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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 22 (1983), S. 557-568 
    ISSN: 1435-1528
    Keywords: Viscoelastic constitutive equation ; elongational flow ; elastic recovery ; stress growth ; strain-hardening
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Notes: Abstract The viscoelastic behavior of polymeric systems based upon the Leonov model has been examined for (i) the stress growth at constant strain rate, (ii) the stress growth at constant speed and (iii) the elastic recovery in elongational flow. The model parameters have been determined from the available rheological data obtained either in steady shear flow (shear viscosity and first normal-stress difference as a function of shear rate) or oscillatory flow (storage and loss moduli as a function of frequency in the linear region) or from extensional flow at very small strain rates (time-dependent elongation viscosity in the linear viscoelastic limit). In addition, the effect of the parameter characterizing the strain-hardening of the material during elongation has also been studied. The estimation of this parameter has been based upon the structural characteristics of the polymer chain which include the critical molecular weight and molecular weight of an independent segment. Five different polymer melts have been considered with varying number of modes (maximum four modes). Resulting predictions are in fair agreement with corresponding experimental data in the literature.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 25 (1986), S. 80-94 
    ISSN: 1435-1528
    Keywords: Streamwise integration ; abrupt contraction ; Leonov model ; birefringence ; Weissenberg number ; Deborah number
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Notes: Abstract A numerical scheme based on the Finite Element Method has been developed which uses a relaxation factor in the momentum equation with the stresses being evaluated via a streamwise integration procedure. A constitutive equation introduced by Leonov has been used to represent the rheological behavior of the fluid. The convergence of the scheme has been tested on a 2 : 1 abrupt contraction problem by successive mesh refinement for non-dimensional characteristic shear rates, of 5 and 50 for polyisobutylene Vistanex at 27 °C. The recirculation region is shown to increase in size with non-dimensional characteristic shear rate. Theoretical predictions have been compared with the experimental data which include birefringence and pressure loss measurements. In general, the comparisons have been reasonably good and demonstrates the usefulness of the present numerical scheme and the Leonov constitutive equation to describe real polymer flows.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 32 (1993), S. 270-276 
    ISSN: 1435-1528
    Keywords: Entry flow ; two-dimensional flow ; rubber compound ; contraction or expansion flow ; pressure drop
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Notes: Abstract An automated rheometer based on an injection molding machine is developed for the evaluation of entry flow problems. Several entry flow geometries having different contraction and expansion angles and different channel lengths are tested. Two pressure transducers are flushmounted along the die length and a displacement transducer is installed to measure the screw motion. Signals generated by the pressure transducers and displacement transducer are supplied to an A/D converter and an IBM PC/AT computer. The pressure losses for a rubber compound are measured between two cross-sections along the flow direction. The time evolution of pressure with overshoot during flow before and after the entry region is observed. At low flow rates the pressure drops of the expansion flow are larger than those of the contraction flow. At high flow rates the pressure drops of the contraction flow become higher than those of the expansion flow. A ratio of the pressure drop to absolute pressure before the entry is found to be almost independent of flow rate.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 14 (1975), S. 135-144 
    ISSN: 1435-1528
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Description / Table of Contents: Zusammenfassung Untersucht wurden Zäh- und Elastizitätseigenschaften von Bitumensorten bei zyklischer und kontinuierlicher Schubverformung und bei Vereinigung beider Verformungsbeanspruchungen. Die Versuchsergebnisse zeugen von ähnlichem Zäh- und Elastizitätsverhalten von Bitumen und Polymerstoffen. Es wurde gezeigt, daß für die untersuchten Bitumensorten die Abhängigkeit der dynamischen Viskosität von der Frequenz im linearen Gebiet und die der effektiven Viskosität von der Schubgeschwindigkeit zusammenfallen. Aufgestellt wurden Relaxationsspektren von Bitumen, die vom Vorhandensein einer Relaxationszeitverteilung zeugen, die komplizierter als dieGaußsche Verteilung ist. Es wurde festgestellt, daß die zyklische Verformung mit großen Scheitelwerten eine Abstumpfung vom Niederfrequenzteil der Frequenzabhängigkeiten der Elastizitätsmoduln und der Verluste verursacht, wobei die Abstumpfung um so höher liegt, je größer die Amplitude der Verformungsgeschwindigkeit ist. Die Abhängigkeit des Absolutwertes der komplexen Viskosität von der Amplitude der Verformungsgeschwindigkeit im Bereich der höheren Amplituden ist der Abhängigkeit der effektiven Viskosität von der Schubgeschwindigkeit formähnlich. Jedoch wirkt die Amplitude der Verformungsgeschwindigkeit verschiedenartig auf den Absolutwert der komplexen Viskosität und auf die effektive Viskosität ein. Es wurde eine Vermutung über die komplizierte Art der Änderung des Relaxationsspektrums bei zyklischer Verformung mit großen Amplituden geäußert.
    Notes: Summary Viscoelastic properties of bitumens have been investigated under cyclic and continuous shear deformation and under the joint effect of both types of deformation. The experimental results indicate a similarity between the viscoelastic behaviour of bitumens and polymer systems. It is shown that for the bitumens studied the curves of the loss modulus against frequency in the linear region, and of the shear stress against the shear rate, coincide. Relaxation spectra of bitumens have been constructed, which point to a relaxation-time distribution more complicated than theGaussian distribution. It has been established that cyclic deformation with large amplitudes causes the truncation of the low-frequency portion of the curves of the storage and loss moduli against frequency, which is higher, the greater the deformation rate amplitude. The dependence of the absolute value of the complex viscosity on the deformation rate amplitude, in the range of large amplitudes, is symbatic to the dependence of the apparent viscosity on the shear rate. The deformation rate amplitude, however, affects the absolute value of the complex viscosity and the apparent viscosity in different ways.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 16 (1977), S. 266-281 
    ISSN: 1435-1528
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Description / Table of Contents: Zusammenfassung Das rheologische Verhalten von acht Bitumenproben, die für Asphalt-Zemente Verwendung finden, wird untersucht. Es handelt sich dabei um oxidierte Bitumina verschiedener Zusammensetzung und mit verschiedenem Oxidationsgrad, sowie um eine Probe Rückstands-Bitumen. Diese werden über einen weiten Bereich von Schergeschwindigkeiten, Frequenzen, Deformationsamplituden und Temperaturen untersucht. Das Fließen der Bitumina kann von elastischen Deformationen überlagert werden. Im Bereich niedriger Schubspannungen entspricht die Beziehung zwischen rückstellbarer Deformation und Spannung dem Hookeschen Gesetz für Scherung, mit einem vom Asphalten-Gehalt abhängigen Wert des Moduls. Im Bereich hoher Spannungen erreicht die rückstellbare Deformation der Bitumina etwa 60%. Oxidierte und Rückstands-Bitumina von übereinstimmender Zusammensetzung zeigen ein praktisch übereinstimmendes viskoelastisches Verhalten. Je nach dem Asphaltengehalt variiert das rheologische Verhalten der Bitumina vom typischen Verhalten eines Polymersystems bis zu dem eines dispersen Systems. Es wird experimentell nachgewiesen, daß für Bitumina mit einem schwach entwickelten Struktur-Skelett die Schubspannung bei der stationären Scherung mit dem Verlustmodul einer periodischen Deformation bei kleiner Amplitude übereinstimmt, wenn man die Schergeschwindigkeit mit der Frequenz gleichsetzt. Dagegen ist für strukturierte Bitumina kein derartiger Zusammenhang feststellbar. Die Bitumina sind empfindlich gegenüber Amplitude und Frequenz einer periodischen Deformation. Sowohl die Viskosität als auch die dynamischen Moduln ändern sich unter dem Einfluß von Scherschwingungen großer Amplitude erheblich, wobei wiederum der Asphaltengehalt von Bedeutung ist. Die Frequenzabhängigkeit der dynamischen Moduln ermöglicht eine Abgrenzung zweier charakteristischer Bereiche des viskoelastischen Verhaltens, nämlich des Fließbereichs und des Übergangsbereichs zum Plateau des Speichermoduls. Die Intensität der Veränderung in den dynamischen Moduln hängt in ausgeprägter Weise von der Gruppenzusammensetzung der Bitumina ab. Die Anfangswerte der Relaxationszeit, welche dem undeformierten Zustand korrespondieren, werden aus dynamischen Messungen bei niedrigen Frequenzen extrapoliert. Es handelt sich dabei um einen besonders strukturempfindlichen Parameter. Die Verwendung dieser Relaxationszeit ermöglicht die Herstellung einer verallgemeinerten Relaxationscharakteristik der Bitumina, welche bezüglich der Struktur, dem Verarbeitungsprozeß, der Natur, dem Oxidationsgrad, der Temperatur sowie der Frequenz der Zwangsbean-spruchung invariant ist. Eine verallgemeinerte Temperaturabhängigkeit des Reduktionskoeffizienten in einer Gleichung vom WLF-Typ wird erhalten, wobei als Reduktionstemperatur die Erweichungstemperatur gewählt wird. Es wird eine Methode zur Bestimmung des viskoelastischen Verhaltens der Bitumina vorgeschlagen, wobei außer der für eine einzige Temperatur bekannten Fließkurve nur noch der Erweichungspunkt sowie die Relaxationszeit bei einer einzigen Frequenz benötigt werden.
    Notes: Summary The rheological behaviour of eight specimens of bitumens which can be used in asphalt cements has been studied. These are oxidized bitumens of different group composition and degree of oxidation, as well as a specimen of residual bitumen. They were investigated over a wide range of shear rates, frequencies, deformation amplitudes, and temperatures. The bitumen flow may be attended by accumulation of elastic (recoverable) deformations. In the region of low shear stresses the relationship between recoverable deformation and stress corresponds to the Hooke law in shear, and its value depends on the asphaltene content. In the high-stress region the recoverable deformations of the bitumens investigated may reach 0.6 relative unit. Oxidized and residual bitumens of identical group composition show a practically identical viscoelastic behaviour. Depending on the asphaltene content, the rheological behaviour of bitumens may vary from one typical of polymer systems to one typical of disperse systems. It has been established experimentally that for bitumens with a weakly developed structural skeleton the shear stress in stationary flow coincides with the loss modulus in small-amplitude cyclic deformation provided the shear rate and frequency are numerically equal. For structurized bitumens, no correlation of this kind is observed. Bitumens are systems sensitive to the amplitude and frequency of oscillatory shear. The viscosity and dynamic characteristics of bitumens change substantially under the effect of large-amplitude oscillatory shear. The scope of these changes is essentially determined by the asphaltene content. The frequency dependence of the dynamic characteristic enables one to isolate two characteristic regions of viscoelastic behaviour of bitumens: the flow region and that of transition to the storage modulus plateau. The intensity of the change in the dynamic characteristics in the flow region strongly depends on the group composition of the bitumens. The initial values of the relaxation time of bitumens corresponding to their initial undeformed state have been determined from dynamic measurements at low frequencies by extrapolating to the zero frequency. This is one of the most structure-sensitive parameters of the bitumens. The use of this relaxation time makes it possible to plot a generalized relaxation characteristic of the bitumens, which is invariant to the structure and type of the bitumen, the manufacturing process, the nature, the depth of oxidation, the temperature and frequency of coercion. A generalized temperature dependence of the reduction coefficient described by an equation of the type of Williams-Landell-Ferry has been obtained. The softening point of the bitumen was adopted as the reduction temperature. A method for evaluating the viscoelastic behaviour of bitumens on the basis of the known flow curve at a single temperature, the softening point, and the relaxation time at a single frequency is proposed.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 19 (1980), S. 168-182 
    ISSN: 1435-1528
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Description / Table of Contents: Zusammenfassung Das idealisierte Problem einer eindimensionalen, instationären, nicht-isothermen Strömung eines Polymeren zwischen zwei parallelen Platten sowie das der nichtisothermen Relaxation, die auf das Anhalten dieser Strömung folgt, wird mit Hilfe der viskoelastischen Stoffgleichung vonLeonov gelöst. Numerische Ergebnisse werden für die Zeitabhängigkeit der folgenden Größen gegeben: des Druckgradienten, der Verteilung der linearen Geschwindigkeit, der Schergeschwindigkeit, der Schubspannung, der Normalspannungsdifferenzen sowie der Komponenten der Doppelbrechung in verschiedenen Ebenen. Die hier vorliegenden Voraussagen sind bezüglich des Druckgradienten in guter Ubereinstimmung mit denen, die auf dem “inelastischen“ Modell beruhen, unterscheiden sich von diesen aber wesentlich bezüglich der Normalspannungsdifferenzen. Das Modell wird auf den Spritzgußprozeß angewandt. Dieser wird als zweistufiger Prozeß, bestehend aus einer Abfüll- und einer Kühlstufe, behandelt. Numerische Ergebnisse werden für die Verteilung der Restspannungen und der assoziierten Doppelbrechung im Formteil gegeben, so wie sie durch die rheologischen und thermischen Eigenschaften des Polymeren und der Prozeßbedingungen beeinflußt werden. Die theoretischen Voraussagen für die Doppelbrechung werden mit Meßergebnissen aus der Literatur verglichen. Gute Übereinstimmung wird für die Lage und den Wert der maximalen Doppelbrechung in der 1–2 Ebene erzielt, während die Voraussagen für die Werte der Doppelbrechung in den 1–3 und 2–3 Ebenen wesentlich kleiner als die gemessenen Werte ausfallen. Die vorliegende Theorie zeigt an, daß für ein gegebenes Polymer die Schmelzentemperatur und die Einspritzgeschwindigkeit als Hauptfaktoren zu werten sind, die die Restspannungen und die Doppelbrechung beeinflussen. Diese sollen auf dem höchstzulässigen Stand gehalten werden.
    Notes: Summary By using theLeonov viscoelastic constitutive equation, an idealized problem has been solved for onedimensional, unsteady, non-isothermal flow of polymer between two parallel plates and the subsequent non-isothermal relaxation following cessation of flow. Numerical results are presented for the time dependence of the pressure gradient, the gapwise distribution of linear velocity, shear rate, shear stress and normalstress differences, together with the components of birefringence in different planes. Comparison of the present predictions for the pressure gradient with results based upon an “inelastic” model indicate close agreement whereas the corresponding predictions for normal-stress differences are found to be markedly different from those for the “inelastic” case. The model is applied to the injection-molding process which is treated in terms of a filling and a cooling stage. Final results are given in terms of the distribution of residual stresses and associated birefringences in the molded part, as influenced by the rheological and thermal properties of the polymer and the processing conditions. The theoretical predictions are compared with birefringence measurements in the literature. Reasonable agreement is obtained for the position and value of maximum birefringence in the 1–2 plane although the birefringence predictions in the 1–3 and 2–3 planes are found to be markedly smaller than the measured values. The present theory indicates that, for a given polymer, the main factors affecting residual stresses and birefringence are melt temperature and flow rate, both of which should be held at the highest permissible levels.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 28 (1989), S. 176-189 
    ISSN: 1435-1528
    Keywords: Shear flow ; orthogonalsuperposition ; largeamplitudeoscillations ; harmonicanalysis ; Leonov model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Notes: Abstract The orthogonal superposition of small and large amplitude oscillations upon steady shear flow of elastic fluids has been considered. Theoretical results, obtained by numerical methods, are based on the Leonov viscoelastic constitutive equation. Steady-state components, amplitudes and phase angles of the oscillatory components of the shear stress, the first and second normal stress differences as functions of shear rate, deformation amplitude and frequency have been calculated. These oscillatory components include the first and third harmonic of the shear stresses and the second harmonic of the normal stresses. In the case of small amplitude superposition, the effect of the steady shear flow upon the frequency-dependent storage modulus and dynamic viscosity has been determined and compared with experimental data available in literature for polymeric solutions. The predicted results have been found to be in fair agreement with the experimental data at low shear rates and only in qualitative agreement at high shear rates and low frequencies. A comparison of the present theoretical results has also been made with the predictions of other theories. In the case of large amplitude superposition, the effect of oscillations upon the steady shear flow characteristics has been determined, indicating that the orthogonal superposition has less influence on the steady state shear stresses and the first difference of normal stresses than the parallel superposition. However, in the orthogonal superposition a more pronounced influence has been observed for the second difference of normal stresses.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 36 (1997), S. 66-81 
    ISSN: 1435-1528
    Keywords: Key words Filled polymer ; rheological model ; yield function ; structure function ; steady flow ; transient flow ; oscillatory flow
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Notes: Abstract A viscoelastic plastic model for suspension of small particles in polymer melts has been developed. In this model, the total stress is assumed to be the sum of stress in the polymer matrix and the filler network. A nonlinear viscoelastic model along with a yield criterion were used to represent the stresses in the polymer matrix and the filler network, respectively. The yield function is defined in terms of differential equations with an internal parameter. The internal parameter models the evolution of structure changes during floc rupture and restoration. The theoretical results were obtained for steady and oscillatory shear flow and compared with experimental data for particle filled thermoplastic melt. The experimental data included the steady state shear strress over a wide range of shear rates, the transient stress in a start up shear flow, stress relaxation after cessation of a steady state shear flow, the step shear and the oscillatory shear flow at various amplitudes.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 32 (1993), S. 181-191 
    ISSN: 1435-1528
    Keywords: Two-dimensional flow ; viscoelastic plastic medium ; yield surface ; contraction and expansion flow ; pressure drop
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Notes: Abstract The present study is concerned with finite element simulation of the planar entry flow of a viscoelastic plastic medium exhibiting yield stress. The numerical scheme is based on the Galerkin formulation. Flow experiments are carried out on a carbon black filled rubber compound. Steady-state pressure drops are measured on two sets of contraction or expansion dies having different lengths and a constant contraction or expansion ratio of 4:1 with entrance angles of 90, 45 and 15 degrees. The predicted and measured pressure drops are compared. The predicted results indicate that expansion flow has always a higher pressure drop than contraction flow. This prediction is in agreement with experimental data only at low flow rates, but not at high flow rates. The latter disagreement is possibly an indication that the assumption of fully-developed flow in the upstream and downstream regions is not realistic at high flow rates, even for the large length-to-thickness ratio channels employed. The evolution of the velocity, shear stress, and normal stress fields in the contraction or expansion flow and the location of pseudo-yield surfaces are also calculated.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Rheologica acta 20 (1981), S. 443-457 
    ISSN: 1435-1528
    Keywords: Viscoelastic constitutìve equation ; birefringence ; stress growth ; stress relaxation ; elastic recovery ; polymeric fluid
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology , Physics
    Description / Table of Contents: Zusammenfassung Das viskoelastische Verhalten von Polymer-Systemen wurde auf der Basis des Leonov-Modells für die folgenden Fälle untersucht: (i) Spannungsaufbau und Relaxation bei unterbrochener Scherströmung, (ii) abklingende Spannung nach einem Sprung in der Scherdeformation und (iii) elastische Erholung nach voraufgegangener Scherung. Eine große Zahl von Relaxations-Elementen konnte einbezogen werden, wobei die Modellparameter von herkömmlichen rheologischen Daten mit Hilfe der Methode der kleinsten Quadrate ermittelt wurden. Bei einer genügend großen Zahl von Relaxations-Elementen stimmen die gewonnenen Vorhersagen sehr gut mit experimentellen Daten aus der Literatur überein. Dies gilt vor allem auch für solche Daten, die mit Hilfe von optischen Methoden für die Fälle (i) und (ii) ermittelt worden sind. Die hier vorgelegte Theorie bestätigt ebenfalls die Lodge-Meissner-Beziehung (τ 11 −τ 22)/τ 12 =γ 0 bei einer sprunghaften Änderung der Schergeschwindigkeit. Durchweg führt das Leonov-Modell in diesen Testfällen zu Ergebnissen, die mit Wagners Theorie vergleichbar sind. Leonovs Modell ist jedoch weitaus einfacher anzuwenden und bietet somit die Möglichkeit, auch praktisch bedeutsame Strömungsprobleme zu analysieren.
    Notes: Summary The viscoelastic behavior of polymeric systems based upon the Leonov model has been examined for (i) stress growth and relaxation with intermittent shear flow, (ii) stress relaxation after a step in the shear strain and (iii) elastic recovery after shear flow. A large number of modes have been conveniently incorporated through the determination of the model parameters from conventional rheological data by using an effective least-square procedure. With a sufficient number of modes, the predictions are in very good agreement with corresponding experiments in literature, including the recent data for cases (i) and (ii) obtained by optical methods. The present theory agrees also with the Lodge-Meissner relation (τ 11 −τ 22)/τ 12 =γ 0 in a step-shear experiment. In general, the Leonov model leads to results which, in these test cases, are comparable to those from Wagner's theory. It is, however, considerably less difficult to apply, thus offering the possibility of analysing flow problems of practical interest.
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