Fractional Kelvin-Voigt Model Research Articles

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.

Comprehensive Characterization of the Viscoelastic Properties of Bovine Submaxillary Mucin (BSM) Hydrogels and the Effect of Additives.

This study presents a comprehensive characterization of the viscoelastic and structural properties of bovine submaxillary mucin (BSM), which is widely used as a commercial source to conduct mucus-related research. We conducted concentration studies of BSM and examined the effects of various additives, NaCl, CaCl2, MgCl2, lysozyme, and DNA, on its rheological behavior. A notable connection between BSM concentration and viscoelastic properties was observed, particularly under varying ionic conditions. The rheological spectra could be well described by a fractional Kelvin-Voigt model with a minimum of model parameters. A detailed proteomics analysis provided insight into the protein, especially mucin composition within BSM, showing MUC19 as the main component. Cryo-scanning electron microscopy enabled the visualization of the porous BSM network structure. These investigations give us a more profound comprehension of the BSM properties, especially those pertaining to viscoelasticity, and how they are influenced by concentration and environmental conditions, aspects relevant to the field of mucus research.

Thermoviscoelastic response of skin tissue via fractional Pennes bioheat and fractional viscoelasticity

In this paper, the thermoviscoelastic response of skin tissue, as a typical class of viscoelastic biological tissue, subject to thermal shock is studied based on the fractional Kelvin-Voigt model and fractional Pennes bioheat conduction. A time-fractional biothermoviscoelastic governing equation is derived and initial-boundary value conditions are given for monolayer skin tissue of finite thickness. The temperature change and thermal stress of viscoelastic skin tissue are obtained by the Laplace transform and its inversion using matlab software. The influences of viscosity coefficient, relaxation times, fractional-order viscosity/heat parameter, and blood perfusion rate of skin tissue on the temperature distribution and the elastic displacement are analyzed. The numerical results show that fractional-order viscosity does not affect temperature, whereas fractional-order heat flow does. Moreover, fractional-order heat flow affects the sensitivity of blood perfusion rate to the thermodynamic response.

Pea and soy protein isolate fractal gels: The role of protein composition, structure and solubility on their gelation behaviour

The gelation behaviour of two different pea protein isolates and one soy protein isolate were investigated with a focus on the role of the protein properties. Protein solubility was the lowest in pH 3 citrate-phosphate buffer (<10% w/w), increased in pH 7.4 phosphate-buffered saline (12–21% w/w), and was the highest in pH 7.6 MilliQ water (∼20–40% w/w). Heat-induced gelation conditions for the protein sources were sensitive to both the soluble and the insoluble fractions as obtained during extraction. At low protein concentrations (≤5% w/v), the proteins started to lose their viscoelastic behaviour and exhibited predominantly viscous properties. Fitting of the fractional Kelvin-Voigt model to the frequency sweeps showed an increase in the fractal gel strength with increasing protein concentration. Secondary structures of the soluble species showed mostly unordered proteins, suggesting that the proteins were denatured during the commercial extraction process although gelation has to date been suggested to be highly dependent on the denaturation of soluble proteins. Synchrotron Radiation Circular Dichroism measurements of the insoluble proteins showed a significant amount of ordered protein structures. SEM imaging of the gels also suggested a new gelation pathway in which insoluble proteins act as dispersed fillers within a continuous matrix of soluble proteins. This research elucidated the role of different protein fractions, globulins and albumins, and their secondary structure in the formation of a gel network and how this affects their viscoelastic behaviour.

Chaotic dynamics of fractional viscoelastic PET membranes subjected to combined harmonic and variable axial loads

Nonlinear chaotic vibrations of fractional viscoelastic PET (polyethylene terephthalate) membranes subjected to combined harmonic and variable axial loads is investigated in this paper. Axial tension variations arise from the machine disturbances of the processing line of roll-to-roll manufacturing. The viscoelasticity of PET membrane is characterized by the fractional Kelvin-Voigt model. Based on the Hamilton principle, the equation of motion of the membrane is established with the consideration of geometric nonlinearity, and the Galerkin procedure is employed to discretize the resulting governing equation. For the solution, the finite difference method is utilized in conjunction with the Caputo-type fractional derivative to reliably estimate the nonlinear response of fractional viscoelastic PET membrane. The reliability of this numerical strategy is proved by the available results of the fractional system and comparison examples. The influence of system parameters on chaotic behaviors is described by the bifurcation diagram and the detailed responses at the set bifurcation parameters. The fractional model together with the analysis provides a fundamental framework for the control of viscoelastic substrates in flexible manufacturing.

Consolidation of multilayered soil with fractional derivative viscoelasticity due to surface loading and internal pumping

Classical Terzaghi’s solution provides a useful tool to predict the one-dimensional consolidation of homogeneous elastic soil layer with fully drained/undrained boundary condition under instantaneous loading. Due to the sedimentation process and complex internal structure, natural soils usually exhibit multilayered inhomogeneity and viscoelastic behavior. In practical cases, the drainage capacity of the boundaries can change with time. Thus, these practical factors should be well in considered for accurate prediction of the consolidation. This paper extends the classical Terzaghi’s solution to multilayered viscoelastic soil with continuous drainage boundaries and subjected to time dependent loading. The fractional derivative Kelvin-Voigt model is used to describe the viscoelasticity of soil. Two loading cases namely surface loading and internal pumping are considered. The consolidation problem is solved by Laplace transform technique and a transfer matrix formulation. Analytical solutions for excess pore water pressure, effective stress and settlement are given. The present solution is general and can reduce to existing solutions available in the literatures. Numerical studies are conducted to investigate the effects of viscoelastic parameters, boundary drainage capacity and loading rate on the consolidation behavior. It is shown that the multilayered soil system with large viscosity coefficient and small fractional order can consolidate fast. For the consolidation due to internal pumping, the drainage parameters have no influence on the consolidation process.

Orthotropic plates resting on viscoelastic foundations with a fractional derivative Kelvin-Voigt model

The objective of this paper is to propose an analytical model for orthotropic plates resting on viscoelastic foundations using a fractional derivative Kelvin-Voigt model. The viscoelastic foundation offers the advantage of capturing time-dependent behavior. The incorporation of the fractional derivative enhances the accuracy of the model in predicting the viscoelastic response. The behavior of plate is described using classical plate theory, while the viscoelastic foundation is simulated by modifying the Kelvin-Voigt model. The governing equation is established by utilizing the principle of virtual work. The analytical solution of the simply supported rectangular plate is obtained applying Navier’s approach and the Mittag-Leffler function. Subsequently, the influences of the parameters involved in this model on the bending responses are thoroughly studied. The results reveal the following key findings: (1) the order of fractional derivative determines the characteristics of foundation, specially whether it exhibits elastic or viscoelastic behaviors; (2) the creep time controls both the rate of the incremental deflection and the position of characteristic point; and (3) the stiffnesses of the spring and springpot exert a dominant influence on the bending responses. Furthermore, numerical results calculated from the proposed analytical solution will serve as a benchmark for future research on numerical validation.

Effect of the photothermal Moore–Gibson–Thomson model on a rotating viscoelastic continuum body with a cylindrical hole due to the fractional Kelvin-Voigt model

Effect of the photothermal Moore–Gibson–Thomson model on a rotating viscoelastic continuum body with a cylindrical hole due to the fractional Kelvin-Voigt model

Dynamic response of fractionally damped two-layered viscoelastic plate

The vibration response of the plates attached with the free viscoelastic damping layer is investigated to predict the performance of free surface damping treatment. The frequency-dependent dynamic properties of the viscoelastic layer are expressed by fractional Kelvin-Voigt model. A simple and efficient identification approach based on the gradient-based optimization principle is employed to evaluate the viscoelastic model parameters from the experimental data. Both classical and fractional Kelvin-Voigt models are employed to describe the viscoelasticity, and the corresponding model parameters are identified. Further, the application of these identified parameters in the modeling and analyzing of real viscoelastic composite structures is demonstrated through quantitative examples. Step load-induced transient responses of the viscoelastic structures modeled using the classical and fractional model parameters are investigated and compared, to demonstrate the consequences of unknowingly using the classical Kelvin-Voigt model. The effects of the viscoelastic layer thickness on the transient response of the two-layered plate are also studied.

Dynamic viscoelastic model for rock joints under compressive loading

The mechanical properties of rock masses are significantly affected by the deformated behaviors of discontinuities, such as joints and faults. Rock masses in nature are also often subjected to dynamic disturbances, including explosive and seismic loads. Therefore, it is crucial to understand the deformed response of rock joints under dynamic loads. In this study, a series of dynamic compression expriments are carried out on rock joints with different joint roughness using the SHPB equipment with digital image correlation (DIC) technique. It found the dynamic compressive bebaviors of rock joints have viscoelstic features. Then a variable-order fractional Kelvin-Voigt (VFKV) model is proposed to simulate this behaviour, which consists of a variable-order fractional viscous dashpot and a spring in parallel. The model is verified to effectively represent the dynamic compressive process of rock joints. Finally, based the proposed model, the influence of model parameters, loading rate, and joint contact area ratio on the viscoelastic behaviour and mechanical property of rock joints are discussed. The results show the viscoelasticity behaviour of joints is significantly affected by the loading rate and joint contact area ratio, high loading rate and joint contact area may strengthen the elastic properties of the joints, while the joint roughness has the opposite effect.

Fractional viscoelastic models with novel variable and constant order fractional derivative operators

This paper deals with the application of a novel variable-order and constant-order fractional derivative without singular kernel of Atangana-Koca type to describe the fractional viscoelastic models, namely, fractional Maxwell model, fractional Kelvin-Voigt model, fractional Zener model and fractional Poynting-Thomson model. For each fractional viscoelastic model, the stress relaxation modulus and creep compliance are derived analytically under the variable-order and constant-order fractional derivative without singular kernel. Our results show that the relaxation modulus and creep compliance exhibit viscoelastic behaviors producing temporal fractality at different scales. For each viscoelastic model, the stress relaxation modulus and creep compliance are derived analytically under novel variable-order and constant-order fractional derivative with no singular kernel.

Experimental workflow to estimate model parameters for evaluating long term viscoelastic response of CO2 storage caprocks

Understanding the time-dependent behavior of reservoir and sealing formations for geologic carbon storage is critical to assessing geomechanical risks since time-dependent deformation strongly influences the mechanical response of some rock types. Many studies have evaluated the risk of CO2 leakage and induced seismicity by assuming poroelastic rheology in sealing formations. Few have considered viscoelastic or other time-dependent responses, and much of the literature on the long-term mechanical behavior of rocks examines only a 1D uniaxial response. This is primarily because, to date, the general form of a reasonable 3D time-dependent model for rocks remains unclear. In this paper, we propose an approach to address this by using a new workflow to estimate constitutive modeling parameters for the evaluation of a 3D viscoelastic model. The proposed approach uses a 1D power-law response to extrapolate several-hour-long experimental data to the decades-long time frames of interest in geologic carbon storage. Experimental data were obtained by conducting multi-level loading/unloading triaxial relaxation tests with four different rock types. The experimental results showed that the maximum load relaxation observed is approximately 49%, with some rock types considered showing as little as 1.4%. Using the proposed workflow, two linear 3D viscoelastic models, i.e., generalized Maxwell (GM) model and fractional Kelvin-Voigt (FKV) model, were evaluated and their model parameters were chosen with the extrapolated 30-year data such that a maximum deviation from the assumed power-law response for these two 3D models was 2 MPa in axial stress and 7 MPa in radial stress. We provided reasonable ranges for the model parameters to be later used for 3D modeling of rock time-dependent responses. Our results also showed that when GM is selected to analyze the rocks considered here, the relaxation time has a general range of 1–1010 s. This time scale captures the time-dependent behavior as long as centuries, which is much longer than a 10–30 years-long time frame envisioned for CO2 injection projects.

Improved indirect measurement of the dynamic stiffness of a rail fastener and its dependence on load and frequency

The dynamic stiffness of rail fasteners has a significant effect on the noise radiated by the rails as well as the vibration transmitted to the underlying structures, such as bridges, tunnels or track at grade. This study investigates the load- and frequency-dependence of a WJ-2A fastener which is commonly used in urban rail transit systems in China. This is a two-stage fastener with a rail pad and a baseplate pad separated by a steel plate. Results are obtained using the indirect measurement method in the frequency range 30 to 1000 Hz. The dynamic stiffness of the individual components is investigated as well as that of the whole fastener system and the combined stiffness of the components is verified by comparison with the whole fastener system. A numerical model of the test rig is used to provide understanding of various artefacts that are observed in the measurements and corrections are proposed to minimise their effects on the measured results. These allow for the differences between the response at the measurement positions and at the ideal positions at the interfaces between the fastener system and the rig. The stiffness magnitude of the rail pad, baseplate pad, and whole fastener system increases strongly with increasing static preload and increases weakly with increasing frequency; it is important to take these effects into account in prediction models for noise and vibration. The damping loss factors are not strongly dependent on preload or frequency. To describe the frequency-dependence, a fractional derivative Kelvin-Voigt (FDKV) model is introduced and is fitted to the dynamic stiffness with the help of a genetic algorithm method. The dynamic stiffness of the whole fastener system is influenced by both the rail pad and the lower baseplate pad. It is important to take account of both of them, as well as the internal resonance of the baseplate assembly which appears at around 1 kHz.

Analysis model of restoring force of a rubber air spring

Not only has the dynamic stiffness of a rubber air spring been inherited by effects of the compressed air, but it has also been affected by hysteresis behaviors caused by the friction, viscoelasticity of bellow material. Hence, this paper will analyze comprehensively the stiffness model of a commercial rubber air spring. One of the first works is to predict the structure parameters including effective area, volume and their change rate. Then, the restoring force generated by compressed air will be analyzed and built through the theory of thermodynamics. The hysteresis model of the rubber bellow will be obtained based on the Berg’s frictional model connecting in parallel with fractional Kelvin-Voigt model. Next, an experimental apparatus is set up to identify the parameters of this model as well as evaluate the proposed restoring force model of the rubber air spring. The study results show that the analysis model of the rubber air spring matches well the measured data. This work will offer a helpful insight in the design of the vibration isolation system using rubber air springs as elastic elements.

Effective Acoustic Equations for a Layered Material Described by the Fractional Kelvin-Voigt Model

The paper is devoted to the construction of effective acoustic equations for a two-phase layered viscoelastic material described by the Kelvin–Voigt model with fractional time derivatives. For this purpose, the theory of two-scale convergence and the Laplace transform with respect to time are used. It is shown that the effective equations are partial integro-differential equations with fractional time derivatives and fractional exponential convolution kernels. In order to find the coefficients and the convolution kernels of these equations, several auxiliary cell problems are formulated and solved

Position Control of the Dielectric Elastomer Actuator Based on Fractional Derivatives in Modelling and Control

Successful control of a dielectric elastomer actuator (DEA) can be a challenging task, especially if no overshoot is desired. The work presents the first use of the PIλDμ control for a dielectric elastomer actuator to eliminate the overshoot. The mathematical model of the dielectric elastomer was established using the fractional Kelvin-Voigt model. Step responses are first tested in the Laplace domain, which gave the most satisfactory results. However, they did not represent the real model. It cannot have negative force acting on the dielectric elastomer actuator. Simulations in Matlab/Simulink were performed to obtain more realistic responses, where output of the PIλDμ controller was limited. Initial parameters for a PID control were obtained by the Wang–Juang–Chan algorithm for the first order plus death time function approximation to the step response of the model, and reused as the basis for the PIλDμ actuator control. A quasi-anti-windup method was introduced to the final control algorithm. Step responses of the PID and the PIλDμ in different domains were verified by simulation and validated by experiments. Experiments proved that the fractional calculus PIλDμ step responses exceeded performance of the basic PID controller for DEA in terms of response time, settling time, and overshoot elimination.

A method for the experimental identification of equivalent viscoelastic models from vibration of thin plates

Aim of the present study is the identification of equivalent viscoelastic models for layered thin walled structures, obtained from vibration measurement only, able to fit the experimental data on a relatively wide frequency range by means of a minimum number of parameters. A novel approach is proposed, based on a definition of an equivalent modal damping ratio applied to the circle-fit technique, to overcome the difficulties related to the identification of modal parameters when adopting non-conventional viscoelastic models. When the structural internal dissipative effects are dominant, this procedure identifies the parameters of an equivalent Young’s modulus in the frequency domain, representing the viscoelastic properties of a homogenized structure as a scalar function with frequency-dependent real and imaginary parts. The proposed procedure is applied to the analysis of Aluminum plates coated by damping pads and of plates made by Quiet Aluminum. To fit the experimentally found equivalent modal damping ratios, several viscoelastic models are adopted and compared (viscous, hysteretic, generalized Maxwell, fractional derivative damping, and in particular the Fractional Kelvin-Voigt model), assessing the accuracy of the identified parameters by comparison of numerically simulated with experimentally measured frequency response functions.

Internal resonances in nonlinear fractionally damped plates of the Uflyand-Mindlin type

In the present paper, the non-linear free vibrations of fractionally damped plates are investigated for the case when its equations of motion involve the rotary inertia and shear deformations via five coupled nonlinear differential equations in terms of three mutually orthogonal displacements and two angles of rotation. The damping features of the surrounding medium are described by the fractional detivative Kelvin-Voigt model. The procedure resulting in decoupling linear parts of governing equations has been proposed, what allows one to utulyze effectively the fractional derivative expansion method for solving the non-linear equations of motion. The occurrence of the internal or combinational resonances has been revealed and classified for plates of the Uflyand-Mindlin type.

A refined stiffness model of rolling lobe air spring with structural parameters and the stiffness characteristics of rubber bellows

An accurate model of rolling lobe air spring (RLAS) is the key to achieve the design, matching and optimization of air suspension. A refined stiffness model of RLAS with structural parameters and the stiffness characteristics of rubber bellows is developed in this paper. Prediction models of structural parameters, including effective area and its change rate, effective volume and its change rate, are obtained by geometrical and mechanical analysis. A nonlinear hysteresis model of rubber bellows is also proposed, which is composed of a fractional Kelvin-Voigt model and a smooth friction model in parallel. The structural parameters were identified by the experiments, including static stiffness and dynamic stiffness tests using the MTS370. The conclusions suggest that amplitude dependency and frequency dependency of RLAS can be completely characterized by the refined stiffness model. The refined stiffness model provides theoretical support for mechanical property match of RLAS.

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.