Generalized Zener Model Research Articles

A hyperbolic generalized Zener model for nonlinear viscoelastic waves

A macroscopic model describing nonlinear viscoelastic waves is derived in Eulerian formulation, through the introduction of relaxation tensors. It accounts for both constitutive and geometrical nonlinearities. In the case of small deformations, the governing equations recover those of the linear generalized Zener model (GZM) with memory variables, which is widely used in acoustics and seismology. The structure of the relaxation terms implies that the model is dissipative. The chosen family of specific internal energies ensures also that the model is unconditionally hyperbolic. A Godunov-type scheme with relaxation is implemented. A procedure for maintaining isochoric transformations at the discrete level is introduced. Numerical examples are proposed to illustrate the properties of viscoelastic waves and nonlinear wave phenomena.

Effect of compatibilizer concentration on dynamic rheological behavior and morphology of thermoplastic starch/polypropylene blends

ABSTRACTThe reactive compatibilization of blends consisting polypropylene (PP) and thermoplastic starch (TPS) (70/30) with different portions of PP‐grafted maleic anhydride (PP‐g‐MA) is carried out by melt mixing. The esterification reaction between the starch hydroxyl and the PP‐g‐MA groups proved by the FTIR leads to a compatibility improvement. The dynamic rheological properties, morphology, elongation at break, and the impact strength of the blends were studied. The SEM images show that increasing the compatibilizer concentration reduces the dispersed TPS droplet size. The generalized Zener model states that an elastic interface is established (minimum α value) and enables us to predict the dynamic rheological properties of our blends in a longer frequency range to where the current experimental limitation exists. The modified Cross model is implemented to confirm better adhesion between phases when 20 wt % PP‐g‐MA is used (minimum ac value). The increase in the dynamic viscoelastic moduli at concentrations up to 20 wt % and the observed plateau at the elongation at break point at this concentration confirmed that this concentration is the optimum for the maximum stress transfer. © 2019 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2020, 137, 48742.

A unified poroviscoelastic model with mesoscopic and microscopic heterogeneities

The wave-induced fluid flow (WIFF) is considered to be the main cause of dispersion and attenuation of seismic waves in fluid-saturated porous media. Among numerous theories, the mesoscopic and microscopic heterogeneities are considered to be the primary mechanisms causing the WIFF. Furthermore, in most rocks, the mesoscopic and microscopic heterogeneities exist simultaneously and can cause obvious transitions of the fast P-wave velocity, which means it is necessary to consider the influence of the two mechanisms on the dispersion and attenuation simultaneously. Numerous results have shown that the dispersions and attenuations caused by these two mechanisms can be approximated in terms of the Zener model. To combine the two mechanisms into a unified model, we introduce a new generalized Zener model into the Biot poroelasticity theory to obtain a new poroviscoelastic model. Comparisons between the numerical results and two groups of experimental data further confirm the validity of our new model.

Incorporating Full Attenuation Mechanisms of Poroelastic Media for Realistic Subsurface Sensing

Porous materials are ubiquitous in the subsurface formations of the earth where acoustic and seismic waves are used for remote sensing. However, it is not well understood how the dissipation and the dispersion of poroelastic waves are caused by the viscoelastic and viscous properties of the constituents such as solid grains and pore fluid and by the viscoelastic dissipation of the solid frame, as well as the viscodynamic coupling of the pore fluid to the solid frame due to its global and local flows relative to the solid grains. Such attenuation mechanisms have seldom been incorporated in subsurface sensing simulations, although they can be very important to applications. In this paper, we propose a complete attenuation model, including both full stiffness and viscodynamic dissipation, for poroelastic media in seismic wave simulations. Completely based on a generalized Zener model, the effects associated with physical dissipation and frequency-dependent dispersion are accurately simulated by a finite-difference time-domain algorithm. Verifications with analytical solutions show the accuracy, efficiency, and flexibility of our method. Numerical results demonstrate that the attenuation of Biot’s model in the sediment of the seafloor has significant effects on acoustic wave scattering from complex geologic structures.

Wave equation for generalized Zener model containing complex order fractional derivatives

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial-boundary value problem for such materials is formulated and solution is presented in the form of convolution. Two specific examples are analyzed.

Highly accurate stability-preserving optimization of the Zener viscoelastic model, with application to wave propagation in the presence of strong attenuation

This article concerns the numerical modeling of time-domain mechanical waves in vis-coelastic media based on a generalized Zener model. To do so, classically in the literature relaxation mechanisms are introduced, resulting in a set of so-called memory variables and thus in large computational arrays that need to be stored. A challenge is thus to accurately mimic a given attenuation law using a minimal set of relaxation mechanisms. For this purpose, we replace the classical linear approach of Emmerich & Korn (1987) with a nonlinear optimization approach with constraints of positivity. We show that this technique is significantly more accurate than the linear approach. Moreover it ensures that physically-meaningful relaxation times that always honor the constraint of decay of total energy with time are obtained. As a result these relaxation times can always be used in a stable way in a modeling algorithm, even in the case of very strong attenuation for which the classical linear approach may provide some negative and thus unusable coefficients.

A model of the viscoelastic behavior of flowable resin composites prior to setting

ObjectiveThe aim of this study is to develop fractional derivative models for the assessment of viscoelastic properties related to handling characteristics of dental resin composites belonging to two classes: flowable (Revolution Formula 2 and Filtek Ultimate) and posterior “bulk-fill” flowable base (Smart Dentin Replace). MethodsRheological measurements on all materials tested in this study were performed using dynamic oscillatory rheometer at temperature of 23°C. A parallel plates module with a diameter of 20mm was used to measure the properties of the resin composites tested. We developed two models to describe the obtained data: the generalized Newton model and the generalized Zener model (the so-called three parameter model). Both models contain fractional derivatives of Riemann–Liouville type. By determining the parameters of the model we were able to fit experimental data with high accuracy. ResultsOur results show that flowable “bulk-fill” resin-composite material (SDR) has distinct properties as compared to other two flowable resin composite materials (Revolution Formula 2 and Filtek Ultimate). Thus, previously found SDR properties as “bulk-fill” flowable base results in the fact that it is described by generalized Zener model (i.e., it has properties of solid like viscoelastic material). SignificanceOur model may be used to predict behavior of tested composites in different flow conditions. The SDR has initially small almost constant complex viscosity showing that it has good self-leveling property.

Effect of Compatibilizer Concentration on Rheological Behavior, Morphologies, and Mechanical Properties of PVDF/TPU Blends

The rheological behavior, morphologies, and tensile properties of reactively compatibilized PVDF/TPU blends are reported. Using PVDF-g-AAc as the compatibilizer, PVDF/TPU 90/10 and 10/90 blends are prepared. The carboxylic acid groups of PVDF-g-AAc react with the urethane linkages of TPU during melt blending to generate in situ PVDF-g-AAc-g-TPU which leads to compatibilization of PVDF/TPU blends. The introduction of PVDF-g-AAc into the PVDF/TPU blends causes an increase in viscosity. The rheological behavior of the compatibilized PVDF/TPU 90/10 and 10/90 blends are well described by the generalized Zener model. The addition of the compatibilizer PVDF-g-AAc reduces the dispersed-phase domain size and narrows the size distribution. ▪Author: The summary has been shortened to comply with the maximum of 700 characters. Pls check/confirm changes!▪

Numerical modeling of transient two-dimensional viscoelastic waves

This paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids. Dissipation mechanisms are described using the generalized Zener model. No time convolutions are required thanks to the introduction of memory variables that satisfy local-in-time differential equations. By appropriately choosing the relaxation parameters, it is possible to accurately describe a large range of materials, such as solids with constant quality factors. The evolution equations satisfied by the velocity, the stress, and the memory variables are written in the form of a first-order system of PDEs with a source term. This system is solved by splitting it into two parts: the propagative part is discretized explicitly, using a fourth-order ADER scheme on a Cartesian grid, and the diffusive part is then solved exactly. Jump conditions along the interfaces are discretized by applying an immersed interface method. Numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems, such as multiple scattering configurations.

Rheological Behavior and Morphologies of Reactively Compatibilized PVDF/TPU Blends

AbstractUsing the comb copolymer PVDF‐g‐AAc as a compatibilizer, five reactively compatibilized PVDF/TPU blends with different compositions were prepared. During melt blending, the carboxylic acid groups of PVDF‐g‐AAc react with the urethane linkages of TPU to generate the graft copolymers PVDF‐g‐AAc‐g‐TPU, which lead to compatibilization of the PVDF/TPU blends. The rheological behavior of the compatibilized PVDF/TPU blends can be described by a generalized Zener model. PVDF/TPU 50/50 and PVDF/TPU 30/70 blends show a particle network structure, indicating elastic interactions between the grafted shells of compatibilized PVDF domains. SEM shows that almost all compatibilized PVDF/TPU blends exhibit a dispersion‐type morphology. magnified image

A simplified methodology to predict the dynamic stiffness of carbon-black filled rubber isolators using a finite element code

A new and different approach to the inclusion of the amplitude-dependent effect, known as the Fletcher–Gent effect or Payne effect, in a linear viscoelastic rubber material model is presented to predict the dynamic stiffness of filled rubber isolators using a finite element (FE) code. The technique is based on providing a linear viscoelastic model with the adequate material data set, once the dynamic strain amplitude, to which the rubber mount is subjected, is estimated. A generalized Zener model is adopted to describe the frequency-dependent behaviour of the material through the use of hereditary integrals. The dynamic strain amplitude dependence is not modelled through any friction model or plasticity theory, as usually is in literature. It is introduced by considering the frequency-dependent properties of the compound at an adequate strain value, which enforces the estimation of an equivalent strain value. As a first approximation, a quasi-static value is used as the reference value at which material properties should be provided to the linear viscoelastic model. The technique works directly in frequency domain, the dynamic stiffness of the bushing being directly obtained. The methodology is applied to evaluate the dynamic stiffness of a real bushing in working conditions with very satisfactory results. Despite the assumptions made, especially regarding the estimation of the equivalent strain amplitude value, errors of the predictions fall within the limits usually accepted by rubber manufacturers.

Dynamics of a Viscoelastic Rod of Fractional Derivative Type

We study dynamics of a viscoelastic rod with fractional derivative type of dissipation under time dependent loading. First the moment curvature relation for the rod from the generalized Zener model of a standard linear solid is derived. Then, we show that the dynamics of the problem is governed by a single linear differential equation with fractional derivative. The existence and uniqueness of solutions for this equation is studied in detail. The form of the solution for the case of periodic compressive force is also obtained.

Ground radar simulation for archaeological applications1

AbstractThis work presents a new modelling scheme for the simulation of electromagnetic radio waves, based on a full‐field simulator. Maxwell's equations are modified in order to include dielectric attenuation processes, such as bound‐ and free‐water relaxation, ice relaxation and the Maxwell–Wagner effect. The new equations are obtained by assuming a permittivity relaxation function represented by a generalized Zener model. The convolution integral introduced by the relaxation formulation is circumvented by defining new hidden field variables, each corresponding to a different dielectric relaxation. The equations are solved numerically by using the Fourier pseudospectral operator for computing the spatial derivatives and a new time‐splitting integration algorithm that circumvents the stiffness of the differential equations. The program is used to evaluate the georadar electromagnetic response of a Japanese burial site, in particular, a stone coffin‐like structure.

Plane-wave analysis of a hyperbolic system of equations with relaxation in $\mathbb{R}^d$

We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda's block structure condition. Well-posedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.