The hole-pressure problem

There is a need to unify present hypotheses of the nature and role of the hole-pressure,p _e , and thus provide consolidation on which to base future research and understanding. This paper is intended to meet this need. Attention is directed towards the calculation ofp _e from the velocity and stress fields for viscoelastic fluids flowingacross rectangular holes. The constitutive models used are the Newtonian, Second-order and Maxwell models, for values of Reynolds number up to 10 and Weissenberg number up to 0.1.The numerical complications involved are studied through an investigation of the constituent parts ofp _e . Verification of present theory is then sought, from which justification may be derived for the estimation of elasticity fromp _e measurements. Attention is directed towards the predictions of Higashitani and Pritchard and the extension to the Tanner and Pipkin theory for Second-order fluids. The effects of variation of geometric dimensions and flow type uponp _e are also discussed.     

Molecular theories of nonlinear viscoelasticity of polymers

It is emphasized that considerable advances have been made recently in the development of the molecular theories of nonlinear viscoelasticity of concentrated solutions and melts of linear polymers. The new ideas in this exceptionally important field of the rheology of polymers are analysed. The methods by which the constraints (entanglements) imposed on the motion of macromolecules by the polymer environment are taken into account are also considered in the paper. The most detailed discussion is devoted to the model of topological constraints in the form of a tube and to the self-consistent theory of anisotropic micro-viscoelasticity which takes into account the relaxation nature of the interaction of macromolecules with their surrounding medium as well as the anisotropy of their mobility.Invited paper, presented at the XII-th All-Union Symposium on Rheology held in Riga (USSR), December 7–9, 1982.     

Predicting the nonlinear hereditary viscoelasticity of polymers

A mathematical model for the nonlinear hereditary viscoelasticity of polymer materials is proposed to predict deformation processes of various complexity — from simple relaxation and simple creep to complex deformation-relaxation and reverse relaxation processes with alternative loading and unloading. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 147–157, November–December, 2007.     

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.     

Shear flow rheological properties of vinylon- and glass-fiber reinforced polyethylene melts

Shear viscosity, shear stress and first normal-stress difference have been investigated for glass- and vinylon-fiber filled polyethylene melts over a wide range of shear rate by means of three kinds of instruments. The influence of fiber content and fiber properties on the rheological properties is discussed. The viscosity increases with increasing aspect ratio and fiber content, and the influence of these parameters on the flow properties is evident at low shear rates. The first normalstress difference increases more rapidly with increasing glass fiber content, especially at low shear stresses. The influence of vinylon fibers on the first normal stress-difference vs. shear-stress relationship is different from that of glass fibers.     

The rheology of suspensions of vinylon fibers in polymer liquids. I. Suspensions in silicone oil

Summary The steady shear flow properties of suspensions of vinylon fibers in silicone oil were measured by means of a cone-plate type rheometer. Three kinds of vinylon fibers used had no distributions of length and were more flexible than glass fibers and the like. The content of the fibers ranged from 0 to 7 wt.%. Shear viscosity, the first normal-stress difference, yield stress, and relative viscosity were discussed. Shear viscosity and relative viscosity increased with the fiber concentration and the aspect ratio, and depended upon the shear rate. The applicability of Ziegel's equation of viscosity for fiber suspensions was investigated. The first normal-stress difference increased with the fiber concentration, aspect ratio, and shear rate and its relative increase was much larger than for shear stress and viscosity depending on the properties of the characteristic time, The yield stress could be determined by Casson plots for large aspect ratio fiber suspensions even in low concentration comparing with the suspensions of spherical particles or powder. The influence of the flexibility of the fibers for the rheological properties of the fiber suspensions can not be ignored.With 12 figures and 2 tables     

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).     

Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks

Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally, but the lack of studies of real environmental conditions on these absorbers is felt. The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES). In this paper, the Burgers model is assumed for the viscoelasticity in an NES, and a linear oscillator system is considered for inve...     

In-line measurement of rheological properties of polymer melts

Shear viscosity (), first normal stress difference (N ₁), and extensional viscosity ( _E ) of polymer melts measured under processing conditions are important in process modeling, quality control, and process control. A slit rheometer that could simultaneously measure , N ₁, and the planar extensional viscosity ( _p ) was designed and tested by attaching it in-line to a laboratory model single-screw extruder. A tube (circular cross-section) rheometer to measure and the uniaxial extensional viscosity ( _u ) simultaneously was also designed and tested. Two commercial grades of LDPE (low density polyethylene) with melt index values of 6 and 12 were used as test materials for the study. Exit and hole pressure methods were used to estimate N ₁, and the entrance pressure drop method using the analyses of Cogswell, Binding, and Gibson (the last analysis used with the axisymmetric case only) was used to estimate _E .The hole pressure method was considered better than the exit pressure method to estimate N ₁ (due to the greater susceptibility of the latter to experimental errors). From the hole pressure method N ₁ was obtained from 100 kPa to 500 kPa over a range of shear rates from 40 s^–1 to 700 s^–1. Among the analyses used to estimate the extensional viscosity, Cogswell's is recommended due to its simpler equations without loss of much information compared to the other analyses. The range of extension rates achieved was 1 to 30 s^–1. The combination of the hole pressure and entrance pressure drop methods in a slit rheometer is a feasible design for a process rheometer, allowing the simultaneous measurement of the shear viscosity, first normal stress difference and planar extensional viscosity under processing conditions. Similarly, combining the entrance pressure drop measurements with a tube rheometer is also feasible and convenient.     

Non-symmetric viscoelasticity of anisotropic polymer liquids

The liquid crystalline (LC) polymers are considered as anisotropic viscoelastic liquids with nonsymmetric stresses. A simple constitutive equation for nematic polymers describing the coupled relaxation of symmetric and antisymmetric parts of the stress tensor is formulated. For illustration of non-symmetric anisotropic viscoelasticity, the simplest viscometric flows of polymeric nematics in the magnetic field are considered. The frequency and shear rate dependencies of extended set of Miesowicz viscosities are predicted. Received: 23 March 1999/Accepted: 13 December 1999     

Nonlinear viscoelasticity of polymer melts in different types of flow

The nonlinear viscoelastic properties of a fairly large class of polymeric fluids can be described with the factorable single integral constitutive equation. For this class of fluids, a connection between the rheological behaviour in different flow geometries can be defined if the strain tensor (or the damping function) is expressed as a function of the invariants of a tensor which describes the macroscopic strain, such as the Finger tensor. A number of these expressions, proposed in the literature, are tested on the basis of the measuring data for a low-density polyethylene melt. In the factorable BKZ constitutive equation the strain-energy function must be expressed as a function of the invariants of the Finger tensor. The paper demonstrates that the strain-energy function can be calculated from the simple shear and simple elongation strain measures, if it is assumed to be of the shape proposed by Valanis and Landel. The measuring data for the LDPE melt indicate that the Valanis-Landel hypothesis concerning the shape of the strainenergy function is probably not valid for polymer melts.     

An approach to the estimation of polymer melt elasticity

An effective method has been proposed to estimate the primary normalstress difference versus shear rate curves at temperatures relevant to the processing conditions only from the knowledge of the melt flow index, the molecular-weight distribution and the glass transition temperature of the polymer. The method involves the use of a unified curve obtained by coalescing the elastic response curves of various grades in terms of the modified normal-stress coefficient ₁ (MFI)² and a modified shear rate . Unified curves have been reported for low density polyethylene, high density polyethylene, polypropylene and nylon.Nomenclature C ₁ constant in eq. (4) - J _e steady state compliance (cm²/dyne) - proportionality constant in eq. (6) - L load (kg) - L ₁ load (kg) at ASTM test conditions - L ₂ load (kg) at required conditions - MFI melt flow index (gm/10 min) - number average molecular weight - weight average molecular weight - z-average molecular weight - (z+1)-average molecular weight - n slope of the shear stress vs. shear curve on a log-log scale - N ₁ primary normal-stress difference (dynes/cm²) - Q molecular weight distribution expressed as - T ₁is> temperature (K) at condition 1 - T ₂is> temperature (K) at condition 2 - T _g glass transition temperature (K) - T _s standard reference temperature equal toT _g + 50 K - shear rate (s^–1) - ₀ zero-shear viscosity (poise) - apparent viscosity (poise) - density (g/cm³) - ₁₂ shear stress (dynes/cm²) - ₁₁– ₂₂ primary normal-stress difference (dynes/cm²) - _1,0 zero shear rate primary normal-stress coefficient (dynes/cm² · sec²) - ₁ primary normal-stress coefficient (dynes/cm² · sec²) - ₂ secondary normal-stress coefficient (dynes/cm² · sec²) NCL-Communication No. 3106     

A micromechanical analysis of the nonlinear elastic and viscoelastic constitutive relation of a polymer filled with rigid particles

Micromechanical theory is applied to study the nonlinear elastic and viscoelastic constitutive relations of polymeric matrix filled with high rigidity solid particles. It is shown that Eshelby's method can be extended to the case of nonlinear matrix and Eshelby's tensor still exists provided that Poisson's ratio of the nonlinear matrix assumes constant value in deforming process and the rigidity of elastic filling particles is much higher than that of the matrix. A new method for averaging process is proposed to overcome the difficulty that occured in applying the ordinary equivalent inclusion method or the self-consistant method to nonlinear matrices. A rather simple constitutive equation is obtained finally and the strengthening effect of solid particles to composites is investigated. The work supported by the LNM, Institute of Mechanics, Chinese Academy of Sciences and by the National Natural Science Foundation of China     

The “decaying springs” model for irreversibility in polymer melts

In this work, entanglements in a polymer melt are modeled as a system of parallel springs which form and decay spontaneously. The springs are assumed to be nonlinear, and a certain fraction of them is torn apart by a certain strain.Based on these assumptions, a model of behavior in simple shear is developed. This model is shown to predict a behavior comprising that of a Wagner fluid, and is generalized to a tensorial model of single integral type. The integrand depends on a product of a material function, modeling reversible behavior, and a material functional which takes irreversible processes into account.Irreversibility of network disentanglement, which may occur when deformation changes or reverses direction, can be modeled in this way. It is shown that the two well-known Wagner constitutive equations with and without irreversibility assumptions are special cases of the model developed. In case of a deformation which does not change directions, the new material function and the material functional are multiplied to yield Wagner's damping function.When the rate of spring formation is a function of temperature, the developed model is shown to predict thermorheologically simple behavior. A constitutive equation for non-isothermal flow of polymers is developed with this assumption.     

Relaxational interactions and viscoelasticity of polymer melts. Part I: model development

Rheological equations of state for the concentrated solutions and melts of high polymers are derived by applying a structural approach. The dynamics of a macromolecule are considered on the basis of the fundamental model of the polymer chain, e.g., the bead-spring model. The drag forces describing correlations of motion macromolecules are determined by means of the relaxation equations. The oscillatory shearing flow of the melts is studied on the basis of the equations derived. Expressions for the dynamic modulus and relaxation times are determined. As can be judged from the form of the dependence of the dynamic modulus on frequency, the relaxation time distribution is the same as in real materials. The relaxation spectrum of high polymers has a terminal zone with abnormally long relaxation times.     

Step planar extension of polymer melts using a lubricated channel

A technique to do step planar extension on polymer melts has been developed using a rectangular channel with lubricated walls and the linear motor of the Rheometrics System Four rheometer. Using this method we probe the stress relaxation of two polymer melts, a linear low density polyethylene (LLDPE) and a highly branched low density polyethylene (IUPAC X), and compare the step planar extensional data to step shear data. Since a step planar deformation is theoretically equivalent to a step shear in a rotating frame of reference, we expect that the nonlinear modulus for step planar extension should be equivalent to that for step shear. Although we find the time dependence of the stress relaxation modulus to be the same in both shear and planar extension, the strain dependence is surprisingly different for the two experiments.     

SOME IDENTITY RELATIONS BETWEEN PLANE PROBLEMS FOR VISCO-AND ELASTICITY

I.IntroductionThereareimportantapplicationsfortheor}:ofplanea'iscoelasticit}'Inthefieldsofgeology,miningandconstructingetc.,butformostproblemsofviscoelastici[}'.theirsolutionsareobtainedfromthecorrespondingelasticsolutionsb}'"leansofthecorrespondenceprinc…