Fractional Zener Model Research Articles

The effect of temperature dependent tissue parameters on acoustic radiation force induced displacements

Multiple ultrasound elastography techniques rely on acoustic radiation force (ARF) in monitoring high-intensity focused ultrasound (HIFU) therapy. However, ARF is dependent on tissue attenuation and sound speed, both of which are also known to change with temperature making the therapy monitoring more challenging. Furthermore, the viscoelastic properties of tissue are also temperature dependent, which affects the displacements induced by ARF. The aim of this study is to quantify the temperature dependent changes in the acoustic and viscoelastic properties of liver and investigate their effect on ARF induced displacements by using both experimental methods and simulations. Furthermore, the temperature dependent viscoelastic properties of liver are experimentally measured over a frequency range of 0.1–200 Hz at temperatures reaching 80 °C, and both conventional and fractional Zener models are used to fit the data.The fractional Zener model was found to fit better with the experimental viscoelasticity data with respect to the conventional model with up to two orders of magnitude lower sum of squared errors (SSE). The characteristics of experimental displacement data were also seen in the simulations due to the changes in attenuation coefficient and lesion development. At low temperatures before thermal ablation, attenuation was found to affect the displacement amplitude. At higher temperature, the decrease in displacement amplitude occurs approximately at 60–70 °C due to the combined effect of viscoelasticity changes and lesion growth overpowering the effect of attenuation. The results suggest that it is necessary to monitor displacement continuously during HIFU therapy in order to ascertain when ablation occurs.

An equivalence between generalized Maxwell model and fractional Zener model

Both classic rheological models and fractional derivative models have been widely adopted to model the viscoelastic behaviors of materials. In this work, we present a detailed comparison of the performance of the generalized Maxwell model and fractional Zener model. We first describe a method to determine the parameters of the generalized Maxwell model from the fractional Zener model based on the equivalence of complex modulus in the frequency domain of the two models. The two models are then applied to investigating the stress response under constant strain rate, stress relaxation, cyclic and random loading conditions. The simulation results of the two models show excellent quantitatively equivalence. This finding can provide insight into choosing the more suitable model for specific conditions.

A general approach for viscoelastic model validation applied on the analyses of epoxy resin modified by end-functionalized liquid polybutadiene

The present work presents a general approach for viscoelastic material characterization. The standard solid model and the fractional Zener model are taken into consideration. Model calibration is performed under the formalism of the inverse problems. Model validation and model selection are guided by the philosophy of the verification and validation field. Comparisons between measured data and model predictions take model uncertainties into account through Monte Carlo simulation analysis. This approach is used to calibrate models for two diglycidyl ether of bisphenol A (DGEBA) based epoxy systems modified by the incorporation of hydroxyl-terminated polybutadiene (HTPB), namely: DGEBA/HTPB blend (EPH) and DGEBA/HTPB blend modified with NCO groups (EPI). Three different types of tests are considered and a large group of test specimens are analyzed. The constitutive models are used to provide predictions for the loss factor of the materials as well as their confidence intervals. Predictions indicate that EPI presents better performance concerning mechanical energy dissipation than EPH which is highly correlated with the fact that EPI presents rubber particles distributed uniformly in nanometric scale within the epoxy matrix.

Fitting stress relaxation experiments with fractional Zener model to predict high frequency moduli of polymeric acoustic foams

This paper presents a time domain method to determine viscoelastic properties of open-cell foams on a wide frequency range. This method is based on the adjustment of the stress–time relationship, obtained from relaxation tests on polymeric foams’ samples under static compression, with the four fractional derivatives Zener model. The experimental relaxation function, well described by the Mittag–Leffler function, allows for straightforward prediction of the frequency-dependence of complex modulus of polyurethane foams. To show the feasibility of this approach, complex shear moduli of the same foams were measured in the frequency range between 0.1 and 16 Hz and at different temperatures between −20 °C and 20 °C. A curve was reconstructed on the reduced frequency range (0.1 Hz–1 MHz) using the time–temperature superposition principle. Very good agreement was obtained between experimental complex moduli values and the fractional Zener model predictions. The proposed time domain method may constitute an improved alternative to resonant and non-resonant techniques often used for dynamic characterization of polymers for the determination of viscoelastic moduli on a broad frequency range.

Application of a fractional model for simulation of the viscoelastic functions of polymers

ABSTRACTIn the present work the dynamic behavior of two representative polymeric materials, experimentally studied in previous works, has been analyzed by a fractional derivative model. It is shown that the well‐known fractional derivative Zener model, in its simplest form as a four‐parameter model is capable of capturing the main features of the dynamic moduli of the polymeric structures examined. Furthermore, the time dependent viscoelastic functions, namely the compliance and the relaxation modulus could be simulated with the same model parameter values, indicating this way that the fractional model can provide a method of interconversion between viscoelastic material functions. The model's inadequacy of describing the loss modulus peak asymmetry, exhibited by the materials, has been encountered by the five‐parameter version of the fractional Zener model. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016, 133, 43505.

Prediction of Dissipative Properties of Flax Fibers Reinforced Laminates by Vibration Analysis

This study proposes an experimental-numeric method to identify the viscoelastic properties of flax fibres reinforced composite laminate (flax/epoxide). The used method consists in identifying the evolutions of both loss factor and stiffness when vibrational frequency changes. In this way, several free-free symmetrically guided beams are excited on a dynamic range of 10 to 4000 Hz with sweep sine excitation focused around the 4-first’s modes. Fractional derivative Zener model is used to identify the on-axis ply complex moduli and describe the laminate dissipative linear behavior with the classical laminate theory. Results obtained on a quasi-isotropic laminate show that this model adequately predicts the vibrational behavior of the tested laminates.

A fractional derivative Zener model for the numerical simulation of base isolated structures

A fractional derivative Zener (FDZ) model connected in parallel with a linear viscous damper and a Coulomb friction slider is used to numerically simulate the mechanical behavior of a base isolated (BI) building tested under free vibration conditions in Solarino, Sicily. This hybrid BI system comprises high damping rubber bearings in combination with low friction sliding bearings. A comparison study of the present model with previous ones appearing in the literature, namely the bi-linear and the tri-linear models defined in the time domain, is carried out here. Furthermore, the linear viscoelastic solid, namely the classical Zener model, is also implemented and evaluated. The rheological models representing all the above BI systems are analyzed and, for the first time, the rheological formulation for the tri-linear model is presented. The present comparison study shows that the FDZ model is capable of describing the complex nonlinear response of BI systems.

Transient modeling of ultrasonic guided waves in circular viscoelastic waveguides for inverse material characterization

In this contribution, we present an efficient approach for the transient and time-causal modeling of guided waves in viscoelastic cylindrical waveguides in the context of ultrasonic material characterization. We use the scaled boundary finite element method (SBFEM) for efficient computation of the phase velocity dispersion. Regarding the viscoelastic behavior of the materials under consideration, we propose a decomposition approach that considers the real-valued frequency dependence of the (visco-)elastic moduli and, separately, of their attenuation. The modal expansion approach is utilized to take the transmitting and receiving transducers into account and to propagate the excited waveguide modes through a waveguide of finite length. The effectiveness of the proposed simulation model is shown by comparison with a standard transient FEM simulation as well as simulation results based on the exact solution of the complex-valued viscoelastic guided wave problem. Two material models are discussed, namely the fractional Zener model and the anti-Zener model; we re-interpret the latter in terms of the Rayleigh damping model. Measurements are taken on a polypropylene sample and the proposed transient simulation model is used for inverse material characterization. The extracted material properties may then be used in computer-aided design of ultrasonic systems.

On the Melt Rheological Behavior and Microstructure of Nanoclay-Filled Polyethylene/Ethylene Vinyl Acetate (PE/EVA) Blend

Correlation of rheological response with microstructure and also the tensile mechanical properties of nanoclay-filled polyethylene/ethylene vinyl acetate blends were studied. From transmission electron microscopy observations, the same clay localization within polyethylene- and ethylene vinyl acetate-rich blends was revealed. However, a better clay dispersion in the ethylene vinyl acetate-rich blend was found by analyzing the changes in the linear viscoelastic rheological properties discussed by the fractional Zener model and the power law expression. The clay influences on the tensile properties of polyethylene- and ethylene vinyl acetate-rich blends were in good agreement with its microstructure, implied by the rheological studies.

The fractional viscoelastic response of human breast tissue cells

The mechanical response of a living cell is notoriously complicated. The complex, heterogeneous characteristics of cellular structure introduce difficulties that simple linear models of viscoelasticity cannot overcome, particularly at deep indentation depths. Herein, a nano-scale stress-relaxation analysis performed with an atomic force microscope reveals that isolated human breast cells do not exhibit simple exponential relaxation capable of being modeled by the standard linear solid (SLS) model. Therefore, this work proposes the application of the fractional Zener (FZ) model of viscoelasticity to extract mechanical parameters from the entire relaxation response, improving upon existing physical techniques to probe isolated cells. The FZ model introduces a new parameter that describes the fractional time-derivative dependence of the response. The results show an exceptional increase in conformance to the experimental data compared to that predicted by the SLS model, and the order of the fractional derivative (α) is remarkably homogeneous across the populations, with a median value of 0.48 ± 0.06 for the malignant population and 0.51 ± 0.07 for the benign. The cells’ responses exhibit power-law behavior and complexity not associated with simple relaxation (SLS, α = 1) that supports the application of a fractional model. The distributions of some of the FZ parameters also preserve the distinction between the malignant and benign sample populations seen from the linear model and previous results while including the contribution of fast-relaxation behavior. The resulting viscosity, measured by a composite relaxation time, exhibits considerably less dispersion due to residual error than the distribution generated by the linear model and therefore serves as a more powerful marker for cell differentiation.

Dynamic characteristics of multilayered beams with viscoelastic layers described by the fractional Zener model

This paper concerns the dynamic analysis of composite beams containing elastic and viscoelastic (VE) layers. A method for determination of the dynamic characteristics of multilayered beams with VE layers (i.e., natural frequencies, non-dimensional damping ratios and modes of vibration) is presented. The Euler–Bernoulli beam theory and the Timoshenko theory are used to describe the elastic and VE layers, respectively. To describe the mechanical properties of a VE layer, the four-parameter rheological model with fractional derivative is applied. A number of particular rheological models are particular cases of this general model. The virtual work principle and the finite element method together with the Laplace transform are used to derive the equation of motion in the frequency domain. The dynamic characteristics of a beam with VE layers are obtained as the solution to a properly defined nonlinear eigenvalue problem. The continuation method is adopted for solving the nonlinear eigenproblem. Several conclusions concerning the accuracy of the method and variability of results are presented on the basis of numerical studies.

Parameter estimation for the generalized fractional element network Zener model based on the Bayesian method

In the present study, an inverse problem to estimate parameters in the Zener model of viscoelasticity based on the generalized fractional element (GFE) network is studied. The Bayesian method is proposed to obtain the optimal estimation of the viscoelastic parameters. Three examples are performed to certify the validity of the method. All numerical results lead to an excellent fitting between the calculative results and experimental data. It is shown that the Bayesian method is feasible in the inverse problem of parameter estimation for the fractional constitutive equation, and the GFE network Zener model is efficient in the modeling of the viscoelastic behavior.

Earthquake response of adjacent structures with viscoelastic and friction dampers

We study the seismic response of two adjacent structures connected with a dry friction damper. Each of them consists of a viscoelastic rod and a rigid block, which can slide without friction along the moving base. A simplified earthquake model is used for modeling the horizontal ground motion. Energy dissipation is taken by the presence of the friction damper, which is modeled by the set-valued Coulomb friction law. Deformation of viscoelastic rods during the relative motion of the blocks represents another way of energy dissipation. The constitutive equation of a viscoelastic body is described by the fractional Zener model, which includes fractional derivatives of stress and strain. The problem merges fractional derivatives as non-local operators and theory of set-valued functions as the non-smooth ones. Dynamical behaviour of the problem is governed by a pair of coupled multi-valued differential equations. The posed Cauchy problem is solved by use of the Gr?nwald-Letnikov numerical scheme. The behaviour of the system is analyzed for different values of system parameters.

Comparison of approaches to model viscoelasticity based on fractional time derivatives

Two approaches to describe a constitutive fractional Zener model at large strain are presented. Both viscoelastic Zener models consist of a nonlinear elastic spring and a fractional Maxwell element in parallel. The fractional Maxwell element represents the viscoelastic behaviour of the formulation. Here, development of a new fractional viscoelastic material model under consideration of finite strain theory is presented. Additionally, the constitutive equations based on two different algorithmic approaches to capture the fractional time integration within this material model are derived. The consideration of fractional elements enables the characterization of highly inelastic, time dependent materials with relatively few material parameters. For the fractional element, a material parameter α determines the transition of the rheological element’s behaviour between spring (α=0) and dashpot (α=1). Accuracy and efficiency of a classical (non-recursive) and a new recursive algorithm to handle the fractional elements have been verified and validated by the comparison of several finite element (FE) simulations to material test results. The simulations found on finite strains and an implicit time integration scheme. Finally, a FE moulding simulation, found the extended constitutive model for the explicit time integration scheme has been carried out to illustrate the performance for large scale simulations in comparison to a real forming process and found to be quite efficient.

Parameter identification of modified fractional Zener model for thermorheological materials

Parameter identification of an analytical model plays significant role in predicting the dynamic mechanical properties of thermorheological materials over wide frequency range. A deterministic parameter identification method for modified fractional Zener (MFZ) model is studied in this paper. Three of the parameters can be determined directly by deduced formulations, and the other two can be carried out by exhaustive search method within a finite search space. Firstly, parameters of MFZ model are obtained by our method using the dynamic mechanical thermal analysis (DMTA) data. Secondly, the master curves of this model are calculated. The calculated master curves of MFZ model with identified parameters agree well with the DMTA data's master curves obtained by time–temperature superposition principle (TTSP). Finally, the proposed algorithm is compared with genetic algorithm (GA). The results indicate that the presented parameter identification approach of MFZ model is effective. The investigation in this paper has some significant meanings for the industrial applications of thermorheological materials.

Morphological and rheological properties of (low-density polyethylene)/thermoplastic starch blend: investigation of the role of high elastic network

Rheological and morphological properties of low-density polyethylene (LDPE) and thermoplastic starch (TPS) with low-density polyethylene-grafted-maleic anhydride as a compatibilizer were investigated. The results showed that the circularity of the droplets decreased with increasing TPS content. The presence of compatibilizer led to finer morphology and higher continuity. The rheological analyses showed that TPS and compatibilizer can increase elasticity and viscosity of the blend dramatically. In addition, the compatibilizer enhanced the compatibility of the blends, as evidenced by the shifting of the relaxation time peak of TPS to longer times. The rheological properties of the neat components and their blends were discussed by the Carreau-Yasuda and fractional Zener models. The fractional Zener model results proved the existence of network structure in the compatibilized blends. The transient properties of blends showed that TPS and compatibilized blends had strong overshoot compared with the uncompatibilized blend, owing to the formation of high elastic network in their structure. J. VINYL ADDIT. TECHNOL., 20:250–259, 2014. © 2014 Society of Plastics Engineers

Exploring the mechanical behavior of degrading swine neural tissue at low strain rates via the fractional Zener constitutive model

The ability of the fractional Zener constitutive model to predict the behavior of postmortem swine brain tissue was examined in this work. Understanding tissue behavior attributed to degradation is invaluable in many fields such as the forensic sciences or cases where only cadaveric tissue is available. To understand how material properties change with postmortem age, the fractional Zener model was considered as it includes parameters to describe brain stiffness and also the parameter α, which quantifies the viscoelasticity of a material. The relationship between the viscoelasticity described by α and tissue degradation was examined by fitting the model to data collected in a previous study (Bentil, 2013). This previous study subjected swine neural tissue to in vitro unconfined compression tests using four postmortem age groups (<6h, 24h, 3 days, and 1 week). All samples were compressed to a strain level of 10% using two compressive rates: 1mm/min and 5mm/min. Statistical analysis was used as a tool to study the influence of the fractional Zener constants on factors such as tissue degradation and compressive rate. Application of the fractional Zener constitutive model to the experimental data showed that swine neural tissue becomes less stiff with increased postmortem age. The fractional Zener model was also able to capture the nonlinear viscoelastic features of the brain tissue at low strain rates. The results showed that the parameter α was better correlated with compressive rate than with postmortem age.

The Physical and Linear Viscoelastic Properties of Fresh Wet Foams Based on Egg White Proteins and Selected Hydrocolloids

The aim of this work was to evaluate the physicochemical properties of fresh foams based on egg white proteins, xanthan gum and gum Arabic. The distributions of the size of gas bubbles suspended in liquid were determined, as well as density and volume fraction of gas phase of the generated foams. Additionally, the viscoelastic properties in the linear range were measured, and the results were analyzed with the use of the fractional Zener model. It was shown, that foam supplementation with hydrocolloids considerably decreased their volume fraction of gas phase in comparison to pure egg white protein-based foams. Application of gum Arabic did not cause an increase in the size of foam bubbles when compared to pure white egg foam, whereas application of xanthan gum significantly decreased the size of the bubbles. Application of the fractional Zener model allowed to determine the relaxation times, their intensity in analyzed suspensions and also equilibrium module (G e). The increase in the concentration of xanthan gum resulted in the prolongation of the relaxation time and increased its intensity. Gum Arabic, when added, weakened the viscoelastic properties of the mixture as a viscoelastic solid.

The application of fractional Zener model on MCL

The application of fractional Zener model on MCL

Multivalued fractional differential equations as a model for an impact of two bodies

Impact models of two colliding bodies are usually generated by fundamental physical and geometrical principles. Initially, the number of unknown variables is not the same as the number of equations expressing the principles, so the models should be complemented with an appropriate set of constitutive equations which contain enough information on the physical properties of the system and therefore allow accurate predictions of its behavior. Along these lines we here discuss a problem when a block, moving along a line on a dry surface, impinges against another block being at rest, through a deformable straight rod of negligible mass. Among the variety of all possible choices that can be used, we suggest the constitutive model of the viscoelastic body with fractional derivatives of stress and strain, restrictions on the coefficients that follow from Clausius–Duhem inequality, and the Coulomb friction law given in the set-valued form. Owing to the presence of dry friction and the proposed fractional model, known as the fractional Zener model, the problem belongs to the class of set-valued fractional differential equations (or multivalued differential equations of arbitrary real order) leading to the equivalent Cauchy problem given in terms of two coupled integro-differential inclusions, for which the existence result ensuring the contractible solution set exists. By use of the combinatorial analysis of the problem, we identify 11 imaginable scripts and separate 10 feasible ones that were confirmed numerically by a procedure that combines nonlocal and nonsmooth modules.