Continuous Relaxation Spectrum Research Articles

Multi-parameter viscoelastic material model for denture adhesives based on time-temperature superposition and multiple linear regression analysis

BackgroundRestorative solutions designed for edentulous patients such as dentures and their accompanying denture adhesives operate in the complex and dynamic environment represented by human oral physiology. Developing material models accounting for the viscoelastic behavior of denture adhesives can facilitate their further optimization within that unique physiological environment. This study aims to statistically quantify the degree of significance of three physiological variables - namely: temperature, adhesive swelling, and pH - on denture adhesive mechanical behavior. Further, based on these statistical significance estimations, a previously-developed viscoelastic material modelling approach for such denture adhesives is further expanded and developed to capture these variables’ effects on mechanical behavior.MethodsIn this study a comparable version of Denture adhesive Corega Comfort was analysed rheologically using the steady state frequency sweep tests. The experimentally derived rheological storage and loss modulus values for the selected physiological variables were statistically analyzed using multi parameter linear regression analysis and the Pearson’s coefficient technique to understand the significance of each individual parameter on the relaxation spectrum of the denture adhesive. Subsequently, the parameters are incorporated into a viscoelastic material model based on Prony series discretization and time-temperature superposition, and the mathematical relationship for the loss modulus is deduced.ResultsThe results of this study clearly indicated that the variation in both the storage and loss modulus values can be accurately predicted using the oral cavity physiological parameters of temperature, swelling ratio, and pH with an adjusted R2 value of 0.85. The R2 value from the multi-parameter regression analysis indicated that the predictor variables can estimate the loss and storage modulus with a reasonable accuracy for at least 85% of the rheologically determined continuous relaxation spectrum with a confidence level of 98%. The Pearson’s coefficient for the independent variables indicated that temperature and swelling have a strong influence on the loss modulus, whereas pH had a weak influence. Based on statistical analysis, these mathematical relationships were further developed in this study.ConclusionsThis multi-parameter viscoelastic material model is intended to facilitate future detailed numerical investigations performed with implementation of denture adhesives using the finite element method.

Novel hyper-viscoelastic approach to modelling elastomer mechanic behaviour with relaxation spectrum

Elastomers, such as polyurethane, exhibit hyperelastic behaviour and distinct rate dependence under a wide range of strain rates. Herein, a methodology for developing a hyper-viscoelastic constitutive model for succinctly characterising the mechanical behaviour of polyurethane under large deformations and a wide range of strain rates is proposed. The model is constructed using a long-term hyperelastic formulation combined with a rate-dependent viscoelastic model. This model is based on internal state variables and a finite strain formulation. A polyurethane elastomer was used as a sample to develop this model, by way of testing wide strain-rate mechanical behaviour (0.001–5000s−1) and viscoelastic behaviour. The viscoelastic behaviour was obtained using the equivalent time–temperature superposition relation. A parameter-fitting algorithm was adopted to separate the long-term behaviour from the viscoelastic behaviour. The long-term behaviour was modelled using a Mooney–Rivlin hyperelastic model, and its parameters were identified using the least-squares method. The time-dependent viscous effect calculated using the Prony series-based model was approximately determined based on the continuous relaxation spectrum. The characterisation capacity was confirmed through a thorough comparison with experimental curves. This model can be extended to other elastomers, even under tension, without any specific limitations of the internal components.

Improved parameter acquisition of generalized Maxwell model in asphalt mixtures by continuous relaxation spectrum

The generalized Maxwell (GM) model is widely used to describe the viscoelastic behavior and predict performance of asphalt mixtures. However, the existing methods may not provide satisfactory accuracy because relaxation spectral distribution features are not considered. In this study, the methods (area equalization (AE) and peak equalization (PE) method) consider the distribution characteristics of the relaxation spectrum were proposed to obtain the Prony series parameters of the GM model. Two typical asphalt mixtures, HMA (hot mix asphalt mixture) and WMA (warm mix asphalt mixture), were used as case studies. The Prony series parameters of the GM model obtained by the proposed methods and existing methods (traditional fitting (TF) method and Limit equalization (LE) method) were discussed and analyzed. The derived Prony series parameters were also employed in finite element (FE) simulations to inversely predict dynamic modulus, and the simulated results were then compared with the experimental results. The results indicate that the spring stiffnesses of Maxwell elements in the Prony series parameters obtained by different methods have different distribution features. For the TF method, the Maxwell element with larger relaxation time has smaller spring stiffness. In the LE and PE method, the spring stiffness of the Maxwell element first increases and then decreases as the relaxation time increases. The spring stiffness is the same for the different Maxwell elements in the AE method. More importantly, the number of branches of the GM model is increased from 6 to 12, the relative error of the predicted dynamic modulus of HMA by the PE, AE, LE and TF method is reduced from 29.8% to 4.8%, from 49.9% to 25.9%, from 61.4% to 32.7%, from 159.1% to 133.9% respectively. The PE method has the best accuracy in FE analysis, even with a smaller number of branches of the GM model. The similar conclusion can also be summarized in the dynamic modulus prediction of WMA. Therefore, the PE method that considers the rate of change of the relaxation spectrum is preferentially recommended to obtain the Prony series parameters of the GM model.

Development of a tensile relaxation modulus test for identifying the tensile relaxation spectrum of asphalt binder

The tensile properties of asphalt binder such as strength, ductility, and stress-relaxation (relaxation) modulus are not only closely related to binder performance, but the modulus is also a necessary input for accurate material characterisation, modelling, and simulation. The linear behaviour of a homogeneous isotropic material can be fully characterised by both the tensile and shear relaxation moduli or their corresponding relaxation spectra. The shear properties of asphalt binders are often studied through dynamic shear tests. However, its tensile relaxation modulus and spectrum are seldom determined. Instead, they are often derived from the shear relaxation modulus with a prescribed Poisson ratio. However, it remains a question whether the ratio is material-, time- or even loading-independent. A tensile relaxation test is thus developed for determining the tensile relaxation modulus and spectrum directly. The continuous tensile relaxation spectrum is numerically derived based on the integral equation that connects the modulus to the spectrum. Spectrum regularisation and L-curve are used for solving the overfitting issue of the ill-posed equations. Two binders are used to evaluate the experimental method and the computational approach. The data suggest that the Poisson ratio may not be regarded as a constant for different asphalt binders. Hence, it is necessary to characterise asphalt binders using the tensile relaxation spectrum. Cross-checking between tensile and shear test data and between experimental relaxation moduli and the back-calculated ones from the tensile relaxation spectra indicate that the developed methods are robust and reliable.

A Class of Algorithms for Recovery of Continuous Relaxation Spectrum from Stress Relaxation Test Data Using Orthonormal Functions.

The viscoelastic relaxation spectrum provides deep insights into the complex behavior of polymers. The spectrum is not directly measurable and must be recovered from oscillatory shear or relaxation stress data. The paper deals with the problem of recovery of the relaxation spectrum of linear viscoelastic materials from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test. A class of robust algorithms of approximation of the continuous spectrum of relaxation frequencies by finite series of orthonormal functions is proposed. A quadratic identification index, which refers to the measured relaxation modulus, is adopted. Since the problem of relaxation spectrum identification is an ill-posed inverse problem, Tikhonov regularization combined with generalized cross-validation is used to guarantee the stability of the scheme. It is proved that the accuracy of the spectrum approximation depends both on measurement noises and the regularization parameter and on the proper selection of the basis functions. The series expansions using the Laguerre, Legendre, Hermite and Chebyshev functions were studied in this paper as examples. The numerical realization of the scheme by the singular value decomposition technique is discussed and the resulting computer algorithm is outlined. Numerical calculations on model data and relaxation spectrum of polydisperse polymer are presented. Analytical analysis and numerical studies proved that by choosing an appropriate model through selection of orthonormal basis functions from the proposed class of models and using a developed algorithm of least-square regularized identification, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and robust on measurement noises; small model approximation errors are obtained. The identification scheme can be easily implemented in available computing environments.

Mapping the role of oral cavity physiological factors into the viscoelastic model of denture adhesives for numerical implementation.

Physiological parameters of the oral cavity have a profound impact on any restorative solutions designed for edentulous patients including denture adhesives. This study aims to mathematically quantify the influence of three such variables, namely: the temperature, pH, and the swelling of such adhesives under the influence of saliva on its mechanical behavior. The mathematical quantification is further aimed to implement a material model for such adhesives which considers the impact of such physiological factors. The denture adhesive is experimentally investigated by means of rheological steady state frequency sweep tests to obtain the relaxation spectrum of the material. The relaxation behavior is measured for a wide range of oral cavity temperatures and pH. Also, the adhesive is hydrated and upon swelling to different levels again tested to understand the impact of swelling on the mechanical behavior. The experimentally measured continuous relaxation spectrum is modeled as a viscoelastic material using a discrete set of points based on the Prony series discretization technique. The relaxation spectrums for various temperatures are compared and the possibility of a time-temperature superposition is explored for the model. Similarly, the measured values of Storage and loss modulus are investigated to understand the role of pH and swelling. The results in this study clearly indicated a horizontal shift in the relaxation behavior with increase in temperature. And hence, the time-temperature shift factor was calculated for the adhesive. The relaxation spectrum also showed a strong correlation with swelling of the adhesive and the pH. The influence of these two parameters were captured into the model based on the relaxation time parameter in the Prony series approach. Based on this study the impact of these parameters could be appreciated on the performance and mechanical behavior of denture adhesives and implemented into a Prony series based viscoelastic material model which can be used with numerical simulations.

Inter-conversion of the generalized Kelvin and generalized Maxwell model parameters via a continuous spectrum method

• Constructed the continuous spectra ( H τ and L ρ ) via the direct and indirect methods. • Obtained the optimal ranges of retardation time and relaxation time. • Proposed a spectrum method to establish the inter-conversion between E k and D j Accurate determination of the Prony series parameters of the generalized Maxwell (GM) model and the generalized Kelvin (GK) model is crucial to the characterization of linear viscoelastic (LVE) properties of asphalt mixture and the prediction of in-service performance of asphalt pavement. However, existing methods may not be able to provide satisfactory accuracy due to their weakness of fully complying with the LVE theory or the involvement of unsolvable high-order nonlinear systems. To identify the Prony series parameters more efficiently and accurately, this paper developed a continuous spectrum method to establish the inter-conversion between the GM and GK model parameters. In the newly developed method, the complex modulus mater curves are first constructed through the conventional master curve models. The continuous relaxation and retardation spectra are then established using the proposed direct or indirect methods. The time and strength constants of the GM and GK models are subsequently calculated via an extended search method, based on which the Prony series parameters and the LVE variables of the asphalt mixture can be finally determined. Accuracy of the new method in determining the LVE variables was validated through a comparison of the calculated and measured LVE variables of asphalt mixture, while the equivalence between the GM and GK models established based on the new method was verified in different domains. Besides, the success implementation of the new method to the generalized Sigmodal model demonstrates the universality of the newly developed continuous spectrum method in determining the LVE variables of asphalt mixture.

An improved method to establish continuous relaxation spectrum of asphalt materials

The relaxation spectrum plays a vital role in the study of linear viscoelastic (LVE) properties of asphalt materials; however, the current developed using methods are not accurate enough to characterize the LVE properties of asphalt materials. This paper aims to develop an improved method to determine the continuous relaxation spectrum of asphalt materials. A modified Christensen–Anderson–Marasteanu (CAM) model is first selected as the master curve model of the storage modulus, upon which the master curve model of loss modulus is then established using the numerical form of the exact Kramers-Kronig relation between the storage modulus and loss modulus. According to LVE conversions between the relaxation spectrum and storage modulus, the model of the continuous relaxation spectrum is subsequently derived and finally determined based on the same model parameters of the storage modulus and loss modulus models. This study demonstrates that the relaxation spectrum of both mixtures and binders constructed by the proposed method is compliant with LVE theory and physical causality. The generated relaxation spectrum is verified to be accurate enough to compare the storage modulus and loss modulus values recalculated by the relaxation spectrum and the corresponding values predicted by master curve models. The results also show that the approximate Kramers-Kronig relation between the storage modulus and loss modulus cannot be used to establish the relaxation spectrum of asphalt materials, especially for asphalt binders. In addition, the generated relaxation spectrum was utilized to construct the relaxation modulus of mixtures and to develop the stiffness modulus of binders, respectively. Since the relaxation modulus and creep stiffness are determined from the relaxation spectrum, both are consistent with LVE theory and physical causality.

Continuous spectrum approach to evaluate the effect of binder modification and aging

ABSTRACT The present work proposes a novel mathematical approach to directly obtain the continuous spectrum of asphalt binder from frequency domain data. Using the asymmetric sigmoidal function to obtain continuous spectrum offers advantages of flexibility and improvement in describing the spectrum well. Initially, complex modulus, storage modulus, and storage compliance master curves were constructed using experimental data. In the next stage, the continuous relaxation and retardation spectrum were constructed directly from the storage modulus and storage compliance master curves, respectively. Statistical indicators like mean, variance, and skewness were used to describe the shape of these finalised continuous spectra. To check the efficacy of the proposed approach, asphalt binders altered with modifiers (at different dosages) were aged under laboratory conditions. Critical evaluation of the results indicated that the underlying changes could be captured using the continuous spectra (which were not obvious with respective master curves or black space diagrams). Continuous spectra of binders constructed (with different dosages of modifier and under different aging conditions) were compared using the statistical indicators. Thus, the spectral shape parameters could capture the effect of aging and modifier dosage. These changes can be attributed to molecular changes, chemical reactions occurring with the addition of modifiers and aging.

Fast-Acquiring High-Quality Prony Series Parameters of Asphalt Concrete through Viscoelastic Continuous Spectral Models.

Prony series representations have been extensively applied to characterizing the time-domain linear viscoelastic (LVE) material functions for asphalt concrete. However, existing methods that can generate high-quality Prony series parameters (i.e., discrete spectra) mostly involve complicated programming algorithms, which poses a challenge for quick access of Prony series parameters. Also, very limited research has been devoted to establishing methods for simultaneously determining both retardation and relaxation spectra. To resolve these issues, this study presented a practical approach to fast acquiring high-quality Prony series parameters for both relaxation modulus and creep compliance of asphalt concrete by using the complex modulus test data. The approach adopts the analytical representations of the continuous relaxation and retardation spectra from the Havriliak-Negami (HN) and 2S2P1D complex modulus models to directly determine the discrete spectra, and the elastic constants, Ee and Dg, for both LVE modulus and compliance functions are further calculated by fitting the corresponding generalized Maxwell model representations to smoothed data from the storage modulus representations of the HN and 2S2P1D complex modulus models. In this way, all the procedures in the proposed method can be easily implemented in Microsoft Excel. The results showed that the HN and 2S2P1D models yielded slightly different continuous spectral patterns at shorter relaxation times and longer retardation times. However, at the region covered by the test data, the continuous spectra of the two complex modulus models were very close to each other. Thus, the two models can generate comparable Prony series parameters within the time or frequency range covered by the test data. Considering that the quality of the resulting Prony series parameters are closely related to the master curve models used for presmoothing, the HN and 2S2P1D models were compared with the conventional Sigmoidal model. Additionally, the Black diagram was recommended for examining the quality of the complex modulus test data before constructing the master curves.

A triaxial linear viscoelastic characterization framework for asphalt concrete based on the 2S2P1D model

Asphalt concrete materials are known to exhibit pressure dependence in response to mechanical loading. However, for conventional dense-graded asphalt mixtures, only weak pressure sensitivity was noted in the relaxation spectrum, time-temperature shift factors, and low-temperature moduli. Given these facts, a triaxial linear viscoelastic characterization framework was developed, which consisted of a convenient experimental protocol for the confined dynamic modulus test and a modeling approach based on the 2S2P1D model. In this framework, the same relaxation spectrum and time-temperature equivalence were applied across all different confining conditions, while the pressure dependence was fully assigned to the equilibrium modulus. The resultant prediction errors in dynamic modulus were in general below or comparable to typical test variability. Numerical implementation of the modeling method in the time domain was accomplished via Prony discretization of the continuous relaxation spectrum, and was demonstrated via application to triaxial relaxation and creep data. The model's satisfactory performance suggested that the confining condition had an insignificant impact on the molecular processes occurring in the asphalt phase during relaxation, owing to the low aggregate mobility and thus minimal asphalt flow as restrained by the dense structure. Limitations of the proposed modeling scheme as well as methods available from the literature were revealed lastly upon asphalt concrete materials with unconventional gap and open gradations.

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.

Dynamic Modulus Characteristics of Mixtures Containing Recycled Asphalt Pavements, Warm Mix Additives, and Antistrip Agents

Abstract The utilization of reclaimed asphalt pavement (RAP) has proven to be a promising technology to solve some of the environmental issues and resource shortages in many parts of the world. The objective of this study was to investigate dynamic modulus and phase angle of surface type mixtures using two warm mix asphalt (WMA) technologies (i.e., warm mix additive E and foaming), and two liquid antistripping agents (ASAs). A total of 20 mixtures were designed and tested. Viscoelastic master curve modeling, statistical analysis, as well as continuous relaxation spectrums were conducted to obtain a better understanding of the properties of the mixtures. The results implied that RAP or WMA technologies had significant effects on mechanical performances, which should be carefully considered during mix design, but ASA agents, statistically, did not have such effects. In addition, the influence of RAP depended on WMA technology type, and the effects of ASA agents relied on aggregate and RAP source. The rheological index showed that both WMA technologies and ASA materials had negative impacts on the rheological properties of the mixtures, but the addition of RAP mitigates this impact only for mixtures containing ASA additives. Furthermore, the continuous relaxation spectrum confirmed the negative influences of WMA technologies and no obvious effects of ASA materials.

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.

A framework for linear viscoelastic characterization of asphalt mixtures

The master curve of a viscoelastic variable is of significance due to its capability of characterizing the linear viscoelastic (LVE) property in an extended time or frequency range. However, the master curves constructed using the traditional approach fail to strictly comply with the LVE theory, leading to inaccurate predictions in the extended range. In order to address this issue, a framework was developed for the LVE characterization of asphalt mixtures. The generalized logistic sigmoidal model was adopted as the master curve model of storage modulus. A numerical model of loss modulus was established in relation to the continuous relaxation spectrum, whose mathematical model was derived in light of its relationship with the storage modulus. The model parameters determined using the storage modulus and loss modulus test data were employed to construct the master curves of storage modulus, loss modulus, dynamic modulus and phase angle. Then the relaxation modulus master curve was generated by establishing a numerical model. Afterwards, the continuous retardation spectrum was solved numerically based on its relationship with the continuous relaxation spectrum. The master curves of storage compliance, loss compliance and creep compliance were obtained using the corresponding numerical models that were established with respect to the continuous retardation spectrum. The interrelationship among the viscoelastic variables was then employed to obtain the dynamic compliance and phase angle master curves. It was demonstrated that the developed framework ensured the master curves of all viscoelastic variables complied with the LVE theory.

A multiple approach in determination of interfacial tension of biodegradable melt-mixed PBAT/EVOH blends: Correlation of morphology, rheology and mechanical properties

The interfacial tension of biodegradable melt-mixed blends of poly(butylene adipate-co-terephthalate), PBAT, and poly(ethylene-co-vinyl alcohol), (EVOH), was measured by breaking thread (BT), imbedded fiber retraction (IFR), and rheological methods. The PBAT-rich blends were prepared under different melt mixing conditions in order to investigate the effect of mixing conditions and possibility of reactive mixing between the blend components on the blend morphology, rheology, mechanical properties and interfacial tension values. The conditions were varied based on a Taguchi design of experiment using four factors namely EVOH content (0–30 wt%), mixing time (2–15 min), rotor speed (50–90 rpm), and mixing temperature (185–200 °C), each varying at three levels. The average size of EVOH droplets in PBAT matrix was determined for each blend by a field emission-scanning electron microscopy technique. The values of interfacial tension of PBAT/EVOH were found to be 2.57 ± 0.22 and 2.73 ± 0.30 mN m−1 by the BT and IFR methods, respectively. The Palierne, Gramespacher, and Bousmina models were fitted to the rheological data to verify the interfacial tension of the blends. The continuous relaxation spectrum of the blends was determined in order to obtain the relaxation time of the EVOH droplets in the PBAT matrix. The Taguchi analysis revealed that the most effective factor is the EVOH content, and other factors do not play a significant role in the ultimate properties of the blends. Finally, based on the obtained mechanical properties, the possibility of reactive mixing under the applied mixing conditions was ruled out by means of repeated differential scanning calorimetry (DSC) and rheological measurements.

Characterizing the aging of asphaltic materials through the evolution of continuous relaxation spectrum

Similar to other construction materials, asphaltic materials used for pavements are susceptible to environmental conditions. Subsequently, the environmental factors such as aging should be considered in the design process of flexible pavements. In this study, the possibility of the use of relaxation spectrum for viscoelastic materials to capture the effect of aging has been explored. Different types of asphalt binders and asphalt mixtures were tested and analyzed to ensure the validity of the results. The results indicate that through characterizing the evolution of spectrum using statistical parameters, the effect of aging on physical properties of asphaltic materials can be captured.

Improved methodology for identifying Prony series coefficients based on continuous relaxation spectrum method

The Prony series models have been widely utilized in characterizing the linear viscoelastic properties of asphalt mixtures. However, the determination of the time coefficients, as a crucial step in identifying the Prony series coefficients, is based on empirical adjustments and time-consuming selections. In order to overcome the drawbacks of the existing methods, this study proposes a method for determining the optimal relaxation time range (ORTR) based on the continuous relaxation spectrum method, which is then utilized to identify the Prony series coefficients. There are four major steps in the proposed method: (1) establish the continuous relaxation spectrum by applying the inverse Stieltjes transform to the storage modulus model; (2) set trial groups of the relaxation time range on the basis of the relaxation time corresponding to the peak value of the continuous relaxation spectrum; (3) identify the ORTR through comparing the characteristic number of terms of each trial group; and (4) determine the Prony series coefficients based on the relationship between the relaxation strength and the relaxation time. The proposed method is validated by using complex modulus data of two types of asphalt mixtures at four test temperatures and six loading frequencies. The results show that: (1) the midpoint of the ORTR is close to the relaxation time corresponding to the peak point of the continuous relaxation spectrum; and (2) the total number of terms in the Prony series model developed from the ORTR is less than 30 and the model errors are less than 3.2%.