Discrete Relaxation Spectrum Research Articles

Experimental evaluation and modelling of low temperature failure properties of asphalt binders by means of the monotonic torsional loading test

• The MTL test entails the use of a dynamic shear rheometer with 4 mm parallel plates. • Low temperature failure properties of asphalt binders were assessed with the MTL. • An improved modelling approach was used to derive discrete relaxation spectra. • Brittleness master curves and critical failure temperatures were determined. • The proposed approach highlights the enhanced toughness of polymer modified binders. A recent study showed the effectiveness of the monotonic torsional loading (MTL) test in evaluating low temperature failure properties of asphalt binders. The study described in this paper, which combines laboratory testing and data modelling, aimed at taking a step forward in the development of such a methodology. The MTL test, which involves using the dynamic shear rheometer (DSR) with 4 mm parallel plates, was employed to assess the performance of a set of selected asphalt binders of different types and origin in a wide range of testing conditions. Experimental results were processed with an improved modelling approach to derive the discrete relaxation spectra of investigated materials. Moreover, the time–temperature superposition principle was used to construct brittleness master curves, from which critical failure temperatures can be determined for any loading time. Outcomes of the investigation showed that the MTL test is extremely effective in evaluating and discriminating the low temperature failure properties of asphalt binders.

Mechanism behind Time Dependent Elasticity of Crumb Rubber-Nano-Asphalt Hybrids Using Discrete Relaxation Spectrum

Crumb rubber powder is a successfully used renewable material obtained from waste tire rubber, which has been incorporated into paving asphalt since the 1930s due to its good resistance to deformation and fatigue as well as its eco-friendly performance. In this study, carbon nanotubes and nano silica were incorporated into the terminal blend crumb rubber modified asphalt technology to remedy the issues of excessive desulfurization and degradation of ground tyre rubber with this technology. The mechanism behind the high temperature delayed elastic properties of the crumb rubber-nano-asphalt hybrids was experimentally investigated based on discrete relaxation spectrum. Development of the discrete relaxation spectra was accomplished by fitting on the 60°C storage modulus data tested by the dynamic shear rheometer using the generalized Maxwell model. Subsequently, the feasibility of characterizing delayed asphalt elasticity using main relaxation time was verified by test results from the 60°C creep and recovery test. Results indicated that the crumb rubber-nano-asphalt hybrids exhibited arrheodictic behavior and the asphalt elasticity was strengthened by two nano agents. Moreover, the elasticity reinforcement with carbon nanotubes was greater than with nano silica. Additionally, a good correlation was observed between the 60°C zero shear viscosity and main relaxation time, and greater 60°C zero shear viscosity was correlated to longer main relaxation times. Furthermore, longer main relaxation time of the asphalt was related to greater average recovery rate in the creep and recovery test. This research is expected to shed some light on the mechanism behind time-dependent elasticity of crumb rubber modified asphalt from the perspective of polymer physics.

Spectral method for time-strain separable integral constitutive models in oscillatory shear

The time-strain separable Kaye–Bernstein–Kearsley–Zappas model (tssKBKZM) is a popular integral constitutive equation that is used to model the nonlinear response of time-strain separable materials using only their linear viscoelastic properties and damping function. In oscillatory shear, numerical evaluation of tssKBKZM is complicated by the infinite domain of integration, and the oscillatory nature of the integrand. To avoid these problems, a spectrally accurate method is proposed. It approximates the oscillatory portion of the integrand using a discrete Fourier series, which enables analytical evaluation of the resulting integrals for the Maxwell model. The spectral method is generalized for arbitrary discrete and continuous relaxation spectra. Upper bounds for quadrature error, which can often be driven to machine precision, are presented. The Doi–Edwards model with independent-alignment approximation (DE-IA) is a special case of tssKBKZM; for DE-IA, the spectral method is compared with trapezoidal rule to highlight its accuracy and efficiency. The superiority of the proposed method is particularly evident at large strain amplitude and frequency. For continuous relaxation spectra, the spectral method transforms the double integral corresponding to the tssKBKZM to a single integral. Solutions computed to a specified level of accuracy using standard numerical libraries show that the spectral method is typically two to three orders of magnitude faster. Extensions to fractional rheological models, materials with nonzero equilibrium modulus, stretched exponential models, etc., are also discussed.

The Discrete and Continuous Retardation and Relaxation Spectrum Method for Viscoelastic Characterization of Warm Mix Crumb Rubber-Modified Asphalt Mixtures.

To study the linear viscoelastic (LVE) of crumb rubber-modified asphalt mixtures before and after the warm mix additive was added methods of obtaining the discrete and continuous spectrum are presented. Besides, the relaxation modulus and creep compliance are constructed from the discrete and continuous spectrum, respectively. The discrete spectrum of asphalt mixtures can be obtained from dynamic modulus test results according to the generalized Maxwell model (GMM) and the generalized Kelvin model (GKM). Similarly, the continuous spectrum of asphalt mixtures can be obtained from the dynamic modulus test data via the inverse integral transformation. In this paper, the test procedure for all specimens was ensured to be completed in the LVE range. The results show that the discrete spectrum and the continuous spectrum have similar shapes, but the magnitude and position of the spectrum peaks is different. The continuous spectrum can be considered as the limiting case of the discrete spectrum. The relaxation modulus and creep compliance constructed by the discrete and continuous spectrum are almost indistinguishable in the reduced time range of 10−5 s–103 s. However, there are more significant errors outside the time range, and the maximum error is up to 55%.

Relaxation spectra using nonlinear Tikhonov regularization with a Bayesian criterion

Nonlinear Tikhonov regularization within a Bayesian framework is incorporated into a computer program called pyReSpect, which infers the continuous and discrete relaxation spectra from oscillatory shear experiments. It uses Bayesian inference to provide uncertainty estimates for the continuous spectrum h(τ) by propagating the uncertainty in the regularization parameter λ. The new algorithm is about 6–9 times faster than an older version of the program (ReSpect) in which the optimal λ was determined by the L-curve method. About half of the speedup arises from the Bayesian formulation by restricting the window of λ explored. The other half arises from the nonlinear formulation for which the spectrum is a weak function of λ, allowing us to use a coarse mesh for λ. The program is tested and validated on three examples: a synthetic spectrum, a H-polymer, and an elastomer with a nonzero terminal plateau.

Modelling of a visco-hyperelastic polymeric foam with a continuous to discrete relaxation spectrum approach

The prediction of compressive properties of foams at large strains and in a wide range of strain rates is still an open issue. In this work we propose a visco-hyperelastic formulation, suitable for large finite strain applications, for the prediction of the compressive response of foams that takes into account the viscoelasticity of the polymer, nonlinear damping, nonlinear behaviour of the cellular structure and effect of gas permeability through the pores at high strain rates. A mathematical expression of the continuous relaxation spectrum is proposed to model the viscoelastic behaviour of the polymer. The relaxation spectrum is then discretized with the desired accuracy required for the subsequent numerical simulations. The model parameters are identified by coupling dynamic measurements at small strain with static ones at large strain. The results are validated by comparing numerical predictions with experimental data from compressive tests up to 50% strain performed at strain rates spanning over 6 degrees of magnitude.

Rheological measurement of the nonlinear viscoelasticity of the ABS polymer and numerical simulation of thermoforming process

The thickness distribution of thermoformed products is greatly affected by the viscoelastic behavior of the extruded polymer sheet. In this work, linear and nonlinear rheological experiments are carried out to characterize the viscoelastic properties of acrylonitrile-butadiene-styrene sheets under thermoforming conditions including a wide range of temperatures, strains, and strain rates. First, aspects of linear viscoelasticity such as the storage modulus and loss modulus are measured by small-amplitude oscillatory shear experiments. The discrete relaxation spectra and the Williams-Landel-Ferry parameters are obtained from the constructed linear master curves. Then, nonlinear time-dependent extensional viscosity is measured by uniaxial extensional experiments. The parameters of the damping function are evaluated using an optimization method. In addition, the effect of the orientation of the polymer is analyzed. The uniaxial extensional stress and viscosity in the extruder direction demonstrate higher resistance against tearing and extreme thickness reduction during processing. Finally, the linear and nonlinear input parameters for the numerical simulation are prepared. Numerical simulations are performed using the Wagner model with the obtained nonlinear viscoelasticity. The thickness distribution in thermoformed ABS sheets, obtained numerically, shows good agreement with the experimentally obtained values.

Establishment of linkages between empirical and mechanical models for asphalt mixtures through relaxation spectra determination

Empirical algebraic models as Generalized Sigmoidal model (GSM) and Havriliak-Negami model (HNM) are selected as the pre-smoothing models for comparison. Asphalt mixtures with raw, modified and resin blended binders are tested for original dynamic moduli and phase angles within the linear viscoelastic region by the simple performance test. Three approaches of GSM are evaluated to enhance the accuracy of constructing master curves. The results indicate that modulus master curves are asymmetric with shape parameters less than 1 for thermo-plastic mixtures, and larger than 1 for thermo-setting mixtures, while the most accurate modeling phase angles are from HNM due to its exact Kramer-Kronig relations. The discrete relaxation spectra are derived from pre-smoothed data through the Windowing methods (WMs) with Generalized Maxwell model (GMM) based Storage Algorithm and Loss Algorithm (GM-SA and GM-LA). Windows regions of algorithms are modified to be suitable for iterations. Continuous spectra are determined to verify the accuracy of obtained discrete spectra through inverse Fourier transforms. The results indicate that the discrete spectra with high spectrum points density r = 2 from HNM are more consistent with the discretized continuous spectra than those from GSM with low density r = 1. Moreover, the WMs with empirical pre-smoothing models take obvious advantages in parameters adjustments and relaxation times distribution selections. GM-LA rather than GM-SA exhibit higher accuracy and convergence rate in spectra determinations, since the kernel functions. In addition, the comparisons between the GMM recovered and GSM or HNM measured modulus with fit goodness statistics show that low density of spectrum points per decade is the main reason in the GMM loss modulus oscillations. However, the deviations are less obvious between HNM pre-smoothed moduli and GMM generated moduli than those between GMM-GSM moduli. Therefore, the linkages between empirical and mechanical models are established on relaxation spectra determination by modified WM using GM-LA with denser spectrum distribution r = 2.

PyReSpect: A Computer Program to Extract Discrete and Continuous Spectra from Stress Relaxation Experiments

AbstractAn open‐source software package “pyReSpect” is presented to extract relaxation spectra from stress relaxation experiments. It employs nonlinear Tikhonov regularization to obtain the continuous relaxation spectrum (CRS), and robust new algorithm to automatically determine a discrete relaxation spectrum (DRS) with a parsimonious number of modes. The new algorithm uses the CRS to guess the location of the modes, a nonlinear least squares optimization to fine‐tine the guess, and an information criterion to determine an optimal number of modes. The program is subjected to three validation tests, where data are generated from synthetic spectra, and three additional tests drawn from a variety of macromolecular architectures and sources. On the validation tests, pyReSpect is able to extract the original spectra. In all cases, the DRS follows the shape of the CRS, and offers additional regularization. Overall, pyReSpect is an excellent choice to obtain the DRS when the number and placement of modes is not known in advance.

Applications of Monte Carlo method to nonlinear regression of rheological data

In rheological study, it is often to determine the parameters of rheological models from experimental data. Since both rheological data and values of the parameters vary in logarithmic scale and the number of the parameters is quite large, conventional method of nonlinear regression such as Levenberg-Marquardt (LM) method is usually ineffective. The gradient-based method such as LM is apt to be caught in local minima which give unphysical values of the parameters whenever the initial guess of the parameters is far from the global optimum. Although this problem could be solved by simulated annealing (SA), the Monte Carlo (MC) method needs adjustable parameter which could be determined in ad hoc manner. We suggest a simplified version of SA, a kind of MC methods which results in effective values of the parameters of most complicated rheological models such as the Carreau-Yasuda model of steady shear viscosity, discrete relaxation spectrum and zero-shear viscosity as a function of concentration and molecular weight.

Characterizing and Modeling the Rheological Performances of Potassium Silicate Solutions

By using rheological measurements, a quantitative modeling method, as well as fitting of data, the main rheological properties of potassium silicate solutions (PSS) were determined. The results show that the steady state shear viscosities of PSS rapidly increase with concentration in the tested domain and molar ratios. However, they decrease with temperature from 15 to 60 °C. The newly improved Krieger–Dougherty expression, which inherited the principal properties of both the Krieger–Dougherty and the Kaminsky equations, can describe the measured viscosities with respect to concentration. The blends of potassium and sodium silicate solutions reveal a positive synergistic interaction for the viscosity at fixed concentration and molar ratio. The storage and loss moduli of PSS principally increased with angular frequency, and it was possible to fit to the discrete relaxation spectrum model of these two quantities with relative error less than 3.6%.

Frictional force between a rotationally symmetric indenter and a viscoelastic half‐space

The paper presents a numerical study of the tangential contact between a rigid indenter and an elastomer with linear rheology. Occurring friction forces are caused exclusively by internal dissipative losses. Especially the combinations of a conical or paraboloid indenter and the Maxwell or standard body with fixed ratio of moduli are studied. One result is, that by the use of proper chosen dimensionless variables, for each of the combinations, the contact forces depend explicitly on the indentation depth only. Curves describing this dependencies are given as Bézier function fits. Application of these methods to an indenter with arbitrary shape and materials with a discrete relaxation spectrum described by Prony series is possible. Although only stationary solutions are analysed here, the method enables us to examine the transient process as well.

A systematic approximation of discrete relaxation time spectrum from the continuous spectrum

Most of viscoelastic models contain linear and nonlinear viscoelastic parameters and the linear viscoelastic parameters correspond to relaxation time spectra of materials. The relaxation spectra can be classified into continuous and discrete ones. Discrete relaxation time spectrum is more convenient to simulate multi-mode models than continuous one because it demands shorter calculation time. It has been demonstrated that the continuous spectrum is uniquely determined in views of theoretical (Fuoss and Kirkwood, 1941; Davies and Anderssen, 1997) [1,23] as well as empirical approaches (McDougall et al., 2014) [5]. Whereas, it is reported that different algorithms for discrete spectrum infer the different results from the same data (Malkin and Masalova, 2001) [14]. This is the study on a systematic method for discrete spectrum on the basis that a discrete spectrum must be consistent with the continuous one. We suggest a simple method to extract discrete relaxation spectrum as a systematically approximated continuous spectrum by means of the Levenberg–Marquardt method. The new algorithm is tested and compared with previous algorithms using synthesized model spectra and experimental data.

Dynamic properties of polyurea-milled glass composites Part I: Experimental characterization

Polyurea (PU) is an elastomer, which exhibits unique thermo-mechanical properties. It is synthesized from a di-functional amine, e.g. Versalink P-1000 and a diisocyanate, e.g. Isonate143L. In this study, various volume fractions milled glass (MG) was added to create polyuria-milled glass composites (PU-MG). Milled glass is a micro fiber with cylindrical shape. The distribution of the milled glass in the polyurea matrix was observed under the scanning electron microscope. The dynamic properties of pure polyurea and the PU-MG composites were measured by dynamic mechanical analysis (DMA) in the low frequency range (1–20 Hz) and by ultrasonic wave measurement in the high frequency range (0.5–1.5 MHz). Both experiments show that increasing the milled glass volume fraction drastically increases both the storage and loss moduli of the composites. DMA results show that dynamic Young’s modulus increases with increasing frequency. However, longitudinal and shear moduli from ultrasonic wave measurement appears to be insensitive to frequency within the range of 0.5–1.5 MHz. The experimental dynamic moduli master curves of PU and PU-MG composites were constructed and compared. The relaxation function or creep compliance are generally useful than dynamic moduli for modeling of material response under complex histories. It is of practical use to convert dynamic mechanical data from the frequency domain into the time domain. The discrete relaxation spectra of the composites were calculated by fitting Prony series to the master curves, using least square nonlinear regression. Retardation spectra were then calculated using the interrelation between relaxation modulus and creep compliance in Laplace domain. Finally, the time domain relaxation modulus and creep compliance for each composite were obtained from the two spectra. In order to extend our understanding of the dynamic behavior of the PU-MG composite, micromechanical models have been created and are discussed in the accompanying paper Nantasetphong (2016a).

Characterization of asphalt concrete linear viscoelastic behavior utilizing Havriliak–Negami complex modulus model

This study proposed a new procedure for characterizing linear viscoelastic (LVE) behavior of asphalt concrete using the Havriliak–Negami (HN) complex modulus model and corresponding continuous relaxation spectrum model. To extend its application to modeling retardation behavior of asphalt concrete, the continuous retardation spectrum was analytically derived from the HN model by performing an inverse Fourier–Laplace transform followed by a variable substitution. The two continuous spectra of the HN model allow the accurate construction of all the modulus and compliance function master curves in time and frequency domains as well as the efficient determination of the high-quality discrete relaxation and retardation spectra. The advantages of the proposed approach over existing ones were demonstrated through two different complex modulus test data sets. Also, a simple and practical numerical interconversion method was presented based on the LVE relations between the continuous relaxation and retardation spectra, which can effectively compute one continuous spectrum from the other for any characterization model. Further, for practical considerations, a two-step method for determining reduced master curves with fewer discrete spectrum lines was suggested.

A unified procedure for rapidly determining asphalt concrete discrete relaxation and retardation spectra

This paper presents a unified procedure that can rapidly determine the discrete relaxation and retardation spectra of asphalt concrete. The new procedure involves three consecutive steps: (1) pre-smoothing the complex modulus data with the Havriliak–Negami (HN) model, (2) identifying the discrete relaxation spectrum from the smoothed data with a modified windowing method (MWM) and (3) converting the obtained spectrum into the corresponding discrete retardation spectrum. The HN model adopted furnishes reasonable analytical representations for all the complex modulus components and allows an asymmetrical inflection point for the dynamic modulus and storage modulus master curves on the log–log scale, effectively overcoming the drawbacks of the conventional sigmoidal function in characterizing the asphalt concrete linear viscoelastic (LVE) behavior. Also, the HN model can well predict the phase angle from the dynamic modulus data, substantially extending the application of the old data lacking the phase angle information in the LVE analysis. Additionally, the MWM offers a more appropriate estimation approach for the glassy modulus, successfully avoiding the undesirable spectrum oscillations and negative spectrum lines. A distribution of the time constants with 0.5 decade intervals was implemented in this procedure, completely excluding the waviness in the bell-shaped master curves of the generalized Maxwell (GM) and generalized Voigt (GV) models. Very few parameters are required for the initial inputs in the procedure and no empirical adjustments or selections are required for the GM or GV model constants during the whole computation process. The convergence of the iterations for determining the discrete spectra proved to be very fast.

Polyester fiber spinning analyzed with multimode Phan Thien-Tanner model

The fiber spinning of polyester is analyzed with an algorithm using the Phan Thien-Tanner model. The discrete relaxation spectrum, necessary for the multimode model, is determined with the storage and loss moduli. Experimental values of the moduli are extrapolated with the help of the Cox-Merz rule in such a way that they cover a frequency range of 106s-1. This allows determining the very short relaxation times necessary to handle the extreme deformation rates of 10,000s−1 during high speed spinning. The algorithm correctly predicts the swelling of the polyester after exiting the spinneret, the elongation of the melt, and for high spinning speeds the crystallization as function of spinning stress. The spinning stress on its part is influenced by temperature and crystallinity. The computations therefore take into account the non-uniformity of the temperature profile in the melt strand. The same set of parameters in the algorithm is used without any change for intermediate and high spinning speeds. The algorithm is capable of simulating realistically the polyester spinning process. Good agreement was obtained with experimental data from industrial trial runs.

Numerical and Experimental Study on Disk Flutter Reduction Using Disk-Spacer Dampers in Hard Disk Drives

This paper presents a disk-spacer damper (DSD), a new damper structure, for reducing disk flutter. Three types of DSD structures, which consist of an aluminum disk spacer having grooves embedded with viscoelastic (VE) films, have been developed in this paper. First, we investigated the material properties of the VE film, such as shear modulus and discrete relaxation spectra, and then carried out numerical analysis of the damping effect using the material properties. Further, we measured the damping effect of each DSD structure and confirmed that the proposed DSD reduces the disk flutter amplitude between the (0,1) mode and the (0,4) mode by 20%-30%.

Study on discrete relaxation time spectrum for PBO/PPA liquid crystal polymer solution

Based on KBKZ integral type rheological equation, discrete relaxation spectra were attained by use of the least square linear and non-linear regressions. The results show that the number of relaxation mode ranging from 9 to 11 for linear regression method has little effect on the non-linear regression method. Apart from that the relaxation modulus coefficient related with the different time range was gained, the relaxation time range had little effect on the time spectrum. The trend for relaxation time spectrum of PBO solution is similar to common polymer solution except that there is not a platform modulus region for high elastic state. The relaxation time spectrum at 160°C was obviously different from that at 170°C. However, the difference diminished when the temperature exceeded 170°C, and no significant distinction was found at glass transition region. While at 170°C, the influence on the movement of the chain segment was trivial as the molecular weight increases, mainly resulting in the movement restriction of polymer chains.

Elastin-like recombinamer catalyst-free click gels: Characterization of poroelastic and intrinsic viscoelastic properties

Elastin-like recombinamer catalyst-free click gels (ELR-CFCGs) have been prepared and characterized by modifying both a structural ELR (VKVx24) and a biofunctionalized ELR-bearing RGD cell-adhesion sequence (HRGD6) to bear the reactive groups needed to form hydrogels via a click reaction. Prior to formation of the ELR-CFCGs, azide-bearing and cyclooctyne-modified ELRs were also synthesized. Subsequent covalent crosslinking was based on the reaction between these azide and cyclooctyne groups, which takes place under physiological conditions and without the need for a catalyst. The correlation among SEM micrographs, porosity, swelling ratio, and rheological measurements have been carried out. The storage and loss moduli at 1Hz are in the range 1–10kPa and 100–1000Pa, respectively. The linear dependence of |G∗| on f½ and the peak value of tan δ were considered to be consistent with a poroelastic mechanism dominating the frequency range 0.3–70Hz. The discrete relaxation spectrum was obtained from stress relaxation measurements (t>5s). The good fit of the relaxation modulus to decrease exponential functions suggests that an intrinsic viscoelastic mechanism dominates the transients. Several recombinamer concentrations and temperatures were tested to obtain gels with fully tuneable properties that could find applications in the biomedical field.