Fractional Derivative Viscoelastic Model Research Articles

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.

Fractional derivative viscoelastic response of high-viscosity modified asphalt

To explain the dynamic viscoelastic characteristics of high-viscosity modified asphalt (HVMA) more comprehensively and accurately, one type of finished HVMA and two types of laboratory-produced HVMA were chosen as the research objects. Firstly, a dynamic shear rheometer was employed for the strain sweep test to define the linear viscoelastic range of three different types of HVMA. Then, the frequency sweep test was performed in the linear viscoelastic range under different high-temperature conditions, and the viscoelastic properties of three different types of HVMA were analysed. Finally, an HVMA fractional derivative viscoelastic model was built and compared to the traditional Sigmoid function model. The results demonstrated that the traditional Sigmoid function model has limitations and cannot effectively illustrate the changes in the HVMA over a wide frequency range. The viscoelastic properties of HVMA can be fully described by the 1S1A1D fractional derivative model, including the storage modulus, loss modulus, complex shear modulus, and phase angle variations. For the 1S1A1D fractional derivative model, one Abel dashpot can illustrate the viscoelastic properties of HVMA, and increasing the number of Abel dashpots no significance on the performance.

Study on Dynamic Characteristics of Circular Tunnel Lining in Fractional Derivative Viscoelastic Radially Inhomogeneous Soil

Considering the heterogeneous and viscoelastic characteristics of the soil around the tunnel, the dynamic characteristics of circular tunnel lining in fractional derivative viscoelastic radially inhomogeneous soil are studied. To accurately describe the viscoelasticity of the soil around the circular tunnel, the fractional derivative viscoelastic model is used to describe the stress-strain relationship of the soil. To consider the heterogeneous characteristics of the soil around the circular tunnel caused by construction and in situ stress, the soil around the tunnel lining is divided into undisturbed area and disturbed area, and the disturbed area is divided into multiple thin concentric rings. A dynamic interaction model of fractional derivative viscoelastic radially inhomogeneous soil-circular tunnel lining is established. By solving the vibration of the soil in undisturbed area and each circle and considering the continuity conditions of the radial displacement and radial stress of the soil at the junction of each circle, the transfer matrix of the radial displacement and radial stress of the soil at the outer and inner boundaries of disturbed area is obtained. Based on the transfer matrix of the radial displacement and radial stress of soil in the disturbed area and considering the boundary conditions and continuity conditions of the problem, the solution of the dynamic response of circular tunnel lining in fractional derivative viscoelastic radially inhomogeneous soil is obtained in the frequency domain. The results show that the fractional derivative viscoelastic inhomogeneous soil-circular tunnel lining system also has resonance under the action of simple harmonic internal pressure, and the curves of the radial displacement and radial stress of soil varying with frequency obtained by the fractional derivative viscoelastic model fluctuate greatly. When studying the dynamic response of circular tunnel lining in inhomogeneous soil, the influence of viscoelasticity of the soil around the tunnel should not be ignored. The shear modulus ratio of soil in the affected area around the tunnel has a great influence on the dynamic characteristics of circular tunnel lining in fractional derivative viscoelastic radial inhomogeneous soil, and the inhomogeneous characteristics of soil should not be ignored. The greater the thickness and shear modulus of the tunnel lining, the greater the stiffness of tunnel lining and the smaller the radial stress and radial displacement of the soil.

Research on dynamic stiffness with the high-order fractional derivative model for rubber bushing

A hybrid strategy for predicting the dynamic stiffness of rubber bushing is proposed in this study to analyze the axial amplitude-frequency dependence with the high-order fractional derivative model. Firstly, the finite element model (FEM) is constructed to calculate the force-displacement hysteresis curves under certain conditions. Secondly, a parametric model composed of the elastic element, the friction element, and the high-order fractional derivative viscoelastic element in parallel is proposed to describe the axial amplitude-frequency dependent behavior of rubber bushing. Then, based on the hysteresis curve obtained by the finite element analysis, the parameters of the high-order fractional derivative viscoelastic model are identified by using an adaptive chaos particle swarm optimization (ACPSO). Finally, the effectiveness of the proposed strategy is evaluated through simulations under sinusoidal excitation in the amplitude range of 0.1–2.0 mm and the frequency range of 0.1–25.0 Hz. The simulation results show that the parametric model can accurately estimate the dynamic stiffness under the condition of large amplitude and wide frequency. The hybrid strategy is practically useful for the dynamic stiffness estimate of rubber bushing, and reduces time and computational resources compared with the finite element method.

Theoretical and Numerical Analysis of Soil-Pipe Pile Horizontal Vibration Based on the Fractional Derivative Viscoelastic Model

To describe the mechanical properties of the system of pipe pile-soil reasonably and accurately, the constitutive relations of the soil around pile and pile core soil are characterized by the fractional derivative viscoelastic model. We assume that the radial and circumferential displacements of the soil around the pile and pile core soil are the functions of r, θ, and z. The horizontal dynamic control equations of soil layers are derived by using the fractional derivative viscoelastic model. Considering the fractional derivative properties, soil layer boundary condition, and contact condition of pile and soil, the potential function decomposition method is used to solve the radial and circumferential displacements of the soil layer. Then, the force of unit thickness soil layer on the pipe pile and the impedance factor of the soil layer are obtained. The horizontal dynamic equations of pipe pile are established considering the effect of soil layers. The horizontal dynamic impedance and horizontal-swaying dynamic resistance at the pile top are obtained by combining the pipe pile-soil boundary conditions and the orthogonal operation of trigonometric function. Numerical solutions are used to analyze the influence of pile and soil parameters on the soil impedance factor and horizontal dynamic impedance at pile top. The results show that the horizontal impedance factors of the soil layer and horizontal dynamic impedance of pipe pile by using the fractional derivative viscoelastic model can be degraded to those of the classical viscoelastic model and the elastic model. For the fractional derivative viscoelastic model of soil layer, the influence of soil around pile on the dynamic impedance is greater than that of pile core soil. The model parameter TOa, the inner radius of pipe pile, and the pile length have obvious effects on the horizontal impedance of the soil layer and pipe pile, while the influence of the pile core soil on the pile impedance is smaller.

A quadrature-free Legendre polynomial approach for the fast modelling guided circumferential wave in anisotropic fractional order viscoelastic hollow cylinders

Compared to the traditional integer order viscoelastic model, a fractional order derivative viscoelastic model is shown to be advantageous. The characteristics of guided circumferential waves in an anisotropic fractional order Kelvin–Voigt viscoelastic hollow cylinder are investigated by a quadrature-free Legendre polynomial approach combining the Weyl definition of fractional order derivatives. The presented approach can obtain dispersion solutions in a stable manner from an eigenvalue/eigenvector problem for the calculation of wavenumbers and displacement profiles of viscoelastic guided wave, which avoids a lot of numerical integration calculation in a traditional polynomial method and greatly improves the computational efficiency. Comparisons with the related studies are conducted to validate the correctness of the presented approach. The full three dimensional spectrum of an anisotropic fractional Kelvin–Voigt hollow cylinder is plotted. The influence of fractional order and material parameters on the phase velocity dispersion and attenuation curves of guided circumferential wave is discussed in detail. Moreover, the difference of the phase velocity dispersion and attenuation characteristics between the Kelvin–Voigt and hysteretic viscoelastic models is also illustrated. The presented approach along with the observed wave features should be particularly useful in non-destructive evaluations using waves in viscoelastic waveguides.

Rate-dependent damping properties of recycled aggregate concrete from creep perspective

Creep and damping, which are two important static and dynamic properties of concrete material, have similar mechanical characteristics and have not been studied integratedly. This paper aims to develop a new method to investigate the creep and damping properties of recycled aggregate concrete (RAC) from viscoelastic perspective. In this research, a fractional order derivative viscoelastic creep model of RAC under different load levels was proposed. Then, with the consideration of recovery coefficient and time-dependent behavior, the damping properties of RAC based on interconversion between viscoelastic functions were investigated. The creep tests of reinforced RAC beams, accompanied by loading and unloading tests, were conducted to study the viscoelastic properties, meanwhile, free-vibration tests at different ages were carried out to analyze the time-dependent damping behavior of RAC. Test results indicate that compared with three parameter model and generalized three parameter model, fractional three parameter model is a feasible viscoelastic model to characterize both creep and damping behavior of RAC. The creep compliance of RAC is higher than that of NAC, and increases with the increasing of load level. The damping property of RAC decreases with the increasing of load frequency and the decreasing of load level, meanwhile, the damping property of RAC is related to recoverable instantaneous deformation. In addition, the damping property of RAC decreases with the increasing age when it is not considering load, but increases when it is subjected to sustained load. The proposed damping model was validated with a database of RAC damping in published literatures.

Experimental and numerical research of stress relaxation behavior of magnetorheological elastomer

The paper presents experimental research and viscoelastic modeling of stress relaxation response of isotropic magnetorheological elastomer (MRE). The isotropic MRE has been prepared based on silicone matrix filled by magnetically micro-sized carbonyl iron particles. Effects of constant strain level and external magnetic field on the stress relaxation behavior of the MRE were carefully investigated by single- and multi-step relaxation tests in shear mode using double-lap shear specimens. Results revealed that the stress relaxation response of the MRE was dependent on the applied constant strain and external magnetic field. The relaxed stress and modulus of the MRE increased with increasing the constant strain level. In addition, the values of absolute stress and modulus in the relaxation periods enhanced with the rise of magnetic flux density. A four-parameter fractional derivative viscoelastic model was used to describe the stress relaxation behavior of the MRE. The studied model was fitted well to experimental data of the MRE in both single- and multi-step relaxation tests. The fitting of shear-stress relaxation modulus for the MRE is in a very good agreement with the experimental one. Effects of applied constant strain and magnetic field intensity on the fitted parameters were discussed. Moreover, the model can be applied to predict accurately the long-term relaxation behavior of the MRE.

The fractional Kelvin-Voigt model for circumferential guided waves in a viscoelastic FGM hollow cylinder

Compared to the traditional integer order viscoelastic model, a fractional order derivative viscoelastic model is shown to be more accurate. A thorough knowledge of the dispersive characteristics of such model is very essential to the application of guided wave testing technique. In this paper, the guided waves in a fractional Kelvin-Voigt viscoelastic FGM hollow cylinder with material changing in the thickness direction are investigated. The Weyl definition of fractional order derivatives and the extended Legendre polynomial approach are employed for the derivations of the governing equations. The presented approach has the advantage that the solution of the complex partial differential wave equations with variable coefficients is reduced to an eigenvalue problem, which overcomes the shortcomings of the existing iterative methods such as the Newton downhill method and improves computation efficiency. The previous methods for dealing with viscoelastic guided wave transform the wave equations into a matrix determinant problem solved by the iterative methods, which has a very slow calculation speed. Comparisons with the related studies are conducted to validate the correctness of the presented approach, and the convergence of the approach is discussed. The full three dimensional spectrum, phase dispersion curves and attenuation curves are illustrated for various fractional order viscoelastic FGM hollow cylinders. The influences of fractional order, grade field and radius-thickness ratio on dispersion and attenuation curves are illustrated. The difference of the dispersion characteristics between the viscoelastic model and the elastic one is discussed. The influences of fractional order on displacement distributions are also studied.

An experimentally validated rubber shear spring model for vibrating flip-flow screens

Vibrating flip-flow screens (VFFS) provide an effective solution for screening highly moist and fine-grained minerals, and the dynamic response of the main and the floating screen frames largely accounts for a VFFS’s screening performance and its processing capacity. An accurate dynamic model of the rubber shear springs inserted between the frames of the VFFS is critical for its dynamic analysis but has rarely been studied in detail. In this paper, a variance-based global sensitivity analysis is applied to actually illustrate that the rubber shear spring is the most important component for the dynamics of VFFS. Then a nonlinear rubber shear spring model is proposed to predict its amplitude and frequency dependency, which is described by a friction model and a fractional derivative viscoelastic model, respectively, and the elasticity is predicted by a nonlinear spring. The reasonability of the proposed model is verified by experimental cyclic tests of the rubber shear spring. Comparisons between the newly proposed model and other classic models, including the Generalized Maxwell model, adopted for the dynamic analysis of the VFFS are carried out, and experimental tests of an industrial VFFS’s dynamic response show that dynamics of the VFFS can be better described using the proposed model than the existing models. Furthermore, the method of the global sensitivity analysis is also applied to the newly VFFS dynamic model to calculate the sensitivities of model outputs caused by the input parameters. The results reveal that the dynamic response of an operating VFFS is most sensitive to changes in the stiffness of the rubber shear spring, followed by the mass of the floating screen frames.

Experimental characterization and viscoelastic modeling of isotropic and anisotropic magnetorheological elastomers

The paper presents experimental research and numerical modeling of dynamic properties of magnetorheological elastomers (MREs). Isotropic and anisotropic MREs have been prepared based on silicone matrix filled by micro-sized carbonyl iron particles. Dynamic properties of the isotropic and anisotropic MREs were determined using double-lap shear test under harmonic loading in the displacement control mode. Effects of excitation frequency, strain amplitude, and magnetic field intensity on the dynamic properties of the MREs were examined. Dynamic moduli of the MREs decreased with increasing the strain amplitude of applied harmonic load. The dynamic moduli and damping properties of the MREs increased with increasing the frequency and magnetic flux density. The anisotropic MREs showed higher dynamic moduli and magnetorheological (MR) effect than those of the isotropic ones. The MR effect of the MREs increased with the rise of the magnetic flux density. The dependence of dynamic moduli and loss factor on the frequency and magnetic flux density was numerically studied using four-parameter fractional derivative viscoelastic model. The model was fitted well to experimental data for both isotropic and anisotropic MREs. The fitting of dynamic moduli and loss factor for the isotropic and anisotropic MREs is in good agreement with experimental results.

Active Control of Nonlinear Transient Vibration of Laminated Composite Beams Using Triangular SCLD Treatment With Fractional Order Derivative Viscoelastic Model

Abstract This paper is concerned with the analysis of the effectiveness of triangular shaped smart constrained layer damping (SCLD) treatment in attenuating geometrically nonlinear transient vibrations of laminated composite beams. The SCLD treatment is comprised of an advanced vertically reinforced 1–3 piezoelectric composite (PZC) as the constraining layer and an isotropic viscoelastic layer as the constrained layer, which is modeled using a two-dimensional fractional order derivative (FOD) model with Grünwald definition of the FODs. A nonlinear meshfree model of the smart composite beam is developed for analyzing its nonlinear transient response within the framework of a layerwise shear and normal deformation theory considering von Kármán type geometric nonlinearity. Cantilever type composite beams having different lamination sequences integrated with regular rectangular/triangular type of SCLD treatments are considered for presenting the numerical results. For comparison purpose, a geometrical constraint has been imposed such that both the rectangular and triangular shaped SCLD treatments will cover the equal area on the top surface of the beam. The numerical analyses demonstrate the effectiveness of the triangular shaped SCLD patches over the rectangular SCLD treatment in controlling the nonlinear vibration of laminated composite beams. The two-dimensional FOD model of the viscoelastic material has been efficiently implemented for the active damping analysis of smart composite beam.

A One-Dimensional Fractional-Derivative Viscoelastic Model for the Aquitard Consolidation of an Aquifer System

Land subsidence resulting from the withdrawal of groundwater is one of the problems of major importance to geosciences and geomechanics. In order to analyze the phenomena of land subsidence, this study attempts to constitute a more reliable prediction model to simulate the consolidation process of the aquifer system under the drawdown of groundwater. A fractional-derivative viscoelastic model is introduced to characterize the rheological properties of the aquitard, and the equation governing the consolidation process is established on the basis of the one-dimensional consolidation theory. The semianalytical solutions for the pore-water pressure, the settlement of aquitard, and the degree of consolidation are deduced using both finite sine transform and Laplace transform. Then, the comparisons of two special cases are performed to validate the correctness of the proposed model. The variations in the degree of consolidation and settlement of the aquitard are simulated to analyze the consolidation behavior of the aquifer system, and the influences of model parameters and the drawdown speed of groundwater are investigated.

Fractional Derivative Viscoelastic Response Model for Asphalt Binders

AbstractThe main goal of this paper is to establish an appropriate fractional derivative model with fewer parameters that can model all the viscoelastic characteristics of asphalt binders. Based on...

Characterization of elastomeric scaffolds developed for tissue engineering applications by compression and nanoindentation tests, μ-Raman and μ-Brillouin spectroscopies.

In tissue engineering, porous biodegradable scaffolds are developed with morphological, chemical and mechanical properties to promote cell response. Therefore, the scaffold characterization at a (sub)micrometer and (bio)molecular level is paramount since cells are sensitive to the chemical signals, the rigidity, and the spatial structuring of their microenvironment. In addition to the analysis at room temperature by conventional quasi-static (0.1-45 Hz) mechanical tests, the ultrasonic (10 MHz) and μ-Brillouin inelastic light scattering (13 GHz) were used in this study to assess the dynamical viscoelastic parameters at different frequencies of elastomeric scaffolds. Time-temperature superposition principle was used to increase the high frequency interval (100 MHz-100 THz) of Brillouin experiments providing a mean to analyse the viscoelastic behavior with the fractional derivative viscoelastic model. Moreover, the μ-Raman analysis carried out simultaneously during the μ-Brillouin experiment, gave the local chemical composition.

A new three-dimensional magneto-viscoelastic model for isotropic magnetorheological elastomers

In this work, a four-parameter fractional derivative viscoelastic model was developed to describe the dynamic shear behaviour of magnetorheological elastomers (MREs) as a function of the matrix, particle content and magnetic field. The material parameters were obtained from experimental data measured with a Physica MCR 501 rheometer from the Anton Paar Company, equipped with a magnetorheological cell. The synthetised isotropic MRE samples were based on room-temperature vulcanising silicone rubber and spherical carbonyl iron powder micro particles as fillers, and seven volumetric particle contents were studied. The influence of particle contents was included in each parameter of the four-parameter fractional derivative model. The dependency of the storage modulus as a function of an external magnetic field (magnetorheological (MR) effect) was studied, and a dipole–dipole interaction model was used. A new three-dimensional magneto-viscoelastic model was developed to couple the viscoelastic model, the particle-matrix interaction and the magneto-induced modulus model, which predicts the influence of the magnetic field and the particle content in the MR effect of isotropic MREs.

Application of Fractional Calculus to Frontal Crash Modeling

Fractional derivative viscoelastic model is used in the analysis of a frontal impact of a vehicle against a rigid obstacle. The frontal part of the vehicle is first modeled as a viscoelastic fractional rod and then it is modeled as two different viscoelastic fractional rods with a different length. In the second model also the friction is taken into account. A motion is analyzed during several phases because of both different lengths of the rods and the presence of a dry friction force in the later model. Governing systems of differential equations together with the corresponding initial conditions are derived. Parameter identification is done on the basis of the existing experimental results using the solution of a posed impact problem. What makes the problem more complex, regarding the second model, is the fact that it belongs to the class of nonsmooth fractional order systems, which require special treatment when dealing with deformation history during different motion phases.

Dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load

In the paper, the dynamic response of a simply supported viscoelastic beam of the fractional derivative type to a moving force load is studied. The Bernoulli-Euler beam with the fractional derivative viscoelastic Kelvin-Voigt material model is considered. The Riemann-Liouville fractional derivative of the order 0 < α ˂ 1 is used. The forced-vibration solution of the beam is determined using the mode superposition method. A convolution integral of fractional Green’s function and forcing function is used to achieve the beam response. Green’s function is formulated by two terms. The first term describes damped vibrations around the drifting equilibrium position, while the second term describes the drift of the equilibrium position. The solution is obtained analytically whereas dynamic responses are calculated numerically. A comparison between the results obtained using the fractional and integer viscoelastic material models is performed. Next, the effects of the order of the fractional derivative and velocity of the moving force on the dynamic response of the beam are studied. In the analysed system, the effect of the term describing the drift of the equilibrium position on the beam deflection is negligible in comparison with the first term and therefore can be omitted. The calculated responses of the beam with the fractional material model are similar to those presented in works of other authors.

Torsional Vibration of a Single Pile in Transversely Isotropic Soil Based on Fractional Derivative Viscoelastic Model

A single pile in soil is modeled as a vertical circular elastic prismatic bar, and the soil around the pile is regarded as transversely isotropic medium. The relationship between stress and displacement of the soil is described by fractional derivative viscoelastic model. Ignoring the radial and vertical displacement, the torsional vibration equation of the soil is built. It is solved by using the method of separation of variables and the boundary conditions of the soil. The torsional vibration of the single pile is also obtained. The torsional complex stiffness at pile head is investigated in particular. The results indicate that anisotropic parameters and model parameters of the soil have effects on the torsional complex stiffness, and the influence rules are different with the homogeneous soil.

A nonlinear and fractional derivative viscoelastic model for rail pads in the dynamic analysis of coupled vehicle–slab track systems

A nonlinear and fractional derivative viscoelastic (FDV) model is used to capture the complex behavior of rail pads. It is implemented into the dynamic analysis of coupled vehicle–slab track (CVST) systems. The vehicle is treated as a multi-body system with 10 degrees of freedom, and the slab track is represented by a three layer Bernoulli–Euler beam model. The model for the rail pads is one dimensional, and the force–displacement relation is based on a superposition of elastic, friction, and FDV forces. This model takes into account the influences of the excitation frequency and of the displacement amplitude through a fractional derivative element, and a nonlinear friction element, respectively. The Grünwald representation of the fractional derivatives is employed to numerically solve the fractional and nonlinear equations of motion of the CVST system by means of an explicit integration algorithm. A dynamic analysis of the CVST system exposed to excitations of rail harmonic irregularities is carried out, pointing out the stiffness and damping dependence on the excitation frequency and the displacement amplitude. The analysis indicates that the dynamic stiffness and damping of the rail pads increase with the excitation frequency while they decrease with the displacement amplitude. Furthermore, comparisons between the proposed model and ordinary Kelvin model adopted for the CVST system, under excitations of welded rail joint irregularities and of random track irregularities, are conducted in the time domain as well as in the frequency domain. The proposed model is shown to possess several modeling advantages over the ordinary Kelvin element which overestimates both the stiffness and damping features at high frequencies.