Parameter identification for viscoelastic materials

Starting from the general stress-strain relation for a linear Boltzmann-Volterra material, which is in agreement with the principle of inertia, a new identification procedure is proposed. Instead of running one long-range relaxation experiment, following asingle suitably specified deformation history, material characterization is done using the data ofn short relaxation experiments followingn different deformation histories. To interpret these data a direct non-iterative algorithm has been developed. Compared with other methods, for example curve fitting by using Gauss' method, this direct method is numerically stable and allows a simple direct evaluation of the error due to the scattering of experimental data. The method has been applied to the determination of the relaxation times of an unsaturated polyester material.     

Extremalprinzipe der Thermodynamik und Mechanik für stabile Gleichgewichtslagen von Kontinua

Übersicht In dieser Arbeit soll der Zusammenhang zwischen den Extremalprinzipien der Thermodynamik und der Mechanik für stabile Gleichgewichtszustände eines Kontinuums studiert werden. Als Nebenprodukt ergeben sich dabei zwanglos Verallgemeinerungen der klassischen Extremalprinzipe der Elastizitätstheorie (des Minimum-Energie-Prinzips und des Prinzips von Castigliano) auf multipolare und chemisch reagierende Materialien.

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Summary The aim of this paper is to study the relations between the variational principles of thermodynamics and mechanics for stable equilibrium states of a deformable body. As a by-product one obtains generalisations of the classical principles of elasticity-theory (the minimum-energy-principle and Castigliano's theorem) to multipolar and chemically reacting materials.

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Herrn Prof. Dr.-Ing. K. Klotter zum siebzigsten Geburtstag gewidmet.     

The “decaying springs” model for irreversibility in polymer melts

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

Marangoni effect in the Couette flow

Viscous heating in an axisymmetric creeping flow of a second-order fluid with free surface between two coaxially mounted cylinders produces a radial temperature gradient in the fluid. The dependence of the surface tension upon temperature is the cause for a secondary flow in the meridional plane of the flow field. This secondary flow (Marangoni effect), and its influence upon the shape of the free surface are studied. The deformation of the free surface caused by the Marangoni effect is compared with the deformation caused by inertia and normal stress differences.     

Ein rheometer zur untersuchung von flüssigen polymermaterialien in einachsiger und zweiachsiger Dehnung

Zusammenfassung Wir stellen ein neues Dehnrheometer vor, mit dem sowohl einachsige als auch zweiachsige Dehnexperimente an flüssigen Polymermaterialien durchgeführt werden können. Bei der Apparatur wird das Prinzip der atmenden Blase verwendet: Über eine Düse wird in das zu untersuchende Material eine kugelförmige Blase aus einer niederviskosen, inkompressiblen, unmischbaren Flüssigkeit injiziert. Wachstum bzw. Schrumpfen dieser Blase führt zu einer zweiachsigen bzw. einachsigen Dehndeformation der umgebenden Polymerflüssigkeit. Der Injektionsdruck dient als Meßgröße zur Bestimmung der viskoelastischen Eigenschaften des Probenmaterials. Bei vergleichsweise niederviskosen Silikonölen gelingt die Messung der Nullviskosität bzw. der linear viskoelastischen Eigenschaften in guter Übereinstimmung mit Scherexperimenten. Bei Materialien mit ausgeprägt nichtlinear viskoelastischem Verhalten kann unter Anwendung des Wagner-Modells die Dämpfungsfunktion für ein- und zweiachsige Dehnung ermittelt werden. Unsere Ergebnisse für das Polyisobutylen Oppanol B 15 stimmen mit Messungen von Demarmels und Meißner überein, die mit der Methode der Rotierenden Klemmen durchgeführt werden. Für drei Polyisobutylene unterschiedlichen Molekulargewichts ergibt sich im Rahmen der Meßgenauigkeit die gleiche Dämpfungsfunktion.Vortrag auf der Jahrestagung 1991 der Deutschen Rheologischen Gesellschaft e.V. in Berlin.     

Dilithiation of Bis(benzene)molybdenum and subsequent isolation of a molybdenum-containing paracyclophane

The homoleptic sandwich complex bis(benzene)molybdenum, [Mo(eta6-C6H6)2], was successfully dilithiated by employing an excess of BuLi in the presence of N,N,N',N'-tetramethylethylenediamine (up to 6 equiv each) at slightly elevated temperatures furnishing the highly reactive, ring metalated species [Mo(eta6-C6H5Li)2].tmeda in high yields. Alternatively, this compound was synthesized upon prolonged sonication with 5 equiv of tBuLi/tmeda without heating. An X-ray crystal structure determination revealed a symmetrical, dimeric composition in the solid state, i.e., a formula of [Mo(eta6-C6H5Li)2]2.(thf)6, where the six-membered rings are connected by two pairs of bridging lithium atoms. The synthesis of an elusive ansa-bridged complex failed in the case of a [1]bora and a [1]sila bridge due to the thermal lability of the resulting compounds. Instead, reverse addition of the dilithio precursor to an excess of the appropriate element dihalide facilitated the isolation of several unstrained, 1,1'-disubstituted derivatives, namely, [Mo{eta6-C6H5(BN(SiMe3)2X)}2] (X = Cl, Br) and [Mo{eta6-C6H5(SiiPr2Cl)}2], respectively. However, the incorporation of a less congesting [2]sila bridge was accomplished. In addition to the formation of [Mo{(eta6-C6H5)2Si2Me4}], a molybdenum-containing paracylophane complex was isolated and characterized by means of crystal structure analysis. The ancillary formation of 1 equiv of bis(benzene)molybdenum strongly suggests that this species is generated by deprotonation of the ansa-bridged complex by the dilithiated precursor and subsequent reaction with a second equivalent of the disilane.     

Determination of axial forces during the capillary breakup of liquid filaments – the tilted CaBER method

The capillary breakup extensional rheometry (CaBER) is a versatile method to characterize the elongational behavior of low-viscosity fluids. Commonly, data evaluation is based on the assumption of zero normal stress in axial direction ( $\upsigma_{\rm zz}=0$ ). In this paper, we present a simple method to determine the axial force using a CaBER device rotated by 90° and analyzing the deflection of the filament due to gravity. Forces in the range of 0.1–1,000?μN could be assessed. Our study includes experimental investigations of Newtonian fructose solutions and silicon oil mixtures (viscosity range, 0.9–60?Pa s) and weakly viscoelastic polyethylene oxide (PEO, $M_{\rm w}=10^{6}$ ?g/mol) solutions covering a concentration range from c?≈?c* (critical overlap concentration) up to c?>?c _e (entanglement concentration). Papageorgiou’s solution for the stress ratio $\upsigma_{\rm zz}/\upsigma_{\rm rr}$ in Newtonian fluids during capillary thinning is experimentally confirmed, but the widely accepted assumption of vanishing axial stress in weakly viscoelastic fluids is not fulfilled for PEO solutions, if c _e is exceeded.     

On the Modelling of the Electro-Hydrodynamic Flow Field in Electrostatic Precipitators

Electro-hydrodynamic (EHD) flows are investigated theoretically and numerically in this paper and results are presented for the flow field in model electrostatic precipitators (EPs). The resulting flow fields are shown in various representations and explained qualitatively. Numerical calculations with different flow models (non-turbulent and RANS) were conducted to investigate the influence of the flow model on the resulting secondary flows. Furthermore, a perturbation analysis is presented, leading to a simple differential equation of the Helmholtz type. This allows a more detailed view of the important mechanisms forming the secondary flows as well as being able to obtain a very fast estimation of the resulting flow field. The calculations reveal a strong influence of a vortex formation at the beginning of the precipitation zone on the whole flow field. Furthermore, a strong effect of the boundary conditions of the electric field and the operating parameters is shown. This revised version was published online in July 2006 with corrections to the Cover Date.     

The normal stress behaviour of suspensions with viscoelastic matrix fluids

 We investigate the variations in the shear stress and the first and second normal stress differences of suspensions formulated with viscoelastic fluids as the suspending medium. The test materials comprise two different silicone oils for the matrix fluids and glass spheres of two different mean diameters spanning a range of volume fractions between 5 and 25%. In agreement with previous investigations, the shear stress–shear rate functions of the viscoelastic suspensions were found to be of the same form as the viscometric functions of their matrix fluids, but progressively shifted along the shear rate axis to lower shear rates with increasing solid fraction. The normal stress differences in all of the suspensions examined can be conveniently represented as functions of the shear stress in the fluid. When plotted in this form, the first normal stress difference, as measured with a cone and plate rheometer, is positive in magnitude but strongly decreases with increasing solid fraction. The contributions of the first and the second normal stress differences are separated by using normal force measurements with parallel plate fixtures in conjunction with the cone-and-plate observations. In this way it is possible for the first time to quantify successfully the variations in the second normal stress difference of viscoelastic suspensions for solid fractions of up to 25 vol.%. In contrast to measurements of the first normal stress difference, the second normal stress difference is negative with a magnitude that increases with increasing solid content. The changes in the first and second normal stress differences are also strongly correlated to each other: The relative increase in the second normal stress difference is equal to the relative decrease of the first normal stress difference at the same solid fraction. The variations of the first as well as of the second normal stress difference are represented by power law functions of the shear stress with an unique power law exponent that is independent of the solid fraction. The well known edge effects that arise in cone-and-plate as well as parallel-plate rheometry and limit the accessible measuring range in highly viscoelastic materials to low shear rates could be partially suppressed by utilizing a custom- designed guard-ring arrangement. A procedure to correct the guard-ring influence on torque and normal force measurements is also presented. Received: 20 December 2000 Accepted: 7 May 2001     

Einige Erhaltungssätze der Kontinuumsmechanik vom J-Integral-Typ

Übersicht Aus der Translations- und Rotationsinvarianz der allgemeinen Bilanzgleichung der Kontinuums-mechanik werden Erhaltungssätze in Form von Oberflächenintegralen hergeleitet. Für die Energiebilanz folgen daraus Erhaltungsgrößen vom J-Integral-Typ, die sich in den Fällen der Statik, der stationären ebenen Welle und der Hyperelastizität besonders vereinfachen. Sind bewegliche Energiequellen, wie Defekte vorhanden, so wird durch die Erhaltungsintegrale die Kraft auf die Quelle beschrieben.

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Summary Conservation laws in terms of surface integrals are derived from the translation respectively rotation invariance of general balance equations of continuum mechanics. From them conservation quantities of J-Integral type follow for the energy balance. They reduce to simpler forms in the cases of statics, of stationary plane waves and of hyperelasticity. When there are mobile energy sources like defects, then the force acting on the source is described by the conservation integrals.

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Herrn Prof. E. Mettler zum 70. Geburtstag gewidmet