, the Maxwell model expressed with a second-order Prony series, and the Maxwell model expressed with a third-order Prony series.The graph reveals a sudden drop in stress for the prepreg during the initial stages of the relaxation experiment.As time progresses, the magnitude of stress reduction gradually diminishes.It is evident that a single Scott-Blair element and a Maxwell element represented Figure 5 illustrates the stress reduction ratio obtained from the fitting results of different models.The calculation formula for the reduction ratio is as follows: Where, s ( ) t denotes the compressive stress on the prepreg measured at a given moment, and s t 0 is the compressive stress on the prepreg measured at the initiation of the experiment.From figure 5, it is evident that the PFOV model and the Maxwell model expressed with a third-order Prony series closely align with the experimental stress reduction rate results.This suggests that these two models accurately capture the stress relaxation behavior of the prepreg.Conversely, the Scott-Blair model and the Maxwell model expressed with a second-order Prony series deviate significantly from the actual situation, as they fail to precisely represent the abrupt initial drop in prepreg stress.Therefore, in subsequent experiments, the PFOV model and the Maxwell model expressed with a third-order Prony series are utilized for fitting the stress relaxation behavior of the prepreg.
Figure 6 illustrates the relaxation behavior for samples with different layer counts, specifically 5, 10, and 15 layers.Figure 7 presents the fitting results of stress relaxation in the prepreg using two different models: the Maxwell model with third-order Prony series and the PFOV model.Table 1 presents the parameters for the fitting results of stress relaxation curves using the Maxwell model with third-order Prony series.These fits consistently exhibit a coefficient of determination (R 2 ) is 0.99.Table 2 provides the specific parameters for the fitting results of stress relaxation curves using the PFOV model.The R 2 for these fits consistently is 0.99.The fitting results demonstrate that both the Maxwell model and the PFOV model effectively capture the stress relaxation behavior of the prepreg.However, the description of stress relaxation behavior using a Maxwell model expressed with a third-order Prony series involves 7 parameters, whereas the PFOV model requires only 4 parameters.From table 2, it is evident that as the number of layers in the prepreg increases, the order parameters (b 1 and b 2 ) of the fractional-order model show an upward trend.In Scott-Blair elements, a higher order parameter β indicates behavior closer to that of a Newtonian fluid.This implies that prepreg with fewer layers demonstrates compression behavior more closely resembling that of an elastic body.During the compression process, with a consistent compression amount, samples with fewer layers experience greater strain.When unpressed, the fibers in the prepreg are relatively loose, and there are many voids within and between layers.However, as the sample undergoes significant strain, the voids in the prepreg are filled with resin, and the fibers begin to bear the pressure.As the fibers in the prepreg experience more pressure, their compression behavior tends to resemble that of an elastic body.Hence, samples with larger strain and fewer layers demonstrate stress relaxation behavior more inclined towards an elastic body, leading to a decrease in the order parameters of the PFOV model with a reduction in the number of layers in the sample.
Figure 8 presents the relaxation behavior curves of prepreg samples under various compression speeds, specifically, 0.1, 0.5, 1, 5, and 10 mm min −1 .Figure 9 displays the fitting results for the stress relaxation behavior of prepreg under different compression speeds using the Generalized Maxwell model expressed with a thirdorder Prony series and the PFOV model.Similarly, from tables 3 and 4, it is evident that the Generalized Maxwell model provides a high degree of fit with the experimental results, with all R 2 values reaching 0.99.From table 4, it is evident that the order parameter (β) in the PFOV model decreases with an increase in the compression speed of the prepreg.At higher compression speeds, the time for fiber misalignment and slippage in the prepreg is shorter.This increased probability of fibers bearing pressure results in the mechanical behavior being closer to that of an elastic body compared to low-speed compression.Thus, the order parameter in the PFOV model decreases with an increase in speed.The outcomes from tables 3 and 4 demonstrate that both the Maxwell model and the PFOV model effectively capture the stress relaxation behavior of prepreg under various compression speeds.However, in comparison to the Maxwell model, the PFOV model requires fewer parameters and offers a clear physical interpretation, making it more suitable for describing the viscoelastic behavior of prepreg.

Conclusion
In summary, the study introduced a Parallel Fractional-Order Viscoelastic (PFOV) model comprising two Scott-Blair elements to analyze the stress relaxation behavior of unidirectional prepreg materials during compression.Comparative analysis with other models, including the Scott-Blair model and the Maxwell model with second and third-order Prony series, underscores the superior performance of the PFOV model in accurately depicting stress relaxation dynamics, with all R 2 values reaching 0.99.Remarkably, the PFOV model requires fewer parameters than the Maxwell model, facilitating a clearer physical interpretation of prepreg material viscoelastic behavior.The investigation reveals that as compression speed increases, the order parameter (β) in the PFOV model decreases, suggesting a shift towards behavior resembling that of an elastic body.Similarly, a reduction in the number of prepreg layers leads to an upward trend in order parameters (β), indicative of compression behavior closer to that of a Newtonian fluid.These observed trends in parameter variations provide theoretical insights into the stress relaxation mechanisms of unidirectional prepreg materials.Future research endeavors could focus on predicting prepreg material compression behavior at different temperatures, utilizing insights gleaned from this study.By expanding investigations to encompass temperature-dependent properties, researchers can refine predictive models and optimize manufacturing processes of prepreg a like composite material hot-stamp molding, hot diaphragm forming, liquid composite molding, and similar manufacturing scenarios.
, b is the order of the fractional derivative, indicating the degree of viscoelasticity exhibited by the material, where 0 b 1. b c denotes the viscoelastic coefficient, which characterizes the material's resistance to deformation.Smaller values of b indicate that the material exhibits behavior closer to that of an elastic solid.As shown in figure 2(A), when b equals 0, equation (3) represents the expression for a linear elastic model and b c corresponds to the elastic modulus.Conversely, larger values of β suggest that the material approaches the characteristics of a Newtonian fluid.When b equals 1, equation (3) corresponds to the expression for a Newtonian fluid and b c represents the viscosity coefficient in the ideal viscous model.While the Scott-Blair model offers a concise description of the viscoelastic behavior of materials with fewer parameters, it falls short in accurately representing the substantial stress drop observed during the relaxation of unidirectional prepreg.As illustrated in figure 2(B), this study utilizes the PFOV model to characterize the stress relaxation behavior of unidirectional prepreg.The model is composed of parallel Scott-Blair elements, and the expression is as follows:

Figure 1 .
Figure 1.(A) Compress device mounted on the tensile testing machine; (B) The force history of the prepreg during the experimental process.

Figure 4 .
Figure 4.The fitting results of different models: (A) Scott-Blair model; (B) PFOV model; (C) Maxwell model expressed with a thirdorder Prony series; (D) Maxwell model expressed with a third-order Prony series.

Figure 5 .
Figure 5.The reduction ratio of different models.

Figure 6 .
Figure 6.Relaxation curves of samples at different number of prepreg layers, with a compression distance of 0.2 mm, a temperature of 80 °C and a velocity of 1 mm min −1 .(Shaded areas indicate measurement errors).

Figure 7 .
Figure 7.The theoretical fitting for the stress in the relaxation phase for the different number of prepreg layers: (A) Five layers; (B) Ten layers; (C) Fifteen layers.

Figure 8 .
Figure 8. Relaxation behavior of prepreg at different compression velocities, with a compression distance of 0.2 mm, a temperature of 80 °C and ten layers.(Shaded areas indicate measurement errors).

Table 1 .
The parameters of the maxwell model for different number of layers.

Table 2 .
The parameters of the PFOV model for different number of layers.

Table 3 .
Parameters of the Maxwell model fitting at different compression velocities.

Table 4 .
Parameters of the PFOV model fitting at different compression velocities.