Fractional Maxwell Model Research Articles

Dynamic Response Analysis of Ballastless Tracks Considering the Temperature-Dependent Viscoelasticity of Cement-Emulsified Asphalt Mortar Based on a Vehicle–Track–Subgrade Coupled Model

This study aims to explore the dynamic response of ballastless tracks under various temperatures of the cement-emulsified asphalt (CA) mortar layer and other environmental factors. CA mortar is the key material in the ballastless track structure, exhibiting notably temperature-dependent viscoelastic properties. It can be damaged or even fail due to the continuous loads from trains. However, the dynamic behaviors of ballastless tracks considering the temperature-dependent viscoelasticity of CA mortar have been insufficiently studied. This paper captures the temperature-dependent viscoelastic characteristics of CA mortar by employing the fractional Maxwell model and applying it to finite element simulations through a Prony series. A vehicle–track–subgrade (VTS) coupled CRTS I ballastless track model, encompassing Hertz nonlinear contact and track irregularity, is established. The model is constrained symmetrically on both of the longitudinal sides, and the bottom is fixed on the infinite element boundary, which can reduce the effects of reflected waves. After the simulation outcomes in this study are validated, variations in the dynamic responses under different environmental factors are analyzed, offering a theoretical foundation for maintaining the ballastless tracks. The results show that the responses in the track subsystem will undergo significant changes as the temperature rises; a notable effect is caused by the increase in speed and fastener stiffness on the entire system; the CA mortar layer experiences the maximum stress at its edge, which makes it highly susceptible to damage in this area. The original contribution of this work is the establishment of a temperature-dependent vehicle–track–subgrade coupled model that incorporates the viscoelasticity of the CA mortar, enabling the investigation of dynamic responses in ballastless tracks.

Numerical analysis of double-fractional PDEs in MHD hybrid nanofluid blood flow with slip velocity, heat source, and radiation effects

Abstract The study of blood flow in cylindrical geometries resembling small arteries is crucial for advancing drug delivery systems, cardiovascular health, and treatment methods. However, Conventional models have failed to capture the complex memory effects and non-local behavior inherent in blood flow dynamics, which hinders their accuracy in predicting critical flow and heat transfer properties for medical applications. To overcome these limitations, this research introduces a novel fractional-order magnetohydrodynamic model for blood flow, incorporating a ZnO and Fe 3 O 4 hybrid nanofluid. The model uniquely integrates boundary slip velocity effects within the double fractional Maxwell model (DFMM) rheology framework and utilizes the dual fractional phase lag bioheat model (DFPLM) applied to a porous cylindrical structure. Fractional-order time derivatives in the thermal and momentum equations are formulated using the Caputo approach, with numerical solutions derived via finite difference methods leveraging L1 and L2 approximations for Caputo fractional derivatives. The study examines the effects of fractional orders, relaxation time, and phase lags for heat and temperature, along with parameters such as thermal radiation, wall slip velocity, and porosity. These factors are analyzed for their impact on velocity, temperature, skin friction, and the Nusselt number. Results indicate that the hybrid nanofluid enhances heat transfer compared to blood or mono-hybrid nanofluids, while also reducing skin friction. Furthermore, fractional-order models provide more reliable and realistic predictions under varying flow conditions. The DFMM shows smoother transitions in velocity and friction, while the DFPLM predicts higher temperatures and greater heat transfer enhancement compared to classical and single-phase lag models. By integrating fractional calculus, this model offers improved simulation of complex transport phenomena in small arteries, contributing to the development of more effective cardiovascular treatments.

Viscoelastic study of cement and emulsified asphalt mortar under temperature variations based on a novel variable-order fractional Maxwell model

Viscoelastic study of cement and emulsified asphalt mortar under temperature variations based on a novel variable-order fractional Maxwell model

A Description of the Isothermal Ageing Creep Process in Polymethyl Methacrylate Using Fractional Differential Models.

Fractional differential viscoelastic models can describe complex material behaviours and fit experimental data well; however, the physical significance of model parameters is difficult to express. In this study, the fractional differential Maxwell, Kelvin, and Zener models were used to fit the short-term creep compliance curves of polymethyl methacrylate at different ageing times. The model fits were in good agreement with the experimental data. As the ageing time increased, the fractional differential Zener model showed a relative increase in the modulus parameter of the spring and a relative decrease in the modulus parameter reflecting the viscosity of the spring-pot, which indicated that physical ageing made the material more elastic. The relaxation time of the material increased, which indicated that the physical ageing reduced the free volume of the material, hindered the movement of molecules/segments, and increased the time required for the material to reach equilibrium. The fractional order of the model decreased, which reflected the phenomenon that physical ageing reduced the creep compliance of the material. Using the relaxation time as the time scale, the creep curves at different ageing times under the same stress level could be superimposed, naturally presenting the time-ageing time equivalence principle.

A modified parametric model to predict visco-elastic properties of magneto-rheological elastomers at non-LVE region

The magneto-rheological elastomer is mostly used in vibration isolation, for which higher modulus and lower Payne effect factors are crucial parameters. Strain amplitudes and frequencies influence many applications under dynamic modes. In this work, the dynamic viscoelastic properties of MRE, fabricated using electrolyte iron (EI) particles, were measured for varying strain amplitude, magnetic field and frequency. A fractional Kelvin-Voigt (KV) model is used in a frequency region from 0.01 to 40 Hz to predict the rheological behaviour. However, the available models failed to explain the observed behaviour at low frequencies and high magnetic fields and increasing strain amplitude (i.e. in the non-viscoelastic region). Therefore, a new modified KV model is proposed in this work to incorporate the drawbacks and hence can validate for varying frequency, magnetic field and strain amplitudes. The added terms can also be used in the fractional derivative Maxwell model to explain the effect of strain amplitude and magnetic field at various frequencies. The proposed model significantly improves the quality of experimental prediction in the low-frequency range, corresponding to a slow dissipative process at different strain amplitudes.

An Adoption of the Fractional Maxwell Model for Characterizing the Interfacial Dilational Viscoelasticity of Complex Surfactant Systems

In this communication, the single-element version of the fractional Maxwell model (single FMM) is adopted to quantify the observed behaviour of the interfacial dilational viscoelasticity. This mathematical tool is applied to the results obtained by the oscillating drop method for aqueous solutions of ethyl lauroyl arginate (LAE). The single FMM adequately fits the experimental results, fairly well characterizing the frequency dependence of the modulus and the inherent phase-shift angle of the complex physical quantity, i.e., the interfacial dilational viscoelasticity. Further speculations are envisaged to apply the FMM to step perturbations in the time domain, allowing for the same parameter set as in the frequency domain.

Integrating classical and fractional calculus rheological models in developing hydroxyapatite-enhanced hydrogels

This study presents a novel method for comprehending the rheological behavior of biomaterials utilized in bone regeneration. The focus is on gelatin, alginate, and hydroxyapatite nanoparticle composites to enhance their mechanical properties and osteoconductive potential. Traditional rheological models are insufficient for accurately characterizing the behavior of these composites due to their complexity and heterogeneity. To address this issue, we utilized fractional calculus rheological models, such as the Scott-Blair, Fractional Kelvin-Voigt, Fractional Maxwell, and Fractional Kelvin-Zener models, to accurately represent the viscoelastic properties of the hydrogels. Our findings demonstrate that the fractional calculus approach is superior to classical models in describing the intricate, time-dependent behaviors of the hydrogel-hydroxyapatite composites. Furthermore, the addition of hydroxyapatite not only improves the mechanical strength of hydrogels but also enhances their bioactivity. These findings demonstrate the potential of these composites in bone tissue engineering applications. The study highlights the usefulness of fractional calculus in biomaterials science, providing new insights into the design and optimization of hydrogel-based scaffolds for regenerative medicine.

Optimizing heat transfer with nano additives: A mathematical approach

Optimizing heat transfer with nano additives: A mathematical approach

Non-local Maxwell model for ultraslow relaxation of concrete under different normal stress levels

Non-local Maxwell model for ultraslow relaxation of concrete under different normal stress levels

Experimental study and multi‐scale refinement model of high damping acrylic polymer matrix VEDs for civil structural seismic retrofit

AbstractThe viscoelastic dampers (VEDs), which can provide both stiffness and damping, have been recently introduced into the field of structural vibration control for seismic enhancement of civil engineering structures. In this study, a kind of high damping acrylic polymer matrix VEDs (HDPVED) is developed independently, and this innovative HDPVED can solve the significant problem of service performance under medium‐high temperature environments, as well as low‐frequency vibration control under earthquake actions for civil engineering structures. To systematically investigate the influencing rules of frequency, temperature, and displacement amplitude on the mechanical properties and damping dissipation performance of HDPVEDs, a series of dynamic mechanical performance tests of the developed HDPVEDs are carried out at different frequencies, temperatures, and displacement amplitudes. The research results show that HDPVED exhibits excellent damping dissipation capability and adaptability under medium‐high temperature environments and low‐frequency excitations. The mechanical properties and energy dissipation performance present a strong correlation with frequency, temperature and displacement amplitude, and there is an obvious coupling effect between the three influencing factors. Based on the macroscopic mechanical property research of HDPVED, the microscopic damping mechanism and microscopic mechanical properties of HDPVED are then investigated. High‐order fractional derivative fraction Voigt and Maxwell model in parallel (FVMP) models are preferred to characterize the combined hyper‐elasticity and viscoelasticity owned by networked molecular chains and free molecular chains. The breaking and reconstruction theory of microphysical bonds is used to assess the effect of packing particles, and the time‐temperature equivalence principle is introduced to assess the effect of temperature. The multi‐scale refinement model is proposed, and the validity and accuracy of this model are verified by testing data of HDPVED. The study results show that the proposed model can accurately describe the effects of frequency, temperature, displacement amplitude, and microstructure on the multi‐scale mechanical properties of HDPVED. It provides a theoretical basis for the multi‐scale design and development of high damping VEDs.

On viscoelastic blood in a locally narrow artery with magnetic field: application of distributed-order time fractional Maxwell model

Purpose. The study of hemorheology of heterogeneous vessels is of importance for the therapy of cardiovascular diseases. The present paper aims to investigate the effects of the height parameter of the stenosis, the Reynolds number, the relaxation time and the Hartmann number on the blood flow by applying distributed order time fractional Maxwell model. Methodology. Distributed order time fractional Maxwell constitutive fluid model is used to simulate viscoelastic blood flowing in a narrowing blood vessel with uniform magnetic field. The continuity and momentum equations are established in two-dimensional cylindrical coordinate system for calculation. With vorticity-stream function method, the finite difference technique, combined with the fractional derivative L1 interpolation approximation, is utilized to obtain numerical solutions of velocity and wall shear stress. The effectiveness of the numerical method is verified. Outcomes. It reveals the choice of relaxation time in the governing equation has an influence on the results. The numerical results show that an appropriate increase of Reynolds number can reduce the possibility of vessel damage. Furthermore, when the height parameter of the stenosis increases, the damage to the vessel wall is getting serious. Besides, the wall shear stress along the blood vessel increases with the increase of the magnetic field, but the wall shear stress in the narrowest part of the blood vessel changes little.

Sampling Points-Independent Identification of the Fractional Maxwell Model of Viscoelastic Materials Based on Stress Relaxation Experiment Data.

Considerable development has been observed in the area of applying fractional-order rheological models to describe the viscoelastic properties of miscellaneous materials in the last few decades together with the increasingly stronger adoption of fractional calculus. The fractional Maxwell model is the best-known non-integer-order rheological model. A weighted least-square approximation problem of the relaxation modulus by the fractional Maxwell model is considered when only the time measurements of the relaxation modulus corrupted by additive noises are accessible for identification. This study was dedicated to the determination of the model, optimal in the sense of the integral square weighted model quality index, which does not depend on the particular sampling points applied in the stress relaxation experiment. It is proved that even when the real description of the material relaxation modulus is entirely unknown, the optimal fractional Maxwell model parameters can be recovered from the relaxation modulus measurements recorded for sampling time points selected randomly according to respective randomization. The identified model is a strongly consistent estimate of the desired optimal model. The exponential convergence rate is demonstrated both by the stochastic convergence analysis and by the numerical studies. A simple scheme for the optimal model identification is given. Numerical studies are presented for the materials described by the short relaxation times of the unimodal Gauss-like relaxation spectrum and the long relaxation times of the Baumgaertel, Schausberger and Winter spectrum. These studies have shown that the appropriate randomization introduced in the selection of sampling points guarantees that the sequence of the optimal fractional Maxwell model parameters asymptotically converge to parameters independent of these sampling points. The robustness of the identified model to the measurement disturbances was demonstrated by analytical analysis and numerical studies.

Analysing the microscopic creep characteristics of carbonate rocks using nanoindentation experiments

ABSTRACT In recent years, the depth of oil drilling has exceeded 9000 m, and the testing and evaluation of mechanical properties of ultra-deep rocks face great challenges. Obtaining complete core samples has lots of serious problems through ultra-deep drilling, so it is difficult to process standard rock samples for general tests to obtain the mechanical properties. In order to study the creep behaviour of deep rocks, we conducted continuous step holding nanoindentation creep experiments on carbonate rocks from the Tarim Oilfield in Xinjiang, China. We first investigated the effects of load and holding time on the creep behaviour; Furthermore, by calculating the stress exponent, the deformation mechanism is revealed; At last, four traditional creep models and fractional order Maxwell model were used for regression to explore the applicability of models in describing the creep behaviour. The results show the creep displacement is positively correlated with the load and holding time. Creep curves with different shapes were obtained in nanoindentation (named A1 and A2 respectively). The Burgers and the two-dashpot Kelvin model are in good agreement with experimental data under long holding time. The stress exponents of A2 are greater than 3, indicating that the creep mechanism of A2 is dislocation creep.

The Application of Fractional Derivative Viscoelastic Models in the Finite Element Method: Taking Several Common Models as Examples

This paper aims to incorporate the fractional derivative viscoelastic model into a finite element analysis. Firstly, based on the constitutive equation of the fractional derivative three-parameter solid model (FTS), the constitutive equation is discretized by using the Grünwald–Letnikov definition of the fractional derivative, and the stress increment and strain increment relationship and Jacobian matrix are obtained by using the difference method. Subsequently, we degrade the model to establish stress increment and strain increment relationships and Jacobian matrices for the fractional derivative Kelvin model (FK) and fractional derivative Maxwell model (FM). Finally, we further degrade the fractional derivative viscoelastic model to derive stress increment and strain increment relationships and Jacobian matrices for a three-component solid model and Kelvin and Maxwell models. Based on these developments, a UMAT subroutine is implemented in ABAQUS 6.14 finite element software. Three different loading modes, including static load, dynamic load, and mobile load, are analyzed and calculated. The calculations primarily involve a convergence analysis, verification of numerical solutions, and comparative analysis of responses among different viscoelastic models.

Generalized vs. fractional: a comparative analysis of Maxwell models applied to entangled polymer solutions.

Fractional viscoelastic models provide an excellent description of rheological data for polymer systems with power-law behaviour. However, the physical interpretation of their model parameters, which carry fractional units of time, often remains elusive. We show that for poly(ethylene oxide) (PEO) solutions, the fractional Maxwell model (FMM) requires much fewer model parameters than the classical generalized Maxwell model for a good description of the data and that it can be applied consistently to solutions with varying degrees of viscoelasticity. Despite their fractional units, the parameters exhibit scaling laws similar to classical parameters as a function of polymer concentration.

The distributed order models to characterize the flow and heat transfer of viscoelastic fluid between coaxial cylinders

The distributed order fractional derivatives can describe complex dynamic systems. In this paper, considering the periodic pressure gradient and magnetic field, the time distributed order fractional governing equations are established to simulate the two-dimensional flow and heat transfer of viscoelastic fluid between coaxial cylinders. Numerical solutions are obtained by the L1 approximation for the Caputo derivative (L1-scheme) and the finite difference method, and the effectiveness of numerical method is verified by a numerical example. Results demonstrate that the time distributed fractional Maxwell model can promote the flow while the distributed Cattaneo model can weaken heat transfer than the fractional Maxwell and Cattaneo model, and different weight coefficients have different effects on the fluid. The effect of physical parameters, such as the relaxation time of velocity and temperature λ 1, λ 2, the magnetic parameter M, the amplitude P 0 and frequency w of pressure gradient, and the Prandtl number Pr on velocity and temperature are discussed and analysed in detail.

On Applicability of the Relaxation Spectrum of Fractional Maxwell Model to Description of Unimodal Relaxation Spectra of Polymers.

The relaxation time and frequency spectra are vital for constitutive models and for insight into the viscoelastic properties of polymers, since, from the spectra, other material functions used to describe rheological properties of various polymers can be uniquely determined. In recent decades the non-integer order differential equations have attracted interest in the description of time-dependent processes concerning relaxation phenomena. The fractional Maxwell model (FMM) is probably the most known rheological model of non-integer order. However, the FMM spectrum has not yet been studied and used to describe rheological materials. Therefore, the goal of the present paper was to study the applicability of the relaxation spectrum of FMM to the description of the relaxation spectra of polymers. Based on the known integral representation of the Mittag-Leffler two-parameter function, analytical formulas describing relaxation time and frequency spectra of FMM model were derived. Monotonicity of the spectra was analyzed and asymptotic properties were established. Relaxation frequency spectrum grows for large frequencies with a positive power law, while the relaxation time spectrum decays for large times with a negative power of time. Necessary and sufficient conditions for the existence of the local extrema of the relaxation spectra were derived in the form of two trigonometric inequalities. A simple procedure for checking the existence or absence of the spectra extrema was developed. Direct analytical formulas for the local extrema, minima, and maxima are given in terms of model fractional and viscoelastic parameters. The fractional model parameters, non-integer orders of the stress and strain derivatives of FMM uniquely determine the existence of the spectrum extrema. However, the viscoelastic parameters of the FMM, elastic modulus, and relaxation time affect the maxima and minima of the relaxation spectra and the values of their local peaks. The influence of model parameters on their local extrema was examined. Next, the applicability of the continuous-discrete spectrum of FMM to describe Baumgaertel, Schausberger and Winter (BSW) and unimodal Gauss-like relaxation spectra, commonly used to describe rheological properties of various polymers, was examined. Numerical experiments have shown that by respective choice of the FMM parameters, in particular by respective choice of the orders of fractional derivatives of the stress and strain, a good fit for the relaxation modulus experiment data was obtained for polymers characterized both by BSW and Gauss-like relaxation spectra. As a result, a good approximation of the real spectra was reached. Thus, the viscoelastic relaxation spectrum of FMM, due to the availability of the two extra degrees of freedom (non-integer orders of the stress and strain derivatives), provides deep insights into the complex behavior of polymers and can be applied for a wide class of polymers with unimodal relaxation spectra.

Fractional rheology-informed neural networks for data-driven identification of viscoelastic constitutive models

Developing constitutive models that can describe a complex fluid’s response to an applied stimulus has been one of the critical pursuits of rheologists. The complexity of the models typically goes hand-in-hand with that of the observed behaviors and can quickly become prohibitive depending on the choice of materials and/or flow protocols. Therefore, reducing the number of fitting parameters by seeking compact representations of those constitutive models can obviate extra experimentation to confine the parameter space. To this end, fractional derivatives in which the differential response of matter accepts non-integer orders have shown promise. Here, we develop neural networks that are informed by a series of different fractional constitutive models. These fractional rheology-informed neural networks (RhINNs) are then used to recover the relevant parameters (fractional derivative orders) of three fractional viscoelastic constitutive models, i.e., fractional Maxwell, Kelvin-Voigt, and Zener models. We find that for all three studied models, RhINNs recover the observed behavior accurately, although in some cases, the fractional derivative order is recovered with significant deviations from what is known as ground truth. This suggests that extra fractional elements are redundant when the material response is relatively simple. Therefore, choosing a fractional constitutive model for a given material response is contingent upon the response complexity, as fractional elements embody a wide range of transient material behaviors.

A hierarchical creep model for cement paste: From decoding nano-microscopic C-S-H creep to considering microstructure evolution

A hierarchical creep model for cement paste: From decoding nano-microscopic C-S-H creep to considering microstructure evolution

Pipeline flow of double fractional Maxwell fluids based on the rheological experiment of xanthan gum

The purpose of this paper is to investigate the pipeline flow of fractional Maxwell fluids based on the rheological experiment of xanthan gum solutions. The rheological properties of xanthan gum solutions at different concentrations are measured. Based on the experimental data, a grid-search method is used to determine the optimal parameters. The fitting results show that the double fractional Maxwell model is suitable to describe the flow constitutive relationship of xanthan gum solutions. Moreover, the governing equation of the pipeline flow of fractional Maxwell fluids is solved by the numerical difference method. The effects of fractional stress, fractional strain and relaxation parameter on the pipeline flow are discussed. The increase in fractional stress parameter and relaxation parameter has a negative effect on the flow process, while the increase in strain parameter enhances the flow of solutions in the pipeline.