Fractional Kelvin-Voigt Model Research Articles

Vibration analysis of pipes conveying fluid resting on a fractional Kelvin-Voigt viscoelastic foundation with general boundary conditions

In this paper, the stability of pipes conveying fluid with viscoelastic fractional foundation is investigated. The pipe is fixed at the beginning while the pipe end is constrained with two lateral and rotational springs. The fluid flow effect is modeled as a lateral distributed force, containing the fluid inertia, Coriolis and centrifugal forces. The pipe is modeled using the Euler-Bernoulli beam theory and a fractional Kelvin-Voigt model is employed to describe the viscoelastic foundation. The equation of motion is derived using the extended Hamilton’s principle. Presenting the derived equation in Laplace domain and applying the Galerkin method, a set of algebraic equations is extracted. Calculating the determinant of the coefficients of the extracted algebraic equations results in the stability margin of the pipe. Some run is done and effects of some physical parameters such as stiffness and damping of fractional viscoelastic foundation, fractional order parameter and the end lateral and rotational stiffness of end springs on the stability boundary of the pipe are considered and some conclusions are drawn.

Force driven vibrations of fractionally damped plates subjected to primary and internal resonances

In the present paper, nonlinear force driven vibrations of thin plates in a viscoelastic medium have been studied, when its damping features are described by the fractional derivative Kelvin-Voigt model. The motion of the plate is governed by a set of three coupled nonlinear differential equations subjected to the different conditions of the internal resonance accompanied by the external resonance, resulting in the interaction of three modes corresponding to the mutually orthogonal displacements. Nonlinear sets of resolving equations in terms of amplitudes and phase differences have been obtained. The influence of fractional order viscosity and external vertical harmonic force on the energy exchange mechanism between the coupled modes has been analyzed.

Frequency domain identification of the fractional Kelvin-Voigt’s parameters for viscoelastic materials

In this work, a new innovative method is used to identify the parameters of fractional Kelvin-Voigt constitutive equation. These parameters are: the order of fractional derivation operator, 0 ≤ α ≤ 1, the coefficient of fractional derivation operator, CV, and the stiffness of the model, KV. A particular dynamic test setup is developed to capture the experimental data. Its outputs are time histories of the excitation and excited accelerations. The investigated specimen is a polymeric cubic silicone gel material known as α-gel. Two kinds of experimental excitations are used as random frequencies and constant frequency harmonic excitations. In this study, experimental frequency response functions confirm that increasing the static preload changes the behavior of the investigated viscoelastic material and the fractional Kelvin-Voigt model loses its validity by increasing the precompression. It is shown that, for a random frequencies excitation, by transforming from the time domain to the frequency domain the mentioned parameters can be identified. Using the identified parameters, analytical frequency response functions are so close to their experimental counterparts. Also, analytically produced time histories of the test setup’s output for steady-state experimental tests are so close to the captured time histories. The mentioned results validate the procedure of identification.

Nonlinear vibrations of a cylindrical shell in a fractional viscoelastic medium with combinational internal resonances of the second order

Non-linear damped vibrations of a cylindrical shell embedded into a fractional derivative medium are investigated numerically for the case of the combinational internal resonance of the second order, using two different numerical methods with further comparison of the results obtained. The damping properties of the surrounding medium are described by the fractional derivative Kelvin-Voigt model utilizing the Riemann-Liouville fractional derivatives. A good agreement in results is declared.

Rheological characterization of thermal hydrolysed waste activated sludge

Rheological properties are important in the design and operation of sludge-handling process. Despite this, the rheology of sludge in thermal hydrolysis processes (TH) is not well described.In-situ measurements were performed to characterize the flow behaviour of various concentrations (7–13 wt%) of waste activated sludge (WAS) at TH conditions. Equations were presented for predicting in-situ rheological parameters (high-shear viscosity, η∞,i, consistency index, ki, and yield stress, σc,i) under various treatment conditions, which are useful for design of process units. The equations enable convenient estimation of in-situ properties based on ambient rheological measurements. Results suggested that the proportion of sludge solubilization and its rate were unaffected by varying sludge concentration.Thermally treated sludge still exhibited gel-like, viscoelastic characteristics similar to untreated sludge; however, the storage (G′) and loss (G”) moduli decreased with higher treatment temperatures. Frequency and creep responses were described by a fractional derivatives Kelvin-Voigt (FKV) model, which showed increasing viscous characteristics of treated sludge. These equations can be utilised in CFD models. Results obtained from oscillatory measurements can also approximate steady-shear behaviour by comparing dynamic viscosity, η′(ω), and steady-shear viscosity, η(γ̇), whose values were very similar. This enables convenient estimation of steady-shear behaviour of sludge from oscillatory measurements, which is found to be a non-destructive technique for measuring flow behaviour of highly concentrated sludge. Yield stress can also be predicted from the product of modified Cox-Merz shift factors and storage modulus (G’). Practical engineering implications of the rheological observations were discussed.

Impact response of a nonlinear viscoelastic auxetic plate

Impact induced geometrically non-linear vibrations of a viscoelastic plate are investigated for the case when the shear operator is governed by the fractional derivative Kelvin-Voigt model in conjunction with the time-independent coefficient of volume extension-compression. Such a model could describe the behavior of so-called auxetic materials with negative Poisson’s ratios. The modified method of multiple time scales involving the expansion of the fractional derivative in terms of a small parameter has been utilized for solving nonlinear governing equations of motion. The asymptotic behavior of the roots of the characteristic equation for determining the frequencies of the system as a function of the retardation time is studied and three modes of vibration in the process of impact interaction are described.

Numerical analysis of nonlinear forced vibrations of a cylindrical shell with combinational internal resonance in a fractional viscoelastic medium

Nonlinear force driven vibrations of thin cylindrical shells in a viscoelastic medium, the damping properties of which are described by the fractional derivative Kelvin-Voigt model, are studied. A comparative analysis has been carried out for numerical results of both free and forced shell’s oscillations using two different numerical methods for the case of the combination internal resonance.

NUMERICAL ANALYSIS OF NON-LINEAR VIBRATIONS OF A FRACTIONALLY DAMPED CYLINDRICAL SHELL UNDER THE ADDITIVE COMBINATIONAL INTERNAL RESONANCE

Non-linear damped vibrations of a cylindrical shell subjected to the additive type combinational internal resonance are investigated numerically using two different numerical methods. The damping features of the surrounding medium are described by the fractional derivative Kelvin-Voigt model involving the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential are estimated using numerical solution of nonlinear multi-term fractional differential equations by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the additive combinational internal resonance. A good agreement in results is declared

Transient vibrations of a fractional Kelvin-Voigt viscoelastic cantilever beam with a tip mass and subjected to a base excitation

In the paper, transient vibrations of a Bernoulli-Euler cantilever beam with a rigid mass attached at the end and subjected to base motion are presented. The tip mass centre of gravity coincides with the endpoint of the beam. The viscoelastic properties of the beam material are described using a fractional Kelvin-Voigt model. The Riemann–Liouville fractional derivative of an order of 0 < γ ≤ 1 is used. Exact relationships for the natural frequencies and mode shapes of the beam are derived. Moreover, a method of calculating the damped natural frequencies of the analyzed beam is presented. The forced-vibration solution of the beam is derived using the mode superposition method. Transient movement of the base is described by an oscillating function with a linearly time-varying frequency. A convolution integral of the fractional Green's and forcing functions is used to achieve the beam response. The Green's function is formulated by two terms. The first term describes damped vibrations around the drifting equilibrium position, while the second describes the drift of the equilibrium position. The dynamic responses are calculated numerically. The impact of a fractional order derivative on transient responses when passing through resonance regions of the analyzed beam is shown. The beam responses obtained using a fractional viscoelastic material model are compared with those obtained using an integer viscoelastic material model. In the system analyzed, the influence of the term describing the drift of the equilibrium position on the beam deflection is relatively low and may be neglected. The results obtained indicate that an increase in the derivative order causes a decrease in the vibration amplitudes of the beam. The calculations reveal that the vibration amplitudes in the region of the second resonance are larger than in the region of the first resonance for some parameters of the system under consideration, which is not the case when using integer order derivatives in the viscoelastic material model.

Concentration-independent mechanics and structure of hagfish slime

The defense mechanism of hagfish slime is remarkable considering that hagfish cannot control the concentration of the resulting gel directly; they simply exude a concentrated material into a comparably “infinite” sea of water to form a dilute, sticky, cohesive elastic gel. This raises questions about the robustness of gel formation and rheological properties across a range of concentrations, which we study here for the first time. Across a nearly 100-fold change in concentration, we discover that the gel has similar viscoelastic time-dependent properties with constant power-law exponent (α=0.18±0.01), constant relative damping tanδ=G′′/G′≈0.2–0.3, and varying overall stiffness that scales linearly with the concentration (∼c0.99±0.05). The power-law viscoelasticity (fit by a fractional Kelvin-Voigt model) is persistent at all concentrations with nearly constant fractal dimension. This is unlike other materials and suggests that the underlying material structure of slime remains self-similar irrespective of concentration. This interpretation is consistent with our microscopy studies of the fiber network. We derive a structure-rheology model to test the hypothesis that the origins of ultra-soft elasticity are based on bending of the fibers. The model predictions show an excellent agreement with the experiments. Our findings illustrate the unusual and robust properties of slime which may be vital in its physiological use and provide inspiration for the design of new engineered materials. Statement of SignificanceHagfish produce a unique gel-like material to defend themselves against predator attacks. The successful use of the defense gel is remarkable considering that hagfish cannot control the concentration of the resulting gel directly; they simply exude a small quantity of biomaterial which then expands by a factor of 10,000 (by volume) into an “infinite” sea of water. This raises questions about the robustness of gel formation and properties across a range of concentrations. This study provides the first ever understanding of the mechanics of hagfish slime over a very wide range of concentration. We discover that some viscoelastic properties of slime are remarkably constant regardless of its concentration. Such a characteristic is uncommon in most known materials.

Parameters identification method for viscoelastic dielectric elastomer actuator materials using fractional derivatives

Since dielectric elastomer actuators are commonly used as artificial muscles, different approaches in material parameters identification have been used. Most dielectric actuators are parametrized with the help of continuum mechanics. Alternatively, rheological models can be used. Unfortunately, in basic rheological models, frequency dependence of viscoelastic materials cannot be obtained with a single equation. In order to obtain frequency dependences of viscoelastic material, fractional Kelvin-Voigt model is used. Fractional Kelvin-Voigt model can be applied for parameters identification in case of dynamical or cyclical excitation with different frequencies and for creep and stress relaxation analysis. Basic Kelvin-Voigt is limited on specific frequency and on creep. It cannot provide stress relaxation. Fractional Kelvin-Voigt model can further be used for fractional control. All previous work dealing with this subject is focused on analysis of material properties. Our contribution and novelty is in preparing governing equations for development control algorithms for dielectric elastomer actuators.

Computational Analysis of the New Approach for Studying Non-Linear Dynamics Response of a Thin Fractionally Damped Cylindrical Shell With Combinational Internal Resonance of the Second Order

Non-linear Non-linear damped vibrations of a cylindrical shell embedded into a fractional derivative medium are investigated for the case of the combinational internal resonance of the second order numerically, using two different numerical methods with further comparison of the results obtained. The damping properties of the surrounding medium are described by the fractional derivative Kelvin-Voigt model utilizing the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential are estimated using numerical solution of nonlinear multi-term fractional differential equations by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the combinational internal resonance of second order. A good agreement in results is declared.

Computing the linear viscoelastic properties of soft gels using an optimally windowed chirp protocol

We use molecular dynamics simulations to investigate the linear viscoelastic response of a model three-dimensional particulate gel. The numerical simulations are combined with a novel test protocol (the optimally windowed chirp or OWCh), in which a continuous exponentially varying frequency sweep windowed by a tapered cosine function is applied. The mechanical response of the gel is then analyzed in the Fourier domain. We show that (i) OWCh leads to an accurate computation of the full frequency spectrum at a rate significantly faster than with the traditional discrete frequency sweeps, and with a reasonably high signal-to-noise ratio, and (ii) the bulk viscoelastic response of the microscopic model can be described in terms of a simple mesoscopic constitutive model. The simulated gel response is in fact well described by a mechanical model corresponding to a fractional Kelvin-Voigt model with a single Scott-Blair (or springpot) element and a spring in parallel. By varying the viscous damping and the particle mass used in the microscopic simulations over a wide range of values, we demonstrate the existence of a single master curve for the frequency dependence of the viscoelastic response of the gel that is fully predicted by the constitutive model. By developing a fast and robust protocol for evaluating the linear viscoelastic spectrum of these soft solids, we open the path toward novel multiscale insight into the rheological response for such complex materials.

Power law attenuation of shear waves in fractal media

There are two mechanisms for attenuation of mechanical waves: absorption and multiple scattering. We explore the second mechanism for shear waves in the context of MR elastography. The theory for attenuation was first given in the seismic field (O’Doherty & Anstey, Geophys. Prosp. 1971). Later phase was included, and the mean field concept was applied (Banik et al., Geophys. 1985). Then, a more rigorous theory which also deals with random fractals was developed (Solna, SIAM J ApplMath, 2003; Garnier & Solna, Multiscale Model Simul, 2009). We have developed this theory further by allowing fractality over a limited length scale and also shown that it gives results that are indistinguishable from those of power law absorption models like the fractional Kelvin-Voigt model (Holm & Sinkus, JASA, 2010). This makes it a challenge to understand whether attenuation in a real medium is due to one or the other mechanism or both. Measurement of dispersion has been done for polystyrene microspheres embedded in agarose which is nearly lossless up to 1000 Hz. The phase velocity has a power law variation with exponent up to d = 0.24 (Lambert et al., PRL 2015). According to the theory this is consistent with a fractional dimension which is d less than the Euclidean dimension.

Rheological properties of kuzu starch pastes with galactomannans.

The paper describes the effects of galactomannans on viscoelastic properties of commercial Japanese white kuzu starch pastes. The study included morphological, thermal and rheological analyses of the biopolymer. The results obtained in the form of storage modulus G'(ω) and loss modulus G″(ω) were described by the modified fractional Kelvin-Voigt model with two springpot-type elements, created on the basis of differential calculus of fractional order and Fourier transform. It allowed to determine 17 material parameters providing a lot of additional information about structure and viscoelastic properties of the biopolymer in comparison to the classical analysis of oscillatory and creep tests. The study led to the conclusion that commercial Japanese white kuzu starch was so-called type II starch with a high pasting temperature of 75°C and an average granule diameter equal to 10.9μm. Rheological properties of the pastes depended on the galactose-to-mannose ratio in galactomannan molecule. The larger substitution degree, the higher viscosity, characteristic relaxation times, polydispersity index, gel stiffness, and the lower cross-linking density and average molecular weights. The presence of galactose side groups favored the hydration and immobilization of water molecules.

Numerical analysis of non-linear vibrations of a fractionally damped cylindrical shell under the conditions of combinational internal resonance

Non-linear damped vibrations of a cylindrical shell embedded into a fractional derivative medium are investigated for the case of the combinational internal resonance, resulting in modal interaction, using two different numerical methods with further comparison of the results obtained. The damping properties of the surrounding medium are described by the fractional derivative Kelvin-Voigt model utilizing the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential equations of the second order are estimated using numerical solution of nonlinear multi-term fractional differential equations by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the combinational internal resonance. A good agreement in results is declared.

The fractional Kelvin-Voigt model for Rayleigh surface waves in viscoelastic FGM infinite half space

In this paper, we use the fractional Kelvin-Voigt model to investigate the propagation behavior of Rayleigh waves along the surface of viscoelastic functionally graded material (FGM) half space. Fractional derivatives combing with the power series technique are for the first time employed for the derivations of the governing equations. The influences of the viscous parameter, integer order, and fractional order derivative variations on Rayleigh wave dispersion relations are discussed. Numerical results show that the viscous parameter has little effect on the dispersion relations and attenuation curves of the Rayleigh waves in infinite half space. Furthermore, the rule regarding the trend of attenuation curves for various types of modes are studied when the Rayleigh wave is in normal dispersion. In general, the attenuation in the first mode is very small, so we can omit it. Moreover, the second mode is significantly different from other higher order modes in numerical value and rule of change. These results may provide theoretical guidance for the fabrication of new materials and the nondestructive evaluation in engineering applications, and might also serve to explain some phenomena surrounding seismic wave propagation.

Comparison between classical Kelvin-Voigt and fractional derivative Kelvin-Voigt models in prediction of linear viscoelastic behaviour of waste activated sludge

Appropriate sewage sludge rheological models are essential for computational fluid dynamic simulation of wastewater treatment processes, in particular aerobic and anaerobic digestions. The liquid-like behaviour of sludge is well documented but the solid-like behaviour remains poorly described despite its importance for dead-zone formation. In this study, classical Kelvin-Voigt model, commonly used for sludge in literature, were compared with fractional derivative Kelvin-Voigt model regarding their predictive ability for describing the solid-like behaviour.Results showed that the fractional Kelvin-Voigt model best fitted the experimental data obtained from creep and frequency sweep tests. Whereas, classical Kelvin-Voigt could not fit the frequency sweep data as this model is not a function of angular velocity. Also, the Kelvin-Voigt model was unable to predict the creep data at low stresses.

Experimental investigation into temperature- and frequency-dependent dynamic properties of high-speed rail pads

In order to investigate the temperature- and frequency-dependent dynamic properties of high-speed rail pads excited by rail vehicle’s large-amplitude quasi-static loads due to bogie interval and wheelset spacing, the soft pads from WJ-7, WJ-8 and Vossloh 300 fasteners frequently used in China’s high-speed railway track systems are taken as test subjects. At first, a series of dynamic tests of the three high-speed rail pads at a specific frequency of 0.3Hz and under various low temperatures ranging from −60 to 20°C are carried out. Then, the Time-Temperature Superposition (TTS) and Williams-Landel-Ferry (WLF) formula are used to predict their dynamic performance at frequencies higher than the test frequency, and the Fractional Derivative Kelvin-Voigt (FDKV) model is proposed to represent their frequency-dependent dynamic properties. Finally, the frequency-dependent dynamic performance of the three high-speed rail pads under rail vehicle’s small-amplitude dynamic loads due to the surface roughness of rail and wheel is discussed. It is found that the storage stiffness of WJ-7, WJ-8 and Vossloh 300 rail pads is sensitive to low temperatures, as well as much larger than their corresponding loss stiffness. The glass transformation temperatures of WJ-7, WJ-8 and Vossloh 300 rail pads are −40, −45 and −45°C, respectively. The FDKV model can basically give a satisfactory fitting with the experimental results within 100Hz, especially at ordinary temperatures. The dynamic properties of rail pads excited by rail vehicle’s small-amplitude dynamic loads are time-variable and related to rail vehicle’s large-amplitude quasi-static load values on rail pads. Thus, it is very important that the temperature- and frequency-dependent dynamic parameters of the polymer materials in vehicle-track systems must be measured according to the specific operation conditions, since otherwise an accurate prediction of the broadband train-induced vibration and noise at different temperatures is difficult to achieve.

To the question on the correctness of fractional derivative models in dynamic problems of viscoelastic bodies

The question is discussed on the possibility of utilization of such simplest fractional derivative viscoelastic models as Kelvin-Voigt and standard linear solid models for the description of the dynamic response of viscoelastic bodies, namely: beams and plates. It has been shown that the adoption of the Kelvin-Voigt model with the fractional derivative in conjunction with the time-independent Poisson’s ratio for the description of the dynamic behavior of a viscoelastic Kirchhoff-Love plate results in the fact that the given problem is reduced to the equivalent problem on dynamic response of an elastic Kirchhoff-Love plate in a viscoelastic medium with fractional damping. If Young’s operator is preassigned via the fractional derivative Kelvin-Voigt model in conjunction with the time-independent coefficient of volume extension-compression for solving the same problem, what seems to be more logical from the point of view of experimental data, then such a model tunes out to be inapplicable at all, while if the shear operator is governed by the fractional derivative Kelvin-Voigt model, then such a model could describe the behavior of so-called auxetic materials with negative Poisson’s ratios. As for the fractional derivative standard linear solid model, then it remains to be correct both with and without considering the volume relaxation.