Stress relaxation after steady shear flow in immiscible model polymer blends

Stress relaxation in immiscible blends is studied for a well defined shear history, i.e. after prolonged steady state shearing. Model systems are used that consist of quasi-Newtonian liquid polymers. Hence the relaxation is dominated by changes in the morphology of the interface. Both shear stress and the first normal stress are considered. The measurements cover the entire concentration range. For dilute blends the interfacial contribution to the stress relaxation compares well with model predictions. Deviations occur when the matrix phase is slightly elastic. In that case the similarity between the relaxation of shear and normal stresses is also lost. The latter is attributed to a wider drop size distribution.Increasing the concentration of the disperse phase results in a complex evolution of the characteristic relaxation times. The normal stresses relax systematically slower than the shear stresses and the concentration curve includes two maxima. Even for equiviscous components the concentration curves are not symmetrical. It is concluded that even a slight degree of elasticity in the matrix phase drastically affects the morphology and the interfacial relaxation of such blends.     

The use of a generalized model based on the relaxation spectrum for calculation of the first normal stress difference from a flow curve

In this work a derivation has been made of a model based on the relaxation spetrum. This model can be used for the calculation of the first normal stress difference from a flow curve. The model has been generalized in order to describe better shear rate dependence of the first normal stress difference. Predictions of the generalized model are compared with experimental data. The comparison has been made for materials with dependencies taken from the literature, and also for materials investigated by the authors of this work.     

Elastic properties of polyethylene melts at high shear rates with respect to extrusion

At high shear rates a steady state of shear flow with constant shear rate, constant shear stress, and constant recoverable shear strain is observed in the short-time sandwich rheometer after some few shear units already. The melt exhibits rather high elastic shear deformations and the recovery occurs at much higher speed than it is observed in the newtonian range. The ratio of first normal stress difference and twice the shear stress, being equal to the recoverable strain in the second-order fluid limit, significantly underestimates the true elastic shear strains at high shear rates. The observed shear rate dependence of shear stress and first normal stress difference as well as of the (constrained) elastic shear strain is correctly described on the basis of a discrete relaxation time spectrum. In simple shear a stick-slip transition at the metal walls is found. Necessary for the onset of slip is a critical value of shear stress and a certain amount of elastic shear deformation or orientation of the melt.     

Time-dependent,non-Newtonian behavior of viscoelastic materials

Summary The present paper is a further development in our effort to describe non-Newtonian flow behavior. As in the past, the thixotropic shear stress or shear rate is characterized by the kinetic theory and currently a modified form ofOldroyds model provides the elastic contribution. Presented here are further modifications of the model in order to predict the first normal stress difference and logical changes to make the model more internally consistent.In order to test the approach, transient and steadystate data have been obtained with aWeissenberg rheogoniometer on a solution of polymethylmethacrylate in diethylphthalate. Both constant stress and constant shear rate data have been taken over a broad range of the parameters involved. To facilitate data acquisition, a PDP-15, model 30 computer was programmed to gather data at rates of 1000 points per second. Of the data obtained there is a great deal of uncertainty about the normal stress results.By optimization methods, the constants of the model were evaluated from steady-state data and transient results on stress growth and shear rate growth. Comparisons of the model with stress growth results as well as the prediction of the first normal stress difference are given. The inadequacies of the analysis are emphasized and possible points for improvement are made.With 8 figures     

Stress relaxation as a microstructural probe for immiscible polymer blends

Relaxation has been investigated in immiscible blends that consist of slightly viscoelastic components. Both the shear and normal stresses have been measured after cessation of steady shear flow as well as after transient shear histories. The latter can generate a fibrillar structure which can relax by either retraction or break-up via end-pinching or Rayleigh instabilities. Each of these three relaxation mechanisms is reflected in the shape of the stress curves, from which also the corresponding structural time scales can be deduced. The experimental results have been used to evaluate the Doi-Ohta and Lee-Park models for immiscible blends. The scaling relations by Doi-Ohta are confirmed by the experimental results, but none of the existing models can correctly predict the complex relaxation behaviour observed for a highly deformed droplet phase. In the present study an alternative approach has been proposed. The stress relaxation due to fibril break-up via Rayleigh instabilities has been predicted successfully by combining physical models for the structural changes with the basic approach of the Doi-Ohta model.     

A constitutive equation derived from Lodge's network theory

A constitutive equation is presented, derived from Lodge's molecular network theory. The equation is of a form similar to the one presented previously by Carreau. The rate of creation of junctions and the probability of loss of junctions depend here also on the second invariant of the rate-of-strain tensor. The dependence, however, is through simple exponential functions, resulting in easy-to-use equations.Material functions are presented for the viscosity, the primary normal stress coefficient, the complex viscosity, the stress relaxation after cessation of steady simple shear, the stress growth after onset of steady simple shear and for elongational flow. The usefulness of the simplified (series truncated) equations is discussed and the model is evaluated with typical viscoelastic data.     

Influence of the flow history on stress growth and structure changes in the thermotropic liquid crystalline polymer Vectra B950

The structure changes in the start-up flow of the thermotropic liquid crystalline polymer Vectra 8950 are probed by performing transient experiments after various flow histories. The shear and normal stress growth curves of a squeezed sample and of a randomly oriented sample show a pronounced overshoot at low strains, whereas the stress growth curve of a sample pre-sheared until steady state shows a gradual increase. This first peak is associated with the re-orientation of the director into the shearing plane. All stress transients show a second broad maximum at large strains that results from the generation of a steady defect network. The effect of varying the relaxation period after pre-shearing is reflected in the appearance of two peaks in the subsequent stress growth curves. One of these peaks shifts linearly with re laxation period and the other is more or less fixed in position. The orientation of the molecules during steady shear flow is on average in the flow direction. Intermediate orientation levels may exist in the transient depending on the amount of strain. The material is able to maintain the flow-induced orientation distribution for a long time after cessation of flow. This is reflected in a similar fashion in the initial magnitudes of the stresses and the dynamic moduli after various preshear strains. Moreover, the dynamic moduli decrease with time after cessation of steady shear flow, indicating that the orientation increases during relaxation.     

Transient shear flow properties of fiber-filled polyethylene melts

The growth and relaxation of shear and normal stresses have been investigated for glass and carbon fiber-filled polyethylene melts over a wide range of shear rates and temperatures by means of a cone-and-plate rheogoniometer. Flow parameters and flow curves characterizing the stress overshoot and relaxation phenomena of the fiber-filled systems were determined experimentally. The influence of fiber loading, fiber size and temperature on the transient flow parameters are discussed.Predictions by the Meister and Bogue constitutive equations were compared with the experimental data for the transient shear and normal stresses. These equations predict satisfactorily the non-linear transient shear flow of polymer melts and its fiber-filled systems.     

Large-amplitude oscillatory shear flow from the corotational Maxwell model

Using the single relaxation time corotational Maxwell fluid, we derive explicit analytical expressions for the first, third, and fifth harmonics of the alternating shear stress response in large-amplitude oscillatory shear (LAOS). We also derive corresponding expressions for the zeroth, second, and fourth harmonics of both the first and second normal stress differences. These harmonics are found to depend upon just two dimensionless groups: the Deborah and Weissenberg numbers, each of which causes non-Newtonian behavior. The form of the solution for the corotational Maxwell model in LAOS matches the forms of the analytical solutions for two molecular models for dilute solutions and one for concentrated solutions or melts. We also derive an analytical solution for the corotational Maxwell model after startup of LAOS. For this we find that both small and large amplitude cases approach a periodic limit cycle (alternance) at the same rate for both the shear stress response and for the normal stress differences. For molten high density polyethylene that is lightly filled with carbon black, we find good quantitative agreement with measured LAOS behavior when our analytical solution is superposed for multiple relaxation times.     

Exact static solutions to piezoelectric smart beams including peel stresses. II. Numerical results,comparison and discussion

This part presents the numerical results, comparisons and discussion for the exact static solutions of smart beams with piezoelectric (PZT) actuators and sensors including peel stresses presented in Part I. (International Journal of Solids and Structures, 39, 4677–4695) The actuated stress distributions in the adhesive and the adhesive edge stresses varying with the thickness ratios are firstly obtained and presented. The actuated internal stress resultants and displacements in the host beam are then calculated and compared with those predicted by using the shear lag model. The stresses in the adhesive caused by an applied axial force, bending moment and shear force are calculated, and then used to compute the sensing electric charges for comparison with those predicted using the shear lag model. The numerical results are given for the smart beam with (a) one bonded PZT and (b) two symmetrically bonded PZTs, with a comparison to those predicted using the shear lag model. Novel, simple and more accurate formulas for the equivalent force and bending moment induced by applied electric field are also derived for the host beam with one PZT or two symmetrically bonded PZTs. The symmetric shear stress and the anti-symmetric peel stress components caused by a shear force are discussed. In addition, in the case of PZT edge debonding, the stress redistribution in the adhesive and the self-arresting mechanism are also investigated.     

Is there a relation between the relaxation time measured in CaBER experiments and the first normal stress coefficient?

We investigate a variety of different semidilute polymer solutions in shear and elongational flow. The shear flow is created in the cone-plate-geometry of a commercial rheometer. We use capillary thinning of a filament that is formed by a polymer solution in the Capillary Breakup Extensional Rheometer (CaBER) as an elongational flow. We compare the relaxation time measured in the CaBER with relaxation times based on the first normal stress difference and the zero shear polymer viscosity that we measure in our rheometer. All of these three measurable quantities depend on different fluid parameters—the viscosity of the solvent, the polymer concentration within the solution, and the molecular weight of the polymers—and on the shear rate (in the shear flow measurements). Nevertheless, we find that the first normal stress coefficient depends quadratically on the CaBER relaxation time. Several scaling laws are presented that could help to explain this empirical relation.     

On the relaxation of shear and normal stresses of viscoelastic fluids following constant shear rate experiments

By means of a cone and plate rheometer the relaxation of the shear stress and the first normal stress difference in polymer liquids upon cessation of a constant shear rate were examined. The experiments were conducted mostly in a high shear rate region of relevance for the processing of these materials. The relaxation behavior at these shear rates can only be measured accurately under extremely precise specifications of the rheometer. To determine under which conditions the integral normal thrust is a convenient measure for the relaxing local first normal stress difference the radial distribution of the pressure in the shear gap was measured. The shape of relaxation of both the shear stress and the first normal stress difference could be closely approximated for the entire measured shear rate and time range by a two parameter statistical function. In the range of measured shear rates, one of the parameters, the standard deviationS, is equal for the shear and the normal stress, and is independent of the shear rate within the limit of experimental error. The second parameter, the mean relaxation timet _50, of the shear stress andt _50, of the first normal stress difference, can be calculated approximately from the viscosity function and only a single relaxation experiment.     

Shear relaxation characteristics of rock joints under stepwise loadings

The stress relaxation characteristic of rock mass is an important aspect of rheology and has important practical significance for rock engineering. In order to investigate the relaxation characteristic of rock joints with different slope ratios and normal stresses, a series of shear stress relaxation tests were conducted on artifical rock joints poured by cement mortar. Test results show that the relaxation curves can be divided into three stages, i.e. instantaneous relaxation stage, attenuation relaxation stage, and stable relaxation stage. Furthermore, the nonlinear Maxwell relaxation equation was obtained by using the relation between the viscosity coefficient and time, and the theoretical curves based on the empirical equation agreed well with the test results. Moreover, the change law of the initial viscosity coefficient was investigated. Accordingly, a stress relaxation method, termed as relaxation stress peak method, was proposed to determine the long-term strength of rock joints.     

Friction relaxation model for fast transient flows application to water hammer in two-phase flow – The WAHA code

The paper deals with the problem of the wall shear stress during rapid transient 1D flows in a piping system caused by water hammers in two-phase flow induced by a fast valve closure. The evolution of the transient wall shear stress is interpreted in terms of two steps. The first step is a sudden and dramatic change of the wall shear stress due to the passage of the pressure wave. The second step is a relaxation process of the shear stress which is modeled from the Extended Irreversible Thermodynamics theory. The friction relaxation model (FRM) presented in the first part of this paper describes both steps of the evolution of the wall shear stress during water hammers. The second part of the paper deals with the application of the FRM model as a closure law in the WAHA code. The main purpose of the WAHA code is to predict various situations relative to single- and two-phase water hammer transients in piping systems. The last part of the paper deals with the simulation of several cases from the UMSICHT databank using the adapted WAHA computer code with the FRM model. The results of these simulations are systematically compared with the experimental data. It is concluded that the new FRM model has a clear effect on water hammer pressure wave damping and on the pressure wave propagation velocity.     

Rheological properties of poly(dimethylsiloxane) filled with fumed silica: II. Stress relaxation and stress growth

The steady-state viscosity, stress relaxation, and stress growth function were measured for a 3.0% by weight dispersion of fumed silica in poly(dimethylsiloxane); using a two-step shear rate history. For short rest times the stress growth function η⁺ increased monotonically towards steady-state, while at long rest times η⁺ exhibited large overshoot followed by a monotonic decrease to steady-state. For intermediate rest times η⁺ exhibited more complicated transient behaviour. The effect of both shear rates in the two-step shear-rate history on η⁺ have been studied, and the resulting rheological behaviour is discussed in terms of phenomenological microstructural models.     

A constant-contour-length reptation model without independent alignment or consistent averaging approximations for chain retraction

A reptation model for the primitive chain that does not assume independent alignment or consistent-averaging for the retraction process, or equilibrium relaxation for the reptation process is proposed and compared to the analytical expressions of Doi and Edwards in single-step, double-step strains and steady-state shear flow. The Doi and Edwards model with independent alignment approximation underpredicts the magnitude of the relaxation modulus by 25%, and consistently overpredicts the magnitude of the damping function; for steady shear flow, it predicts the correct shape for the steady-state viscosity and the first normal stress difference coefficient, although the magnitude is incorrect. The analytical expressions of Doi and Edwards without independent alignment approximation are excellent approximations to the damping function. In double-step strains, the expressions of Doi assuming consistent averaging, but no independent alignment, predict well the stress decay following the second strain. Linear response theory is found to be invalid for describing the stress relaxation following single-step strain for the models considered. Similar to the Doi and Edwards model, no overshoot for the first normal stress difference is observed for the simulation model. Unlike the Doi equation derived without the independent alignment approximation but restricted to double-step strains, the simulation model proposed here can be easily generalized to complex flow fields. No contour length fluctuation or constraint release is considered in this model, and chain retraction is assumed to be instantaneous.     

The measurement of the yield stress of liquids

An analysis has been made of different methods of measuring the yield stress of liquids. In the experimental program, a comparison is made of measurements of the yield stress using an Instron 3250 Rheometer in several geometries (cone-plate, parallel plate and eccentric disk) in shear flow and stress relaxation, a laboratory vane and a cone penetrometer. Good agreement has been obtained between the shear flow data and the laboratory vane, while stress relaxation appears to underestimate the yield stress.     

Possibility of a master plot for material functions of high-polymer solutions

A new systematic method has been developed for estimating rheological properties (such as relaxation time, viscosity, primary normal stress difference) for various polymer solutions form the zero-shear-rate specific viscosity and a dimensionless shear rate.A linear relation has been found between the relaxation time and the zero-shear-rate specific viscosity on a double-logarithmic plot, and a new master plot for the primary normal stress difference has been established. In addition the applicability of the FENE-P dumbbell model to high-polymer solutions is discussed.     

Theoretical and experimental investigation of the flow of a viscoelastic fluid in the gap between two rotating disks

A theory of the nonlinear viscoelastic behavior of polymer fluids has been constructed in [1]. The theory was used in [2] to investigate the motion of a nonlinear viscoelastic medium under steady and unsteady deformation rates in simple shear flow, and a comparison was made with experiment. The experiments in [2], which were performed on a cone-plate Weissenberg rheogoniometer, indicate that this arrangement is unsuitable for measurements of normal stresses under unsteady conditions in fluids with a fairly high viscosity. Below, we will show the suitability of using a disk-disk Weissenberg rheogoniometer to measure normal stresses in this case for unsteady conditions (transition to steady flow and stress relaxation). In this regard, a theoretical study of the flow of a viscoelastic fluid in the gap between rotating disks is needed. Note that in this case new information will be obtained from a comparison with simple uniform shear flow, since in the flow of a polymer between two disks all three normal stress components contribute to the axial force, while in the gap between a cone and a plate only the first normal stress difference contributes to the normal force.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 25–30, March–April, 1976.     

Normal stress differences in large-amplitude oscillatory shear flow for the corotational ��ANSR�� model

Using the integral form of a nonlinear corotational model, we derive explicit analytical expressions for the zeroth, second, and fourth harmonics of the second normal stress difference in large-amplitude oscillatory shear (LAOS). This model yields an arbitrary normal stress ratio (ANSR) in any simple shearing deformation, including LAOS. This corotational ANSR model adds one parameter to the corotational Maxwell model, a time constant ?? ₀ controlling the ratio ??₂/??₁ for both the real and imaginary parts of each harmonic of the normal stress difference. The explicit analytical expressions for all harmonics of the alternating shear stress and first normal stress difference responses in LAOS match those obtained previously for the corotational Maxwell model. We evaluate the corotational ANSR model for the case of a single Maxwell relaxation time fluid.