Free damped vibrations of linear viscoelastic materials

Summary A mechanical system consisting of an inert component, attached to a linear viscoelastic spring, is studied theoretically. Basic assumptions about the viscoelastic material areBoltzmann's superposition principle and a positive discrete relaxation spectrum. The equation of motion and its formal solution for free damped vibrations are discussed.The theory focusses on the determination of the complex dynamic modulus, defined for undamped sinusoidal vibrations, by free damped vibrations. Simple approximation formulae to calculate the dynamic modulus from free vibration data, i. e. eigen frequency and logarithmic decrement, are given; upper limits for the approximation errors could be derived.Paper read at the Annual Meeting of the German Rheologists, Berlin-Dahlem June 7–10, 1966.     

Dynamic rheology of the immiscible blends of liquid crystalline polymers and flexible chain polymers

Immiscible blends containing liquid crystalline polymers (LCP) as dispersed phases show different dynamic rheological properties than those composed of flexible polymers. The widely used Palierne’s model was shown by many authors to be insufficient to describe the frequency dependence of dynamic modulus of such blends. A new model was presented to describe the dynamic rheology of the immiscible blend containing LCP as a dispersed phase. The flexible chain polymer matrix was assumed to be a linear viscoelastic material under small amplitude oscillatory shear flow, and the LCP was assumed to be an Ericksen’s transversely isotropic fluid. The Rapini-Papoular equation of anisotropic interfacial energy was used to account for the effect of nematic orientation on the interfacial tension. It was found that the orientation of the director and the anchoring energy greatly influenced the storage modulus at the “shoulder” regime. The overall dynamic modulus of the blend can be well described by the model with suitable choice of the orientation of the director and anchoring energy of LCP.     

Theory of thermal instability of the flow of a viscoelastic fluid

The problem of the stability of the flow of viscoelastic fluids has fundamental importance for the technology of the production of polymer products and viscosimetry. This problem is not reduced only to classical inertial turbulence. A number of other mechanisms leading to flow instability are known [1, 2]. A thermal mechanism based on the allowance for dissipative heating and elastic properties within the framework of a linear model of a viscoelastic fluid was drawn upon to explain this phenomenon in [1]. The possibility of a self-oscillatory mode of flow was demonstrated on the basis of a qualitative analysis of the theological equation and the equation of heat balance in application to simple shear flow and uniform stretching. A theoretical analysis of the self-heating of flowing systems possessing viscoelastic properties is carried out in the present report. The main laws of the thermal instability of viscoelastic fluids discovered in [1] are described.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 115–122, May–June, 1979.     

Transient delivery movement of a fluid in pipes made of a viscoelastic material

The propagation of perturbation waves in an infinite viscoelastic pipeline is examined in [1], where the initial equations describing the movement of a fluid in pipelines made of viscoelastic materials are given. The subject of this paper is the transient movement of a fluid in viscoelastic pipes of finite length, a topic which has been already partially investigated in [2, 3] for a Maxwell model of a standard linear body. Solutions are given below for the problem of the movement of a fluid in pipes of constant diameter and in a pipeline made up of pipes of different diameters, and actual models of the pipe materials are examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 178–182, March–April, 1976.     

Characterization of the Damping Behavior of a Nanoindentation Instrument for Carrying Out Dynamic Experiments

A new displacement modulation based dynamic indentation method is demonstrated and shown to be effective for viscoelastic characterization of a glassy polymer. The analysis of dynamic experiments requires a complete understanding of the measuring system’s dynamic characteristics especially the damping. Accordingly, an improved method, based on the use of a wire spring, is developed for determining the damping characteristics. In general, damping in an indentation instrument is contributed by two elements: the eddy current damping from the electromagnetic loading coil and the squeeze film damping from the capacitive displacement transducer. Therefore, a method to determine the relative contribution from the different damping elements present in the system is demonstrated and the results are compared with the calibration obtained from the wire spring method. Finally, dynamic indentation tests are carried out on a glassy polymer to obtain the complex modulus; the values of which are compared with those obtained from bulk dynamic mechanical analysis (DMA) tests. Storage modulus values are found to be in good agreement with bulk data but some divergence in the case of loss modulus is observed. The calibration procedure of the measuring instrument is critically examined in view of these observations. Overall, displacement modulation based dynamic indentation is shown to be a promising method for viscoelastic characterization at the micron length scales. An erratum to this article can be found at     

The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model

A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail.     

Shear wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers

The attenuation and dispersion of elastic waves in fluid-saturated rocks due to the viscosity of the pore fluid is investigated using an idealized exactly solvable example of a system of alternating solid and viscous fluid layers. Waves in periodic layered systems at low frequencies are studied using an asymptotic analysis of Rytov’s exact dispersion equations. Since the wavelength of shear waves in fluids (viscous skin depth) is much smaller than the wavelength of shear or compressional waves in solids, the presence of viscous fluid layers necessitates the inclusion of higher terms in the long-wavelength asymptotic expansion. This expansion allows for the derivation of explicit analytical expressions for the attenuation and dispersion of shear waves, with the directions of propagation and of particle motion being in the bedding plane. The attenuation (dispersion) is controlled by the parameter which represents the ratio of Biot’s characteristic frequency to the viscoelastic characteristic frequency. If Biot’s characteristic frequency is small compared with the viscoelastic characteristic frequency, the solution is identical to that derived from an anisotropic version of the Frenkel–Biot theory of poroelasticity. In the opposite case when Biot’s characteristic frequency is greater than the viscoelastic characteristic frequency, the attenuation/dispersion is dominated by the classical viscoelastic absorption due to the shear stiffening effect of the viscous fluid layers. The product of these two characteristic frequencies is equal to the squared resonant frequency of the layered system, times a dimensionless proportionality constant of the order 1. This explains why the visco-elastic and poroelastic mechanisms are usually treated separately in the context of macroscopic (effective medium) theories, as these theories imply that frequency is small compared to the resonant (scattering) frequency of individual pores.     

Dynamic viscoelastic behaviour of wheat flour doughs

Summary The general dynamic mechanical properties of wheat flour doughs have been investigated for a range of strains over a frequency spectrum of 0.032–32 Hz. An approximately linear stress-strain relationship was found to hold at the lower range of strains imposed. Departures from the linear response of the dough are explained in terms of a protein chain-starch granule interaction and the accompanying time dependence in terms of a stress-biased diffusion process. The influence of water and temperature on the dynamic moduli are shown, indicating that a frequency-temperature and a water absorption-frequency superposition principle is applicable. The transient stress relaxation modulus is calculated from the dynamic moduli to demonstrate the ability of dynamic methods to investigate the time scale inaccessible to transient techniques.     

Slow motions of a rigid sphere in incompressible viscoelastic fluids

Problems concerned with the force of resistance and the moment of forces acting from the side of a viscoelastic fluid on a sphere moving with acceleration are considered in a linear formulation. Fairly simple relations are obtained for a fluid with a single relaxation time or a single after-action time. A discussion of the asymptotic expressions is given for a fluid with a large number of times.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–16, March–April, 1976.     

Temperature Dependent Viscoelastic Properties of Polymers Investigated by Small-Scale Dynamic Mechanical Analysis

The temperature-dependent viscoelastic properties of polymers were investigated by small-scale dynamic mechanical analysis in the range of −100°C to 200°C. The polymers tested included glassy polymer (atactic polystyrene), semicrystalline polymer (high-density polyethylene) and rubbery polymer (polyisobutylene). The small-scale dynamic mechanical analyses were performed by using a flat-tip indenter with an oscillating displacement of amplitude 36 nm. The force amplitude and phase angle were measured, from which the storage modulus E′ and loss tangent tanδ were calculated. The results obtained from indentation experiments are consistent with those obtained from conventional dynamic mechanical analyzer (DMA). It is thus demonstrated that the indentation technique can quantitatively measure the temperature-dependent viscoelastic properties of polymers at small dimensions.     

Influence of an acoustic wave on a system of two spheres in a viscous fluid

In this article we formulate and solve the problem of the influence of radiation forces (forces created by the radiation pressure) on two spheres in a viscous fluid during the transmission of an acoustic wave. On the basis of these forces we investigate the nature of the interaction between the spheres as determined by the mutual disturbance of the flow fields around them as a result of interference between the primary and secondary waves reflected from the spheres. A previously proposed [2] approach is used in the investigations. The radiation force acting on one of the spheres is filtered by averaging the convolution of the stress tensor in the fluid with the unit normal to the surface of the sphere over a time interval and over the surface of the sphere. The stresses in the fluid are represented, to within second-order quantities in the parameters of the wave field, in terms of the velocity potentials obtained from the solution of the linear problem of the diffraction of the primary wave by the free spheres. The diffraction problem is formulated and solved within the framework of the theory of linear viscoelastic solids [6]. The case of an ideal fluid has been studied previously [3–5, 7]. Radiation forces are one of the causes of the relative drift of solid particles situated in a fluid in an acoustic field.S. P. Timoshenko Institute of Mechanics, Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 33–40, February, 1994.     

Small perturbation spectrum for plane-parallel flows of viscoelastic fluids

The investigation of the occurrence of a transition from the laminar to the turbulent flow regime in weak polymer solutions is of great practical interest. Experimental data indicate both an increase in flow stability and an occurrence of early turbulence [1]. Paper [2] explains the discrepancy in the experimental data for the numerical investigation of the first-mode symmetric perturbations, which are unstable for a Newtonian fluid. Paper [3] shows that other modes also become unstable in the case of the flow of a viscoelastic Maxwellian fluid in a channel. These features of the hydrodynamic stability of viscoelastic fluids indicate a significant rearrangement of the small perturbation spectrum. In the present paper, the perturbation spectrum for plane-parallel flows of viscoelastic Oldroyd and Maxwellian fluids is investigated at small Reynolds numbers, and at large and small wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 164–167, November–December, 1984.     

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.     

Viscoelastic stresses in the stagnation flow of a dilute polymer solution

In this paper, we consider viscoelastic stresses T₁₁, T₁₂ and T₂₂ arising in the stagnation flow of a dilute polymer solution; in particular, we consider an upper convected Maxwell (UCM) fluid. We present exact solutions to the coupled partial differential equations describing the viscoelastic stresses and deduce the results for the stress T₂₂ of Becherer et al. [P. Becherer, A.N. Morozov, W. van Saarloos, Scaling of singular structures in extensional flow of dilute polymer solutions, J. Non-Newtonian Fluid Mech. 153 (2008) 183–190]. As we considered the viscoelastic stresses over two spatial variables, we are able to study the effect of variable boundary data at the inflow. As such, our results are applicable to a wider range of fluid flow problems.     

Conversion of dynamic into transient properties,and vice versa

Summary A short review is given of a recent theory for converting into each other dynamic and transient properties of linear viscoelastic materials.The applicability is illustrated for a polymer in the glass-rubber transition range.

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Zusammenfassung Es wird ein kurzer Überblick über eine neue Theorie gegeben, die es erlaubt, dynamische und statische Kennwerte linear-viskoelastischer Stoffe ineinander zu überführen. Die Anwendbarkeit wird am Beispiel eines Polymeren im Glasübergangsgebiet gezeigt.

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Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968.     

Modelling the Stationary Vibrations and Dissipative Heating of Thin-Walled Inelastic Elements with Piezoactive Layers

A coupled dynamic problem of thermoelectromechanics for thin-walled multilayer elements is formulated based on a geometrically nonlinear theory and the Kirchhoff–Love hypotheses. In the case of harmonic loading, an approximate formulation is given using the concept of complex moduli to characterize the cyclic properties of the material. The model problem on forced vibrations of sandwich beam, whose core layer is made of a passive physically nonlinear material, and face layers, of a viscoelastic piezoactive material, is considered as an example to demonstrate the possibility of damping the vibrations by applying harmonic voltage to the oppositely polarized layers of the beam. Substantiation is given for a linear control law with a complex coefficient for the electric potential, which provides damping of vibrations in the first symmetric mode at the linear and nonlinear stages of deformation. The stress–strain state and dissipative-heating temperature are studied     

波能耗散的结构阻尼损耗因子度量方法

根据波动理论,用Timoshenko梁理论在高频范围内分析能量的耗散,通过重建作为频率函数的色散关系曲线,得到动力粘弹性模量及材料的损失因子。根据动力系统的固有特征方程与材料弹性特性的关系,研究利用阻尼损耗因子定量描述在高频情况下,波在结构中传播时的能量耗散效应以及结构阻尼损耗因子的表示方法,并通过实验利用波的色散关系估计结构的动力粘弹性模量,理论分析和实验结果表明了这种方法的可行性。     

A method for calculating rheological and morphological properties of constant-volume polymer blend models in inhomogeneous shear fields

We show how to formulate two-point boundary-value problems in order to compute fully-developed laminar channel and tube flow profiles for viscoelastic fluid models. The formulation is applied to Couette and pressure-driven flows separately, or a combination of both. The application of this methodology is illustrated analytically for the Upper-Convected Maxwell Model, and it is applied computationally for the Phan-Thien/Tanner and Giesekus Models. Numerical solutions exist for the last two models [J.Y. Yoo, H.C. Choi, On the steady simple shear flows of the one-mode Giesekus fluid, Rheol. Acta 28 (1989) 13–24; P.J. Oliveira, F.T. Pinho, Analytical solution for fully developed channel and pipe flow of Phan-Thien–Tanner fluids, J. Fluid Mech. 387 (1999) 271–280; M.A. Alves, F.T. Pinho, P.J. Oliveira, Study of steady pipe and channel flows of a single-mode Phan-Thien–Tanner fluid, J. Non-Newtonian Fluid Mech. 101 (2001) 55–76], allowing verification of the computational technique. Subsequently, the computational algorithm is applied to the constant-volume polymer blend models of Maffettone and Minale [P.L. Maffettone, M. Minale, Equation of change for ellipsoidal drops in viscous flow, J. Non-Newtonian Fluid Mech. 84 (1999) 105–106 (Erratum), J. Non-Newtonian Fluid Mech. 78 (1998) 227–241] and Dressler and Edwards [M. Dressler, B.J. Edwards, The influence of matrix viscoelasticity on the rheology of polymer blends, Rheol. Acta 43 (2004) 257–282; M. Dressler, B.J. Edwards, Rheology of polymer blends with matrix-phase viscoelasticity and a narrow droplet size distribution, J. Non-Newtonian Fluid Mech. 120 (2004) 189–205]. Rheological and morphological properties of the model blends are thus obtained as functions of the spatial position within the channel, applied pressure drop, and shear rate at the wall.     

The relation between CDM instability and Deborah number in differential type rheological equations

Previous studies have argued that rheological equations of the differential type, such as second-order fluid models, are inadequate because they result in unstable solution after cessation of steady shear. If the sign of the viscoelastic coefficient is selected so that the storage modulus is positive, the fluid velocity increases indefinitely and the flow does not decay by viscous dissipation, in contradiction to thermodynamic laws. This study mitigates this problem by demonstrating that the solution of such equations is actually stable at low values of Deborah number De, where these equations are only valid for other reasons. In fact, second order and higher order differential type equations are applicable only if the relaxation time of the fluid is low relative to a characteristic time of the flow. The study shows how to determine the characteristic time and thus clarifies, in practical terms, the limits of the region where differential type equations can be applied.Presented at the Third European Rheology Conference and Golden Jubilee Meeting of the British Society of Rheology, Edinburgh, Sept. 3–7, 1990.     

Rheology of concentrated microgel solutions

Viscosity, modulus, and yield stress for 0–6 wt% aqueous solutions of Carbopol 941 were investigated using constant shear rate, constant shear stress, and dynamic oscillatory experiments. The microgel character of the polymer was evident from the solid-like behavior of the solutions above 1 wt%. Yield stress increased with concentration, but yield occurred at a critical shear strain of 40%, independent of concentration. The static stress-strain relationship became non-linear at ~ 25% strain, in fair agreement with the onset of non-linear response in the storage modulus at ~ 10% strain. Small strain moduli from static and low frequency measurements agreed rather well; modulus values obtained from the recoverable strain after yielding were 30–40% smaller. Solutions flowed at near-constant stress in the low shear rate regime; at higher rates the stress increases with shear rate more rapidly. The viscosity did not obey the Cox-Merz rule. Steady-state viscosity scaled with polymer concentration to the 3/4 power. Results were interpreted using a cellular, deformable sphere model for the polymer, in analogy to emulsions and foams.