Characterization of viscoelastic properties of polymeric materials through nanoindentation

Nanoindentation testing was used to determine the dynamic viscoelastic properties of eight polymer materials, which include three high-performance polymers and five densities of high-density polyethylene. It was determined that varying the harmonic frequency of nanoindentation does not have a significant effect on the measured storage and loss moduli of the polymers. Agreement was found between these nanoindentation results and data from bulk dynamic mechanical testing of the same materials. Varying the harmonic amplitude of the nanoindentation had a limited effect on the measured viscoelastic properties of the resins. However, storage and loss moduli from nanoindentation were shown to be sensitive to changes in the density of the polyethylene.     

Superposition principles for small amplitude oscillatory shearing of nematic mesophases

The linear viscoelasticty of Leslie-Ericksen monodomain liquid crystals subjected to a bend distortion through a small amplitude oscillatory shear flow driven by harmonic wall stress is analyzed, using numerical and asymptotic methods. The viscoelastic material functions were derived using a new scaling approach that extracts the material parameters that control superposition. Small and high frequency superposition schemes for linear viscoleasticity were derived. The schemes were successfully applied to collapse the predicted loss and storage linear viscoelastic moduli of seven experimental data sets. Comparisons between different shear flows (simple shear and capillary Poiseuille) and different director distortion modes (splay and bend) shows that the superposition schemes are applicable to shear flows in any single director distortion mode.     

On separation of energies in viscoelasticity

Separation of the storaged and dissipated energies in viscoelastic deformation is considered. This is a key problem for the construction of viscoelastic minimum rinciples and for the micromechanics of heterogeneous materials with memory. The notion of the viscoelastic free energy functional is discussed, thermodynamic admissibility conditions are established. An engineering analysis is realized through the method of harmonic strain regimes, influence of the loss and the storage moduli on the dissipation rate is studied. For the Volterra-Frechet integral expansion approach, necessary conditions on the general form of a free energy viscoelastic functional are formulated. The obtained results are used to examine the thermodynamic validity of certain classic viscoelastic models, like that of Staverman-Schwarzl. Through the spectral method, this energy representation is shown to correspond to a generalized Maxwell model.     

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.     

A New Technique for Characterization of Low Impedance Materials at Acoustic Frequencies

Measuring the mechanical properties of low impedance rubbery polymers at acoustic frequencies is a challenging problem due to the small signal amplitudes, relatively high loss, and the long wavelength of stress waves. One such material is solid polyurea (PU), an elastomeric copolymer, which has excellent chemical, thermal, and mechanical properties and is widely used as a coating (e.g. in truck bed lining) or blast protection (advanced helmet designs and concrete structures) material. Moreover, due to its heterogeneous structure, PU has a wide transition of thermo-mechanical behavior from rubber-like to glassy compared to most engineering polymers, which translates to a broader loss spectrum in frequency domain. In this study, we have developed a new test technique by modifying the split Hopkinson pressure bar and using ball impact to measure Young’s storage and loss moduli of polyurea at kHz frequencies. This will therefore fill the frequency gap between the dynamic mechanical analysis (DMA) and ultrasonic (US) wave measurement. The measured Young’s storage and loss moduli from this technique are compared with the master curves of the moduli developed using experimental data of dynamic mechanical analysis and ultrasonic wave measurements. This technique is a direct measurement which provides more reliable data in the kHz frequency range and can be used to evaluate the reliability of other indirect estimations including master curves. The utility of this technique is not limited to polyurea and it can be used to characterize other low impedance materials at kHz frequencies.     

High-bandwidth one-and two-particle microrheology in solutions of wormlike micelles

We have developed a large-bandwidth two-particle microrheology technique to measure loss and storage moduli of viscoelastic materials from 0.1 Hz to about 100 kHz using laser trapping and interferometry. We found that quantitative agreement between one- and two-particle microrheology exist in entangled solutions of wormlike micelles chosen as a simple model viscoelastic system. These results validate both experimental method and data interpretation. The consistent results also prove that in a simple system, where the solution length scales are much smaller than the micron probe size, one-particle microrheology can accurately measure bulk viscoelastic parameters.This paper was presented at the AERC 2005     

Elongational flow behavior of polymeric fluids according to the Leonov model

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.     

An Approach to the Determination of the Deformation Characteristics of Viscoelastic Materials

An efficient method is proposed to determine the deformation function of a viscoelastic material from experimental data. The deformation function is assumed to be an integral operator with Rabotnov's fractional-exponential kernel or a sum of such kernels. This representation enables effective use of the method of operator continued fractions. To illustrate the method, deformation data for polymethylmethacrylate are used. The viscoelastic characteristics of a composite based on this material are obtained using the method of operator continued fractions __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 41–50, August 2005.     

Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers II: oblique anchoring angles

We extend previous work on the linear viscoelastic moduli of heterogeneous nematic polymers in a small-amplitude oscillatory shear flow, focusing on the role of the orientational anchoring conditions at the plates. When tangential or normal anchoring conditions are applied, the Doi–Marrucci–Greco orientation tensor-flow model effectively reduces to the Leslie–Ericksen director-flow model, predicting that director distortions control the dynamic moduli with negligible contributions from tensor-order parameters. In this paper, we examine oblique anchoring angles. We use a combination of analysis and numerical simulation on the generalized tensor-flow system for arbitrary anchoring conditions to show that any oblique anchoring condition induces a nontrivial order parameter contribution to the dynamic moduli, which vanishes only in the limit of tangential or normal anchoring. Our approach reveals that the storage and loss moduli admit an approximate decomposition in terms of two reduced problems that are exactly solvable: the heterogeneous director–flow response plus the monodomain tensor response to an imposed shear. The importance of this result is that we gain scaling properties of the moduli with respect to material parameters and experimental conditions without having to compute and assimilate across the full parameter space. These results provide insight into the relative importance of the distortional vs bulk nematic elastic stress in determining the viscoelastic moduli, predicting that anchoring conditions tune the relative contributions.     

Determination and analysis of the effective relaxation properties of a composite with viscoelastic components

The relaxation properties of a two-component material are determined depending on time, volume fraction, and type of reinforcement, and the relationship among them. The type of reinforcement is determined by the aspect ratio of the ellipsoid of revolution that models the inclusion. The effective moduli of the composite are determined from the relaxation properties of the components. It is assumed that the composite components are made of isotropic viscoelastic materials with volume expansion and shear characteristics described by two Rabotnov’s fractional-exponential functions with different orders of fractionality. To obtain the solution in the time domain, its fractional rational representation in the frequency domain is used. Optimizing the parameters of this representation and transforming the parameters of the solution to the time domain make it possible to obtain solutions in compact form in terms of relaxation kernels     

A method of determining complex moduli of viscoelastic materials

A method is introduced whereby the complex moduli of viscoelastic materials may be determined in a relatively simple and accurate manner by means of calibration of the measuring system using a specimen of known properties. The appropriate data-reduction equations are presented and use of the method is demonstrated for determination of complex moduli for bovine bones over a four-octave frequency range.     

Linear rheology of compressible soft nanocomposites

The influences of interfacial tension and compressibility to the linear viscoelastic properties of nanocomposite and nanoporous materials are considered theoretically. The effective bulk and shear moduli of the systems are calculated within the generalized composite sphere model which takes into account the effect of interfacial tension. It is found that frequency dependence of the effective dynamic shear and bulk moduli of nanocomposites with the compressible elastic matrix and viscous inclusions may be represented in terms of the Zener model comprising of the viscoelastic Kelvin element in series with the elastic spring. The relations of the Zener model parameters with the material characteristics are revealed. The physical interpretation of the frequency behavior of the dynamic shear and bulk moduli against the interfacial tension, component compressibility, viscosity, and inclusion volume fraction is discussed. Victor G. Oshmyan deceased.     

Closed-form solution for horizontal and vertical shiftings of viscoelastic material functions in frequency domain

The closed-form shifting (CFS) algorithm is a simple mathematical methodology which determines the unique solution in the process of constructing master curves at selected reference temperature and pressure conditions. In a previous paper, the CFS algorithm has been fully described for monotonically increasing or monotonically decreasing functions only. This paper presents detailed steps of the generalized CFS methodology for non-monotonic functions, like the loss tangent. Performing shifting on the loss tangent, which does not require vertical shifting, is particularly important for materials which require vertical adjustment of dynamic viscoelastic functions, i.e., loss and storage moduli. Thus, based on horizontal shifting of the loss tangent, the CFS-based procedure of consecutive horizontal-vertical superposition for the storage modulus is proposed and analyzed. The analysis is done on the example of two synthetically generated non-monotonic tan delta segments and corresponding storage modulus segments in respect to different experimental parameters. It has been shown that the error brought by the shifting method into non-monotonic loss tangent and storage modulus master curves is twice smaller than the corresponding experimental noise level.     

Spectral element technique for efficient parameter identification of layered media. Part III: viscoelastic aspects

This article addresses the issues of wave propagation in elastic–viscoelastic layered systems and viscous parameter identification from non-destructive dynamic tests. A methodology that combines the spectral element technique, for the simulation of wave propagation, with the differential operator technique, for stress–strain relationship in viscoelastic materials, is adopted. The compatibility between the two techniques stems from the fact that both can be treated in the frequency domain, which enables naturally the adoption of Fourier superposition. The mathematical formulation of spectral elements for Burger's viscoelastic material model is highlighted. Also, an inverse procedure for the identification of the material's Young's moduli and complex moduli for layer systems is described. It is shown that the proposed methodology enables the substitution of an expensive laboratory testing procedure for the determination of material complex moduli with non-destructive dynamic testing.     

Implications and Constraints of Time-Independent Poisson Ratios in Linear Isotropic and Anisotropic Viscoelasticity

It is proven that time-independent viscoelastic Poisson ratios (PR) can only exist under separation of variable solutions which severely limits the class of applicable problems to quasi-static ones with incompressible homogeneous materials and non-moving boundaries under separable stress or displacement boundary conditions without any thermal expansions. Therefore, composites which are inherently anisotropic and sandwich structures which are nonhomogeneous and anisotropic are generally precluded from having time-independent PRs. Equal time variations for material properties in all directions are shown to be another simultaneous requirement instead of the incompressibility condition for achieving time-independent PRs. However, such restricted models lead to physically unrealistic bulk moduli responses when compared to experimentally determined relaxation moduli and are not generally achievable in current real materials. Consequently, viscoelastic materials are best characterized in terms of relaxation or creep functions, moduli or compliances rather than combinations of the latter with Poisson's ratios. Additionally, the assumption of constant PRs in problems involving thermal and chemical expansions, such as curing and manufacture of viscoelastic composites, is shown to be unjustified and insupportable. The distinct viscoelastic PR definitions, as found in the literature, are examined and classified into five categories. It is further shown that each is inherently unrelated to the others and all are always time-dependent, unless the above extremely limiting conditions are imposed. An extensive literature review indicates that experimental results overwhelmingly confirm the time dependent nature of viscoelastic PRs as no constant experimentally observed PRs were reported.     

Effective properties of a linear viscoelastic composite

The relaxation moduli of a composite are determined. The relaxation of its components is described by various few-parameter kernels: Mittag-Leffler functions of different orders and Rzhanitsyn kernel. It is assumed that the composite components are made of model materials with volume relaxation. The Laplace transform and fractional rational approximation are used to develop an algorithm for reducing the relaxation functions of the composite to one class (series of decreasing exponents or exponents of fractional order). The relaxation moduli of a unidirectionally reinforced composite are determined as an example     

Generalized Maxwell model for the viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading

The linear viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading is investigated in the glass transition range. Using the time-temperature superposition technique, the master curves of the shear dynamic relaxation moduli are obtained at a reference temperature of 566°C. A method to determine the viscoelastic constants from dynamic relaxation moduli is proposed. However, some viscoelastic constants cannot be directly measured from the experimental curves and others cannot be precisely obtained due to non-linearity effects at very low frequencies. The generalized Maxwell model is investigated from the experimental dynamic moduli without fixing the viscoelastic constants. A set of parameters is shown to be in good agreement with the experimental dynamic relaxation moduli, but does not give the correct values of the viscoelastic constants of the investigated glass. The soda-lime-silica glass exhibits a non-linear viscoelastic behavior at very low stress level which is usually observed for organic glasses. This non-linear behavior is questioned.     

High Frequency Measurements of Viscoelastic Properties of Hydrogels for Vocal Fold Regeneration

This report describes a torsional wave experiment used to measure the viscoelastic properties of vocal fold tissues and soft materials over the range of phonation frequencies. A thin cylindrical sample is mounted between two hexagonal plates. The assembly is enclosed in an environmental chamber to maintain the temperature and relative humidity at in vivo conditions. The bottom plate is subjected to small oscillations by means of a galvanometer driven by a frequency generator that steps through a sequence of frequencies. At each frequency, measured rotations of the top and bottom plates are used to determine the ratio of the amplitudes of the rotations of the two plates. Comparisons of the frequency dependence of this ratio with that predicted for torsional waves in a linear viscoelastic material allows the storage modulus and the loss angle, in shear, to be calculated by a best-fit procedure. Experimental results are presented for hydrogels that are being examined as potential materials for vocal fold regeneration.