Viscoelastic measurement of complex fluids using forced oscillating torsion resonator with continuously varying frequency capability

Traditional torsional resonators, often obtaining the viscoelastic moduli of complex fluids only at one or several given discrete frequencies, lack the continuously varying frequency capability. This is an obvious disadvantage of the traditional torsional resonator technique. This paper presents an improved strategy, based on our previous discrete-frequency-measuring method (Wang et al., J Rheol 52:999–1011, 2008), to overcome such restriction and thus accomplish the continuously varying frequency capability of the traditional torsional resonator for measuring the viscoelastic properties of complex fluids. The feasibility of this strategy is demonstrated with the Newtonian fluids (several water–glycerol solutions) of viscosities varying from 10 to 1,400 cp by using our homemade torsion resonator apparatus in the 10 ~ 2,500 rad/s frequency range (continuous frequencies). Some results for typical viscoelastic polymers (two polyethylene oxide (PEO) aqueous solutions) are also given. Additionally, a comparison of the PEO results is made with the common rheometer technique. It is demonstrated that this improved strategy could enable the traditional torsional resonators, with one oscillating resonance mode, to work as the microrheological technique and the common rheometer technique in the continuous frequency range.     

Torsional vibrations of a column of fine-grained material: A gradient elastic approach

The gradient theory of elasticity with damping is successfully employed to explain the experimentally observed shift in resonance frequencies during forced harmonic torsional vibration tests of columns made of fine-grained material from their theoretically computed values on the basis of the classical theory of elasticity with damping. To this end, the governing equation of torsional vibrations of a column with circular cross-section is derived both by the lattice theory and the continuum gradient elasticity theory with damping, with consideration of micro-stiffness and micro-inertia effects. Both cases of a column with two rotating masses attached at its top and bottom, and of a column fixed at its base carrying a rotating mass at its free top, are considered. The presence of both micro-stiffness and micro-inertia effects helps to explain the observed natural frequency shift to the left or to the right of the classical values depending on the nature of interparticle forces (repulsive or attractive) due to particle charge. A method for using resonance column tests to determine not only the shear modulus but also the micro-stiffness and micro-inertia coefficients of gradient elasticity for fine-grained materials is proposed.     

Compliance effects on the torsional flow of a viscoelastic fluid

The effects of transducer compliance on transient stress measurements in torsional flows of a viscoelastic fluid are investigated theoretically. The analysis is based on the torsional flow of an upper-convected Maxwell fluid between a rotating and ‘stationary’ disk, which is allowed to twist and displace axially as a result of the stresses exerted on the disk by the fluid. An approximate analytical solution to the governing equations is obtained using a standard perturbation method. Results of the analysis are used to examine how the fluid velocity is altered by the motion of the stationary disk and to gain insight on how transient stress measurements are affected by transducer compliance. The analysis shows that compliance effects increase with applied shear rate and that the effects of torsional and axial compliance are coupled in measurements of the shear stress and first normal stress difference.     

Determination of material response functions for prestrained rubbers

The mechanical response of two natural rubber compounds is examined in order to determine relevant material parameters by non-linear finite element analysis. The materials are subjected to (a) combined static torsion and extension, and (b) small, steady-state torsional oscillations superposed on a large static simple extension. The materials are assumed to be incompressible and isotropic in their undeformed state and a time-strain separable relaxation modulus tensor is employed in order to characterize the steady-state harmonic viscoelastic response. The combined static torsion and extension experiments are used to determine the basic delayed elastic response functions. A Rivlin-type strain energy expression of third-order accuracy is used for the purpose. The two-constant, Mooney-Rivlin form is found to be adequate for both materials in the relatively limited range of deformation magnitudes considered.The torsional storage and loss moduli are determined under quasistatic conditions as functions of frequency and axial static pre-strain. The time-strain separability is found to be a resonable approximation in a relatively limited range of static prestrain magnitudes and frequencies considered for the natural gum rubbers investigated. The experimental methodology is discussed in some detail.     

Dynamics of coaxial torsional resonators consisting or segments with different radii,material properties and surrounding media

High-Q torsional resonators constitute the most sensitive transducers for high frequency dynamic viscoelastic measurements of dilute polymer solutions. Most such resonators described in the literature are segmented. Because of the need for torque and torsional displacement transducers the Q-value of the individual segments most often differ, but normally all segments have the same radius.A detailed analysis of the dynamics of such resonators when both the radii, material properties and surrounding media may be different for each segment, is presented. For resonators where all segment lengths equal an integer multiple of a quarter of the torsional wavelength, we find that the Q-value of the resonator as a whole is mainly determined by the Q-value of the segment with the smallest radius. We further find that reduction of the radius of the segment surrounded by polymer solution results in a stronger mechanical coupling between the resonator as a whole and the polymer solution. These findings suggest that the segment radii are important optimization parameters of segmented torsional resonators used to measure the high frequency dynamic viscoelastic properties of e.g. polymer solutions.     

粘弹性地基上含孔隙的石墨烯增强功能梯度板的振动

为分析粘弹性地基上含孔隙的石墨烯增强功能梯度板的自由和强迫振动特性,基于三参数粘弹性地基模型及复合材料薄板理论,建立了粘弹性地基上含孔隙石墨烯增强功能梯度板的运动方程,用伽辽金法求解其固有频率和动力响应,并通过数值算例分析了粘弹性地基参数、孔隙率、孔隙类型及石墨烯纳米片分布模式、含量等因素对自由振动和动力响应的影响.结果表明,固有频率随着孔隙率的增大非单调变化,孔隙率对固有频率的影响随着地基参数、孔隙类型的不同而不同.另外,在三种孔隙类型中,上下表面层含有最少孔隙数量的板的动挠度最小,且其动挠度随着孔隙率的增大而微弱提高.     

黏弹性地基上功能梯度材料板的振动分析

为研究黏弹性地基上功能梯度材料板的自由和强迫振动特性,基于Reddy高阶剪切变形理论以及由Shen导得的广义Karman型方程,用双重Fourier级数法推导了三参数黏弹性地基上四边简支功能梯度材料板自由振动和动力响应的解析解,计算了各模态自振频率和半波冲击载荷作用下的动力响应,讨论了材料组分指数、黏弹性地基参数、边厚比等因素对自由振动和动力响应的影响.结果表明,黏弹性地基的剪切和压缩刚度显著提升了功能梯度材料板的振动频率,减小了动力响应;另外,地基的黏性对振动频率和动力响应也有一定的影响.     

Isothermal viscoelastic properties of PMMA and LDPE over 11 decades of frequency and time: a test of time–temperature superposition

Many instruments used to measure viscoelastic properties are only capable of subjecting a sample to a limited range of loading frequencies. For thermorheologically simple materials, it is assumed that a change in temperature is equivalent to a shift of the viscoelastic behavior on the log frequency or time axis. For many materials, time–temperature superposition appears to work well for modulus or compliance curves over three decades of time or frequency, but some deviations are known if the window is expanded to five or six decades. To apply a more stringent test of the validity of time–temperature superposition, broadband viscoelastic spectroscopy is used to isothermally study polymethylmethacrylate and low-density polyethylene at several temperatures in the glassy region. Shear modulus and damping (tan δ) are measured isothermally over a wide range (up to 11 decades) of time and frequency. Results indicate that, while modulus curves can be approximately superimposed, the damping (tan δ) curves change in height and shape with temperature.     

Resonant-Based Identification of the Poisson’s Ratio of Orthotropic Materials

The resonant-based identification of the in-plane elastic properties of orthotropic materials implies the estimation of four principal elastic parameters: E ₁ , E ₂ , G ₁₂ , and ν ₁₂ . The two elastic moduli and the shear modulus can easily be derived from the resonant frequencies of the flexural and torsional vibration modes, respectively. The identification of the Poisson’s ratio, however, is much more challenging, since most frequencies are not sufficiently sensitive to it. The present work addresses this problem by determining the test specimen specifications that create the optimal conditions for the identification of the Poisson’s ratio. Two methods are suggested for the determination of the Poisson’s ratio of orthotropic materials: the first employs the resonant frequencies of a plate-shaped specimen, while the second uses the resonant frequencies of a set of beam-shaped specimens. Both methods are experimentally validated using a stainless steel sheet.     

A numerical investigation of side‐loads resulting from rigid body motions of an overexpanded engine nozzle

A numerical model for fluid–structure interactions is presented. Its purpose, within the context of 2D overexpanded engine nozzles, is to improve understanding of interactions between side‐loads and rigid body rotations, and more generally of the underlying physics between a shock in motion and nozzle movements. The model is based on three different solvers, for fluid, structure and mesh deformation respectively, which are linked to a coupling scheme in a parallel environment. In particular it is shown that the nozzle has a natural torsional frequency for which the measured side‐loads are the greatest. This phenomenon is associated with a transversal wave in the flow between the two internal walls of the nozzle. For free coupling cases, our calculations go some way to explain how the mechanical energy is dissipated with a transfer of energy to the shock that encounters the largest motions to dissipate it. It has also been observed that the compression shock may adopt a quasi‐steady state response with regard to nozzle rotations at low frequencies, whereas this will no longer be the case at higher frequencies, where a phase shift may occur between side‐loads and rotational position. This study is aimed at enhancing the only current aeroelastic stability model for overexpanded nozzles (AIAA, 29th Joint Propulsion Conference and Exhibit, Monterey, CA, 28–30 June 1993). Copyright © 2010 John Wiley & Sons, Ltd.     

Diffraction of torsional waves by a penny-shaped crack in a non-homogeneous thick fibre bonded to a non-homogeneous matrix

In this paper the equation of motion is solved when the shear modulus and density are functions of r and z and the latter part of this paper contains an analysis of the interaction of torsional waves normally with penny-shaped crack located in a thick infinite elastic fibre. The infinite elastic fibre is bonded to an infinite elastic matrix. The matrix and the thick elastic fibre are non-homogeneous and are of dissimilar materials. The solution of the problem is reduced to a Fredholm integral equation of the second kind, which is solved numerically. The numerical solution is used to calculate the stress intensity factor at the rim of the penny-shaped crack. Finally the results of the stress intensity factors are displayed graphically.     

A double-superposition global–local theory for vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates: a complex modulus approach

A higher-order global–local theory is proposed based on the double-superposition concept for free vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates subjected to thermomechanical loads. In contrast to all theories proposed so far for analysis of the viscoelastic plates, the continuity conditions of the transverse shear and normal stresses at the layer interfaces and the nonzero traction conditions at the top and bottom surfaces of the sandwich plates are satisfied. Another novelty is that these conditions may be satisfied for viscoelastic plates with temperature-dependent material properties and nonlinear behaviors subjected to thermomechanical loads. Furthermore, transverse flexibility is also taken into account. Some dynamic buckling/wrinkling analyses of the viscoelastic plates are performed in the present paper, for the first time. Comparisons made between results of the paper and results reported by well-known references confirm the accuracy and the efficiency of the proposed theory and the relevant solution algorithm.     

Rigorous Bounds on the Torsional Rigidity of Composite Shafts with Imperfect Interfaces

We derive upper and lower bounds for the torsional rigidity of cylindrical shafts with arbitrary cross-section containing a number of fibers with circular cross-section. Each fiber may have different constituent materials with different radius. At the interfaces between the fibers and the host matrix two kinds of imperfect interfaces are considered: one which models a thin interphase of low shear modulus and one which models a thin interphase of high shear modulus. Both types of interface will be characterized by an interface parameter which measures the stiffness of the interface. The exact expressions for the upper and lower bounds of the composite shaft depend on the constituent shear moduli, the absolute sizes and locations of the fibers, interface parameters, and the cross-sectional shape of the host shaft. Simplified expressions are also deduced for shafts with perfect bonding interfaces and for shafts with circular cross-section. The effects of the imperfect bonding are illustrated for a circular shaft containing a non-centered fiber. We find that when an additional constraint between the constituent properties of the phases is fulfilled for circular shafts, the upper and lower bounds will coincide. In the latter situation, the fibers are neutral inclusions under torsion and the bounds recover the previously known exact torsional rigidity.      

Physical Principles of Methods for Measuring Viscoelastic Properties

The main methods used to measure viscoelastic properties of materials in a wide range of frequencies from 10^?4 to 10⁶ Hz are reviewed. It is demonstrated that the accuracy of many experimental methods can be increased by taking into account the form factors, which depend on the specimen type. An example of the form factor for a cylindrical specimen is provided, which is determined numerically on the basis of a two-dimensional deformation model taking into account the specimen geometry and Poisson’s ratio. The importance of the precise determination of Poisson’s ratio for rubber-like and complex-structured materials is demonstrated. Requirements to such measurements and a setup satisfying these requirements are described. Two methods for measuring viscoelastic properties of living tissues (compliance and disturbance propagation velocity) are considered. Based on the developed method of measuring these parameters for materials with a fixed thickness, methods for the creation of a unified standard of measurements of viscoelastic characteristics of living tissues are proposed.     

Theoretical and experimental analysis of longitudinal wave propagation in cylindrical viscoelastic rods

Wave propagation in viscoelastic rods is encountered in many applications including studies of impact and fracture under high strain rates and characterization of the dynamic behavior of viscoelastic materials. For viscoelastic materials, both material and geometric dispersion are possible when the diameter of the rod is of the same order as the wavelength. In this work, we simplify the Pochhammer frequency equation for low and intermediate loss viscoelastic materials and formulate corrections for geometric dispersion for both the phase velocity and attenuation. The formulation is then experimentally verified with measurements of the phase velocity and attenuation in commercial polymethylmethacrylate rods that are 12 and in diameter. Without correcting for geometric dispersion, the usable frequency range for determining the phase velocity and attenuation for the rod is about , and about for the rod. Using the correction procedure developed here, it was possible to accurately determine the phase velocity and attenuation up to frequencies exceeding for the rod and for the rod. These corrections are applicable to many polymers and other viscoelastic materials. From thereon, the viscoelastic properties of the material can be determined over a wide range of frequencies.     

Dispersion of elastic guided waves in piezoelectric infinite plates with inversion layers

In this work, we study the dispersion of elastic waves in piezoelectric infinite plates with ferroelectric inversion layers. The motivation is to analyze the effect of ferroelectric inversion layers on wave dispersion and resonant behavior under impulsive line loads. A semi-analytical finite-element (SAFE) method has been adopted to analyze the problem. Two model problems are considered for analysis. In one, the plate is composed of a layer of 36° rotated y-cut LiNbO₃ with a ferroelectric inversion layer. In the other, material is PZT-4 with a ferroelectric inversion layer. Comparison with experimental results, reported in the literature for isotropic materials, shows a very good agreement with theoretical predictions obtained using SAFE method. Furthermore, comparison of the resonance frequencies of the S₁ modes, calculated using KLM approximation (f₀ = C_d/2h) and SAFE method, are illustrated for each problem. The frequency spectra of the surface displacements show that resonant peaks occur at frequencies where the group velocity vanishes and the phase velocity remains finite, i.e., a minimum in the dispersion curve below the cut-off frequency. The effect of the ratio of the thicknesses of the inversion layer (IL) and the plate on the frequencies and strength of the resonant peaks is examined. It is observed that for PZT-4 with 50% IL to plate thickness ratio the frequency for the second resonant peak is about twice that for the first one. Results are presented showing the dependence of resonant frequencies on the material properties and anisotropy. Materials selection for single-element harmonic ultrasound transducers is a very important factor for optimum design of transducers with multiple thickness-mode resonant frequencies. The theoretical analysis presented in this study should provide a means for optimum ultrasound transducer design for harmonic imaging in medical applications.     

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.     

Modeling vibration behavior of embedded graphene-oxide powder-reinforced nanocomposite plates in thermal environment

Abstract

This article deals with the thermal vibration analysis of the graphene-oxide powder-reinforced (GOPR) nanocomposite plates, once the plate is embedded on the viscoelastic substrate. The structure is subjected to thermal loadings with various temperature rises such as sinusoidal temperature rise (STR), linear temperature rise (LTR), and uniform temperature rise (UTR). Damping behavior of the GOPR nanocomposite plates is investigated based on the variations of the natural frequencies. Four functionally graded (FG) patterns of GOPs’ distribution are taken into account comparatively in order to find out the best model of reinforcing the structure. The homogenization of the materials is carried based on the Halpin-Tsai micromechanical scheme. The governing equations of the motion have been derived through the combination of refined higher-order shear deformation theory and Hamilton’s principle. The accuracy of this modeling is validated with those reported in the open literature. Finally, the influences of different parameters on the natural frequencies of the embedded GOPR nanocomposite plates are investigated based on the damping coefficient of the viscoelastic substrate. The graphical results reveal that the free vibrational behavior of the structure is remarkably affected by the variations of these parameters.     

Unsteady motion of a non-linear viscoelastic fluid

In this paper, we study the unsteady flow of a generalized second grade fluid. Specifically, we solve numerically the linear momentum equations for the flow of this viscoelastic shear-thinning (shear-thickening) fluid surrounding a solid cylindrical rod that is suddenly set into longitudinal and torsional motion. The equations are made dimensionless. The results are presented for the shear stresses at the wall, related to the drag force; these are physical quantities of interest, especially in oil-drilling applications.     

Nonlinear dynamics of a Jeffcott rotor with torsional deformations and rotor-stator contact

The dynamics of a modified Jeffcott rotor is studied, including rotor torsional deformation and rotor-stator contact. Conditions are studied under which the rotor undergoes either forward synchronous whirling or self-excited backward whirling motions with continuous stator contact. For forward whirling, the effect on the response is investigated for two commonly used rotor-stator friction models, namely, the simple Coulomb friction and a generalized Coulomb law with cubic dependence on the relative slip velocity. For cases with and without the rotor torsional degree of freedom, analytical estimates and numerical bifurcation analyses are used to map out regions in the space of drive speed and a friction parameter, where rotor-stator contact exists. The nature of the bifurcations in which stability is lost are highlighted. For forward synchronous whirling fold, Hopf, lift-off, and period-doubling bifurcations are encountered. Additionally, for backward whirling, regions of transitions from pure sticking to stick-slip oscillations are numerically delineated.