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Fractional viscoelastic models for power-law materials
Alessandra Bonfanti, Jonathan Louis Kaplan, Guillaume Charras, Alexandre J Kabla
TL;DR
Power-law responses arise across complex soft materials, while fractional viscoelasticity remains underused and difficult to access mathematically. This review explains fractional models, compares them with traditional approaches, and shows that they capture diverse relaxation and creep behaviours with parameters representing distributed time-scales. The framework supports systematic comparison across materials, but remains limited for nonlinear large-deformation and failure behaviour.
Problem
Complex materials exhibit power-law viscoelastic responses, but fractional viscoelasticity remains underused because of its intricate formalism and numerical implementation difficulty.
Method
The review introduces fractional viscoelasticity, explains its mathematical and network foundations, and revisits empirical and spring-dashpot models across practical applications.
Results
Fractional viscoelastic models accurately capture rheological responses across a broad range of materials and account for the wide distribution of time-scales in power-law behaviour.
Takeaways & Limitations
Fractional models provide systematic material parameters that can be compared across studies and support unified analysis of power-law materials.
Takeaways & Limitations
The models are limited to linear response and do not themselves explain the underlying mechanisms producing macroscopic power-law behaviour.
Abstract
from arXiv · showhide
Soft materials often exhibit a distinctive power-law viscoelastic response arising from broad distribution of time-scales present in their complex internal structure. A promising tool to accurately describe the rheological behaviour of soft materials is fractional calculus. However, its use in the scientific community remains limited due to the unusual notation and non-trivial properties of fractional operators. This review aims to provide a clear and accessible description of fractional viscoelastic models for a broad audience, and to demonstrate the ability of these models to deliver a unified approach for the characterisation of power-law materials. The use of a consistent framework for the analysis of rheological data would help classify the empirical behaviours of soft and biological materials, and better understand their response.
1 Introduction
Rheology supports industrial, food, biological, and materials applications, but many complex materials show power-law responses that challenge conventional viscoelastic descriptions. The review presents fractional viscoelasticity as an accessible, consistent framework for modelling these behaviours.
- Rheological modelling links material deformation to classification, comparison, prediction, and engineering or manufacturing decisions.Applications include additive manufacturing, food processing, product design, and inferring polymer composition or microstructure.
- Cells and soft tissues have mechanical properties associated with development, disease progression, and changes under external stimuli.Measuring these changes can provide insights into disease evolution and guide diagnostic-tool development.
- Viscoelastic materials combine elastic and viscous behaviour and respond over time through relaxation and creep.Relaxation follows imposed deformation, whereas creep follows imposed stress; creep can be undesirable for durability but useful in extrusion and cavity filling.
- Many complex materials exhibit power-law signatures in creep, relaxation, and spectral behaviour, challenging commonly used models.Examples span biological materials, gels, polymers, concrete, asphalt, ice, and food materials.
- Fractional viscoelasticity extends linear viscoelasticity with non-integer-order calculus and remains underused despite applications across geological, construction, polymeric, food, and biological materials.A consistent formalism could support direct comparison of parameters across studies.
- The review explains fractional models, their limit behaviours, and their advantages over spring-dashpot approaches through applications and revisited studies.It also aims to make the formalism accessible to researchers without specialised fractional-calculus knowledge.
2 Introduction to linear viscoelasticity
Linear viscoelasticity relates stress, strain, and time under superposition, providing a framework for relaxation, creep, and oscillatory analyses. Its response functions and moduli are interconnected, enabling predictions across loading modes.
- Linear viscoelasticity assumes proportional scaling and superposition, allowing responses to arbitrary histories to be assembled from individual perturbations.This framework supports predictions of stress and strain under arbitrary loading conditions.
- Small-deformation linearity is a manageable approximation because many materials become nonlinear mainly at large deformations.The linear theory also provides a starting point for studying more complex nonlinear responses.
- In relaxation tests, a constant strain step produces stress proportional to its amplitude, with G(t) describing the monotonically decreasing relaxation modulus.In creep tests, strain is proportional to the applied stress step and J(t) is a monotonically increasing creep modulus.
- Oscillatory loading preserves frequency while producing a phase difference between stress and strain that lies between elastic and viscous limits.The storage modulus G′ represents the real response component, while the loss modulus G′′ represents the imaginary component associated with dissipation.
- Relaxation, creep, and dynamic moduli are directly related in the Laplace domain, so one behavioural mode can predict another.The relationship is expressed as eG(s)eJ(s) = s^-2.
3 Characteristics of traditional linear viscoelastic models
Traditional spring–dashpot models build linear viscoelastic behavior from series and parallel combinations, but their discrete exponential time-scales only approximate power-law responses. Increasing model complexity improves coverage while making parameters costly, difficult to interpret, and unreliable outside the fitted time window.
- Model construction: Traditional models combine Hookean springs and Newtonian dashpots in series or parallel to derive creep, relaxation, and complex moduli.The Maxwell model is the series combination, while the Kelvin–Voigt model is the parallel combination.
- Model behavior: Spring–dashpot constitutive equations produce exponential creep and relaxation moduli with discrete characteristic time-scales.For the Maxwell model, τ = η/k is the time required for stress to fall to 1/e of its initial value.
- Power-law limitation: Power-law materials require a continuous distribution of relaxation and creep time-scales, so spring–dashpot models can only approximate their behavior.Adding more exponential terms brings the approximation closer to the power-law response.
- Power-law approximation: In the generalized Maxwell model, n = 1, 2, and 4 arms approximate σ(t) = 5 · t^-0.5, with fitted relaxation time-scales increasingly covering the power-law spectrum.The optimal Maxwell time-scales are equally spaced on a logarithmic scale, and additional arms broaden the covered distribution.
- Practical disadvantages: More spring–dashpot parameters increase computational expense and complicate physical interpretation.Fitted characteristic times mainly reflect the data time window rather than intrinsic material properties.
- Empirical alternatives: Empirical power-law ansätze may describe one measured modulus but can require numerical prediction of other loading responses and hinder parameter comparison across studies.Their ad-hoc parameters may not be easily comparable between studies.
4 Fractional viscoelastic models
The springpot is introduced as a fractional viscoelastic element for representing power-law behavior with few parameters. It can be combined in series and parallel with other elements to model more complex power-law signatures.
- Modeling approach: The springpot captures power-law behavior with a minimum number of parameters and can be combined with other elements in series or parallel.The paper presents these combinations as a way to represent complex power-law signatures in different contexts.
4.1 The springpot captures power-law behaviour
The springpot uses a fractional derivative to provide a two-parameter constitutive description whose relaxation, creep, and spectral responses follow power laws. Its exponent β continuously interpolates between elastic and viscous behaviour while preserving distinctive recovery and modulus relationships.
- Fractional constitutive description: G(t) = At^-β leads to a fractional constitutive equation that describes power-law materials with two parameters, c_β and β.The coefficient is redefined using the gamma function to obtain the springpot equation.
- Fractional constitutive description: The springpot reduces to a spring at β = 0 and a dashpot at β = 1, with intermediate β values producing behaviour between elastic and viscous limits.Its parameter c_β has units of Pa s^-β and is often interpreted as material firmness.
- Fractional calculus: The Caputo derivative is non-local because its value at time t depends on integration from the initial time t = 0.The review selects it because initial conditions are naturally specified using integer-order derivatives.
- Power-law responses: The springpot’s relaxation modulus and relaxation spectrum are power laws with exponent β, allowing power-law material behaviour to be captured by design.The creep modulus follows from the Laplace-domain relationship between relaxation and creep moduli.
- Power-law responses: After unloading, springpot strain ultimately returns to zero, while larger β values require more time for complete recovery.This gives the springpot shape memory despite energy dissipation during deformation.
- Oscillatory behaviour: The storage and loss moduli follow the same power-law exponent, match at β = 0.5, and exchange dominance across that value.For β < 0.5 storage exceeds loss, whereas for β > 0.5 loss exceeds storage.
4.2 Generalised fractional viscoelastic models
Generalised fractional viscoelastic models combine springpots with traditional elements to represent diverse, time-limited power-law behaviours more concisely than large spring-dashpot networks. Two springpots in series or parallel provide fractional analogues of the Maxwell and Kelvin-Voigt models, with exponent-dependent responses across time scales.
- Motivation: Power-law responses arise from continuous distributions of time-scales, whereas spring-dashpot networks approximate them using discrete exponential terms.More exponential terms improve the approximation, but increase model complexity.
- Model configurations: Combining springpots with springs and dashpots extends network models to capture power-law regimes across a diversity of material behaviours.The two simplest configurations are series and parallel combinations of two springpots.
- Series configuration: In series, the lower-exponent springpot dominates short times, while the higher-exponent springpot controls long-time relaxation and creep responses.This applies when α > β and is illustrated with α = 0.8 and β = 0.2.
- Parallel configuration: In parallel, the higher-exponent springpot dominates short times, whereas the lower-exponent springpot controls long-time relaxation and creep responses.This reverses the qualitative ordering observed for the series configuration.
- Complex networks: These exponent-based heuristics extend to more complex networks, where different branches can dominate short, intermediate, or long time scales.In a fractional Standard Linear Solid example, the springpot dominates short times, the dashpot intermediate times, and a spring long times.
- Applications: Fractional models can match empirical power-law functions and complex material responses while using fewer parameters than spring-dashpot models.Examples include a three-parameter fractional Maxwell fit replacing a five-parameter two-time-scale Standard Linear Solid for tomato cells.
5 Conclusions
The review finds that fractional viscoelastic models capture power-law rheology across many materials while identifying parameters associated with broad time-scale distributions. Their scope remains limited to linear response, and they do not by themselves explain the mechanisms underlying macroscopic power laws.
- 5 Conclusions: Fractional viscoelastic models accurately capture rheological responses across a broad range of materials and identify parameters accounting for distributed time-scales.The review reports consistent agreement with empirical relaxation and creep responses.
- 5 Conclusions: Large deformations and failure of complex materials remain outside the reach of linear rheological models.The review identifies these regimes as challenges for future work.
- 5 Conclusions: The review uses qualitative springpot responses, simple network compositions, an annex of models, and software to make fractional models more accessible.The software supports fitting experimental data, predicting power-law behaviours, and handling complex loading patterns.
- 5 Conclusions: Fractional models do not independently explain the underlying mechanisms producing observed macroscopic power-law behaviour.The review calls for more systematic theoretical analysis of experimental data to provide physical underpinning.
Appendix: Fitted parameters
The appendix materials identify fitted parameters for several fractional and generalized viscoelastic models, with each table linked to a corresponding figure.
- Appendix: Fitted parameters: Table 2 lists fitted parameters for generalized Maxwell models from figure 3 (b).
- Appendix: Fitted parameters: Table 3 lists fitted parameters for a fractional Kelvin-Voigt model from figure 8.
- Appendix: Fitted parameters: Tables 4 and 5 list parameters for empirical functions and three-element fractional models from figures 9 (a) and 9 (b).
- Appendix: Fitted parameters: Figure 12 presents fitted parameters for the fractional model from figure 10.