Fractional Zener Model Research Articles

A thermodynamically‐based fractional model combined viscoelastic‐viscoplastic‐ductile damage with application to fiber‐reinforced polymer composites

AbstractA thermodynamically‐based fractional viscoelastic‐viscoplastic‐damage constitutive model combined with continuous damage mechanics (CDM) theory was established, in order to describe the rate‐dependent nonlinear behavior of fiber‐reinforced polymer composites (FRPCs). The fractional Helmholtz free energy consists of four contributions: viscoelastic (VE), viscoplastic (VP), hardening and damage, in which the VE and VP parts are constructed by fractional Zener and Scott‐Blair (SB) element forms respectively. The constitutive equation is obtained through Helmholtz free energy for the fractional Zener model, and plastic flow and hardening evolution law are all derived in the process. The ductile damage, coupled to both VE and VP free energy parts, is introduced through fractional damage energy release rates to model the degradation of material properties. The corresponding strain energy release rate and dissipation contributions are also derived. The fractional implicit time integration algorithms of proposed model are presented. The model is applied to validate tests of FRPCs under various loading conditions. The model validation and comparison are presented by simulating experimental data and existing models in the literature. And the corresponding evolution of dissipated energy is discussed to further valid the characterization ability of the model.Highlights A thermodynamical fractional constitutive model was developed for FRPCs. The Helmholtz free‐energy potential for fractional Zener model is adopted. The physical significance of fractional order parameters is explored. Fractional implicit integration algorithm of proposed model is implemented. The validation and comparison of the model are presented under various loads.

Experimental and numerical analysis of the vibrational behaviour of metallic structures filled with damping materials

This work focuses on characterizing and modeling the vibrational behavior of hollow metallic structures filled with damping material. Vibration tests are first carried on a hollow structure, then two filling materials are considered: a cast viscoelastic material and a granular material composed of epoxy and polyurethane beads. Modal parameters (natural frequencies, modal damping and mode shapes) are extracted from the Frequency Response Functions with experimental modal analysis tools. Damping ratios greater than 5~\% are measured. Given the difference in the nature of the filling materials, dissipation mechanisms are different. The effectiveness of a damping material depends on the type of structural modes (bending, torsion or wall mode) to be damped, and dissipation in filling materials cannot be modelled by a constant loss factor or viscous damping. Consequently, a finite element modeling of the structure filled with viscoelastic material is proposed. Numerical modal parameters of the structure are correlated with experimental measurements. By extension, a bead mix material law, based on a fractional Zener model, is proposed, whose parameters are obtained by inverse identification.

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.

Investigation of Dynamic Viscoelastic Asymmetric Response of PA6 Film Based on Fractional Rheological Model.

Polyamide 6 (PA6) film as a typical viscoelastic material, satisfies the time-temperature superposition (TTS), and demonstrates obvious dynamic strain amplitude and frequency correlation under dynamic load. The investigation of the dynamic mechanical behavior of PA6 film is essential to ensure the safety of these materials in practical applications. In addition, dynamic mechanical property testing under conventional experimental conditions generally focuses on the short-term mechanical performance of materials. Therefore, the dynamic viscoelasticity of PA6 film was tested using a dynamic thermo-mechanical analyzer (DMA) in this study, and the complex modulus master curve was constructed based on time-temperature superposition (TTS) to realize the accelerated characterization of long-term mechanical properties. Furthermore, according to experimentally obtained asymmetric characteristics of the Cole-Cole diagram and the loss modulus master curve of the PA6 film, the parameter distribution of the fractional Zener model and the modified fractional Zener model were compared, and the asymmetric dynamic viscoelastic response of PA6 film under different conditions was systematically investigated using these models. The results indicate that the modified fractional Zener model can truly describe the dynamic asymmetric characteristics of PA6 film, verify the feasibility and advantages of the modified fractional rheological model, and provide some theoretical guidance for exploring the tensile rheological mechanism of PA6 film.

Establishment and evaluation of dynamic viscoelasticity constitutive model for asphalt mixtures based on PCA model: Utilizing coal-based synthetic natural gas slag and phosphogypsum whisker as substitute fillers

To promote the secondary utilization of solid waste in asphalt pavement and reduce resource waste and environmental damage, the fine solid waste powder from the secondary treatment of solid waste was used as a substitute filler to prepare asphalt mixtures. To more precisely characterize the effects of filler type and substitution ratio on the dynamic viscoelastic mechanical characteristics of asphalt mixtures, the dynamic modulus test was conducted on asphalt mixtures, while the dynamic modulus master curve of asphalt mixtures was established based on the time-temperature equivalent principle and the Sigmoidal function. The improved generalized Sigmoidal (IGS) model, five-parameter fractional Zener (FFZ) model, six-parameter fractional Zener (SFZ) model and ten-parameter fractional Zener (TFZ) model were used to construct the shrinkage frequency curves of different asphalt mixtures under different constitutive relationships, whereas the applicability of the constitutive models was further evaluated by the principal component analysis (PCA) model. The research results showed that the dynamic modulus and phase angle of the mixture exhibited a similar change rule with the variation of temperature and frequency, and the use of solid waste-based fillers could effectively reduce the effect of frequency variation on the viscoelasticity of the mixture. The dynamic modulus master curve constructed by Sigmoidal function could accurately describe the functional relationship between the dynamic modulus with loading frequency and temperature, and the variation trend of dynamic modulus in a wider frequency range was flattened with the increase in frequency, and this change rule was not affected by the substitution ratio of the filler, while the use of solid waste-based filler could reduce the dependence of the mixture on the temperature. Compared to several fractional constitutive models, the IGS model exhibited a higher fitting accuracy for the dynamic viscoelasticity of several mixtures, and with the use of solid waste-based fillers, the deviation of fitting curves gradually decreased. A comparative study showed that the fractional constitutive model was difficult to accurately describe the dynamic viscoelasticity of the mixture, and the IGS model was preferred to be recommended as a calculation model to investigate the constitutive relationship on the dynamic viscoelasticity of the mixture.

Dynamic viscoelastic properties of ultra-large particle size asphalt mixture based on fractional derivative constitutive model

The aim of this paper was to accurately grasp and precisely characterize the dynamic viscoelastic properties of Ultra-Large Particle Size Asphalt Mixture (LSAM-50), and fully explore the features and simplified forms of the generalized fractional derivative Zener (GFDZ) model. The constitutive relations of the GFDZ model with n Abel elements were first given, and a modified GFDZ (mGFDZ) model was proposed. Then the master curves of LSAM-50, AC-13 and AC-20 mixtures were constructed, and the expression effects of different models and the dynamic viscoelastic behaviors of different asphalt mixtures were compared. The results showed that LSAM-50 was a typical viscoelastic material similar to traditional mixture. Both dynamic modulus and energy storage modulus decreased with increasing temperature. The phase angle increased first and then decreased with temperature increasing, and the peak value appeared near 40℃ at low frequency. The loss modulus increased first and then decreased with the temperature increasing. When the loading frequency was 0.1 Hz∼25 Hz, the peak value appeared at 0℃∼20℃. The rutting factors of LSAM-50 became larger with temperature reduction and frequency enhancement. When the number of Abel elements reached 2 in the GFDZ model or reached 3 in the mGFDZ model, the model fitting effects would no longer change significantly. The 2-mGFDZ model with only 5 parameters fitted master curves data satisfactorily and the physical meanings of parameters were clear. Comparing among three mixtures, LSAM-50, with the lowest asphalt content, showed significantly superior elasticity and better deformation resistance at medium and high temperatures. At low temperatures, the viscosity behavior of LSAM-50 was slightly inferior, but it may have pronounced plate properties considering the effect of aggregate skeletonization. The research results would promote new pavement structural design and mechanical response analysis.

A fractional Hawkes process model for earthquake aftershock sequences

Abstract A new type of Hawkes process, known as the fractional Hawkes Process (FHP), has been recently introduced. This process uses a Mittag-Leffler density as the kernel function which is asymptotically a power law and so similar to the Omori–Utsu law, suggesting the FHP may be an appropriate earthquake model. However, it is currently an unmarked point process meaning it is independent of an earthquake’s magnitude. We extend the existing FHP, by incorporating Utsu’s aftershock productivity law and a time-scaling parameter from the fractional Zener Model to a marked version so that it may better model earthquake aftershock sequences. We call this model the ‘Seismic Fractional Hawkes Process’ (SFHP). We then estimate parameters via maximum likelihood and provide evidence for these estimates being consistent and asymptotically normal via a simulation study. The SFHP is then compared to the epidemic type aftershock sequence and FHP models on four aftershock sequences from Southern California and New Zealand. While it is inconclusive if the seismic fractional Hawkes process performs better in a retrospective predictive performance experiment, it does perform favourably against both models in terms of information criteria and residual diagnostics especially when the aftershock clustering is stronger.

Investigating competition of strong interfacial interaction and chain scission in PBAT/EVOH/GO composite by rheological measurements

The blends of poly(butylene-adipate-co-terephthalate)/poly(ethylene-vinyl alcohol) (PBAT/EVOH) with varying amounts of graphene oxide (GO) were prepared by melt mixing. The localization of GO in PBAT and at the interface was confirmed by morphological evaluation. The dual effect of GO in degradation of PBAT phase and interface improvement of PBAT/EVOH was examined by rheological measurements and models. The results showed that the improved interfacial interaction induced by GO dominates its degradation effect in PBAT. Rheological analysis revealed that the interfacial elasticity originating from GO dominates the total elasticity of the system, resulting in an increase in the final elasticity. Generalized Fractional Zener (GFZ) model was used to analyze elasticity of the polymer blend and its nanocomposites with a well fit to the experimental results. Also, the Lee–Park model was used to distinguish the effects of particle interactions as well as interface strengthening from deterioration of matrix modulus due to PBAT degradation by GO. The increased elasticity of the interface showed that the strengthening effect of GO at the interface overcomes its degradation effect. Also, the application of Coran model to analyze phase homogeneity revealed a linear increase in interfacial interaction with GO concentration.

Wave propagation in three-dimensional fractional viscoelastic infinite solid body

In order to analyze the wave propagation in three-dimensional isotropic and viscoelastic body, the Cauchy initial value problem on unbounded domain is considered for the wave equation written as a system of fractional partial differential equations consisting of equation of motion of three-dimensional solid body, equation of strain, as well as of the constitutive equation, obtained by generalizing the classical Hooke’s law of three-dimensional isotropic and elastic body by replacing Lamé coefficients with the relaxation moduli to account for different memory kernels corresponding to the propagation of compressive and shear waves. By the use of the method of integral transforms, namely Laplace and Fourier transforms, the displacement field, as a solution of the Cauchy initial value problem, is expressed through Green’s functions corresponding to both compressive and shear wave propagation. Two approaches are adopted in solving the Cauchy problem: in the first one the displacement field is expressed through the scalar and vector fields, obtained as solutions of wave equations which are consequences of decoupling the wave equation for displacement field, while in the second approach the displacement field is obtained by the action of the resolvent tensor on the initial conditions. Fractional anti-Zener and Zener model I+ID.ID, as well as fractional Burgers model VII, as representatives of models yielding infinite and finite wave propagation speed, are used in numerical calculations and graphical representations of Green’s functions and its short and large time asymptotic behavior.

Fractional derivative modeling for dynamic response of layered viscoelastic media under moving time-harmonic loads

The dynamic response of layered viscoelastic media under moving time-harmonic loads is an important issue in transportation geotechnics. However, the existing research often has limitations in terms of the constant creep rate or time-independent relaxation modulus. To fill this gap, this study introduces a fractional Zener model to simulate the viscoelastic properties of media. First, the governing equations of the elastic medium are converted into ordinary differential equations using an integral transform. Subsequently, by introducing the fractional viscoelastic model and extended precise integration method, a solution to the dynamic response of layered viscoelastic media under moving time-harmonic loads is obtained. After comparing the results with those in existing literature, the reliability and accuracy of the proposed solution were verified. Finally, a parametric analysis was performed to further explore the influence of load velocity and fractional order on the dynamic response of the viscoelastic media.

Investigating nonlinear dynamic properties of a swing arm rubber joint and their effects on the dynamic behavior of high-speed trains

The rubber joints mounted on swing arms provide the main part of the yaw stiffness for wheelsets and thus have a significant influence on the dynamic behaviour of railway vehicles. However, their nonlinear dynamic properties concerning the frequency, amplitude, and temperature were not fully considered in previous studies, most of which adopt simple models such as the Kelvin–Voigt model to represent rubber components. This study aims to investigate the radial nonlinear properties of swing arm rubber joints under various conditions and assess their effects on the dynamic performance of high-speed trains. A nonlinear rubber spring model which combines the fractional derivative Zener model with Berg’s friction model is established. To achieve a better fit to the measurements of a swing arm rubber joint, an optimisation-based method is employed to identify the model parameters. The viscous and frictional effects of the rubber joint are compared at different frequencies, amplitudes, and temperatures, by determining the predominant element of the nonlinear model. Additionally, the nonlinear rubber spring model is integrated into a high-speed vehicle dynamic model to investigate the effects of the nonlinear properties of swing arm rubber joints on the vehicle dynamic behaviour through the MATLAB/SIMPACK co-simulation method.

Modeling the effects of hydration on viscoelastic properties of nucleus pulposus tissue in shear using the fractional Zener model

Nucleus pulposus (NP) tissue in the intervertebral disc (IVD) is a viscoelastic material exhibiting both solid- and fluid-like mechanical behaviors. Advances in viscoelastic models incorporating fractional calculus, such as the Fractional Zener (FZ) model, have potential to describe viscoelastic behaviors. The objectives of this study were to determine whether the FZ model can accurately describe the shear viscoelastic properties of NP tissue and determine if the fractional order (α) is related to tissue hydration. 30 caudal IVDs underwent equilibrium dialysis in 5% or 25% polyethylene glycol solutions to alter tissue hydration. Excised NP tissue underwent stress relaxation testing in shear and unconfined compression. Stress relaxation data was fitted to the FZ model to obtain viscoelastic properties. In both loading modes, the initial modulus was greater for the less hydrated 25% equilibrated samples compared to 5% with no change in the equilibrium modulus. Samples with lower water content (25% samples) had shorter relaxation times in shear and longer time constants in compression, highlighting the different interactions between the fluid and solid matrix in loading modes. Samples with lower water content had α values closer to 0, indicating that less hydrated samples behaved more solid-like on the viscoelastic spectrum. Tissue hydration correlated with α values for 25% samples in shear. This study demonstrates that the FZ model may be used to describe IVD tissue behavior under both loading modes; however, the greatest utility of the FZ model is in describing flow-independent shear behaviors, and α may inform tissue hydration in shear.

THE FRACTAL RHEOLOGY OF MAGNETORHEOLOGICAL ELASTOMERS DESCRIBED THROUGH THE MODIFIED ZENER MODEL AND THE COLE–COLE PLOT

The main objective of this study focuses on using, for the first time, the Cole–Cole semilogarithmic fractal plot to study the fractal rheological behavior of magnetorheological elastomer materials using the fractional Zener model. Experimental data and theoretical results show that the Cole–Cole semilogarithmic fractal plot predicts the Zener mathematical model’s material parameters. This paper elucidates the benefits of using Cole–Cole semilogarithmic fractal plots to accurately predict material rheological properties of composite elastomer reinforced with nano or microiron particles observed during experimental tests.

Frequency-dependent Q simulation and viscoacoustic reverse time migration based on the fractional Zener model

Seismic attenuation is a basic physical property of the earth, which significantly affects the characteristics of seismic wavefields. Accurately simulating wave propagation in the earth is essential to image subsurface structures. Some prevailing methods (e.g., the standard linear solid and fractional Laplacian equation) to describe seismic wave propagation in attenuating media are mainly based on the constant- Q model (CQM), which is valid at room temperature and pressure. However, laboratory measurements suggest that the quality factor Q is a function of frequencies in some regions. To simulate the frequency-dependent Q effect, we derive a viscoacoustic wave equation from the stress-strain relationship of the fractional Zener model (FZM) with variable fractional orders. During the implementation, we separate the real and imaginary parts of the modulus and introduce a low-rank decomposition method to solve the FZM equation. Because the amplitude dissipation and phase dispersion are decoupled, we establish a compensated reverse time migration ( Q-RTM) algorithm to mitigate adverse effects caused by seismic attenuation and improve the quality of seismic migration in frequency-dependent attenuating media. A two-layer model and the BP gas chimney model are used to perform Q-RTM tests. A low-pass filter with a Tukey window function is applied to suppress numerical instability during the compensation. Numerical results demonstrate that our FZM Q-RTM approach can produce high-resolution images with corrected reflector positions and amplitudes. Because the CQM equation ignores the frequency dependence of Q, it may lead to overcompensation in Q-RTM.

The application of a novel variable-order fractional calculus on rheological model for viscoelastic materials

Viscoelastic materials have become popular for novel multifunctional materials in emerging modern technologies, which have complex elastic-viscous-plastic behaviors. The variable-order fractional derivative has been gradually applied to characterize such variable memory effects of complex physical systems, while existing definitions are somewhat controversial. The new Scarpi’s variable-order fractional calculus has rigorous mathematical logic compared with the existed definitions. However, its practicality and feasibility in engineering are ambiguous due to the fewer applications up to now. In this paper, we aimed at introducing the Scarpi’s variable-order fractional derivative into the rheological applications of viscoelastic materials. Based on the Scarpi’s variable-order fractional derivative, the fractional Zener model and fractional viscoelastic-visoplastic models are established, and the parameters sensitivity on stress-strain relation is discussed. Then, the proposed models are then applied to depict different rheological behaviors of polymers at various temperatures and strain rates, and the results show that they have excellent agreement for experimental data compared to the reference model. It is observed that the order increases linearly with strain of the viscoelastic model and decreases exponentially with strain of the viscoplastic model. Furthermore, the physical significance of the corresponding fractional order parameters is investigated. It is found that they have the variation relation with temperatures and strain rates. It reveals that the variable fractional order can be regarded as an index to characterize the evolution of mechanical behaviors of polymers.

Dynamic response of fractional poroviscoelastic layered media subjected to moving loads

This study investigates the dynamic response of fractional poroviscoelastic layered media subjected to a moving harmonic load. The material viscosity includes two parts: the flow-dependent viscosity between the skeleton and pore water and the skeleton viscosity. Firstly, the governing equations of poroelastic media are derived based on Biot's theory and then transformed into ordinary differential equations by using the double Fourier integral transform. By incorporating fractional calculus theory and the dynamic correspondence principle, the applicability of these equations is extended to fractional poroviscoelastic media. The extended precise integration method (PIM) is employed to obtain solutions. Subsequently, the fractional Zener model and the presented method are validated. Finally, parametric studies are conducted to analyze the effects of fractional order, moving velocity, load frequency, and stratification on the dynamic behaviors of the media.

Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model

A class of generalized fractional Zener-type viscoelastic models with general fractional derivatives is considered. Two integral representations are derived for the corresponding relaxation modulus. The first representation is established by applying the Laplace transform to the constitutive equation and using the Bernstein functions technique to justify the change of integration contour in the complex Laplace inversion formula. The second integral representation for the relaxation modulus is obtained by applying the subordination principle for the relaxation equation with generalized fractional derivatives. Two particular examples of the considered class of models are discussed in more detail: a model with fractional derivatives of uniformly distributed order and a model with general fractional derivatives, the kernel of which is a multinomial Mittag-Leffler-type function. To illustrate the analytical results, some numerical examples are presented.

Dynamic Characteristics of a Composite Beam with Viscoelastic Layers under Uncertain-But-Bounded Design Parameters

In this study, dynamic characteristics of a composite beam with uncertain design parameters are analyzed. Uncertain-but-bounded parameters change only within certain specified limits. This study uses interval analysis to investigate a composite beam with viscoelastic layers whose behavior is described using the fractional Zener model. In general, parameters describing both elastic and viscoelastic layers can be uncertain. Several methods have been studied to determine the lower and upper bounds of the dynamic characteristics of a structure. Among them, the vertex method is a comparative method in which the lower and upper bounds of the dynamic characteristics are approximated using the first- and second-order Taylor series expansion. An algorithm to determine the critical combination of uncertain design parameters has also been described. Numerical examples demonstrate the effectiveness of the presented methods and the possibility of applying them to the analysis of systems with numerous uncertain parameters and high uncertainties.

Modified fractional-Zener model—Numerical application in modeling the behavior of asphalt mixtures

To seek a constitutive model that can both describe the viscoelastic response of asphalt mixtures well and be applied to finite-element simulation, a modified fractional-order Zener model (MFZM) was adopted from the mathematical expressions describing viscoelastic materials based on fractional calculus theory in this study. The MFZM was verified by the generalized fractional-Zener model, and the corresponding model configuration was provided. In this model, the dynamic response and relaxation-modulus function were derived. Based on the relaxation-modulus function, a numerical algorithm for the MFZM that can be implemented in a finite-element environment was proposed. The ability of the fractional-Zener model (FZM) and MFZM to characterize the dynamic response of asphalt mixtures was compared using dynamic modulus test. The dynamic modulus and semicircular bending (SCB) tests of the asphalt mixture were simulated using the proposed numerical algorithm in the software ABAQUS. The results demonstrate that, unlike FZM, MFZM can characterize the asymmetric characteristics of viscoelastic response of the asphalt mixture in the frequency domain. The simulation result of the dynamic modulus test agreed with the analytical solution of the MFZM in the frequency domain, which illustrated the availability of the proposed numerical method for the response of the asphalt mixture in the frequency domain. Moreover, the consistency between the actual experimental and virtual results of SCB proved the accuracy of the model and applicability of the proposed numerical method, allowing the model to be used in a finite-element environment.

Thermomechanical Characterization and Modeling of NiTi Shape Memory Alloy Coil Spring.

Today, shape memory alloys (SMAs) have important applications in several fields of science and engineering. This work reports the thermomechanical behavior of NiTi SMA coil springs. The thermomechanical characterization is approached starting from mechanical loading-unloading tests under different electric current intensities, from 0 to 2.5 A. In addition, the material is studied using dynamic mechanical analysis (DMA), which is used to evaluate the complex elastic modulus E* = E' - iE″, obtaining a viscoelastic response under isochronal conditions. This work further evaluates the damping capacity of NiTi SMA using tan δ, showing a maximum around 70 °C. These results are interpreted under the framework of fractional calculus, using the Fractional Zener Model (FZM). The fractional orders, between 0 and 1, reflect the atomic mobility of the NiTi SMA in the martensite (low-temperature) and austenite (high-temperature) phases. The present work compares the results obtained from using the FZM with a proposed phenomenological model, which requires few parameters for the description of the temperature-dependent storage modulus E'.