Advanced creep modelling for polymers: A variable-order fractional calculus approach

TL;DR

The paper tackles long-term creep prediction in polymers, where traditional fixed-order rheological models fail to capture the observed power-law and evolving viscoelastic behavior. It introduces a single variable-order springpot, governed by a time-varying order $\beta(t)$, to model continuous transitions from glassy to rubbery states, with six material and transition parameters $(E,\eta,\beta_0,\beta_\infty,\gamma,\delta)$ calibrated via cross-entropy optimization. The framework successfully fits creep data for polypropylene and PVC at 20°C across multiple stress levels and demonstrates predictive accuracy for unseen conditions, while revealing stress-dependent trends in stiffness and fractional-order evolution. This VO approach offers a parsimonious yet flexible tool for reliable, long-term predictions of creep in structural polymer components and can be extended to other viscoelastic materials and loading histories.

Abstract

Polymer-based plastics exhibit time-dependent deformation under constant stress, known as creep, which can lead to rupture or static fatigue. A common misconception is that materials under tolerable static loads remain unaffected over time. Accurate long-term deformation predictions require experimental creep data, but conventional models based on simple rheological elements like springs and dampers often fall short, lacking the flexibility to capture the power-law behaviour intrinsic to creep processes. The springpot, a fractional calculus-based element, has been used to provide a power-law relationship; however, its fixed-order nature limits its accuracy, particularly when the deformation rate evolves over time. This article introduces a variable-order (VO) springpot model that dynamically adapts to the evolving viscoelastic properties of polymeric materials during creep, capturing changes between glassy, transition and rubbery phases. Model parameters are calibrated using a robust procedure for model identification based on the cross-entropy (CE) method, resulting in physically consistent and accurate predictions. This advanced modelling framework not only overcomes the limitations of the fixed-order models but also establishes a foundation for applying VO mechanics to other viscoelastic materials, providing a valuable tool for predicting long-term material performance in structural applications.

Figures (16)

• Figure 1: Molecular structure of a polymer alongside common products and structures made from polymeric materials. These materials are widely used in various applications due to their exceptional mechanical properties and resistance to stress and fatigue. They are also recyclable, adding to their environmental significance in industrial applications.
• Figure 2: Schematic representations of some classic constitutive models for viscoelastic materials like polymers, including the Maxwell, Kelvin-Voigt, and Burgers models, illustrating the combination of spring and dashpot elements used to describe the material's creep behavior.
• Figure 3: Schematic representation of the springpot model, illustrating its intermediate behavior between a spring (purely elastic) and a dashpot (purely viscous). The springpot is used to model viscoelastic behavior, where deformation is both time-dependent and elastic.
• Figure 4: The creep curve of PP at a temperature of $20\text{ºC}$ under a load of $\sigma_0 = 1.4 \text{ MPa}$ is compared to the constant-order springpot model. The parameters were estimated based on the time interval between 10 and 140 seconds, yielding $\beta=0.0260$ and $C_\beta$ = 1460 $\text{MPa} \cdot \text{s}^\beta$. In (a), it is evident that the model does not provide a good fit for long-term creep behavior. However, (b) shows that the model fits well within the specified interval.
• Figure 5: The creep curve of PP at a temperature of $20\text{ºC}$ under a load of $\sigma_0 = 1.4 \text{ MPa}$ is compared to the constant-order springpot model. The parameters were estimated based on the time interval between 8 and 46 min, yielding $\beta = 0.0738$ and $C_\beta$ = 2000 $\text{MPa} \cdot \text{s}^\beta$. In (a), it is evident that the model does not provide a good fit for long-term creep behavior. However, (b) shows that the model fits well within the specified interval.