Fractional Derivative Maxwell 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.

Non-local Maxwell model for ultraslow relaxation of concrete under different normal stress levels

Stress relaxation is widely present in concrete structures and has a significant impact on the long-term mechanical properties of materials and structural safety. The previous stress relaxation models for concrete follow exponential or power-law law, which cannot accurately describe the ultraslow relaxation phenomenon with a logarithmic law. Thus, this paper proposes a non-local Maxwell model, i.e. Hadamard fractional derivative Maxwell (HFM) model with a normalized logarithmic kernel, and derives the constitution and relaxation modulus. A set of relaxation experimental data of cement concrete under different normal stresses is used to test the HFM model. Compared with the power-law fractional Maxwell model, the HFM model shows reasonable accuracy and can effectively capture the ultraslow relaxation in concrete. The fitting results indicate that the larger derivative order in the HFM model represents faster relaxation. Interestingly, the results also show that the normal stress in the concrete specimen is positively correlated with the derivative order in the HFM model.

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.

Viscoelastic Behavior and Constitutive Relation of Heavy Crude Oils.

Heavy crude oil exhibits very complex viscoelastic behaviors due to its complex composition of resins, asphaltenes, saturates, and aromatics. It has a great influence on oil production and transportation. In this work, the viscoelastic behaviors of three different heavy crude oils were measured using a rotational rheometer. In conclusion, all of these heavy crude oils display linear viscoelastic behaviors in the experimental range. The loss modulus (E″) of the three crude oils decreased as the experimental temperature increased, and the variation trends of the three crude oils were basically the same. However, the experimental temperature has almost no effect on the storage modulus (E′), which always retained a constant value of 0.4 Pa. Furthermore, the storage modulus (E′) and loss modulus (E″) increase as the angular frequency increases. To describe the physical deformation characteristics of viscoelastic materials, the generalized Maxwell model and the fractional derivative Maxwell model are used to establish the constitutive relation of heavy crude oil. In conclusion, the generalized Maxwell model and the fractional derivative Maxwell model can predict the experimental results very well. All of the square of the correlation coefficient (R2) values are greater than 0.95. However, the number of fitting parameters for the fractional derivative Maxwell model is less than that for the fourth-order generalized Maxwell model which can save the calculating time. Therefore, the fractional derivative Maxwell model is suggested to describe the viscoelastic behavior of heavy crude oil in industrial applications.

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.

Deformation analysis of polymers composites: rheological model involving time-based fractional derivative

A modeling approach to time-dependent property of Glass Fiber Reinforced Polymers (GFRP) composites is of special interest for quantitative description of long-term behavior. An electronic creep machine is employed to investigate the time-dependent deformation of four specimens of dog-bond-shaped GFRP composites at various stress level. A negative exponent function based on structural changes is introduced to describe the damage evolution of material properties in the process of creep test. Accordingly, a new creep constitutive equation, referred to fractional derivative Maxwell model, is suggested to characterize the time-dependent behavior of GFRP composites by replacing Newtonian dashpot with the Abel dashpot in the classical Maxwell model. The analytic solution for the fractional derivative Maxwell model is given and the relative parameters are determined. The results estimated by the fractional derivative Maxwell model proposed in the paper are in a good agreement with the experimental data. It is shown that the new creep constitutive model proposed in the paper needs few parameters to represent various time-dependent behaviors.

Demonstration and validation of constructive initialization method for neural networks to approximate nonlinear functions in engineering mechanics applications

This paper is a sequel paper to Pei et al. (Nonlinear Dyn 71(1–2):371–399, 2013) by the authors and their co-authors published in this journal. The main contribution lies in using complex data collected from real-world problems to validate the methodology and techniques given in Pei et al. (Nonlinear Dyn 71(1–2):371–399, 2013) and others to initialize neural networks in a systematic and constructive manner. Two sets of real-world laboratory data are analyzed: One set is from testing a full-sized nonlinear viscous fluid damper with harmonic displacements at different frequencies, and the other set a well-controlled complex nonlinear mechanism involving either rotational or torsional motion under random excitations. The force-state mapping formulation is adopted to analyze these data sets. In addition, we use the fractional derivative Maxwell model for the nonlinear viscous fluid damper application. This study is focused on determining the number of hidden nodes and the initial values of weights and biases for both multilayer feedforward or tapped delay line neural networks, before training them using the well-established backpropagation approach. This critical but subjective design issue is handled in a transparent manner that directly utilizes either the feature of the data, or the governing mathematical expression, and is carried out in a nearly deterministic manner. This contrasts sharply with Nguyen and Widrow (1990), a random initialization approach that prevails in training neural networks for function approximation. The initialization methodology and techniques developed in our prior and current work starting with Pei (2001) are explained and demonstrated, in the context of nonlinear problems encountered in typical applied mechanics applications, by using the two sets of data, and validated by examining the generalization capability of the trained neural networks. Future work and relevant technical hurdles are identified.

Fluid viscous damper in mitigation of structural vibration effect due to underground blast

Blast induced ground vibration, caused by underground explosions, can result in substantial damage to nearby structures. In the present study, an investigation is made on the possible reduction of structural vibration effect due to underground blast by using the fluid viscous damper (FVD). The blast load is modelled by an exponentially decaying function representative of a typical rock blast. The cylindrical pot FVD is considered for which the fractional-derivative Maxwell model is adopted. The dynamic response analysis of the structure-FVD system is performed in the time domain by an equivalent single degree of freedom (SDOF) viscous oscillator approach which is also compared with the response obtained by the discrete Fourier transform (DFT) approach. It is generally observed that the peak structural displacement due to underground blast can be effectively reduced by the FVD while a limited amount of reduction in the peak absolute acceleration of the structure is obtained.

점성 유체 감쇠기의 크기 변화에 따른 성능 변화 예측

To reduce the vibration in industrial settings, the viscous fluid dampers have been widely used. Since the damper shows a viscoelastic behavior, many methods to predict the behavior have been investigated. But the methods did not consider a change of damper size that is important factor for practical design engineer. In this study, to predict a change of damper ability with respect to a change of damper size, the dynamic experiment were conducted with fixed aspect ratio and gap. The damping coefficient at zero frequency was computed through theoretical and experiment approach in order to fit the experimental results using fractional derivative Maxwell model.

Well posedness of a linearized fractional derivative fluid model

The one-dimensional fractional derivative Maxwell model (e.g. Palade, et al., Rheol. Acta 35 (1996) 265), of importance in modeling the linear viscoelastic response in the glass transition region, has been generalized in Palade, et al., Internat. J. Engrg. Sci. 37 (1999) 315, to objective three-dimensional constitutive equations (CEs) for both fluids and solids. Regarding the rest state stability of the fluid CE, in Heibig and Palade, J. Math. Phys. 49 (2008) 043101, we gave a proof for the existence of weak solutions to the corresponding boundary value problem. The aim of this work is to achieve the study of the existence and uniqueness of the aforementioned solutions and to present smoothness results.

Study of oscillating flow in rolling motion with the fractional derivative Maxwell model

The flowing characteristics in rolling motion are investigated theoretically, with the fractional derivative Maxwell method. The velocity in rolling motion is derived. The effects of rolling motion on Newtonian and Non-Newtonian fluids are analyzed. The variations of peak velocity and velocity gradient on the wall of these two kinds of fluids are also investigated. The velocity profile of Non-Newtonian fluid is parabolic, with its peak velocity at the centerline. Compared with Non-Newtonian fluid, the effect of rolling motion on Newtonian fluid is more averaged. As to the Newtonian fluid, the peak velocity fluctuates between the wall and the tube center, and it has a peak at the middle of the wall and the centerline. When the peak velocity is at its minimum, there are two peak velocities next to the wall and centerline. Their absolute values are the same, but their symbols are different. The oscillation of velocity gradient on the wall is similar to that of the velocity. Their oscillation period is half of the rolling period. The dimensionless velocity gradient on the wall, whose amplitude is between zero and two, is determined by dimensionless frequency. The effects of rolling period, tube radius and fluid viscosity to the dimensionless velocity gradient on the wall are similar.

Study of Oscillating Flow of Viscoelastic Fluid With the Fractional Maxwell Model

The oscillating flow of the viscoelastic fluid in cylindrical pipes has been applied in many fields, such as industries of petroleum, chemistry, and bioengineering. It is studied using the fractional derivative Maxwell model in this paper. The exact solution is obtained utilizing a simpler and more reasonable technique. According to this velocity solution, the time-velocity profile of one kind of viscoelastic fluid is analyzed. From analysis, it is found that the flow behaves like the Newton fluid when the oscillating frequency is low, and the flow reversal occurs when the oscillating frequency is high. Moreover, two series approximations for the velocity are obtained and analyzed for different model parameters. In one series approximation, the velocity is parabolic in profile, while in the other series approximation, the velocity presents three characteristics: (1) it is independent of radius and at the centerline is smaller than that of steady Poiseuille flow, (2) the phase lags about 90deg with respect to the imposed pressure gradient, and (3) the Richardson annular effect is found near the wall.

On the rest state stability of an objective fractional derivative viscoelastic fluid model

The one-dimensional fractional derivative Maxwell model has been found very useful in modeling the linear viscoelastic response in the glass transition and α-relaxation regions. That motivated further work on generalizations to nonlinear viscoelastic fractional constitutive equations. In the work of Palade et al. [Int. J. Eng. Sci. 37, 315 (1999)], a fully objective constitutive equation for an incompressible fluid—reducible to the linear fractional derivative Maxwell model under the small deformations hypothesis—is given, together with a state of rest stability analysis. In the stability section of the aforementioned work, for some physically aceptable perturbation inputs, it has been shown that no continuously differentiable solutions may be found to the corresponding equations of motion. In this work, we extend the stability analysis framework to L2 Hilbert functional space and prove the existence of solutions to the corresponding equations of motions.

Fractional-derivative viscoelastic model of the shock interaction of a rigid body with a plate

The impact of a rigid body upon an elastic isotropic plate is investigated for the case when the equations of motion take rotary inertia and shear deformation into account. The impactor is considered as a mass point, and the contact between it and the plate is established through a buffer involving a linear-spring–fractional-derivative dashpot combination, i.e., the viscoelastic features of the buffer are described by the fractional-derivative Maxwell model. It is assumed that a transient wave of transverse shear is generated in the plate, and that the reflected wave has insufficient time to return to the location of the spring’s contact with the plate before the impact process is completed. To determine the desired values behind the transverse-shear wave front, one-term ray expansions are used, as well as the equations of motion of the impactor and the contact region. As a result, we are led to a set of two linear differential equations for the displacements of the spring’s upper and lower points. The solution of these equations is found analytically by the Laplace-transform method, and the time-dependence of the contact force is obtained. Numerical analysis shows that the maximum of the contact force increases, tending to the maximal contact force when the fractional parameter is equal to unity.

Viscoelastic Behavior Analysis and Application of the Fractional Derivative Maxwell Model

A mathematical model of the viscoelastic phenomenon, employing the fractional derivative Maxwell Model (FDMM) is analyzed in order to determine its consistency with thermodynamic principles. In particular, the development of constraints on the parameters of the model guarantees that the FDMM will predict a nonnegative rate of energy dissipation and a nonnegative internal work. The creep compliance and relaxation modulus of the FDMM are obtained in an easier way. It is found that the monotonic non-decreasing creep compliance and monotonic non-increasing relaxation moduli prove a well-behaved viscoelastic phenomenon of the FDMM. The analysis of relaxation modulus indicates that the FDMM represents viscoelastic fluid behavior, with arbitrary fractional derivatives of stress and strain, only if the thermodynamic constraints are satisfied. The steady state sinusoidal response expressions derived in this article are verified using comparisons with experimental force-displacement loops.

Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel

The unsteady flow of viscoelastic fluid with the fractional derivative Maxwell model (FDMM) in a channel is studied in this note. The exact solutions are obtained for an arbitrary pressure gradient by means of the finite Fourier cosine transform and the Laplace transform. Two special cases of pressure gradient are discussed. Some results given by the classical models with integer-order are included in this note.

Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models

Viscoelastic materials like amorphous polymers or organic glasses show a complex relaxation behavior in the softening dispersion region, i.e. from glass transition to the α relaxation zone. It is known that a uni-dimensional Maxwell model, modified within the conceptual framework of fractional calculus, has been found to predict experimental data in this range of temperatures. After developing a fully objective constitutive relation for an incompressible fluid, it is shown here that the fractional derivative Maxwell model results from the linearization of this objective equation about the state of rest, when some assumptions about the memory kernels are made. Next, it is demonstrated that the three dimensional, linearized version of the frame indifferent equation exhibits anomalous stability characteristics, namely that the rest state is neither stable nor unstable under exponential disturbances. Also, the material cannot support purely harmonic excitations either. Consequently, it appears that fractional derivative constitutive equations may be used to study a very limited category of flows in rheology rather than the whole spectrum.

Seismic response prediction of HDR bearings using fractional derivative Maxwell model

Shaking table tests are conducted for one type of high damping rubber (HDR) bearing in this study. The bearing is strained up to a maximum shear strain of approximately 200%. The equivalent linear characteristics are determined according to AASHTO specifications based on the steady-state responses of sinusoidal tests. An equivalent linear analysis is performed to predict the seismic responses of the test structure and the results are compared with the measured responses. Besides, the fractional derivative Maxwell model of the bearing is formulated and an explicit integration scheme is implemented for solving the equation of motion of the test structure subject to ground motions. Two sets of numerical parameters involved in the fractional derivative Maxwell model are identified with respect to the best fit to the dynamic amplification function and phase angle determined from the sinusoidal tests. According to the comparison of the predicted and measured seismic responses of the test structure, it is found that the parameters should be determined from the best fit to the phase angle rather than the dynamic amplification function. For practical applications, the fractional derivative Maxwell model is equipped with an iteration procedure.

Dynamic Analysis of Generalized Viscoelastic Fluids

A general boundary-element formulation is presented for the predic- tion of the dynamic response of fluids with viscoelastic behavior. The fluid is modeled by a generalized constitutive relation that contains either complex-valued parameters and complex-order derivatives or real-valued parameters and fractional- order derivatives, These models are consistent with basic theories and are not arbitrary constructions. The models are valid for linear viscoelastic fluid behavior and are limited to fluid motions with infinitesimally small displacement gradients. The governing equations are transformed into the Laplace domain and the infinite space fundamental solution is derived. The resulting integral equations are then solved by numerical procedures. The method is applied in the prediction of the dynamic mechanical properties of a viscous damper containing a viscoelastic fluid in the form of silicon gel. The fluid is modeled by a fractional derivative Maxwell model. The predicted mechanical properties of the device are found to be in ex- cellent agreement with experimental results.

Fractional‐Derivative Maxwell Model for Viscous Dampers

A fractional-derivative Maxwell model is proposed for viscous dampers, which are used for vibration isolation of piping systems, forging hammers, and other industrial equipment, as well as for vibration and seismic isolation of building structures. The development and calibration of the model is based on experimentally observed dynamic characteristics. The proposed model is validated by dynamic testing and very good agreement between predicted and experimental results is obtained. Numerical algorithms for the solution of the constitutive relation in either the frequency or the time domain are presented. Some analytical results for a single-degree-of-freedom viscodamper system are presented. These results are useful to the design of vibration-isolation systems. Furthermore, an equivalent viscous oscillator is defined whose response is essentially the same as that of the viscodamper isolator. Finally, the model is employed in the analysis of a base-isolated model structure that has been tested on a shake table.