Determination of parameters in convected Maxwell model from linear viscoelastic parameters

Owing to its simplicity and general ability to correctly portray a number of important flow phenomena, the two-parameter Maxwell model has been employed in a number of important engineering studies. The relation of this equation to linear viscoelasticity and to some molecular theories is considered. A new rule is developed which shows how the shear-dependent relaxation time and viscosity of the Maxwell model can be determined from linear viscoelastic parameters. It is thus shown that the two-parameter Maxwell model may be more general than earlier anticipated.     

Using truncated relaxation spectra in the simulation of viscoelastic flows

The simulation of polymer flows is a complicated problem. One of the limiting factors is the degree to which elasticity plays an important role in the flow. For a given constitutive equation and numerical technique, a higher material relaxation time will result in a lower maximum deformation rate for which convergence is possible. Since industrial processes typically involve large deformation rates, a practical way to allow their simulation is to truncate the longest relaxation times. In this work the Leonov constitutive equation was used to simulate flow through an abrupt contraction. The possibility of truncating the relaxation spectrum was explored and the impact of using truncated spectra on simulations is described. Finally, we propose a technique for obtaining a truncated relaxation spectrum that will be useful for flow simulations.     

Computation of the linear viscoelastic relaxation spectrum from experimental data

Accurate and reliable determination of the linear viscoelastic relaxation spectrum is a critical step in the application of any constitutive equation. The experimental data used to determine the relaxation spectrum always include noise and are over a limited time or frequency range, both of which can affect the determination of the spectrum. Regularization with quadratic programming has been used to derive the spectrum; however, because both the experimental data and the spectrum change by more than an order of magnitude, the input data and the spectrum are normalized in order for the numerical procedure to be accurate. Accurate determination of the relaxation spectrum requires that the spectrum extend about two logarithmic decades on either side of the frequency range of the input data. The spectrum calculated from G′ alone is more accurate at shorter relaxation times, while that from G′ data alone is obtained from a combination of G′ and G′ data, blended in the manner described herein. Comparison with existing methods in the literature shows a consistently improved performance of the present method illustrated with both model as well as experimental data. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 64: 2177–2189, 1997     

An analytical solution of different configurations of the linear viscoelastic normal and frictional-elastic tangential contact model

In the current study an analytical solution describing the impact of a spherical particle on a rigid wall is derived. The contact is linear viscoelastic in normal and frictional-elastic in tangential direction. Due to its simplicity, the model combination considered is one of the most common in the framework of the Discrete Element Method especially for large-scale simulations. The linear viscoelastic normal model is realized according to Zhang and Whiten [1996. The calculation of contact forces between particles using spring and damping models. Powder Technology 88, 59-64.] assuming a contact to be ceased when the normal force attains a value of zero. In literature the frictional elastic tangential model is employed in three different configurations following Cundall and Strack [1979. A discrete numerical model for granular assemblies. Geotechnique 29, 47-65], Di Maio and Di Renzo [2004. Analytical solution for the problem of frictional-elastic collisions of spherical particles using the linear model. Chemical Engineering Science 59 (16), 3461-3475] and Brendel and Dippel [1998. Lasting contacts in molecular dynamics simulations. In: Herrmann, H.J., Hovi, J.-P., Luding, S. (Eds.), Physics of Dry Granular Media. Kluwer Academic Publishers, Dordrecht, 1998, p. 313]. The differences among these configurations are briefly explained, whereas the focus is set on the two most accurate model formulations given in the first two papers. All important final collision properties as positions and velocities are derived in an analytical form. Based on these results a comparison with experimental data of particle/wall and particle/particle collisions by Foerster et al. [1994. Measurements of the collision properties of small spheres. Physics of Fluids 6(3), 1108-1115], Lorenz et al. [1997. Measurements of impact properties of small, nearly spherical particles. Experimental Mechanics 37(3), 292-298] and Gorham and Kharaz [2000. The measurement of particle rebound characteristics. Powder Technology 112(3), 193-202] is performed, showing very good agreement especially for the model configuration of Di Maio and Di Renzo. The analytical solution as derived here helps understanding the complex collision process and can be applied for the evaluation of integration methods or in the context of an event-driven discrete element method.     

An analytical model for the fractional efficiency of a uniflow cyclone with a tangential inlet

An analytical model was developed to predict the fractional efficiency of a uniflow cyclone with a tangential inlet. The analysis showed that the separation efficiency is a function of particle Stokes number and the geometry of the cyclone body. Six sets of experiments were conducted under different conditions to validate the model. The experimental fractional efficiencies were determined by the total mass efficiency and the corresponding size distributions measured by using an offline particle sizer. Overall the experiments agreed with the modeling results well. Both model and experiments showed that the efficiency of this cyclone reached 99.5% and above when Stk > 1.0.     

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

In 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.     

Relaxation spectra and viscoelastic properties of the binary linear flexible polymers

A binary blending law was proposed for a mixture which was composed of two narrowly distributed linear polymer fractions with different molecular weights and the high entanglement density. The relaxation spectrum was used to express the binary blending law and it was derived from the chain relaxation processes based on tube model theories as well as from the earlier observations on the viscoelasticity of linear polymers with bimodal molecular weight distribution. The proposed law showed good agreements with experimental data over a wide range of time or frequency for the viscoelastic properties such as shear stress relaxation modulus and dynamic moduli. It also showed good-fitted results for the compositional dependences of some viscoelastic constants such as zero-shear viscosity and recoverable compliance.     

Relaxation time spectra of linear and crosslinked polymers

In this work, an attempt has been made to define a unique relaxation time spectrum for different types of polymeric materials. Empirical models for the relaxation spectrum, proposed for linear flexible polymers in the literature, have been used. A systematic determination of the parameters defining the relaxation time spectrum has been made from dynamic mechanical data. It has been shown that the resulted expression for the relaxation modulus could then be used to calculate the stress response of the materials to various deformations. The strain rate dependence of yield behavior could also be predicted. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 67: 679–684, 1998     

Genetic algorithm for the determination of linear viscoelastic relaxation spectrum from experimental data

Conventional procedures employed in the modeling of viscoelastic properties of polymer rely on the determination of the polymer's discrete relaxation spectrum from experimentally obtained data. In the past decades, several analytical regression techniques have been proposed to determine an explicit equation which describes the measured spectra. With a diverse approach, the procedure herein introduced constitutes a simulation‐based computational optimization technique based on non‐deterministic search method arisen from the field of evolutionary computation. Instead of comparing numerical results, this purpose of this paper is to highlight some subtle differences between both strategies and focus on what properties of the exploited technique emerge as new possibilities for the field. In oder to illustrate this, essayed cases show how the employed technique can outperform conventional approaches in terms of fitting quality. Moreover, in some instances, it produces equivalent results with much fewer fitting parameters, which is convenient for computational simulation applications. The problem formulation and the rationale of the highlighted method are herein discussed and constitute the main intended contribution. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

The exact discrete model of a system of linear stochastic differential equations driven by fractional noise

Abstract. This paper derives the exact discrete model (EDM) of a kth‐order system of stochastic differential equations driven by a vector fractional noise under fixed initial conditions. The EDM can be used for the Gaussian estimation and forecasting with long‐memory discrete‐time equispaced data. Detailed formulae which are necessary for the construction and numerical evaluation of the Gaussian likelihood under two observation schemes are established. State variables can be observed either at equispaced points in time or as integrals over the observational interval.     

Influence of long-chain branching on linear viscoelastic flow properties and dielectric relaxation of polycarbonates

The dynamic viscoelastic property, creep and creep recovery behavior, and dielectric relaxation of long-chain branched Bisphenol A polycarbonates were measured in parallel plate rheometer and dielectric analyzer. The linear polycarbonate (PC-L) as reference and three branched polycarbonates (PC-Bs) have similar molecular weights and molecular weight distributions, while the PC-Bs have different branching degrees, below 0.7 branch points/chain and above twice of M_c. The long-chain branched polycarbonates exhibit higher zero-shear viscosities, more significant shear shinning, higher flow activation energies, and much longer relaxation times. It was also found that long-chain branches increase the elasticity of melt characterized by the steady-state recoverable compliance and the storage modulus. The ‘dissident’ rheological behavior of long-chain branching exhibiting mainly in addition polymers such as polyolefin, is confirmed in condensation polymers. These behaviors resulted from additional molecular entanglements of long-chain branches can be understood qualitatively in terms of the tube model for topological constraints. The dielectric α-relaxation of linear polycarbonate and branched polycarbonates has been fitted with Vogel-Fulcher-Tammann-Hesse (VFTH) equation and the shape of relaxation time curves is also analyzed. The long-chain branched polycarbonates present longer relaxation times, but divergent α-relaxation temperatures, because the latter is dominated by the free volume.     

Effect of filler on the relaxation time spectra of filled polymers

The temperature and frequency dependences of the complex shear modulus G^* and tan δ of mechanical losses of epoxy compositions with various fillers were studied. The method of Ninomiya-Ferry applied to the reduced curves of frequency dependence of the effective part of the shear modulus was used to draw up relaxation time spectra for specimens with various concentrations of the filler. Regularities in the change of type and position of the spectral curves with increase in filler concentration were indicated. The findings make it possible to draw conclusions about the effect of the filler on the properties of the polymer matrix in the boundary layer and about changes in the conditions of the deformation of the polymer interlayers between the filler particles as compared with the deformations in bulk specimens.     

Piecewise linear relaxation of bilinear programs using bivariate partitioning

Several operational and synthesis problems of practical interest involve bilinear terms. Commercial global solvers such as BARON appear ineffective at solving some of these problems. Although recent literature has shown the potential of piecewise linear relaxation via ab initio partitioning of variables for such problems, several issues such as how many and which variables to partition, which partitioning scheme(s) and relaxation model(s) to use, placement of grid points, etc., need detailed investigation. To this end, we present a detailed numerical comparison of univariate and bivariate partitioning schemes. We compare several models for the two schemes based on different formulations such as incremental cost (IC), convex combination (CC), and special ordered sets (SOS). Our evaluation using four process synthesis problems shows a formulation using SOS1 variables to perform the best for both partitioning schemes. It also points to the potential usefulness of a 2‐segment bivariate partitioning scheme for the global optimization of bilinear programs. We also prove some simple results on the number and selection of partitioned variables and the advantage of uniform placement of grid points (identical segment lengths for partitioning). © 2009 American Institute of Chemical Engineers AIChE J, 2010     

Multiphysics simulations of cure residual stresses and springback in a thermoset resin using a viscoelastic model with cure‐temperature‐time superposition

Simulations of evolution of cure‐induced stresses in a viscoelastic thermoset resin are presented. The phenomenology involves evolution of resin modulus with degree of cure and temperature, the development of stresses due to crosslink induced shrinkage, and the viscoelastic relaxation of these stresses. For the simulations, the detailed kinetic and chemo‐thermo‐rheological models for an epoxy‐amine thermoset resin system, described in Eom et al. (Polym. Eng. Sci. 2000, 40, 1281) are employed. The implementation of this model into the simulation is facilitated by multiphysics simulation strategies. The trends in simulated cure‐induced stresses obtained using the full‐fledged viscoelastic model are compared with those obtained from two other equivalent material models, one involving a constant elastic modulus, and the other involving a cure‐dependent (but time‐invariant) elastic modulus. It is observed that the viscoelastic model not only results in lower estimates of cure‐induced stresses, but also provides subtle details of the springback behavior. © 2012 Wiley Periodicals, Inc. J. Appl. Polym. Sci., 2013     

Conformational relaxation time in polymer solutions by elongational flow experiments: 1. Determination of extensional relaxation time and its molecular weight dependence

The nature of chain extension of high-molecular weight atactic polystyrene in an elongational flow field was investigated. In particular, we studied the sudden onset of high extension which, according to theory, is governed by the expression $\text{?}\text{?}{}_{\mathrm{c}}\text{τ = 1.}$ The molecular weight dependence of $\text{?}\text{?}{}_{\mathrm{c}}$ (the critical strain rate at which high extension occurs) was examined, and the corresponding relaxation time, τ, which would be relevant to the above expression was considered. The experimental measurements of absolute values and the functional dependence on molecular weight of the relaxation times demonstrates that it is the relaxation time associated with the Zimm non-free-draining model which is appropriate to this coil-stretch transition.     

Prediction of phase inversion in agitated vessels using a two-region model

Phase inversion in agitated vessels was studied using a two-region model. In this model, breakup and coalescence were assumed to take place in the vicinity of the impeller and away from that region, respectively. The mechanism of phase inversion was regarded as the result of an imbalance between the breakup and coalescence processes. Hence phase inversion was assumed to occur when the coalescence frequency exceeded that of breakup. In addition, the concept of a radial distribution function was adopted in the model in order to account for droplet coalescence in concentrated dispersions. Using the two-region model, the effect of interfacial tension, viscosity, density and impeller size on the width of the ambivalent range was investigated. The predictions agree well with experimental data particularly for the upper curve of the ambivalent range; however, the organic phase fraction of the lower curve is in some cases underestimated by the model.     

Determination of graessley's relaxation time in concentrated polymer solutions and melts

The method consists of a comparison between experimental data and Graessley's theoretical function of the dependence of viscosity of concentrated polymer solutions and melts on the velocity gradient in a linearized plot using a three-parameter empirical equation. The procedure is objective and allows the use of a computer.     

Calculation of relaxation time in polyurethanes using additive group contributions

The method of additive properties was used to calculate the dynamic mechanical relaxation time for a series of polyurethanes. Calculations were also made of density and glass transition temperature. Group contributions for nine component groups were determined. With these group values, the densities of the 12 polymers used to determine the groups were calculated and found to agree with measured values within an average of 0.2%. Calculated glass transition temperatures also agreed with measured values within 0.2%. The relaxation time, defined as a parameter in the Havriliak-Negami equation, was shown to be correlated with the glass transition temperature, allowing relaxation time to also be expressed as an additive property. Calculated logarithms of relaxation times agree with measured values to within 7% over a range of relaxation times covering many decades. © 1996 John Wiley & Sons, Inc.