Interconversion of Prony series for relaxation and creep

Various algorithms have been proposed to solve the interconversion equation of linear viscoelasticity when Prony series are used for the relaxation and creep moduli, G(t) and J(t). With respect to a Prony series for G(t), the key step in recovering the corresponding Prony series for J(t) is the determination of the coefficients {jk} of terms in J(t). Here, the need to solve a poorly conditioned matrix equation for the {jk} is circumvented by deriving elementary and easily evaluated analytic formulae for the {jk} in terms of the derivative dG(s)/ds of the Laplace transform G(s) of G(t)

Rheological implications of completely monotone fading memory

In the constitutive equation modeling of a (linear) viscoelastic material, the “fading memory” of the relaxation modulusG(t) is a fundamental concept that dates back to Boltzmann [Ann. Phys. Chem. 7, 624 (1876)]. There have been various proposals that range from the experimental and pragmatic to the theoretical about how fading memory should be defined. However, if, as is common in the rheological literature, one assumes that G(t) has the following relaxation spectrum representation: G(t)=∫₀∞ exp(−t/τ)[H(τ)/τ]dτ, t > 0, then it follows automatically that G(t) is a completely monotone function. Such functions have quite deep mathematical properties, that, in a rheological context, spawn interesting and novel implications. For example, because the set of completely monotone functions is closed under positive linear combinations and products, it follows that the dynamics of a linear viscoelastic material, under appropriate stress–strain stimuli, will involve a simultaneous mixture of different molecular interactions. In fact, it has been established experimentally, for both binary and polydisperse polymeric systems, that the dynamics can simultaneously involve a number of different molecular interactions such as the Rouse, double reptation and/or diffusion, [W. Thimm et al., J. Rheol., 44, 429 (2000); F. Léonardi et al., J. Rheol. 44, 675 (2000)]. The properties of completely monotone functions either yield new insight into modeling of the dynamics of real polymers, or they call into question some of the key assumptions on which the current modeling is based, such as the linearity of the Boltzmann model of viscoelasticity and/or the relaxation spectrum representation for the relaxation modulusG(t). If the validity of the relaxation spectrum representation is accepted, the resulting mathematical properties that follow from the complete monotonicity of G(t) allows one to place the classical relaxation model of Doi and Edwards [M. Doi and S. F. Edwards, J. Chem. Soc., Faraday Trans. 2 74, 1789 (1978)], as a linear combination of exp(−t/τ*) relaxation processes, each with a characteristic relaxation time τ*, on a more general and rigorous footing

On the scaling of molecular weight distribution functionals

When formulating a constitutive equation model or a mixing rule for some synthetic or biological polymer, one is essentially solving an inverse problem. However, the data will not only include the results obtained from simple step strain, oscillatory shear, elongational, and other experiments, but also information about the molecular weight scaling of key rheological parameters (i.e., molecular weight distribution functionals) such as zero-shear viscosity, steady-state compliance, and the normal stress differences. In terms of incorporating such scaling information into the formulation of models, there is a need to understand the relationship between various models and their molecular weight scaling, since such information identifies the ways in which molecular weight scaling constrains the choice of possible models. In Anderssen and Mead (1998) it was established formally that the members of a quite general class of reptation mixing rules all had the same molecular weight scaling. The purpose of this paper is to first introduce the concept of a generalized reptation mixing rule, which greatly extends the class examined by Anderssen and Mead, and then show that all such rules have the same molecular weight scaling. The proof is similar to that given by Anderssen and Mead, but uses the implicit function theorem to establish the uniqueness of the mean values which arise when invoking various integral mean-value representations for the molecular weight distribution functionals considered. The rheological significance of the new generalized two-parameter mixing rule, proposed in this paper, is examined in some detail in the conclusions. In particular, it is used to established how one must construct a mixing rule for a general polydispersed polymer where the molecular dynamics involves some single, some double and some higher levels of multiple reptation. The work of Maier et al. (1998) and Thimm et al. (2000) is then utilized to illustrate and validate this proposal

Derivative spectroscopy and the continuous relaxation spectrum

Derivative spectroscopy is conventionally understood to be a collection of techniques for extracting fine structure from spectroscopic data by means of numerical differentiation. In this paper we extend the conventional interpretation of derivative spectroscopy with a view to recovering the continuous relaxation spectrum of a viscoelastic material from oscillatory shear data. To achieve this, the term “spectroscopic data” is allowed to include spectral data which have been severely broadened by the action of a strong low-pass filter. Consequently, a higher order of differentiation than is usually encountered in conventional derivative spectroscopy is required. However, by establishing a link between derivative spectroscopy and wavelet decomposition, high-order differentiation of oscillatory shear data can be achieved using specially constructed wavelet smoothing. This method of recovery is justified when the reciprocal of the Fourier transform of the filter function (convolution kernel) is an entire function, and is particularly powerful when the associated Maclaurin series converges rapidly. All derivatives are expressed algebraically in terms of scaling functions and wavelets of different scales, and the recovered relaxation spectrum is expressible in analytic form. An important feature of the method is that it facilitates local recovery of the spectrum, and is therefore appropriate for real scenarios where the oscillatory shear data is only available for a finite range of frequencies. We validate the method using synthetic data, but also demonstrate its use on real experimental data

Simple moving-average formulae for estimating the relaxation spectrum

Different software packages are available commercially which can be applied to oscillatory shear data to recover an estimate of the relaxation spectrum of the viscoelastic material tested. The underlying algorithms, based on some form of regularization, are indirect and technically involved. Davies and Anderssen [J. Non-Newtonian Fluid Mech. 73, 163–179 (1997)] have derived exact sampling localization results for the determination of elastic moduli from (exact) storage and loss moduli. It is now shown how their results can be exploited to construct simple and explicit moving-average formulae which recover estimates of the relaxation spectrum from oscillatory shear data, with realistic observational errors. Explicit moving-average formulae are presented which experimentalists can apply immediately to appropriately sampled oscillatory shear measurements. The given formulae are validated on noisy data obtained from synthetic relaxation spectra

The Kohlrausch function: properties and applications

In a wide variety of applications, including the modelling of the glassy state of dense matter, non-exponential correlation functions in nuclear magnetic resonance, polymer dynamics, and bone and muscle rheology, Kohlrausch functions have proved to be more appropriate in modelling the associated relaxation and decay processes than the standard exponential function. However, mathematical results about this function, important for both computational and modelling endeavours, are spread over publications in several quite different areas of mathematics and science. The purpose of this paper is to review the key properties of Kohlrausch functions in a unified manner, which motivates its use in the modelling of molecular processes. Some representative applications and related computational issues are discussed

A revised duality proof of sampling localization in relaxation spectrum recovery

The duality proof of sampling localization given by Loy, Newbury, Anderssen and Davies in 2001 contains an oversight, as the classes of functions chosen do not assume the compact support. Here, it is shown how a minor change to the argument there yields a precise conclusion

Exact and explicit probability densities for one-sided Levy stable distributions

We study functions g_{\alpha}(x) which are one-sided, heavy-tailed Levy stable probability distributions of index \alpha, 0< \alpha <1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expression for g_{\alpha}(x), 0 \leq x < \infty, satisfying \int_{0}^{\infty} exp(-p x) g_{\alpha}(x) dx = exp(-p^{\alpha}), p>0, for all \alpha = l/k < 1, with k and l positive integers. We reproduce all the known results given by k\leq 4 and present many new exact solutions for k > 4, all expressed in terms of known functions. This will allow a 'fine-tuning' of \alpha in order to adapt g_{\alpha}(x) to a given experimental situation.Comment: 4 pages, 3 figures and 1 tabl

Discretization of variational regularization in Banach spaces

Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required. Moreover, in general the given data \yd of $y$ are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the $L^\infty$--space