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The Concepts and Applications of Fractional Order Differential Calculus in Modelling of Viscoelastic Systems: A primer

Mohammad Amirian Matlob, Yousef Jamali

arXiv:1706.06446v2physics.bio-phcond-mat.soft

TL;DR

Standard linear equations may not describe complex viscoelastic behavior, particularly in biological materials. This tutorial introduces fractional-calculus concepts and solution methods, then applies fractional models to cell mechanics; its reported models agree with experimental data and compare favorably with previous models.

  • Problem

    Standard linear equations may not describe complex viscoelastic effects such as shape memory during deformation, motivating models for viscoelastic materials.

    Source: Introduction · p. 3, Discussion · p. 27

  • Method

    The tutorial explains fractional calculus and analytical and numerical solution techniques, then applies one-dimensional and two-dimensional fractional models to cell mechanics.

    Source: Introduction · p. 3, Discussion · p. 27

  • Results

    The one-dimensional FOE model has about 3% variance relative to a previous model and 97% precision compared with experimental data; the 2D fractional network model analyzes the same experimental data precisely.

    Source: Discussion · pp. 27–28

  • Takeaways & Limitations

    For the modeled cell responses, the authors report that fractional models match experimental data and can substitute for previously used models.

    Source: Introduction · p. 3, Discussion · pp. 27–28

    The paper gives no fixed rule for selecting a fractional-derivative type; selection depends on physical understanding and consistency with experimental data.

    Source: Laplacean interpretation · p. 10

Abstract

from arXiv · show

Viscoelasticity and related phenomena are of great importance in the study of mechanical properties of material especially, biological materials. Certain materials show some complex effects in mechanical tests, which cannot be described by standard linear equation (SLE) mostly owing to shape memory effect during deformation. Recently, researchers have been applying fractional calculus in order for probing viscoelasticity of such materials with a high precision. Fractional calculus is a powerful tool for modeling complex phenomenon. In this tutorial based paper, we try present clear descriptions of the fractional calculus, its techniques and its implementation. The intention is to keep the details to a minimum while still conveying a good idea of what and how can be done with this powerful tool. We try to expose the reader to the basic techniques that are used to solve the fractional equations analytically and/or numerically. More specifically, modeling the shape memory phenomena with this powerful tool are studied from different perspectives, as well as presented some physical interpretation in this case. Moreover, in order to show the relationship between fractional models and standard linear equations, a fractal system comprising spring and damper elements is considered, and the constitutive equation is approximated with a fractional element. Finally, after a brief literature review, two fractional models are utilized to investigate the viscoelasticity of the cell, and the comparison is made among them, experimental data, and previous models. Verification results indicate that not only does the fractional model match the experimental data well, but it also can be a good substitute for previously used models.

Introduction

Fractional calculus extends ordinary differentiation and integration to non-integer orders, offering a way to model mechanical behavior between ideal springs and dampers. The paper introduces it for complex systems, especially viscoelastic materials, and outlines basic modeling and solution techniques.

Introduction

  • Ordinary differential equations describe many phenomena, but classical calculus may not capture complex mechanical behavior between an ideal spring and an ideal damper.The paper motivates fractional calculus as an approach to modeling these intermediate, potentially nonlinear systems.Source: Introduction · p. 3
  • Fractional calculus generalizes differentiation and integration to arbitrary non-integer orders, raising questions such as what a half-order derivative means.The paper frames this as a route to modeling complex systems, especially viscoelastic materials.Source: Introduction · p. 3
  • The article introduces fractional calculus concepts, fractionalization, physical interpretation, analytical solutions, and viscoelastic-system modeling.It also applies the approach to cell biomechanics and compares model predictions with experimental data and previous spring-dashpot models.Source: Introduction · p. 3

Fractional calculus

The paper presents fractional calculus by extending familiar integer-order operations and then surveys several definitions of fractional derivatives and their properties. It focuses on Grünwald–Letnikov, Riemann–Liouville, and Caputo operators for modeling applications.

Definition of fractional calculus

  • Fractional derivatives have multiple mathematically valid definitions, which agree at integer orders but may differ at non-integer orders and have distinct physical interpretations.The paper introduces common definitions after noting that no single fractional derivative is unique.Source: Fractional calculus · p. 4, Definition of fractional calculus · p. 6

Fractional calculus

  • The fractionalization algorithm extends integer-order operations to real orders, with the Gamma function extending factorials beyond non-negative integers.The paper illustrates the extension through derivative and integral examples, including power functions.Source: Fractional calculus · pp. 3–5

Grünwald–Letnikov derivative

  • The Grünwald–Letnikov derivative extends the natural derivative to fractional order and uses a Gamma-function binomial coefficient.The paper traces its introduction to Grünwald in 1867 and Letnikov in 1868.Source: Grünwald–Letnikov derivative · p. 6

Riemann-Liouville fractional derivative

Caputo derivative

  • The Caputo derivative is defined using an integer-order derivative within a Riemann–Liouville integral, and its value on a constant is zero for non-integer order.The paper focuses on Caputo and Riemann–Liouville derivatives for its viscoelastic-system analysis.Source: Caputo derivative · p. 7

Riemann–Liouville integral and derivative

  • The paper reviews properties of fractional integrals and derivatives, including that the Caputo derivative is a left inverse of the Riemann–Liouville integral.It notes that the semi-group property does not generally hold for the Riemann–Liouville derivative.Source: Riemann–Liouville integral and… · pp. 7–8

The interpretation of fractional calculus

The paper interprets fractional calculus through memory: a system’s present output can depend on earlier inputs. It relates this dependence to fractional derivatives while noting that choosing a derivative definition requires physical understanding and has no fixed rule.

The interpretation of fractional calculus

  • A memory system uses previous input values to determine its current output, unlike a memoryless system that depends only on the current input.The paper gives Markov chains as memoryless examples and human decision making and shape-memory alloys as non-memoryless examples.Source: The interpretation of fractional… · p. 9

Laplacean interpretation

  • A power-weighted sum of earlier inputs makes the output non-memoryless, with memory decaying at a rate determined by the weight function.The paper states that the resulting equation governing system memory is described by a fractional derivative, making such derivatives candidates for modeling memory.Source: Laplacean interpretation · p. 9
  • There is no fixed rule for selecting a fractional-derivative type; the choice depends on understanding the physical phenomenon and consistency with experimental data.The paper identifies this as a modeling constraint.Source: Laplacean interpretation · p. 10

Analytical and numerical methods

The paper demonstrates Laplace-transform solutions for fractional differential equations and presents a discretization algorithm for Caputo derivatives. It compares a numerical solution with an analytical answer while noting that most fractional equations cannot be solved analytically.

Analytical methods

  • The Laplace transform solves an illustrated spring–viscoelastic-element equation by transforming it, solving algebraically in the transform domain, and applying the inverse transform.The example assumes zero initial displacement and yields a solution involving the Mittag-Leffler function.Source: Analytical methods · p. 10

Numerical solution

The Laplace transform of fractional operators

Numerical solution

  • The numerical solution is compared with an analytical answer using the represented mesh and a table of results for a 1100 pN stress.The table caption identifies the numerical results as equation 24 and the analytical answers as equation 25.Source: Numerical solution · p. 13, Figure 1 · p. 14, Table 1 · p. 14

Viscoelastic systems

Viscoelastic materials combine elastic response to deformation with viscous response to deformation rate, so their stress–strain behavior depends on time. Creep and periodic tests characterize that response through deformation over time and phase-shifted cyclic signals.

Viscoelastic systems

  • Hooke’s law relates stress linearly to strain, while the Newtonian-fluid relation makes stress proportional to the rate of deformation.These are the elastic and viscous behaviors that viscoelastic materials combine.Source: Viscoelastic systems · p. 14
  • Viscoelastic materials combine elasticity and viscosity, and linear viscoelasticity assumes stress and strain responses to successive stimuli are additive.The creep compliance J(t) relates strain to a step stress, while the relaxation modulus G(t) describes the corresponding stress response to step strain.Source: Viscoelastic systems · pp. 14–15

Creep test

  • A creep test applies constant tension and records deformation over time; viscoelastic creep exhibits power-law behavior.For a step stress, strain is related to creep compliance by ε(t) = σJ(t).Source: Creep test · p. 15

Periodic test

  • Under sinusoidal loading, viscoelastic strain is periodic at the applied frequency but shifted in phase; storage and loss moduli characterize the response.The storage and loss moduli are denoted G’ and G”.Source: Periodic test · p. 16, Figure 4 · p. 16

Fractional viscoelastic model

The fractional-order element (FOE) models viscoelastic response with an intermediate-order derivative, extending spring-and-damper models. Its limiting orders recover elastic and viscous behavior, while its memory interpretation and periodic response are examined in creep, relaxation, and sinusoidal strain settings.

Fractional viscoelastic model

  • Real-material relaxation modulus G(t) is described as usually deviating from simple exponential behavior, with a power-law response presented as more realistic.The fractional-order element is introduced as a substitute for standard linear viscoelastic models.Source: Fractional viscoelastic model · p. 17
  • The fractional model’s relaxation equation is reported to match experimental results better when obtained through physical interpretation and mathematical methods.The stated time constant is τ = E/η.Source: Fractional viscoelastic model · p. 18
  • At α→0 the fractional equation reduces to Hooke’s law, and at α→1 it becomes the Newtonian-fluid model.The authors conclude from these physical interpretations that the model can be used to investigate viscoelastic mechanical properties.Source: Fractional viscoelastic model · p. 18
  • For creep, the fractional kernel t^(α−1)/Γ(α) embeds prior events, with α=0 representing memory-less behavior and α=1 full memory.The account describes deformation-related information as stored over time according to a power-law function.Source: Fractional viscoelastic model · p. 19
  • For relaxation, α=0 represents full memory and α=1 memory-less behavior, reversing the endpoint interpretation given for creep.Source: Fractional viscoelastic model · p. 19
  • The fractional Maxwell model includes classical Maxwell relaxation as the α=0, β=1 case and extends it to additional fractional behavior.It is constructed from two fractional-order elements.Source: Fractional viscoelastic model · pp. 20–21

Periodic strain

  • For sinusoidal strain, the FOE predicts stress in phase at α=0, 90° out of phase at α=1, and neither relation for intermediate orders.The authors report that this matches the physical interpretation of viscoelastic response.Source: Periodic strain · p. 21

Relation between fractional equations and SLE

The section relates finite spring–damper systems to fractional equations, arguing that an infinite spring–damper system’s constitutive equation reduces to a spring-pot. It derives this relation from a finite mechanical arrangement under stated assumptions.

Relation between fractional equations and SLE

  • An infinite system of springs and dampers has a constitutive equation equal to a spring-pot, offering a fractional representation of viscoelastic systems.The paper notes that increasing SLE elements can improve accuracy, while working with such systems is difficult.Source: Relation between fractional equations… · p. 21
  • The derivation begins with a finite spring–damper circuit and its governing stress–strain equations, then rewrites the result using a fractional equation.The finite arrangement is presented as a way to investigate the relationship between fractional equations and SLEs.Source: Relation between fractional equations… · pp. 21–22
  • The reduction assumes c0 ≪ s as n → ∞; a further derivation sets E0 = 0.The passage defines c0 = E0/η0.Source: Relation between fractional equations… · p. 22
  • The considered system’s output is described as a combination of a fractional element and a damper.The accompanying mathematical discussion places the governing equation in R space and the system output in L1 space.Source: Relation between fractional equations… · pp. 21–22

Application

The application section reviews fractional viscoelastic models across several materials and summarizes their comparisons with experiments or non-fractional alternatives. It reports favorable experimental agreement in some applications, while noting that fractional Maxwell and Kelvin models do not perform equally for polymers.

Application

  • The section outlines a one-dimensional cell-deformation model in the creeping state and a two-dimensional actin-filament network model, concluding that fractional models may replace prior spring–damper models in this context.The authors report a match with experimental data and a simulated model.Source: Application · p. 23
  • A four-parameter fractional model for cell viscoelasticity was compared with magnetic oscillatory cytometry data over 10^-2–10^2 Hz in a periodic test and reportedly matched the data well.Source: Application · p. 23
  • For asphalt mixtures, spring-pot models were proposed because their creep/recovery response is power-law rather than the exponential behavior of spring–damper models, and the proposed models agreed with experimental data.The discussion highlights the mixtures’ sensitivity to environmental and loading conditions.Source: Application · p. 23
  • For pig lung stress relaxation, a fractional standard linear solid matched experimental data far better than two integer standard linear solid generalized Maxwell models.The measured response showed power-law rather than exponential behavior.Source: Application · p. 23
  • For polymer materials, the fractional Maxwell model did not capture behavior, whereas the fractional Kelvin model was judged suitable.Source: Application · p. 23

Two-dimension fractional network model

The section proposes one-dimensional and two-dimensional fractional models for cell mechanics, using fractional elements to represent viscoelastic behavior. The 2D network model matches experimental data better than the previous elastic network, and the 1D FOE model has a lower reported error than the Bausch model.

Two-dimension fractional network model

  • The authors propose two fractional cell models: a 1D model and a more realistic 2D extension.Source: Two-dimension fractional network model · p. 24
  • The 1D FOE model addresses cell deformation in the creeping state, where simple linear viscoelastic models cannot predict cell mechanical properties well, especially under minuscule deformation.The model was compared with experimental data from ref and presented as a possible replacement for the Bausch model.Source: Two-dimension fractional network model · p. 24
  • The 1D FOE model’s reported NRMSE was 3.16%, compared with 10.21% for the Bausch model.The comparison uses normalized root-mean-square deviation against experimental data.Source: Two-dimension fractional network model · p. 25
  • The 2D network represents actin filaments linked by cross-linking proteins; the model treats the filaments as viscoelastic FOE elements with shared fractional order.The model is motivated by prior elastic-network failure to predict bead-micrometry data and by reported actin-filament viscoelasticity.Source: Figure 9 · p. 25, Two-dimension fractional network model · p. 26
  • The 2D network’s domain was estimated from an assumed actin-filament length of 173.2 nm, derived using a CSK pore size of 100 nm.The estimated network area was 4.5464 × 10^6 nm^2.Source: Two-dimension fractional network model · p. 26
  • The 2D fractional network model better matched experimental data than the elastic model under the same stress, making it a candidate for modeling and predicting cell CSK.The comparison is shown in Fig. 11; the experimental data are from ref [51], and the elastic model is represented in ref [53].Source: Two-dimension fractional network model · pp. 26–27, Figure 11 · p. 27

Discussion

The discussion positions fractional calculus as a way to relate complex spring-damper behavior to simpler viscoelastic models, then reports cell-model comparisons and their implications.

  • The paper presents fractional calculus as a tool for systems with long-term memory and spatial interactions, contextualized by a literature review.Source: Discussion · p. 27
  • A large system combining many springs and dampers was approximated using a simple fractional model.Source: Discussion · p. 27
  • 11% data variance relative to previous model contrasted with 3% for FOE; FOE precision was 97% against experimental data.Source: Discussion · p. 27
  • The 2D fractional network model analyzed the same experimental data precisely and included an actin-filament viscosity term.Source: Discussion · p. 28
  • Both models had fractional order α about 0.11, indicating more elastic behavior and high memory; fractional models are described as more efficient than SLE for power-law behavior.Source: Discussion · p. 28

Appendix

The appendix gives derivation steps using Laplace transforms and relates spring-damper stress and strain to fractional-model elements.

  • A Laplace transform with a unit step strain is used to obtain an equation for the example.Source: Appendix · p. 28
  • The appendix obtains a further equation by inverse Laplace transformation and convolution.Source: Appendix · p. 28
  • Spring and damper stress-strain relations are Laplace-transformed; setting E0 to zero yields a series combination of a damper and FOE.Source: Appendix · pp. 30, 33
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