Derivative Kelvin Research Articles

Mathematical Modelling of Viscoelastic Media Without Bulk Relaxation Via Fractional Calculus Approach

In the present paper, several viscoelastic models are studied for cases when time-dependent viscoelastic operators of Lamé’s parameters are represented in terms of the fractional derivative Kelvin–Voigt, Scott-Blair, Maxwell, and standard linear solid models. This is practically important since precisely these parameters define the velocities of longitudinal and transverse waves propagating in three-dimensional media. Using the algebra of dimensionless Rabotnov’s fractional exponential functions, time-dependent operators for Poisson’s ratios have been obtained and analysed. It is shown that materials described by some of such models are viscoelastic auxetics because the Poisson’s ratios of such materials are time-dependent operators which could take on both positive and negative magnitudes.

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.

Damping characteristics of the architectural coated fabric and its influence on the vibration response of membrane structures

The influence of material damping is ignored in the current membrane structure design. Based on the damping characteristics of coated fabric obtained by the dynamic mechanical analyzer, the applicability of classical Maxwell model, generalized Maxwell model, three-parameter liquid model, fractional derivative Maxwell model, and fractional derivative Kelvin model to describe the viscoelastic properties of coated fabric were discussed. The free-falling body excitation vibration test of plane tensioned membrane structure is carried out and the dynamic response of plane membrane structure under impact load was studied. The reasonable viscoelastic damping constitutive model is obtained through a systematic comparison of the existing models then implemented in Abaqus software and subsequently validated with the data obtained from the vibration test. Taking saddle-shaped and umbrella-shaped membrane structures as research objectives, the dynamic response of membrane structures under dynamic wind load considering material damping performance is discussed. The results show that the damping property of the coated fabric significantly affects the dynamic characteristics of the membrane structure. Due to the existence of material damping, the modal frequency of the overall structure is greater than the result of the elastic model. The reduction factor of maximum vertical displacement can reach 67.14%, and the reduction factor of maximum principal stress can reach 68.93%, which makes the structural design safety factor high.

Thermo-viscoelastic properties in a non-simple three-dimensional material based on fractional derivative Kelvin–Voigt model

This paper provides a connection between the theories of thermoelasticity with fractional order, two-temperature, Kelvin–Voigt viscoelasticity and heat conduction with three-phase lags. The proposed model is used to study the thermoelastic vibrations in a homogeneous isotropic three-dimensional solid whose surface is traction free and subject to a time-dependent thermal loading. The problem has been addressed using Laplace and double Fourier transform methods to remove the time variable as well as two space variables. To obtain the numerical results of the investigated fields in the physical domain, we applied a numerical inverse method. We have also concluded some special cases of interest from this work. Furthermore, the results are displayed graphically to illustrate the influence of viscosity, phase lags, two-temperature and fractional-order parameters on all physical fields.

Mixed convolved action for the fractional derivative Kelvin–Voigt model

Based upon the concept of mixed convolved action, a true variational statement for a fractional derivative Kelvin–Voigt model is presented. In this formulation, a single functional is defined as a series of convolution integrals, where the stationarity of this functional leads to all the governing differential equations as well as pertinent initial conditions. Thus, the entire description of a fractional-derivative Kelvin–Voigt model is encapsulated within this framework. This new formulation provides an elegant basis for a development of effective numerical methods involving finite element representation over a temporal domain. Here, the simplest temporal finite element approach is developed, and some computational examples are provided to validate this proposed approach.

Circular arc rules of complex plane plot for model parameters determination of viscoelastic material

We present a new approach to determine the rheological parameters of a mechanical model of viscoelastic materials. The fractional derivative solid model composed of a spring in series with a fractional derivative Kelvin–Voigt element has been employed to characterize the dynamic mechanical response of a real viscoelastic material. In dynamic mechanical analysis (DMA) measurements, the frequency-dependent loss modulus is plotted against the storage modulus in the form of complex plane plot. We find that the complex plane plot of the fractional derivative solid model is a depressed or distorted semicircle with its center below the real axis. The model parameters could be identified graphically via its complex plane curve, from which the spring constants can be obtained through the two intercepts of the extrapolated circular arc with the real axis, whereas the order of fractional dashpot can be estimated by the displacement of the semicircle center. The graphical method allows us to easily find rheological parameters, without using complicated calculation and special algorithms.

Fractional Order Derivative Kelvin Model of Cumulative Pore Pressure under Cyclic Loading

The settlement caused by traffic loading on soft ground has serious effects on subways, tunnels and highway bridges. Therefore, it is important to study the long-term settlement caused by traffic loading. A series of undrained cyclic triaxial tests was conducted on saturated soft clay that involved different initial effective confining pressures, dynamic stress ratios and consolidation modes. The relationship between cumulative pore pressure and cyclic times was obtained. The general Kelvin model and the fractional-order derivative Kelvin model were used to simulate this relationship. Simultaneously, the genetic algorithm was used to optimize the parameters of these two models. When comparing the calculation and test values, it is verified that the general Kelvin model can simulate the cumulative pore pressure only when the dynamic stress ratio is small, and the fractional-order derivative Kelvin model is more suitable for simulating the cumulative pore pressure under cyclic loading.

Shifted-Chebyshev-polynomial-based numerical algorithm for fractional order polymer visco-elastic rotating beam

Abstract In this paper, an effective numerical algorithm is proposed for the first time to solve the fractional visco-elastic rotating beam in the time domain. On the basis of fractional derivative Kelvin–Voigt and fractional derivative element constitutive models, the two governing equations of fractional visco-elastic rotating beams are established. According to the approximation technique of shifted Chebyshev polynomials, the integer and fractional differential operator matrices of polynomials are derived. By means of the collocation method and matrix technique, the operator matrices of governing equations can be transformed into the algebraic equations. In addition, the convergence analysis is performed. In particular, unlike the existing results, we can get the displacement and the stress numerical solution of the governing equation directly in the time domain. Finally, the sensitivity of the algorithm is verified by numerical examples.

Nonlinear Dynamic Analysis of Fractional Damped Viscoelastic Beams

The nonlinear dynamical response of a simply supported viscoelastic beam subjected to transverse harmonic excitations is investigated. The constitutive law of the viscoelastic beam is modeled in the fractional derivative Kelvin sense. The mathematical model is derived and discretized to a set of ordinary differential equations by Galerkin approximation method. The steady-state response of a single-mode system is obtained by the averaging method. Numerical results are obtained by an algorithm based on the fractional-order Grünwald–Letnikov definition, and compared with the analytical ones for verification. A parametric study and singularity analysis are carried out to determine the influence of the coefficients of the material’s constitutive equation on the responses. To study the effect of beam length and nonlinear coefficient on the nonlinear dynamic response, a numerical simulation is carried out. The periodic, multiple periodic, and chaotic responses are determined using Poincaré section bifurcation diagrams of the local maximum displacement. The above analysis allows us to optimize parametric design scheme for the viscoelastic beam.

Investigation on nonlinear and fractional derivative Zener model of coupled vehicle-track system

ABSTRACTThe nonlinear and fractional derivative Zener model of viscoelastic material considering Berg’s friction force model is established, which can be used to describe the frequency-dependent and amplitude-dependent behaviour of rubber isolator. The simplified Berg’s friction force scheme was proposed to improve computational efficiency in the frequency domain. The frequency response of the single degree of freedom system modelled with the fractional derivative Zener model and the proposed simplified Berg’s friction force scheme was investigated, and it agrees well with the results calculated by discrete Fourier transform. In addition, the fractional derivative Zener model and the proposed simplified Berg’s friction force scheme are implemented to model the rail pads and the primary suspension rubber spring in a railway vehicle, and then the nonlinear and fractional derivative Zener model of the coupled vehicle-track system was established. The frequency response calculated by the proposed nonlinear and fractional derivative Zener model of coupled vehicle-track system was compared to the calculation using ordinary Kelvin–Voigt model, and the results have shown that there is little difference in the calculation results between the two models, but the vibration responses of bogies, wheelsets and rails are quite different in time domain and frequency domain. Furthermore, the comparison between the proposed model and the fractional derivative Kelvin–Voigt model were analysed. The difference between the two models is small in the frequency range from 1 to 200 Hz and in the whole time domain, but the difference is large in the high frequency region. The proposed nonlinear and fractional derivative Zener model of the coupled vehicle-track system has high efficiency and accuracy both in time domain and frequency domain.Abbreviations: FDZ: fractional derivative Zener; SBFF: simplified Berg’s friction force; CVTS: coupled vehicle-track system; DFT: discrete Fourier transform; SDOF: single degree of freedom; FDKV: fractional derivative Kelvin–Voigt.

The fractional derivative Kelvin–Voigt model of viscoelasticity with and without volumetric relaxation

The fractional derivative Kelvin–Voigt model of viscoelasticity involving the time-dependent Poisson’s operator has been studied not only for the case of a time-independent bulk modulus, but also when the volumetric relaxation is taken into account. It has been shown that such a model could describe the features of auxetic materials.

Effect of temperature- and frequency-dependent dynamic properties of rail pads on high-speed vehicle–track coupled vibrations

ABSTRACTThe soft under baseplate pad of WJ-8 rail fastener frequently used in China’s high-speed railways was taken as the study subject, and a laboratory test was performed to measure its temperature and frequency-dependent dynamic performance at 0.3 Hz and at −60°C to 20°C with intervals of 2.5°C. Its higher frequency-dependent results at different temperatures were then further predicted based on the time–temperature superposition (TTS) and Williams–Landel–Ferry (WLF) formula. The fractional derivative Kelvin–Voigt (FDKV) model was used to represent the temperature- and frequency-dependent dynamic properties of the tested rail pad. By means of the FDKV model for rail pads and vehicle–track coupled dynamic theory, high-speed vehicle–track coupled vibrations due to temperature- and frequency-dependent dynamic properties of rail pads was investigated. Finally, further combining with the measured frequency-dependent dynamic performance of vehicle’s rubber primary suspension, the high-speed vehicle–track coupled vibration responses were discussed. It is found that the storage stiffness and loss factor of the tested rail pad are sensitive to low temperatures or high frequencies. The proposed FDKV model for the frequency-dependent storage stiffness and loss factors of the tested rail pad can basically meet the fitting precision, especially at ordinary temperatures. The numerical simulation results indicate that the vertical vibration levels of high-speed vehicle–track coupled systems calculated with the FDKV model for rail pads in time domain are higher than those calculated with the ordinary Kelvin–Voigt (KV) model for rail pads. Additionally, the temperature- and frequency-dependent dynamic properties of the tested rail pads would alter the vertical vibration acceleration levels (VALs) of the car body and bogie in 1/3 octave frequencies above 31.5 Hz, especially enlarge the vertical VALs of the wheel set and rail in 1/3 octave frequencies of 31.5–100 Hz and above 315 Hz, which are the dominant frequencies of ground vibration acceleration and rolling noise (or bridge noise) caused by high-speed railways respectively. Since the fractional derivative value of the adopted rubber primary suspension, unlike the tested rail pad, is very close to 1, its frequency-dependent dynamic performance has little effect on high-speed vehicle–track coupled vibration responses.

Parametric bifurcation of a viscoelastic column subject to axial harmonic force and time-delayed control

We investigate the steady state response of a simply supported viscoelastic column subject to axial harmonic excitation. The viscoelastic material is modeled in fractional derivative Kelvin sense. The equation of motion is derived and discretized by the Galerkin approximation resulting in a generalized Mathieu–Duffing equation with time delay. Bifurcations in parametric excitation can be eliminated by appropriate feedback gain and time delay. The bifurcating behavior for various fractional orders and material ratios are also investigated. New criteria of stability determination are established. Based on the Runge–Kutta method, numerical results are obtained and compared with analytical solutions for verification.

Asymptotic analysis of an axially viscoelastic string constituted by a fractional differentiation law

The non-linear creep vibration of an axially moving string constituted by a fractional differentiation law is investigated in this paper. A non-linear partial fractional order differential equation governing the transverse vibration of the string is derived from Newton's second law, the fractional derivative Kelvin constitutive relationship and the Lagrangian strain. Unlike the classic models, the derived governing equation takes creep behavior into account. By using the multi-scale method, the non-linear frequency of the axially moving string is obtained. The results show that the amplitude predicted by the fractional model is larger than that predicted by the viscoelastic solid models.

Stability in parametric resonance of an axially moving beam constituted by fractional order material

The stability of an axially moving beam constituted by fractional order material under parametric resonances is investigated. The governing equation is derived from Newton’s second law and the fractional derivative Kelvin constitutive relationship. The time-dependent axial speed is assumed to vary harmonically about a constant mean velocity. The resulting principal parametric resonances and summation resonances are investigated by the multi-scale method. It is found that instabilities occur when the frequency of axial speed fluctuations is close to two times the natural frequency of the beam or when the frequency is close to the sum of any two natural frequencies. Moreover, Numerical results show that the larger fractional order and the viscoelastic coefficient lead to the larger instability threshold of speed fluctuation for a given detuning parameter. The regular axially moving beam displays a higher stability than the beam constituted by fractional order material.

Quasi-Static Analysis of Beam Described by Fractional Derivative Kelvin Viscoelastic Model under Lateral Load

The beam is assumed to obey a three-dimensional viscoelastic fractional derivative constitutive relations, the mathematical model and governing equations of the quasi-static and dynamical behavior of a viscoelastic Euler-Bernoulli beam are established, the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative model is investigated, and the analytical solution is obtained by considering the properties of the Laplace transform of Mittag-Leffler function and the properties of fractional derivative. The result indicate that the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative viscoelastic model can reduced to the cases of classic viscoelastic and elastic, the order of fractional derivative has great effect on the quasi-static mechanical behavior of Euler-Bernoulli beam.

Unbounded complex modulus of viscoelastic materials and the Kramers–Kronig relations

The Kramers–Kronig (K–K) dispersion relations developed for the complex modulus of elasticity of solid viscoelastic materials connect the frequency dependences of the dynamic modulus and loss modulus. Whether the boundedness of the complex modulus at high (“infinite”) frequency is required, or not, for the applicability of the K–K relations is investigated in this paper. The derivation of the K–K relations developed for the complex modulus is presented by examining the physical background of the relations. It is shown that the K–K relations can be applied even if the complex modulus is unbounded at high frequencies. The fractional derivative Kelvin model is used to demonstrate the application of the K–K relations for a class of unbounded complex modulus.

A fractional derivative model to include effect of ambient temperature on HDR bearings

Based on a previous application of the fractional derivative Kelvin model to the seismic response prediction of high damping rubber (HDR) bearings, the effect of ambient temperature is incorporated into the formulation of the model in this study, considering the variation of ambient temperature may significantly influence the mechanical characteristics of the bearings. Shaking table tests are conducted for a test structure composed of a steel deck isolated by four HDR bearings. The bearings were tested at various temperatures ranging from 0 to 28°C. Identified from the results of sinusoidal tests, the effective shear modulus is expressed in terms of maximum shear strain and temperature of the bearing. The numerical constants involved in the fractional derivative Kelvin model are represented by functions of ambient temperature. The extended model is validated by comparing the predicted seismic responses of the test structure with those measured from the earthquake tests conducted at various temperatures.

Analytical Modeling of High Damping Rubber Bearings

Two analysis models for high damping rubber bearings are proposed based on the results of shaking table tests of a seismically isolated bridge deck. These analysis models are established using the modified Gauss-Newton system identification method and the fractional derivative Kelvin model based on sinusoidal test results. The tests produced a maximum shear strain in the bearing of approximately 100%. Two existing equivalent linear models specified by the American Association of State Highway and Transportation Officials and the Public Works Research Institute of the Japanese Ministry of Construction are also characterized using sinusoidal test results. The predicted seismic responses of the test structure by the proposed models and the two equivalent linear models are compared with the measured responses. It is concluded that the proposed models can predict the seismic responses of the bearing better than the two equivalent linear models. For practical applications, the fractional derivative Kelvin model is implemented into an iteration procedure adopted by the current design practice. An evaluation of the convergence of iteration and the accuracy of prediction is conducted.