Single Integral Constitutive Equation Research Articles

Upgrading the melt flow index to rheogram approach in the low shear rate region

AbstractThe unified viscosity function curves proposed earlier by the authors have an inherent limitation in the low shear rate region. This limitation is the effect of using the melt flow index as a normalizing factor to obtain the coalesced curves, which itself is insensitive to molecular parameters such as molecular weight distribution. A single integral constitutive equation of the BKZ type is used to derive the viscosity function which would be useful in generating unified curves based on the melt flow index but devoid of the limitation of molecular weight distribution effects in the low shear rate region.

A relation between shear flow material functions for a single-integral constitutive equation

This contribution suggests some relations between normal-stress differences and shear stress, both in the steady state and in the transient regimes. It shows that the general form of the BKZ-type equation can be used for the discussion of the relations between several material functions. In addition, a geometric interpretation is offered and some examples are discussed.

Multiaxial elongational flows of polymer melts—classification and experimental realization

For the class of multiaxial elongational flows with constant strain rates, a new classification system is proposed. Ordering the main strain rates such that ∈ . 0  ∈ . 11 ⩾ ∈ . 22 ⩾ ∈ . 33, the constant ratio m  ∈ . 22/ ∈ . 11 characterizes the special type of flow, e.g. in simple elongation m = − 1/2, in equibiaxial elongation m = +1, and in planar elongation (“pure shear”) m=0. For the representation of the test results, three elongational viscosities μ i = A i σ i / ∈ . 0 are defined from the three normal stress differences σ i such that for the linear viscoelastic limit the time-dependent linear viscoelastic shear viscosity follows: μ i ( t)→ ▪ t. For the realization of such multiaxial elongational flows, rotary clamps are used. They are modified such that the force components along and normal to the direction of motion at the clamp can be measured separately. The investigation was performed with polyisobutylene in simple, equiviaxial, and planar elongations. The latter mode of testing is of special interest because two different elongational viscosities are determined simultaneously. The results show that for small deformations the material behaviour is linear viscoelastic. For larger deformations there are different deviations the character of which depends on m, i.e. the type of the elongational flow. From the general single-integral constitutive equation it follows that the predicted stress differences for the different m are very differently connected with the different parts of the equation, as in the case of the two normal stress differences in simple shear flow.

Network theory of the nonlinear behaviour of polymer melts. Simple shear flow

The network model developed in a previous paper is applied to the simple shear flow of polymer melts. The constitutive equation obtained consists of two terms. One of them describes the stress due to the network strands which exist at the onset of the deformation, dissociate during the deformation and result in a single integral constitutive equation with a strain dependent damping function. The formulation of the damping function in invariant form seems to be almost impossible. The second normal stress differenceN 2 of the model is not zero,but has negative values. According to our model this is a consequence of the deformation dependence of the disentanglement process. The theory is compared with experimental data for a LDPE melt. It is found that the model explains the main features of the shear flow behaviour of the LDPE melt investigated preciously.

Large axisymmetric deformation of a non-linear viscoelastic circular membrane

The problem of a viscoelastic circular membrane deforming under its own weight is considered. The membrane, clamped in a horizontal plane at its boundary, is formed from a non-linear viscoelastic material whose behaviour is modelled by a non-linear single integral constitutive equation. The sag of the membrane is considered quasi-static and provision is made in the model for making the relaxation time stretch-dependent. The analysis leads to a system of three equations, two of which are first-order non-linear partial-differential integral equations while the third is an algebraic relation. Time descretization enables the differential equations to be solved as ordinary differential equations which may be dealt with by a familiar step-wise integration process. Numerical solutions are obtained for membranes with both stretch-dependent and stretch-independent relaxation times.

Elongational behaviour of polymer melts in constant elongation-rate, constant tensile stress, and constant tensile force experiments

The extensibility of polymer melts is of great practical importance for polymer processing. On the basis of a single-integral constitutive equation with a straindepent memory functional verified for two similar well-characterized LDPE melts, predictions are made on material behaviour in uniaxial extension under constant strain-rate, tensile stress, and tensile force conditions. It is found that experiments at constant tensile force are more adequately described by assuming purely viscous response of the polymer melts than by assuming Maxwell-model type of behaviour.

Model analysis of nonlinear viscoelastic behaviour by use of a single integral constitutive equation: Stresses and birefringence of a polystyrene melt in intermittent shear flows

A single integral constitutive equation with strain dependent and factorized memory function is applied to describe the time dependence of the shear stress, the primary normal-stress difference, and, by using the stress-optical law, also the extinction angle and flow birefringence of a polystyrene melt in intermittent shear flows. The theoretical predictions are compared with measurements. The nonlinearity of the viscoelastic behaviour which is represented by the so called damping function, is approximated by a single exponential function with one parametern. The damping constantn as well as a discrete relaxation time spectrum of the melt can be determined from the frequency dependence of the loss and storage moduli.

The Irreversibility Assumption of Network Disentanglement in Flowing Polymer Melts and its Effects on Elastic Recoil Predictions

Tanner has suggested a constitutive equation for polymer melts and solutions based on a network rupture hypothesis: entanglements are lost irreversibly in the process of deformation as soon as a limiting strain magnitude is reached. Although this model shows serious defects in describing time‐dependent material properties it has the appealing feature of irreversibility. On the other hand the BKZ model (considered here in the simplified version of a separable memory function), being more useful in describing startup experiments, does not posses this feature of irreversibility, leading to largely erroneous results in recovery calculations. An improved version of a single integral constitutive equation is presented, where the irreversibility of the process of network disentanglement is incorporated. Predictions of elastic recoil behavior in elongation and shear are compared with experimental results for a well‐characterized LDPE melt.

Nonlinear shear creep and constrained elastic recovery of a LDPE melt

Shear creep and constrained elastic recovery experiments on a well characterized low-density polyethylene melt are reported. The temperature dependence of the shear strain and the primary normal stress difference is discussed in detail. Comparison is made with predictions of a strain-dependent single integral constitutive equation, which has already been successfully used for the same polymer melt to describe the stress growth after sudden imposition of a constant shear rate flow and stress relaxation after cessation of steady shear flow. It should be emphasized that this constitutive equation contains no adjustable parameters. The linear-viscoelastic part of the memory function is related to the linear-viscoelastic relaxation spectrum, while the nonlinear, strain-dependent part was determined from rapid-strain experiments. In the case of a prescribed shear stress history the resulting integral equation cannot be solved by closed integration but has to be inverted by numerical methods. Agreement between theoretical predictions and experimental data is rather encouraging for shear strain and primary normal stress difference during creep and retardation tests. Within experimental error, the strain and shear rate dependence of the recoverable portion of the total strain can be correctly predicted.

A constitutive analysis of uniaxial elongational flow data of low-density polyethylene melt

The uniaxial elongational flow data of a well-characterized LDPE melt as measured by Meissner, and Laun and Münstedt are analysed in terms of a single integral constitutive equation, which has already been successfully used to described shear flow experiments of the same polymer melt. It is found, as in the case of shear flow, that within experimental error the memory function of this constitutive equation can be separated into a product of two functions for uniaxial extensional flow, the one expressing the time dependence, the other the strain dependence of the memory function. Calculated values flow data. A steady state of tensile stress is predicted at higher strains in accordance with experimental evidence. Further comparison is made between theoretical predictions and extensional creep and elastic recoil experiments performed on the LDPE melt considered. The resulting nonlinear integral equation was solved by a method of successive integration. Agreement is found to be satisfactory for Hencky strains ϵ up to ϵ = 4, while at higher strains predicted values of recoverable strain lie above experimental results.

Description of the non-linear shear behaviour of a low density polyethylene melt by means of an experimentally determined strain dependent memory function

The applicability of a single integral constitutive equation with strain dependent memory function for the description of the nonlinear shear behaviour of a LDPE melt is examined. The generalized memory function is expressed as a product of Lodge's rubberlikeliquid memory function $$\mathop \mu \limits^ \circ (t - t\prime )$$ and a damping function h(γt, t′). $$\mathop \mu \limits^ \circ $$ characterizes the time dependence of the linear viscoelastic behaviour and is determined by measurements of the frequency dependence of the complex shear modulus. The damping function describes the nonlinearity of the shear behaviour and can directly be determined by measurements of the shear relaxation modulus. From the temperature invariance of the damping function it follows that also in the nonlinear range a variation of temperature only corresponds to a shift in time scale which can be described by the shift factora T (T). By means of the experimentally determined memory function the shear viscosity and the primary normal stress coefficient as functions of shear rate and temperature can be predicted. The time dependence of the shear stress and of the primary normal stress difference in stressing tests and the relaxation behaviour is described correctly.

The prediction of time-dependent non-linear stresses in viscoelastic materials: III. Memory-function non-linearity

The new measure I of strain developed in Part I has been used in a single-integral constitutive equation containing a memory functionμ dependent on both history s and the second strain-rate invariant ii D . Three methods of calculating this non-linear memory function from experimental data are presented, and their relative merits are discussed. When applied to polyisobutylene solution data all three methods yield similar results and show an initially positive memory function decreasing to slightly negative values before finally decaying to zero. At amounts of shear a less than 1.5,μ is equal to the linear value but for a ≈2, μ falls rapidly to much less than the linear value. These results are in agreement with earlier work on superposed oscillatory and continuous shear, which was interpreted in terms of a shear-rate dependent relaxation spectrum truncated at a time inversely proportional to the shear rate. The present results also shed light on the objection that is sometimes raised to memory functions that are dependent on the strain rate.

The prediction of time-dependent non-linear stresses in viscoelastic materials: I. Selection of constitutive equation and strain measure

The theories for the prediction of time-dependent, non-linear stresses in viscoelastic materials such as polymers are reviewed, and it is noted that the commonly observed stress non-linearity may be ascribed either, as is usually done, to memory-function non-linearity or, alternatively, to strain-measure non-linearity. To investigate the latter alternative whilst retaining a general memory-function non-linearity, a single-integral constitutive equation of the Bird—Carreau type is employed but with an arbitrary strain measure I in place of the normally employed Finger tensor F. This model includes as special cases a large proportion of the constitutive equations previously employed for predictive purposes and in particular with a linear memory function it is shown to be indistinguishable, with the normally conducted shear experiments, from the successful BKZ model. In the new model the shear component I 12 of the strain measure can be found from experimental results obtained in the startup of steady shear flow, without specification or restriction of memory-function non-linearity. The form of I 12 found from experiment is quite non-linear in shear a for ¦ a¦> 2, and hence differs from the F tensor for which F 12 = a. The same form for I 12 found for a variety of polymer solutions and a polymer melt and consequently a simple function describing I 12 is proposed as a new, material-independent, strain measure.

Prediction of primary normal stress difference from shear viscosity data using a single integral constitutive equation

Based on a single integral constitutive equation with a strain-dependent memory function, a relation between the primary normal stress functionθ and the shear viscosity functionη is proposed. According to this theory, the primary normal stress functionθ can be obtained from viscosity data by simple differentiation of the viscosity functionη with respect to the shear rate $$\dot \gamma$$ , and multiplication by a factor (−1/n). The material parametern is thereby associated with the strain dependence of the memory function. This relation was compared with the viscosity and primary normal stress data of six polymer melts, three polymer solutions, and an aluminium-soap solution, which were measured by several research groups and are available in the literature. In spite of the vast differences in physical constitution and chemical structure of the melts and solutions considered, agreement between predicted and measured values was encouraging.

Experimental study of the rheological properties of butadiene-styrene gum elastomers and compounds

The linear viscoelastic properties of a polybutadiene and three butadiene-styrene copolymers were determined using stress relaxation measurements. The Tobolsky-Murakami Procedure X was applied to the data to characterize a discrete relaxation spectrum. The nonlinear behavior was studied using transient shear stress buildup experiments. A single integral nonlinear constitutive equation was able to represent both the linear and nonlinear data if a Bogue-type relaxation time was used which involves a time-averaged second invariant of the rate of deformation matrix. The linear viscoelastic behavior is in essence represented by a single function G(t) (or the parameters Gm, τm, Gm−1, τm−1, Gm−2, τm−2, ⋯). Similar studies were carried out on a butadiene-styrene copolymer based tread and sidewall compounds but the shear stress transient data were incompatible with the stress relaxation results. The reasons for this are discussed.

A Generalized Strain Measure in a Single-Integral Constitutive Equation

The Bird-Carreau single-integral constitutive equation is considered with an unspecified and unrestricted form of non-linear memory function, and with an arbitrary strain measure, I, in place of the normally used Finger and Cauchy strain tensors. It is shown that the form of the shear component of the strain measure, I12, can be derived from experimental results obtained in the start-up of steady shear flow; and that normal stresses can be predicted from shear stresses. In contrast to the Finger and Cauchy tensors, the form of I12 obtained from experiment is decidedly non-linear in the amount of shear, and this results in markedly improved normal stress predictions. It is concluded that some of the previous failures of the Bird-Carreau type of equation have been due to a poor choice of strain measure, and not (as has been suggested) to the use of an inappropriate form of memory function. The new strain measure might usefully be applied to other flows.

Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt

A single integral constitutive equation is presented, which gives a reasonable description of the non-linear shear and elongational behavior of a low-density branched polyethylene melt at constant strain rate observed byMeissner. The memory function of this integral equation is a product of the memory function of Lodge's rubber like-liquid theory and a damping functionh = exp [ —n\(h = \exp [ - n\sqrt {I_2 - 3} ]\)], whereI2 is the second invariant of the Finger tensor. Stress- and normal stress overshoot are described as well as shear rate dependence of the transient shear viscosity and the first normal stress coefficient. Tensil stress overshoot and a finite, rate-dependent steady-state elongational viscosity is predicted in simple elongational flow.

Transient and Steady Shear Behavior of SBR Polymers

The low shear rate viscosity behavior of SBR random copolymers is investigated in steady and transient shear flow. The polymers studied have high enough molecular weights to make them of commercial interest. The measurements were performed on an apparatus specially designed but readily available and easy to operate. The results showed that even at very low shear rates, the polymers exhibit non-Newtonian steady shear viscosities. A log-log plot of zero shear rate viscosity vs weight-average molecular weight yields a straight line of slope 3.5. The transient shear data show no stress overshoot at shear rates when the viscosity is non-Newtonian. A theoretical discussion of the independence of these two nonlinear effects is provided. The discussion is based on some popular single-integral constitutive equations. It is shown that a recent modification by Carreau to the Lodge elastic liquid model is very realistic. That is, Carreau's model allows one to obtain separately transient shear stress overshoot and non-Newtonian steady shear viscosity.

Large Axisymmetric Deformation of a Nonlinear Viscoelastic Membrane Due to Spinning

The problem of a viscoelastic membrane undergoing large planar deformation due to spinning is solved. The membrane, rigidly bonded at its inner boundary and traction-free at the outer, consists of a nonlinear viscoelastic solid whose behavior is modeled by a nonlinear single integral constitutive equation. The model includes the possibility that the relaxation time is influenced by the amount of stretching. Spinning is considered to be sufficiently slow so that shear effects can be neglected. The formulation uses radial and circumferential stretch ratios as dependent variables. These satisfy a system of first-order nonlinear partial-differential integral equations. The numerical procedure at each time step obtains the current spatial derivatives of the stretch ratios in terms of the current and previously determined stretch ratios. This gives essentially a first-order nonlinear system of differential equations, of the same structure as that obtained in the elastic version of the problem [1], which is integrated numerically. Solutions are obtained for both strain-dependent and strain-independent relaxation times.

On the orthogonal superposition of simple shearing and small-strain oscillatory motions

Certain limiting high-and low-frequency relations peculiar to the general Kaye-Bernstein-Kearsley-Zapas single integral constitutive equation are developed for orthogonally superposed simple shearing and small-strain time-sinusoidal shearing. The relations do not hold for a general simple fluid nor for single-integral constitutive equations in which arbitrary dependence of the kernel function on the rates of deformation is allowed. It is shown that solutions of polyisobutylene in cetane (4.4%, 6.8%) and solutions of polyethylene-oxide (0.7%) and J-100 (0.1%) in water obey the relations quite closely.