Characterization of polymer dispersions by Fourier transform rheology

Fourier transform rheology is a very sensitive technique to characterize non-linear rheological fluid properties. It has been applied here for the first time to polymer dispersions in water and the results are compared to those from conventional rheology, namely steady and small amplitude oscillatory shear experiments. The investigated systems are mainly based on styrene and n-butylacrylate. A first attempt was made to evaluate how far colloidal parameters like particle volume fraction and ionic strength as well as chemical composition and surface characteristics of the dispersed particles are reflected in FT-rheology spectra. Significantly different non-linearities are observed for highly concentrated dispersions of particles with different T_g. These differences are not detected in linear oscillatory shear and show up in steady shear only at significantly higher shear rates. Particle surface characteristics influence the non-linear response in oscillatory shear significantly and the intensity of the overtones is found to be higher for a dispersion of particles with a “hairy” swollen surface layer as compared to a system of smooth particles, although the solids content was adjusted to match the steady shear viscosity. The intensity of the overtones in FT-rheology strongly decrease upon dilution. At a solid content below 35% no differences are observed in the FT-experiments for the systems investigated here, whereas the differences in steady shear are very pronounced in this concentration range. A significant influence of added salt onto the non-linear response is detected for some systems, which might be correlated to the stability of these systems. The observed phenomena certainly cannot be explained in terms of constitutive equations or microstructural statistical mechanical models at present. Thus, FT-rheology yields information complementary to classical steady or linear oscillatory shear experiments. Received: 11 December 2000 Accepted: 8 April 2001     

The crossover between linear and non-linear mechanical behaviour in polymer solutions as detected by Fourier-transform rheology

 The application of oscillatory shear strain leads, in the non-linear regime, to the appearance of higher harmonic contributions in the shear stress response. These contributions can be analyzed as spectra in Fourier space, with respect to different frequencies, amplitudes and phase angles. In this article, we present an application of this new characterization method to a solution of the linear homopolymer polyisobutylene. The degree of non-linear response during oscillatory shear is quantified using the normalized intensity of the third harmonic contribution. We were able to show experimentally on polyisobutylene that there is an immediate onset of the non-linear response even for very small shear strain amplitudes. Received: 21 June 1999/Accepted: 21 August 1999     

On the identification of non-linear mechanical systems using orthogonal functions

Real world mechanical systems present non-linear behavior and in many cases simple linearization in modeling the system would not lead to satisfactory results. Coulomb damping and cubic stiffness are typical examples of system parameters currently used in non-linear models of mechanical systems. This paper uses orthogonal functions to represent input and output signals. These functions are easily integrated by using a so-called operational matrix of integration. Consequently, it is possible to transform the non-linear differential equations of motion into algebraic equations. After mathematical manipulation the unknown linear and non-linear parameters are determined. Numerical simulations, involving single and two degree-of-freedom mechanical systems, confirm the efficiency of the above methodology.     

Rheological characterization of commercial highly viscous alginate solutions in shear and extensional flows

The rheological properties of sodium alginate in salt-free solutions were studied by steady shear, dynamic oscillatory and extensional measurements. This biopolymer consists of mannuronic and guluronic acid residues that give a polyelectrolyte character. We applied the scaling theories and checked their accordance with polyelectrolyte behaviour for low concentrations with a shift to neutral polymer behaviour at larger concentrations. This nature was supported by the effect of the concentration on the specific viscosity, the relaxation times from steady shear and the longest relaxation times from small amplitude oscillatory shear (SAOS) measurements. To analyze the extensional behaviour of the samples, we conducted a study of dimensionless numbers and time scales where filament thinning driven by viscous, capillary or elastic forces is at play. We conclude that an exponential filament thinning followed by breakup results in the best regimes that describe the experimental data. Besides, the data pointed out that alginate in salt-free concentrated solutions shows strain thinning of the extensional viscosity and chain rigidity, behaviours that cannot be inferred from the shear rheometry.     

Comparison between the dynamic moduli of fully non-linear and quasilinear viscoelastic materials

This paper deals with the non-linear viscoelastic model with multiple hereditary integrals (MHI) in the frequency domain and the conditions that it reduces to single hereditary integral or the quasilinear viscoelastic (QLV) model. It is shown that when the higher order complex moduli are related to the first-order modulus as the MHI model reduces to the QLV model. The coefficients of quasilinearity should be real and independent of amplitude and frequency.     

Numerical issues related to the modelling of electrochemical impedance data by non-linear least-squares

Non-linear least-squares (NLS) fitting is the typical approach to the modelling of electrochemical impedance spectroscopy (EIS) data. In general the application of NLS to EIS models can give rise to ill-posed problems. On the one side, with ill-posed problems it is not possible to prove a priori that a unique solution exists. On the other side, the relevant numerical approximations cannot ensure that a unique solution exists even a posteriori. It is therefore basically pointless to endeavour to achieve one absolute minimum of any objective function for an EIS model in an NLS problem. A lack of awareness of the above-mentioned factors might render numerical approaches tending to locate the absolute minimum questionable.     

On the non-linear instability of fiber suspensions in a Poiseuille flow

The non-linear instability characteristics of fiber suspensions in a plane Poiseuille flow are investigated. The evolution equation of the perturbation amplitude analogous to Landau equation is formulated and solved numerically for different fiber parameters. It is found that the equilibrium amplitude decreases with the increase of the fiber aspect ratio and volume fraction, i.e. the addition of the fibers reduces the amplitude of the perturbation, and leads to the reduction of the flow instability. This phenomenon becomes significant for large volume concentration and aspect ratio. The mechanism of the reduction of the flow instability is also analyzed by taking into account of the modification of the mean flow and the energy transfer from the mean flow to the perturbation flow.     

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency-response curves, stability, and bifurcation points of the system.     

Detecting the position of non-linear component in periodic structures from the system responses to dual sinusoidal excitations

Based on the non-linear output frequency response functions (NOFRFs), a novel method is developed to detect the position of non-linear components in periodic structures. The detection procedure requires exciting the non-linear systems twice using two sinusoidal inputs separately. The frequencies of the two inputs are different; one frequency is twice as high as the other one. The validity of this method is demonstrated by numerical studies. Since the position of a non-linear component often corresponds to the location of defect in periodic structures, this new method is of great practical significance in fault diagnosis for mechanical and structural systems.     

Behavior of a net of fibers linked by viscous interactions: theory and mechanical properties

This paper presents an investigation of the macroscopic mechanical behavior of highly concentrated fiber suspensions for which the mechanical behavior is governed by local fiber-fiber interactions.The problem is approached by considering the case of a net of rigid fibers of uniform length, linked by viscous point interactions of power-law type. Those interactions may result in local forces and moments located at the contacting point between two fibers, and respectively power-law functions of the local linear and angular velocity at this point.Assuming the existence of an elementary representative volume which size is small compared to the size of the whole structure, the fiber net is regarded as a periodic assembly of identical cells. Macroscopic equilibrium and constitutive equations of the equivalent continuum are then obtained by the discrete and periodic media homogenization method, based on the use of asymptotic expansions.Depending on the order of magnitude of local translational viscosities and rotational viscosities, three types of the equivalent continua are proved to be possible. One of them leads to an effective Cosserat medium, the other ones being usual Cauchy media. Lastly, formulations that enable an effective computation of constitutive equations are detailed. They show that the equivalent continuum behaves like an anisotropic power-law fluid.     

On the effects of non-linear elements in the reliability-based optimal design of stochastic dynamical systems

The aim of the present paper is to study the effects of non-linear devices on the reliability-based optimal design of structural systems subject to stochastic excitation. One-dimensional hysteretic devices are used for modelling the non-linear system behavior while non-stationary filtered white noise processes are utilized to represent the stochastic excitation. The reliability-based optimization problem is formulated as the minimization of the expected cost of the structure for a specified failure probability. Failure is assumed to occur when any one of the output states of interest exceeds in magnitude some specified threshold level within a given time duration. Failure probabilities are approximated locally in terms of the design variables during the optimization process in a parallel computing environment. The approximations are based on a local interpolation scheme and on an efficient simulation technique. Specifically, a subset simulation scheme is adopted and integrated into the proposed optimization process. The local approximations are then used to define a series of explicit approximate optimization problems. A sensitivity analysis is performed at the final design in order to evaluate its robustness with respect to design and system parameters. Numerical examples are presented in order to illustrate the effects of hysteretic devices on the design of two structural systems subject to earthquake excitation. The obtained results indicate that the non-linear devices have a significant effect on the reliability and global performance of the structural systems.     

Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics

Two kinds of second-order non-linear ordinary differential equations (ODEs) appearing in mathematical physics and non-linear mechanics are analyzed in this paper. The one concerns the Kidder equation in porous media and the second the gas pressure diffusion equation. Both these equations are strongly non-linear including quadratic first-order derivatives (damping terms). By a series of admissible functional transformations we reduce the prescribed equations to Abel's equations of the second kind of the normal form that they do not admit exact analytic solutions in terms of known (tabulated) functions. According to a mathematical methodology recently developed concerning the construction of exact analytic solutions of the above class of Abel's equations, we succeed in performing the exact analytic solutions of both Kidder's and gas pressure diffusion equations. The boundary and initial data being used in the above constructions are in accordance with each specific problem under considerations.     

Transmissibility of non-linear output frequency response functions with application in detection and location of damage in MDOF structural systems

Transmissibility is a well-known linear system concept that has been widely applied in the diagnosis of damage in various engineering structural systems. However, in engineering practice, structural systems can behave non-linearly due to certain kinds of damage such as, e.g., breathing cracks. In the present study, the concept of transmissibility is extended to the non-linear case by introducing the Transmissibility of Non-linear Output Frequency Response Functions (NOFRFs). The NOFRFs are a concept recently proposed by the authors for the analysis of non-linear systems in the frequency domain. A NOFRF transmissibility-based technique is then developed for the detection and location of both linear and non-linear damage in MDOF structural systems. Numerical simulation results verify the effectiveness of the new technique. Experimental studies on a three-storey building structure demonstrate the potential to apply the developed technique to the detection and location of damage in practical MDOF engineering structures.     

An outline of the non-linear viscoelastic behaviour of wheat flour dough in shear

Creep and creep recovery, stress relaxation and small- and large-amplitude oscillatory shear experiments have been used to study the steady-state flow behaviour and the transient viscoelastic response of wheat flour dough in shear over large ranges of time, stress and strain. The results are discussed with reference to the limited body of reliable literature data. Dough does display a linear viscoelastic domain. The complex character of its non-linear viscoelastic properties is essentially due to the extremely low shear rate limit of the initial Newtonian plateau and to the onset of time-dependent flow behaviour above a certain strain threshold, which explain qualitatively the discrepancies observed in certain cases on a part of the range of the rheological variables explored, despite global self-consistency of the results. Comparison of gluten and dough linear viscoelastic properties shows that dough cannot be viewed simply as a concentrated suspension of starch granules in the hydrated viscoelastic gluten matrix.Paper presented at the second Annual European Rheology Conference (AERC 2005) held in Grenoble, France on April 21–23, 2005.     

Modelling the non-linear interaction of wind and tide: Its influence on current profiles

A single-point model in the vertical is used to examine the coupling between tidal currents and wind-driven flows in shallow near-coastal regions. Calculations using both a linear slip and a no-slip condition at the sea bed clearly show that coupling between tidal and wind-driven currents cannot occur in a linear model with a time-independent eddy viscosity. However with a physically more realistic time-varying viscosity related to the flow field, coupling does occur, the magnitude of this non-linear interaction depending upon the change in eddy viscosity over a tidal cycle and the intensity of shear in the vertical. A point model in the vertical with flow induced by an oscillatory pressure gradient and an additional constant wind stress is used to examine the influence of viscosity parametrization and water depth upon this coupling. The solution in the vertical is accomplished using both a functional approach and a finite difference method. Some conclusions as to the relative merits of these approaches, particularly the use of a transformed grid in the case of high-shear surface and bed boundary layers, are made in the paper.     

Characteristic lengths and maximum entropy estimation from probe signals in the ellipsoidal bubble regime

The bubble size, surface and volume distributions in two and three phase flows are essential to determine energy and mass transfer processes. The traditional approaches commonly use a conditional probability density function of chord-lengths to calculate the bubble size distribution, when the bubble size, shape and velocity are known. However, the approach used in this paper obtains the above distributions from statistical relations, requiring only the moments inferred from the measurements given by a sampling probe. Using image analysis of bubbles injected in a water tank, and placing an ideal probe on the image, a sample of bubble diameter, shape factor and velocity angle are obtained. The samples of the bubble chord-length are synthetically generated from these variables. Thus, we propose a semi-parametric approach based on the maximum entropy (MaxEnt) distribution estimation subjected to a number of moment constraints avoiding the use of the complex backward transformation. Therefore, the method allows us to obtain the distributions in close form. The probability density functions of the most important length scales (D, D₂₀, D₃₀, D₃₂), obtained applying the semi-parametric approach proposed here in the ellipsoidal bubble regime, are compared with experimental measurements.     

Propagation and interaction of non-linear elastic plane waves in soft solids

Using the perturbation method of weakly non-linear asymptotics we analyze the propagation and interaction of elastic plane waves in a model of a soft solid proposed by Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44]. We derive the evolution equations for the wave amplitudes and find analytical formulas for all interaction coefficients of quadratically non-linear interacting waves. We show that in spite of the assumption of almost incompressibility used in Hamilton et al. [Separation of compressibility and shear deformation in the elastic energy density, J. Acoust. Soc. Am. 116 (2004) 41-44], the model behaves essentially like that of a compressible isotropic material. Both the structure of the equations and the interaction patterns are similar. The models differ, however, in the elastic constants that characterize them, and hence the values of the coefficients in the evolution equations and the values of the interaction coefficients differ.     

Comparison of three spectral methods for the Benjamin-Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions

The Benjamin-Ono equation is especially challenging for numerical methods because (i) it contains the Hilbert transform, a nonlocal integral operator, and (ii) its solitary waves decay only as O(1/|x|²). We compare three different spectral methods for solving this one-space-dimensional equation. The Fourier pseudospectral method is very fast through use of the Fast Fourier Transform (FFT), but requires domain truncation: replacement of the infinite interval by a large but finite domain. Such truncation is unnecessary for a rational basis, but it is simple to evaluate the Hilbert Transform only when the usual rational Chebyshev functions TB_n(x) are replaced by their cousins, the Christov functions; the FFT still applies. Radial basis functions (RBFs) are slow for a given number of grid points N because of the absence of a summation algorithm as fast as the FFT; because RBFs are meshless, however, very flexible grid adaptation is possible.     

Stability and non-linear second-order elastic analyses of beam and framed structures with semi-rigid connections using the cross method

The main objective of this publication is to present an extended version of the Moment Distribution Method (MDM) for the stability and non-linear second-order analysis of indeterminate beams and framed structures made of beam-columns of symmetrical cross-section including the combined effects of shear and bending deformations, axial loads, and semi-rigid connections. The proposed method along each member has the following advantages: (1) it can be utilized in the first- and second-order analyses (including buckling analysis) of indeterminate beams and framed structures made of beam-columns with rigid, semi-rigid, and simple end connections; (2) the effects of semi-rigid connections are condensed into the bending stiffness and fixed-end moments without introducing additional degrees of freedom and equations of equilibrium; and (3) it is accurate, powerful, practical, versatile, and an excellent teaching tool. Analytical studies indicate that shear deformations, semi-rigid connections, and axial loads increase the lateral deflections and affect the internal moments and reactions of continuous beams and framed structures. These effects must be taken into account particularly in slender structures and when they are made of beam or columns with relatively low effective shear areas (like laced columns, columns with batten plates or with perforated cover plates, and columns with open webs) or with low shear stiffness (like short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E) making the shear stiffness GA_s of the same order of magnitude as EI/L². These effects become even more significant when the external supports are not perfectly clamped. Three comprehensive examples are included that show the effectiveness of the proposed method.     

Monte Carlo simulation in the stochastic analysis of non-linear systems under external stationary Poisson white noise input

A method for the evaluation of the probability density function (p.d.f.) of the response process of non-linear systems under external stationary Poisson white noise excitation is presented. The method takes advantage of the great accuracy of the Monte Carlo simulation (MCS) in evaluating the first two moments of the response process by considering just few samples. The quasi-moment neglect closure is used to close the infinite hierarchy of the moment differential equations of the response process. Moreover, in order to determine the higher order statistical moments of the response, the second-order probabilistic information given by MCS in conjunction with the quasi-moment neglect closure leads to a set of linear differential equations. The quasi-moments up to a given order are used as partial probabilistic information on the response process in order to find the p.d.f. by means of the C-type Gram-Charlier series expansion.