A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity

This paper presents the development of a physical-based constitutive model for the representation of viscous effects in rubber-like materials. The proposed model originates from micromechanically motivated diffusion processes of the highly mobile polymer chains described within the formalism of Brownian motion. Following the basic assumption of accounting for the elastic and the viscous effects in rubber viscoelasticity by their representation through a separate elastic ground network and several viscous subnetworks, respectively, the kinetic theory of rubber elasticity is followed and extended to represent also the viscous contribution in this work. It is assumed that the stretch probability of certain chain segments within an individual viscous subnetwork evolves based on the movement of the chain endpoints described by the Smoluchowski equation extended in this work from non-interacting point particles in a viscous surrounding to flexible polymer chains. An equivalent tensorial representation for this evolution is chosen which allows for the closed form solution of the macroscopic free energy and the macroscopic viscous overstress based on a homogenization over the probability space of the introduced micro-objects. The resulting model of the viscous subnetwork is subsequently combined with the non-affine micro-sphere model and applied in homogeneous and non-homogeneous tests. Finally, the model capacity is outlined based on a comparison with in the literature available experimental data sets.     

A non-affine network model for polymer melts

In this paper, a network model of polymer melts is proposed in which network junction points move non-affinely. In this non-affine motion, junction points follow particle paths as seen by an observer rotating at the fluid element's net rigid-rotation rate. The speed at which junction points move is reduced as the network segments near their maximum extensions. In order to maintain a frame invariant model, it is necessary that the vorticity be decomposed into two portions, such that, = _R + _D . The deformational vorticity, _D , arises from shear deformation and is frame invariant while the rigid vorticity, _R , is frame dependent. A constitutive equation based on this finitely extensible network strand (FENS) motion is developed. The model illustrates how rotations that cause changes in the relative orientation of a fluid element with its surroundings can be incorporated into a constitutive equation using the deformational vorticity. The FENS model predicts a shear-thinning viscosity, and the Trouton viscosity predicted by the model is finite for all elongation rates. Finally, stochastic simulation results are presented to justify a mathematical approximation used in deriving the constitutive equation.     

Sticky dumbbells: from Hookean dumbbells to transient network

The Green-Tobolsky theory of transient networks is merged to the Hookean dumbbell model by considering Hookean sticky dumbbells, whose beads can randomly be stuck to a network submitted to affine deformation, or be set free from the network and undergo a free diffusive Brownian motion in the solvent. Sticking to and releasing from the network is treated as an instantaneous chemical reaction. This model has a closed-form solution, in which the stress is the sum of two (resp. three) Maxwellian codeformational relaxations for dumbbells with one (resp. two) sticking beads. When Brownian diffusion is faster than the chemical kinetics, one of the modes of two-sticking beads dumbbells is the Green-Tobolsky network relaxation, whereas the other modes correspond to fast configurational relaxations. In the opposite limit of fast chemical kinetics compared to Brownian relaxation, the effect of the network is to slow down the configurational response of Hookean dumbbells. Sticky dumbbells thus realise a continuous transition from Hookean dumbbells to transient networks.     

Rheological characterisation of a high-density polyethylene with a multi-mode differential viscoelastic model and numerical simulation of transient elongational recovery experiments

The rheological characterisation of a high-density polyethylene is performed by means of measurements of the storage and loss moduli, the shear viscosity and the transient uniaxial elongational viscosity, the latter being obtained with the Meissner extensional rheometer. The rheological behaviour of the polymeric material is described by means of a multi-mode Phan Thien-Tanner fluid model, the parameters of which are successively fitted on the basis of the linear and non-linear properties. By using a semi-analytical technique and the finite element method, numerical investigations are performed for the shape recovery of the sample, and the predictions are compared with their experimental counterparts. Surface tension effects are also explored. We discuss the agreement between the experiments and the simulation results. Received: 15 October 1998 Accepted: 22 December 1998     

Constitutive equations for non-affine polymer networks with slippage of chains

A model is derived for isothermal three-dimensional deformation of polymers with finite strains. A polymer fluid is treated as a permanent network of chains bridged by junctions (entanglements). Macro-deformation of the medium induces two motions at the micro-level: (i) sliding of junctions with respect to their reference positions that reflects non-affine deformation of the network, and (ii) slippage of chains with respect to entanglements that is associated with unfolding of back-loops. Constitutive equations are developed by using the laws of thermodynamics. Three important features characterize the model: (i) the symmetry of relations between the elongation of strands and an appropriate configurational tensor, (ii) the strong nonlinearity of the governing equations, and (iii) the account for the volumetric deformation of the network induced by stretching of chains. The governing equations are applied to the numerical analysis of extensional and shear flows. It is demonstrated that the model adequately describes the time-dependent response of polymer melts observed in conventional rheological tests.     

Transient network model for a multiphase polymeric fluid

Rheological behaviors of entangled polymer melts filled with rigid solid particles are phenomenologically described based on a Yamamoto type transient network model. Entanglement points are described by one kind of temporary junctions with a lifetime _p . Presence of fillers introduces another kind of network junctions with a lifetime _s , characteristic of adsorption of polymer chains at the filler surface. The degree of compatibility of filler surfaces with the polymer medium is characterized in terms of the adsorption time _s relative to the disentanglement time _p . By incorporating the effects of interfacial slippage into the deformation mechanism of the transient network, one can explain the variation of such rheological functions as shear viscosity with surface treatment of fillers and with molecular weight of the polymer matrix at a given filler volume fraction.     

Rheological interpretation of pressure anomalies of aqueous dilute polymer solutions (ADPS) in orifice flow

The response of a considerable number of solutions of several polymers (PEO, HPAM, PAM) with concentrations of less than 100 ppm in orifice flow has been investigated. It is shown that the excess pressure (difference between the ADPS and the solvent total pressure drop) behaves linearly as a function of a superficial strain rate (ratio between a velocity and a length scale). In rheological terms this behaviour is interpreted as the result of a constant elongational viscosity whose values are two to three orders of magnitude larger than the shear viscosity. A formal approach to this phenomenological interpretation is suggested.     

Analysis of entry flow to determine elongation flow properties revisited

The minimisation technique proposed by Binding (J. Non-Newtonian Fluid Mech., 27 (1988) 173) was used in our Generalised Engineering Bernoulli Equation framework to relate the entry pressure and stress power. We arrived at a final result similar to Binding's using assumed kinematics. Through subsequent assumptions to the kinematics we finally arrive at a result exactly equivalent to Cogswell's technique (Trans. Soc. Rheol., 16 (1972) 383). Thus, these two techniques are related in this general framework. The techniques were used to predict elongation flow properties of a polymer melt and polymer solution. The results for the polymer melt clearly show Cogswell's technique is adequate at high elongation rates. All these techniques require minimisation of the stress power with respect to the flow volume and discussion is given as to the validity of this minimisation technique. In addition, the approximate variational technique we propose gives clears limits as to when a technique, such as Cogswell's, can be applied.     

The role of transient rheology in polymeric sintering

The initial theory of Frenkel and Eshelby for the coalescence of drops in air (or sintering) of Newtonian fluids, which equated the work of surface tension to the work done by viscous stresses while assuming biaxial extensional flow kinematics, was extended to the case of time-dependent material functions using the Upper Convected Maxwell (UCM) model. A numerical scheme was developed to solve the ordinary differential equations (ODE) for the stresses, which are embedded in the ODE based on the mechanical energy balance. Initial conditions required to solve the set of non-linear ODEs were obtained from visualization experiments of the coalescing drops as the theory for elastic contact gave unrealistically high values of the initial neck radius. The transient model predicted that coalescence was accelerated by increasing the relaxation time, the opposite relationship of what was predicted by the steady-state UCM formulation, and was capable of quantitatively predicting the experimental coalescence rates at times when viscoelasticity was important.     

Extensional flow of HDPE/LDPE blends

Elongational viscosity data, obtained through the converging flow analysis by Cogswell, are presented for two types of HDPE/LDPE blends at various compositions and different temperatures. The results relative to the homopolymer parents compare favourably with literature data obtained also with different and more sophisticated techniques. Those relative to blends show peculiar features for the two cases: when the newtonian viscosity of the LDPE is higher all the blends show a behaviour typical of the LDPE with a maximum in _el / ₀ enhanced at small percentage of HDPE; when the newtonian viscosity of the LDPE is similar to that of the HDPE there is only a gradual change in the properties.     

Squeezing flow of a highly viscous incompressible liquid pressed between slightly inclined lubricated wide plates

The theoretical force-height relationships of Newtonian and pseudo plastic liquids compressed between slightly tilted frictionless plates are compared with those produced when the plates are perfectly parallel. It is shown that a very small inclination angle can distort the flow curve to such an extent that a Newtonian liquid will appear as a pseudo plastic fluid, and a pseudo plastic liquid as having a flow index considerably smaller than its true one. The shape of the biaxial elongational viscosity vs biaxial strain rate relationship is also highly sensitive to the plates' inclination angle. Thus, if an experimental force-height relationship is used to determine a material's biaxial elongational viscosity, an unsuspected slight tilt will result in a considerable underestimate of the viscosity. A slight tilt will also produce an apparent strain rate dependency in a Newtonian liquid, which obviously does not exist. The mathematical model developed to reach these conclusions was tested with commercial mayonnaise, a self-lubricating fluid. A reasonable agreement was found between the predicted force-height relationships and those experimentally determined at tilts of 1°, 3°, and 5°. Received: 4 August 2000 Accepted: 21 August 2000     

Rheological behaviour and molecular structure of long-chain branched semifluorinated thermoplastics

The long-chain branched thermoplastic tetrafluoroethylene–hexafluoropropylene–vinylidenefluoride terpolymers (LCB THV) investigated in this paper are new polymers with a unique combination of properties like a high stability against aging or weathering and a very good chemical resistance. But not much is known about the rheological behaviour of the LCB THV, yet. In this paper, non-linear rheological properties like shear thinning and strain hardening are studied. Two different types of the THV with different contents of comonomers and, therefore, different melting points are examined. The THV with the higher melting point is insoluble. The other with the lower melting temperature is soluble and, therefore, was characterised by size exclusion chromatography coupled with light scattering with respect to its molecular structure. The results of the rheological measurements show a pronounced shear-thinning and strain-hardening behaviour for the long-chain branched materials. Both properties are of great importance for processing operations governed by shear and elongational flows.

A model for viscoelastic fluid behavior which allows non-affine deformation

A continuum theory of viscoelasticity is developed which allows non-affine deformation, defined in an appropriate manner. The constitutive equation is a generalization of that obtained from molecular theory with the addition of one scalar parameter which becomes important for large deformations. The theory is applied to simple shear flows, the scalar parameter being determined to match certain experimental data. The theory shows good agreement with all data examined. The paper concludes with the development of a general non-affine thermodynamic theory.     

Model experiments of the transient response to flooding of the box shaped barge

Coupling of the flooded water and ship motions was studied experimentally. Roll decay tests for one flooded compartment and transient abrupt flooding tests were performed for the box shaped barge model. The tests were conducted to obtain information on the flooding process for the development of numerical tools and to provide validation data. Quantitative values on the effect of flooded water on the roll damping were obtained. Flooded water behaves in a different manner in undivided and divided compartments. Flooded water in divided compartment increases roll damping significantly. In undivided compartment roll damping was high at low amount of flooded water. For higher amounts damping was of the same order as for the intact model. Initial flooding is a complex process where the ship and flooded water motions are coupled. Propagation of the flooding water inside the compartment, at a dam-break type abrupt flooding, was studied by tracking the surface of the flooded water. An image processing algorithm was used to obtain the tracked surface. Flooded water volume and its center of gravity were estimated from the tracked surface. Different internal layouts of the flooded compartment can lead to a totally different roll response. The inflooding jet plays an important role in the response in case of the undivided compartment. While, for a divided compartment, asymmetric flooding due to the obstructions causes high heel angle on the damage side.     

Die Dehnviskosität des Polypropylens

Zusammenfassung Es wurden die Dehnviskositäten von Polypropylen-Schmelzen bei 180°C im Dehngeschwindigkeitsbereich 0,02 bis 1,0 s^–1 gemessen. Dabei wurden Proben von drei verschiedenen Herstellern und mit sehr verschiedenen Molmassen untersucht. Sämtliche Proben zeigten einen monotonen Anstieg der Dehnviskosität mit der Zeit. Die durch Extrapolation gewonnenen Gleichgewichts-Dehnviskositäten betrugen in allen Fällen das Dreifache der Null-Scherviskositäten. In keinem einzigen Fall konnte eine Dehnverfestigung der Schmelzen beobachtet werden. Nur bei unvollständig aufgeschmolzenen Proben (168°C Meßtemperatur) ergab sich der triviale Fall einer Dehnverfestigung als Folge der Dehnkristallisation. Kontrollmessungen an verzweigtem Polyethylen (bei 150°C) ergaben die erwartete Dehnverfestigung, während diese, ebenfalls erwartet, bei linearem Polyethylen ausblieb.

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The elongational viscosities of polypropylene melts were measured at 180°C with different elongation rates between 0.02 and 1.0 s^–1 respectively. Samples were studied from three different manufacturers and some having markedly different molecular masses. All samples showed a monotonic rise of the elongational viscosity with time. The extrapolated steady-state elongational viscosities were always three times the values of the corresponding zero-shear viscosities. In no case strain hardening could be observed. Only incompletely molten samples (temperature of measurement 168°C) showed a kind of strain hardening as a consequence of strain crystallization. Control measurements with branced polyethylene at 150°C showed the expected strain hardening while this, also expected, was not observed in the case of linear polyethylene.

Herrn Professor Dr. Rolf Sammet zum 65. Geburtstag gewidmet     

A new contact & slip model for tracked vehicle transient dynamics on hard ground

This study presents a new general transient contact and slip model for tracked vehicles on hard ground which is simple, accurate, and in agreement with the test results to a satisfactory level. Simulating zero track speed instances become possible with the new contact/shear model which is the major proposed improvement in addition to more accurate results for transient steering and tractive inputs. The model represents a general tracked vehicle having rear or front sprockets, with parameters for center of gravity, wheel positions, number of wheels, and track-pretention. To calculate longitudinal and lateral forces, a transient shear model is used. Shear stress under each track pad is assumed to be a function of shear displacement. The contact time formulation used in shear displacement calculation is improved to gain accuracy for transient and zero track speed conditions.The model is implemented on the Matlab/Simulink platform and verified with a comprehensive program of road tests composed of transient steering and tractive/braking scenarios. The results of the simulations and the road tests are satisfactorily similar for both constant and transient input maneuvers. Moreover, sensitivity simulations for vehicle parameters are conducted to show that the model responses are inline with the expected vehicle dynamics behaviours.     

Elongational properties and molecular structure of polyethylene melts

Summary The viscosity and the recoverable strain in the steady state of elongation have been measured on several polyethylenes of different molecular structures. The elongational viscosity as a function of tensile stress runs through a more or less pronounced maximum in the nonlinear range whereas in the linear range the Trouton viscosity is reached. For low density polyethylenes it could be demonstrated that the maximum of the steady-state elongational viscosity and the elasticity expressed by the steady-state compliances in shear and tension sensitively increase if the molecular weight distribution is broadened by the addition of high molecular weight components. A variation of the weight average molecular weight does only shift the elongational viscosity curve but leaves its shape unchanged. Two of the four high density polyethylenes investigated do not show a maximum of the steady-state elongational viscosity, for the others it is less pronounced than in the case of low density polyethylenes. The influence of branching on the elongational behaviour of polyethylene melts in the steady-state and the transient region is qualitatively discussed.With 11 figures and 4 tables     

Investigation of the rheological properties of short glass fiber-filled polypropylene in extensional flow

The behavior of short glass fiber–polypropylene suspensions in extensional flow was investigated using three different commercial instruments: the SER wind-up drums geometry (Extensional Rheology System) with a strain-controlled rotational rheometer, a Meissner-type rheometer (RME), and the Rheotens. Results from uniaxial tensile testing have been compared with data previously obtained using a planar slit die with a hyperbolic entrance. The effect of three initial fiber orientations was investigated: planar random, fully aligned in the stretching flow direction and perpendicular to it. The elongational viscosity increased with fiber content and was larger for fibers initially oriented in the stretching direction. The behavior at low elongational rates showed differences among the various experimental setups, which are partly explained by preshearing history and nonhomogenous strain rates. However, at moderate and high rates, the results are comparable, and the behavior is strain thinning. Finally, a new constitutive equation for fibers suspended into a fluid obeying the Carreau model is used to predict the elongational viscosity, and the predictions are in good agreement with the experimental data.     

Transient shear and elongational behavior of blends of PET with a LCP

Blends of polyethylene terephthalate (PET) with a liquid crystalline polymer (LCP) and a compatibilizer were produced by twin screw extrusion and injection molding. Transesterification and compatibilization studies were made in a torque rheometer. The morphology of the injection-molded plaques was studied by scanning electron microscopy. The blends shear growth function was measured in a cone and plate rheometer. The elongational growth function was measured in a modified rotational rheometer. Transesterification was observed in the PET/LCP/compatibilizer 95/5/0 blend. The injection-molded plaques displayed the usual “skin-core” morphology. All the blends were highly shear-thinning, even at low shear rates; thus, a zero-shear viscosity could not be calculated. The compatibilized blend had the highest shear viscosity of all the blends, confirming the strong PET/LCP interphase and the effectiveness of the compatibilizing agent. On the other hand, the 90/10/0 blend had the lowest shear viscosity. All the blends showed strain softening behavior, similar to the PET. The 90/10/0 blend had the highest elongational growth function, while the 95/5/0 had the lowest. The compatibilized blend had an intermediate behavior between both blends.     

Influence of molecular structure on the extensional behaviour of polyethylene melts

The elongational behaviour of polyethylene samples having different molecular structure has been tested. Elongational viscosity measurements have been carried out using the isothermal melt spinning technique. The extensional behaviour of the different samples is analysed as a function of total strain. The effect of long-chain branching on elongational viscosities is described. A comparison is presented between elongational viscosity and melt strength data.Some of the results reported here were presented at the VIIIth International Congress on Rheology, Naples, September 1–5, 1980, cf. [16].