1. Introduction
Short-fibre-reinforced composites (SFRCs) are widely used in various engineering fields like the aerospace and automotive industries. Their mechanical properties, including high stiffness and strength, together with their suitability for cheap, large-scale automated production recently increased the interest in SFRC [
1,
2,
3,
4,
5,
6]. SFRC materials consist of two or more different materials that are combined during a mould injection process [
7,
8]. However, due to mould injection, the fibres are spatially distributed within the component [
8]. This results in heterogeneity of mechanical properties at the component level [
9], which is derived from its heterogeneous microstructure [
10,
11]. The probabilistic characteristics of this microstructure are characterised by probability density functions like fibre length, fibre diameter, and fibre orientation [
11,
12].
While thermoplastic polybutylene terephthalate (PBT) serves as a matrix material, the short glass fibres reinforce the structure, leading to increased stiffness and strength. However, polymer-based composites like PBT GF30 (with 30% fibre mass fraction) exhibit time-dependent deformation under sustained loads, known as viscoelasticity or creep. Typically, creep is caused by the matrix material [
13,
14,
15]. To predict the long-term performance of components, it is important to accurately characterise the creep behaviour, especially for the automotive, aerospace, and electronics industries [
1,
2,
3,
4,
5].
Viscoelastic behaviour of SFRC is typically described using the generalised Maxwell model [
16], which combines springs and dampers to represent elastic and viscous elements [
16,
17]. The Maxwell model is standardised and widely used in commercial Finite Element Method (FEM) software packages [
18,
19]. Its implementation in FEM relies on the Prony series [
16,
20,
21], where the Prony series reflects the solution of an Ordinary Differential Equation (ODE), which results from the combination of springs and dampers. As these elements are typically considered to be linear, the ODE solution can be modelled as a sum of exponential functions. The effectiveness of this approach requires a large number of parameters to capture the full spectrum of material behaviour on varying time scales. This increases the complexity of parameter identification, as well as computational cost [
20,
21,
22].
While Prony-based models approximate the non-local nature of creep effects by the superposition of several spring and damper elements, fractional derivative models can inherently model this behaviour. A fractional derivate model, also referred as a fractional element, describes dynamics that lie between a spring and a damper [
16,
23,
24]. However, the practical application of fractional models remains challenging: normally, these models are solved indirectly by transforming them into the Fourier domain to fit experimental data. When established, this approach requires more complex data collection setups as the material behaviour has to be measured across different frequencies in the frequency domain. This makes simple tensile tests insufficient for data collection. Furthermore, Fourier analysis is limited to linear models, which complicates the representation of non-linear behaviour [
25,
26].
Recent advances in machine learning (ML) have enabled us to learn fractional dynamics directly from data. Coelho et al. [
27] introduced neural fractional differential equations (NFDEs). NFDEs combine a solver for Fractional Dynamical Equations (FDEs) with neural nets (NNs) to learn the underlying FDE from arbitrary data. Building on this foundation, Zimmering et al. [
28] improved the solver in accuracy and resource efficiency, enabling the use of the solver in real-world scenarios.
In this study, the need for special experiments that capture the frequency spectrum of SFRCs is overcome by fitting a parametric fractional element model directly to experimental tensile test data. The optimised NFDE solver of Zimmering et al. is employed to perform this fitting. The solver’s general capability to fit arbitrary dynamical models (including NNs) additionally offers flexibility in capturing non-linear creep behaviour from arbitrary data in future studies.
Following this, the contributions of this paper are as follows.
First, the accuracy of the fractional element in describing the creep behaviour of PBT GF0 and PBT GF30 is compared with the generalised Maxwell model. This comparison aims to determine the extent to which fractional derivative models can effectively capture the viscoelastic response of such materials over time.
Second, practical methodologies are studied to efficiently learn the parameters of fractional element models. It is sought to identify computational techniques that enable a stable parameter identification process while maintaining accuracy.
Third, the advantages that fractional derivative models may offer over the Prony models are discussed for SFRCs. It is evaluated if fractional models provide improved predictive capabilities and enhance the representation and interpretability of material properties over extended time scales.
This paper is structured as follows:
Section 2 outlines the theoretical framework for modelling creep behaviour as well as the data fitting methods for the Prony fractional model.
Section 3 details the experimental setup for the data acquisition methods. In
Section 4, the results are presented. Here, it is focused on the determination of Prony coefficients and coefficients derived from fractional models. Finally, in
Section 5, the comparative performance of the models is discussed, and conclusions about their practical applications in material characterisation are drawn.
The developed software is based on Python 3.11 and open-source software libraries and is available along with the experimental data at
https://github.com/zimmer-ing/PBTGF-Creep. This allows others to reproduce the results and apply the presented methods in their own work.
4. Results
This section presents the results of the experimental investigation of creep behaviour for unreinforced (PBT) and fibre-reinforced (PBT GF30) specimens under sustained loading. The experimental setup and conditions are summarised in
Table 2. Additionally, the predictive performance of two modelling approaches, the Prony series and the fractional derivative method, is evaluated to approximate the experimental data.
4.1. Experimental Results of Creep Behaviour
The creep behaviour of PBT and PBT GF30 under sustained load conditions was analysed using data from 20 samples per material type. The results shown in
Figure 5 provide a comparative analysis of the time-dependent deformation characteristics of these two materials. In all samples tested, the initial deformation at the beginning of the creep lies in a comparable range of around
%. This indicates a consistent elastic response prior to viscoelastic deformation. The creep responses of primary and secondary creep stages, as discussed in
Section 2.1.1, are observed in both materials and exhibit characteristics. However, the second stage cannot be clearly separated from the tertiary stage of creep. Clear indicators such as an accelerating strain rate or failure are missing. However, a detailed classification of creep stages or associated mechanisms lies beyond the scope of this study and is therefore not pursued further.
The master curves in
Figure 6 of the total strain
and
Figure 7, where the creep strain
is shown, indicate that PBT exhibits a slightly higher strain rate compared to PBT GF30. These master curves, which are introduced in detail in
Section 2.3, provide more detail on the general trend of deformation compared to the results in
Figure 5. These findings align with previous studies [
52,
53,
54], which reported similar trends in the viscoelastic behaviour of unreinforced and fibre-reinforced materials. The viscoelastic behaviour is governed primarily by the matrix material rather than the reinforcing fibres [
53,
55,
56].
In particular, the master curves in
Figure 7 show a slightly steeper creep rate for PBT compared to PBT GF30. This suggests a reduced resistance to time-dependent deformation in the absence of fibre reinforcement, specific to the PBT system studied here. In contrast, the inclusion of short glass fibres in PBT GF30 results in lower creep rates. These findings emphasise the role of fibre reinforcement in altering the viscoelastic response of PBT under constant load conditions.
4.2. Prony Series Coefficients for PBT and PBT GF30
The viscoelastic properties of the PBT and PBT GF30 materials are characterised by fitting their experimental creep data to the Prony series model using a different number of branches
with respect to the Equation (
7). The coefficients
,
, and
were determined using the fitting procedure detailed in
Section 2.2. The resulting coefficients for each specimen are summarised in
Table A4,
Table A5, and
Table A6, respectively.
The mean values and standard deviations of the Prony series coefficients for both materials are presented in
Table 3. The total stiffness
derived from the Prony series representations in these tables corresponds to the effective stiffness experimentally determined in accordance with the DIN EN ISO 527-1 standard [
57]. This total stiffness characterises the initial elastic response of the materials and serves as a reference for calibrating Prony series models with varying numbers of branches, specifically
,
, and
. The corresponding Young’s moduli, determined in accordance with the DIN EN ISO 527-1 standard, were obtained from tensile tests on 10 samples per material, yielding mean values of
for PBT and
for PBT GF30.
Significant differences in the mean Prony series coefficients between PBT and PBT GF30 underscore the role of fibre reinforcement in the modification of viscoelastic properties. Although the matrix material primarily governs time-dependent deformation, the presence of fibres considerably enhances the overall stiffness and resistance to creep. As noted above, the mean relaxation time
and the stiffness distribution
(see
Table 3) for PBT GF30 are higher compared to PBT, further supporting the non-parallel nature of the master curves shown in
Figure 7. These observations are consistent with findings reported in the literature [
52,
53,
54], which similarly demonstrated the influence of fibre reinforcement on the viscoelastic behaviour of the polymer.
A comparative evaluation of experimental creep data with two predictive modelling approaches, the classical Prony series model and the fractional viscoelastic model, is presented in
Figure 8 for PBT and in
Figure 9 for PBT GF30. These diagrams highlight the ability of each model to reproduce the experimentally observed viscoelastic behaviour over the measured time domain.
For both materials, it is evident that the Prony model with a single branch () does not capture experimental trends with sufficient accuracy. This is particularly apparent in the early and intermediate time regimes, where deviations from the measured data are most pronounced. In contrast, the fractional viscoelastic model provides a significantly improved fit across the full time range, especially for low-order Prony approximations. This can be attributed to the inherent capacity of fractional models to represent broad relaxation spectra with fewer parameters.
However, as the number of branches in the Prony series increases, the model’s ability to replicate the experimental data improves notably. From onward, Prony-based predictions show a better agreement with the measured curves compared to the accuracy of the fractional model. This trend is consistently observed for both material systems, demonstrating the flexibility of the Prony approach when extended with sufficient model complexity. Nevertheless, the trade-off lies in the increased number of fitting parameters and the potential risk of over-fitting or loss of physical interpretability.
In summary, while the fractional model yields an excellent initial approximation of the experimental data with a compact formulation, the Prony series model achieves superior accuracy when configured with multiple branches. This highlights the importance of selecting a modelling strategy based on the required fidelity, parameter efficiency, and application context.
4.3. Fractional Model Coefficients for PBT and PBT GF30
The results for the fractional element in Equation (
8) and fitted according to the methodology in
Section 2.2.3 can be found in
Table 4.
Table 4.
Mean values and standard deviations (Mean ± StdDev) of Fractional Element parameters.
Table 4.
Mean values and standard deviations (Mean ± StdDev) of Fractional Element parameters.
Parameter | PBTGF0 | PBTGF30 |
---|
C | 5.34 × 105 ± 1.63 × 105 | 3.55 × 106 ± 9.60 × 105 |
| 0.35 ± 0.05 | 0.40 ± 0.04 |
Here, C describes the immediate response of the corresponding material, similar to in the previous section. While rises by times for PBT GF30, C rises by 6.65 times, which is significantly higher.
The fractional derivate reflects the mix of elastic () and viscous () behaviour. Its mean values are slightly higher for PBT GF30, but in a similar range. As the corresponding values overlap, due to high standard deviation, it cannot be clearly stated that PBT GF30 shows more viscous behaviour than PBT. The slight increase could be explained by additional micro-slip or friction of the fibres inside the matrix material. Since clear evidence is missing, either more samples or an improved experimental setting is necessary to study the effect in detail.
In summary, the results show that the immediate response C significantly increases for the fibre0enhanced material while the ratio between elastic and viscous behaviour remains similar.
5. Discussion
The measured creep behaviour of fibre-reinforced polymers (PBT GF30) and unreinforced polymers (PBT) is in line with expectations. It confirms that fibre reinforcement plays a role in modulating viscoelastic responses. The PBT-GF30 samples deform less under sustained load than the unreinforced PBT samples, an expected effect, as shown in
Figure 7, as the glass fibres stiffen the matrix and inhibit creep strain [
52,
53,
54]. The master curves shown in
Figure 6 indicate that while both the fractional derivative model and the Prony series provide reasonable approximations of the creep response, there are notable differences in terms of accuracy and parameter stability.
For single-term Prony models (), the approximation of the creep behaviour is less accurate than the fractional model. The fractional model provides a smoother representation of the viscoelastic response. When additional terms are added to the Prony series (), the accuracy improves. This increased number of parameters increases potential over-fitting. The higher standard deviations of the Prony coefficients suggest more variability, which also indicates that the Prony coefficients lack robustness. This variability challenges the physical interpretability of the Prony model, as over-fitting becomes a concern with higher N. In contrast, the fractional derivative model demonstrates better approximation accuracy with a lower number of required parameters. The fractional order parameter () shows stability in different material compositions. This underscores the suitability of fractional calculus-based models to provide a compact and accurate representation of creep behaviour. However, similarly to the Prony series model, greater standard deviations of the parameter C are observed.
Determining the parameters of fractional derivative models presents a challenge due to the complexity of fractional differential equations. This study explores optimisation techniques such as numerical and automatic differentiation-based frameworks to efficiently fit experimental data. While the Prony series benefits from a closed-form solution that simplifies parameter fitting, the fractional model requires solving fractional differential equations numerically, increasing computational demands. To mitigate these challenges, gradient-based optimisation methods are employed, ensuring stable parameter convergence across multiple datasets. Moreover, the application of data-driven techniques, such as NFDE solvers, offers a promising alternative for parameter identification.
Recent developments in the application of artificial intelligence to constitutive modelling have demonstrated significant potential, particularly in areas requiring robust parameter identification from noisy or incomplete data. The novelty of this study lies not in the proposal of new constitutive equations but in the implementation of a machine learning-inspired approach, specifically a neural fractional differential equation (NFDE) solver, to identify the governing parameters of both the Prony and fractional derivative models. This technique combines automatic differentiation with gradient-based optimisation, treating the governing differential equation as an operator constraint. Although the application of AI tools to curve fitting or surrogate modelling in viscoelasticity is a well-established area of research, as evidenced by the work of [
58,
59], the use of a solver to extract interpretable and physically meaningful parameters from time domain data for fractional models represents a novel contribution. The approach demonstrated stable convergence and interpretability across the two material systems studied, highlighting the emerging role of data-driven methods in enhancing classical mechanical modelling.
One of the key challenges in viscoelastic modelling is to determine the model parameters efficiently. The Prony series requires multiple relaxation times and corresponding stiffness coefficients, increasing the complexity of parameter identification. As the number of branches in the Prony model increases, over-fitting becomes a concern, leading to large standard deviations in the model coefficients. In contrast, the fractional derivative model requires fewer parameters, making it computationally more efficient.
The results suggest that existing numerical optimisation techniques, such as gradient-based or evolutionary algorithms, could be refined to improve the robustness of parameter identification for both models. The need for an improved fitting procedure is evident, as the high variability in Prony coefficients indicates the sensitivity of experimental noise data.
Although the test duration was limited to 300 s, the fractional viscoelastic model used allows reliable extrapolation because of its inherent power-law behaviour, as demonstrated in recent studies [
60,
61]. Compared to the Prony series, it offers better long-term predictions from short-term data.
The findings of this study confirm that both the fractional derivative model and the generalised Maxwell model based on the Prony series are viable approaches to modelling the creep behaviour of fibre-reinforced and unreinforced materials. However, the results highlight a critical issue in parameter estimation, particularly for the Prony series model, which exhibits a high standard deviation in parameter values due to over-fitting. The fractional derivative model emerges as a promising alternative, providing a more compact reparametrisation while maintaining accuracy in long-term creep predictions.
It is important to emphasise that the conclusions drawn in this study are specific to the thermoplastic polyester PBT and its short-fibre-reinforced variant PBT GF30. While the viscoelastic modelling strategies developed here may generally be applicable, thermoplastic systems vary greatly in molecular structure, crystallinity, and processing sensitivity. Therefore, caution should be exercised when extrapolating these findings beyond the tested materials.
This emphasises the need for a better or more robust fitting procedure that ensures accuracy while preserving physical interpretability, mitigates over-fitting, and enhances the predictive capability of viscoelastic models. Future research should focus on refining optimisation algorithms, improving the consistency of experimental data, and exploring hybrid modelling approaches that integrate the advantages of both methodologies.
6. Conclusions
The present study investigated the tensile creep behaviour of unreinforced PBT and short-fibre-reinforced PBT GF30 (with a fibre mass fraction of 30%) using two viscoelastic modelling approaches, the classical Prony series and a fractional derivative formulation. A comparative analysis was conducted, which revealed that the fractional derivative model outperforms the Prony series in terms of parameter stability and compactness, especially at low model orders. While the Prony model becomes more accurate with additional terms, it also becomes more susceptible to overfitting and parameter variability. In contrast, the fractional model provides accurate results with fewer parameters and demonstrates increased robustness in predicting larger time ranges.
A central contribution of this work lies in the application of a neural fractional differential equation (NFDE) solver for parameter identification. This approach enables the data-driven extraction of physically interpretable parameters from creep data in the time domain and demonstrates stable convergence, offering a viable alternative to conventional curve-fitting techniques.
Despite the limited experimental duration (300 s), the fractional model enables credible extrapolation due to its inherent power-law behaviour. The methods and conclusions presented are specific to PBT and PBT GF30 and should be cautiously extended to other polymer systems due to variations in morphology and processing sensitivity.
Future work should focus on refining optimisation algorithms for parameter identification, improving data robustness, and exploring hybrid modelling strategies that combine the strengths of fractional and classical viscoelastic representations.