Linear viscoelastic behavior of the Hookean dumbbell with internal viscosity

The linear viscoelastic modulus G(t) predicted by the analytical formulations of Schieber (1993), Wedgewood (1993), Dasbach et al. (1992), and Booij and van Wiechen (1970) for the free-draining Hookean dumbbell with internal viscosity (IV) are compared with exact analytical results and exact numerical results obtained from Brownian dynamics simulations. All of these analytical formulations employ the Booij and van Wiechen expression for the IV force, thereby eliminating errors associated with linearization of the deformational velocity, however the theories differ in the approximations employed to solve configuration moment equations. Comparison with the exact G(t) results provides a means of testing these approximations. The approximate theories all correctly predict the singular part of G(t) at t = 0, providing correct predictions of , however deviations from the exact G(t) are seen in all cases for t > 0.     

The effect of temperature on the viscoelastic behavior of linear low-density polyethylene

Summary Observations are reported on linear low-density polyethylene in isothermal torsional oscillation and relaxation tests at various temperatures ranging from room temperature to 110 C. Constitutive equations are derived for the viscoelastic response of a semicrystalline polymer at small strains. The polymer is treated as an equivalent network of strands bridged by junctions (entanglements, physical cross-links on the surfaces of crystallites and lamellar blocks). The network is thought of as an ensemble of meso-regions with various potential energies for rearrangement of strands. Two types of meso-domains are introduced: active, where strands separate from temporary junctions as they are excited by thermal fluctuations, and passive, where detachment of strands is prevented by the surrounding macromolecules. The time-dependent behavior of the ensemble reflects separation of active strands from their junctions and merging of dangling strands with the network. Stress–strain relations are developed by using the laws of thermodynamics. The governing equations involve six material constants that are found by fitting the experimental data. The study focuses on the effects of (i) temperature, (ii) the deformation mode (torsion versus bending), and (iii) the loading program (oscillations versus relaxation) on the adjustable parameters.This work was partially supported by the West Virginia Research Challenge Grant Program     

The determination of a general time creep compliance relation of linear viscoelastic materials under constant load and its extension to nonlinear viscoelastic behavior for the Burger model

Linear viscoelastic materials yield a creep function which only depends on time if creep experiments are performed under constant stress ₀. In practice, this condition is very difficult to realize, and as a consequence, the experiments are performed under constant force. For small strains the difference between the conditions of constant stress and constant force is negligible. Otherwise, the decrease in cross-section has to be taken into account and leads to increasing stress in the course of time for creep experiments under constant load. The Boltzmann superposition principle is solved under the condition of constant load and for strains . The creep complicance C(t; ₀) defined by the ratio becomes, in principle, dependent on the initial stress ₀. As a consequence, a set of creep compliance curves cannot be approximated with a simple parameter fit. Already the application of the solution on the Burger model yields a creep compliance curve with all three creep ranges. Furthermore, the mathematical structure of the time creep compliance relation of the Burger model allows nonlinear viscoelastic extension via the introduction of the yield strength _max and a nonlinearity parameter n _l . The creep behavior of PBT and PC can be described in the range of long times up to initial stresses ₀, being 75% for PBT and 60% for PC of the yield stress _max with only two or one free fit parameter, respectively.     

A procedure for measuring biaxial viscoelastic behavior of thermoplastics

In order to accurately simulate the thermoforming or blowmolding manufacturing processes using finite elements or some other suitable computational procedure, it is necessary to know the constitutive behavior of the material being formed. In this study, an apparatus was developed to measure the large deformation behavior of thermoplastic sheet at elevated temperatures. The specifications of the test apparatus, as well as sample data measurements are presented. Biaxial viscoelastic material properities of ABS sheets were determined at forming temperatures. In particular, the nonlinear stress-strain relationship of the material was experimentally measured at various temperatures above the glass transition temperature and the data correlated to a time and strain separable viscoelastic material model. The results of this study show that it is possible to recover the underlying nonlinear elastic response of heated ABS sheet material, at finite strains, from tests exhibiting significant viscoelastic behavior.     

Strain localization in an oscillating Maxwell viscoelastic cylinder

The transient rotation responses of simple, axisymmetric, viscoelastic structures are of interest for interpretation of experiments designed to characterize materials and closed structures such as the brain using magnetic resonance techniques. Here, we studied the response of a Maxwell viscoelastic cylinder to small, sinusoidal displacement of its outer boundary. The transient strain field can be calculated in closed form using any of several conventional approaches. The solution is surprising: the strain field develops a singularity that appears when the wavefront leaves the center of the cylinder, and persists as the wavefront reflects to the outer boundary and back to the center of the cylinder. The singularity is alternately annihilated and re-initiated upon subsequent departures of the wavefront from the center of the cylinder until it disappears in the limit of steady state oscillations. We present the solution for this strain field, characterize the nature of this singularity, and discuss its potential role in the mechanical response and evolved morphology of the brain.     

Some remarks on the propagation and non-propagation of discontinuities in linearly viscoelastic liquids

The equations of linear viscoelasticity with a bounded memory kernel have been shown to propagate singularities in a similar way as hyperbolic equations. In this paper, we investigate a model problem for a certain class of unbounded memory kernels. It is shown thatC -solutions are obtained, although there is a discontinuity in the boundary conditions.     

Discrepancies between linear viscoelastic measurements and double-reptation predictions for commercial polystyrene samples

We have found discrepancies between the predictions of two double-reptation models and the corresponding linear viscoelasticity measurements for commercial, polydisperse polystyrene samples with weight-average molecular weights ranging from 140 to 300kg/mol. The accuracy of the experimental data has been confirmed by conducting viscoelasticity measurements in different laboratories using different types of rheometers and by verifying that small distortions in chromatographic measurements cannot account for the discrepancies seen in the viscoelastic response. In addition, we show that the discrepancies between predictions and measurements are systematic, suggesting that gaps in the theory are responsible for the mismatch. Is it concluded that commercial polystyrene resins may pose additional challenges for rheological modeling because of their relatively high polydispersity and low degree of entanglement. The experimental data given here can be used to validate future modeling efforts.     

Concentration dependence of the linear viscoelastic properties of particle suspensions

The response under small amplitude oscillatory deformations of a suspension of non-Brownian spheres dispersed in a viscoelastic fluid is investigated. The correspondence principle of linear viscoelasticity is used to derive a simple constitutive model from a model for a suspension in a Newtonian liquid. The theory predicts that for a specific particulate system the concentration dependence of the viscoelastic properties should collapse to a single master curve when the values are normalized with those of the carrier fluid alone. Measurements with the micro-Fourier rheometer using oscillatory squeeze flow are carried out on two suspensions of 60 and 80 μm sized particles dispersed in polymeric fluid and in silicon oil, and the master curve is verified. Received: 27 April 1999/Accepted: 15 October 1999     

Non-linear viscoelastic behavior of fumed silica suspensions

Suspensions of fumed silica exhibit a wide range of rheological properties depending on the nature and magnitude of the interparticle forces. In a non-polar fluid, the particles interact through hydrogen bonding and can form a three-dimensional network. The microstructure formation is responsible for the non-linear viscoelastic behavior of fumed silica suspensions, even at very small strain. These non-linear rheological properties have been studied in small amplitude oscillatory experiments as a function of particle size, surface treatment of particles, suspending medium polarity and solids concentration. The non-linear viscoelastic behavior is characterized by a non-sinusoidal waveform of the signal response. For suspensions in a non-polar fluid, both the elastic and the loss moduli are shown to be sensitive to the strain amplitude: the elastic modulus is decreasing with increasing strain whereas the loss moduli is initially increasing with strain. We have chosen to examine the dissipated energy which is clearly related to the breakup of the suspension structure. A comparison of model predictions and the experimental data shows the limitations of these models, recently proposed in the literature to describe the behavior of colloidal suspensions. Received: 9 March 1998 Accepted: 17 November 1998     

Quasi-static response of linear viscoelastic cantilever beams subject to a concentrated harmonic end load

This study evaluates the response of a uniform cantilever beam with a symmetric cross-section fixed at one end, and submitted to a lateral concentrated sinusoidal load at the free extremity. The beam material is assumed to be homogeneous, isotropic and linear viscoelastic. Due to the nature of the loading and the beam slenderness, large displacements are developed but the strains are considered small. Consequently, the mathematical formulation only involves geometrical non-linearity. It is also assumed that the beam is inextensible (neutral axis length is constant) and that inertial forces are negligible, i.e., dynamic effects are insignificant and the system can thus be modeled quasi-statically. The beam is therefore subject to oscillations caused by the sinusoidal time-dependent load, leading to a transient response until the material stabilizes and the system exhibits a periodic response, which can be conveniently described in the frequency domain. The time domain solution of this problem is elaborated by considering the quasi-static response for each time interval. The mathematical equations are presented in dimensional and dimensionless forms, and for the latter case, a numerical solution is generated and several case studies are presented. The problem is governed by a set of non-linear ordinary differential equations encompassing functions of space and time that relate the curvature, rotation angle, bending moment and geometrical coordinates. In this study, an elegant solution is deduced using perturbation theory, yielding a precise steady-state solution in the frequency domain with considerable computational economy. The solutions for both time and frequency domain methods are developed and compared using a case study for a series of dimensionless parameters that influence the response of the system.     

Effective viscoelastic behavior of particulate polymer composites at finite concentration

Polymeric materials usually present some viscoelastic behavior. To improve the mechanical behavior of these materials, ceramics materials are often filled into the polymeric materials in form of fiber or particle. A micromechanical model was proposed to estimate the overall viscoelastic behavior for particulate polymer composites, especially for high volume concentration of filled particles. The method is based on Laplace transform technique and an elastic model including two-particle interaction. The effective creep compliance and the stress and strain relation at a constant loading rate are analyzed. The results show that the proposed method predicts a significant stiffer response than those based on Mori-Tanaka's method at high volume concentration of particles.     

Effect of ionic interaction on linear and nonlinear viscoelastic properties of ethylene based ionomer melts

The effect of ionic interaction on linear and nonlinear viscoelastic properties was investigated using poly(ethylene-co-methacrylic acid) (E/MAA) and its ionomers which were partially neutralized by zinc or sodium. Dynamic shear viscosity and step-shear stress relaxation studies were performed. Stress relaxation moduli G(t, y) of the E/MAA and its sodium or zinc ionomers were factorized into linear relaxation moduli G°(t) and damping functions h(y). The relaxation modulus at the smallest strain in each ionomer agreed with the linear relaxation modulus calculated from storage modulus G and loss modulus G. In the linear region, the ionic interaction shifted the relaxation time longer with keeping the same relaxation time distribution as E/MAA. In the nonlinear region, the ionic interaction had no influence on h(y) when the ion content was low. At higher ion content, however, the ion bonding enhanced the strain softening of h(y).     

An approximation method to relate the linear viscoelastic functions

A method based on rational approximations is presented to interpolate the data from sinusoidal experiments in linear viscoelasticity. Bounds to the corresponding dynamical function and a discrete approximation to the spectrum are established. From this approximation the related viscoelastic functions can be computed. The method is checked by considering two theoretical models of physical interest and a satisfactory accuracy is achieved.     

Some aspects of linear and nonlinear viscoelastic behaviour of polymer melts in shear

In linear viscoelastic investigations the frequency dependence of the phase shift between stress and strain appears to be very characteristic of the molecular structure of the material. This function is also a good approximation of the slope of the double logarithmic plot of the absolute value of the shear modulusG _d vs. the angular frequency. The product (G _d /) sin 2 comes very close to the relaxation spectrumH(), with = 1/, in all physical states of the material.The experimentally observed separability of time and strain effects in nonlinear viscoelasticity of highly viscous isotropic polymer fluids imposes restraints to the form of the constitutive equation. A single integral superposition equation of the Boltzmann type containing the product of a time function and a nonlinear strain function gives good results in describing experimental data in shear as well as in elongation. The molecular structure affects both functions in a different way. A universal definition of the nonlinear tensorial strain measure has not yet been developed. There are some indications that a definition on the basis of the principal stretch ratios may be fruitful.Invited paper, presented at the First Conference of European Rheologists at Graz (Austria), April 14–16, 1982.     

Comparison between the dynamic moduli of fully non-linear and quasilinear viscoelastic materials

This paper deals with the non-linear viscoelastic model with multiple hereditary integrals (MHI) in the frequency domain and the conditions that it reduces to single hereditary integral or the quasilinear viscoelastic (QLV) model. It is shown that when the higher order complex moduli are related to the first-order modulus as the MHI model reduces to the QLV model. The coefficients of quasilinearity should be real and independent of amplitude and frequency.     

EIGEN THEORY OF VISCOELASTIC DYNAMICS BASED ON THE KELVIN-VOIGT MODEL

Using the eigen theory of solid mechanics, the eigen properties of anisotropic viscoelastic bodies with Kelvin-Voigt model were studied, and the generalized Stokes equation of anisotropic viscoelastic dynamics was obtained, which gives the three-dimensional pattern of viscoelastical waves. The laws of viscoelastical waves of different anisotropical bodies were discussed. Several new conclusiones are given.     

A model for the viscoelastic behavior of polymers at finite strains

Summary A constitutive model is derived for the isothermal nonlinear viscoelastic response in polymers, which do not possess the separability property. The model is based on the concept of transient networks, and treats a polymer as a system of nonlinear elastic springs (adaptive links), which break and emerge due to micro-Brownian motion of chains. The breakage and reformation rates for adaptive links are assumed to depend on some strain energy density. The viscoelastic behavior is described by an integral constitutive equation, where the relaxation functions satisfy partial differential equations with coefficients depending on the strain history. Adjustable parameters of the model are found by fitting experimental data for a number of polymers in tension at strains up to 400 per cent. To validate the constitutive relations, we consider loading with different strain rates, determine adjustable parameters at one rate of strains, and compare prediction of the model with observations at another rate of strains. Fair agreement between experimental data and results of numerical simulation is demonstrated when the rates of strains differ by more than a decade. Received 1 July 1997; accepted for publication 7 October 1997