Relaxation Modulus Master Curve Research Articles

Characterization of viscoelastic behavior of basalt fiber asphalt mixtures based on discrete and continuous spectrum models.

In order to analyze the differences between the master curves of relaxation modulus E(t) and creep compliance J(t) obtained from discrete and continuous spectrum models, and to comprehensively evaluate the effect of basalt fiber content on the viscoelastic behavior of asphalt mixtures, complex modulus tests were conducted for asphalt mixtures with fiber content of 0%, 0.1%, 0.2% and 0.3%, respectively. Consequently, the master curves of Viscoelastic Parameters of asphalt mixtures were constructed according to the generalized Sigmoidal model(GSM) and the approximate Kramers-Kronig (K-K) relationship. Then, transformation of master curves using discrete and continuous spectrum models to obtain the models of E(t) and J(t) containing all viscoelastic information. Also, the accuracy of the models of E(t) and J(t) was evaluated. The results show that the addition of basalt fibers improves the strength, stress relaxation and deformation resistance of asphalt mixtures. It is worth noting that basalt fibers achieve the improvement of asphalt mixtures by changing their internal structure. Considering the different viscoelastic master curves at four dosages, the optimum fiber dosage was 0.2%. In addition, both discrete and continuous model conversion methods can obtain high accuracy conversion results.

Evaluation on the influence of dynamic water pressure environment on viscoelastic mechanical performance of asphalt mixture using the bending beam rheometer method

With the degraded service performance induced by the erosion of dynamic water pressure environment in summer, asphalt pavement has to subsequently overcome the challenge of low-temperature environment in winter. Considering the unique phenomenon of the gradual disintegration of asphalt mixtures’ components, it is still a lack of systematic research on moisture damage induced by dynamic water pressure environment from the perspective of viscoelastic properties. This study adopted the bending beam rheometer (BBR) method to evaluate the influence of dynamic water pressure environment on viscoelastic mechanical performance for asphalt mixture. The results indicated that both creep stiffness modulus and relaxation modulus of asphalt mixture showed similar downward trends with the increase in severities of different water environments, while values of relaxation modulus were slightly smaller than that of creep stiffness modulus. Master curves of creep compliance gradually moved upward with the increase in severities of dynamic water pressure environments, while master curves of relaxation modulus inversely moved downward. Specially, master curves of both creep compliance and relaxation modulus exhibited irregular trends for the dynamic water pressure environment of 60 ℃ − 0.414 MPa, which implied that significant erosion had been brought into asphalt mixture. A relative larger size of coarse aggregate skeleton could improve the resistance of asphalt mixture to moisture damage. The positive effect on improving resistance of asphalt mixture to moisture damage induced by SBS modified asphalt binder was stronger than the negative effect induced by the gradation with a relatively small nominal maximum aggregate size (NMAS). Differences existed in the reaction mechanisms of different types of asphalt binder subjected to the erosion of different water environments.

New Mechanistic Procedure to Predict the Critical Cracking Temperature of Asphalt Concrete from Bending Beam Rheometer and Indirect Tension Test Data

This study presents a new mechanistic procedure for determining the critical cracking temperature of asphalt concrete (AC) using data from bending beam rheometer (BBR) test of asphalt binder and indirect tension (IDT) test of AC. This new procedure uses BBR creep data to generate the mixture relaxation modulus mastercurve by utilizing the Hirsch model, time-temperature superposition principle, and Prony series-based interconversion method. The Hirsch model parameters are calibrated by comparing creep data from BBR and IDT creep tests performed at the same temperature. Boltzmann hereditary integral and second-order heat equation are then used to calculate thermal stress from the developed relaxation modulus mastercurve. IDT strength data is transferred from test strain rate to thermal strain rate using the viscoelastic continuum damage model. Since a strain gauge is not attached for traditional laboratory IDT strength testing, this study derived an analytical equation based on the Hondros solution to compute the horizontal strain rate from the applied vertical displacement rate. Finally, the critical cracking temperature is determined by coupling the thermal stress and strength profiles. Using the procedure presented in this paper, the critical cracking temperatures of four AC mixtures were predicted from BBR and IDT data. Their actual critical cracking temperatures were measured using thermal stress restrained specimen test performed in the laboratory to validate the method. The predicted critical cracking temperatures are found to be very close to the laboratory measured values. The developed procedure has substantial practical and technical importance in predicting the critical cracking temperature of AC because it utilizes widely available BBR and IDT tests.

Effect of Hydrothermal Aging on Mechanical Properties and Long-Term Durability of Recycled Polysulfone

The present experimental investigation aims to discover the effect of hydrothermal aging on the mechanical properties of virgin and recycled PSU, as well as the effect of hydrothermal aging on the long-term durability including stress relaxation and creep properties of virgin and recycled PSU. Virgin and recycled PSU specimens were exposed to pure hot water at temperatures of 98 °C for 1, 1.5, 3, 4.5, 6, 9, and 12 months, respectively. Mechanical test of tensile, three-point bending, impact, and fracture toughness was carried out on those aged specimens. To obtain the long-term stress relaxation and creep property, a series of short-time stress relaxation and creep tests were performed by using dynamic mechanical analysis (DMA), and the master curve of long-term stress relaxation modulus and creep compliance was drawn by shifting those short relaxation and creep curve based on the time–temperature superposition principle (TTSP). It was finally concluded that the virgin PSU exhibits excellent hydrothermal aging resistance in tensile, flexural, impact and fracture toughness, stress relaxation, and creep properties, even if it was aged in 98 °C hot water for 1 year. Although the tensile and flexural properties of the unaged recycle PSU are similar to those of virgin PSU, the hydrothermal aging resistance in tensile, flexural, stress relaxation, and creep properties of recycled PSU is worse than that of virgin PSU. In addition, the activation energy required for stress relaxation of PSU is same with that for creep.

Experimental, analytical, and analogical study of the effect of RAP extracted binder on rheology of polymer modified binder

This study attempts to scrutinise the effect of RAP on Performance Grade (PG), fatigue, and low temperature performance of polymer modified binder (PMB40). Furthermore, analytical, and analogical modelling was conducted. RAP from two sources was blended with PMB40 in proportions of 15%, 25%, and 40%. Thereafter, PG isochronal temperature sweeps, isothermal frequency sweeps and bending beam rheometer (BBR) testing was conducted. RAP sources affected both high and low temperature PG of PMB40. Viscoelastic transition temperature (TVET ) increased on inclusion of RAP thereby compromising fatigue performance. Based on difference in critical temperatures (ΔTc ), inclusion of RAP compromised the low temperature performance. Stiffness master curves (S(t)) illustrated an enhancement in stiffness on RAP inclusion whereas relaxation modulus master curves (E(t)) depicted a decrease in rate of decay of E(t) thereby compromising the low temperature stress relaxation potential. Both 2S2P1D and Huet models illustrated a reasonably good fit for various RAP blends.

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.

Study on properties and stress relaxation behavior of oil‐extended SBS/LLDPE/EVA blends

ABSTRACTIn order to explore and develop a new sealing material, the oil‐extended styrene‐butadiene‐styrene (SBS)/Linear low density polyethylene (LLDPE)/ethylene‐vinyl acetate (EVA) blends were prepared by melt mixing method and the effects of components on mechanical, rheological properties, and microstructure of the blends were investigated. The results showed that the resilience of the blends was barely changed during three‐time cyclic compression. The increase of oil‐extended SBS content would reduce the hardness of the blends as well as its compatibility and increase the melt flow rate. An increase in the LLDPE content, however, has the opposite effect on the blends. In contrast, EVA has little effect on the mechanical and rheological properties of the blends, but its addition could improve the transparency of samples effectively while promote the phase separation of the sample and complicate its microstructure. The master curve of stress relaxation modulus obtained by time–temperature superposition principle could be used to predict the long‐term (1.5 × 108 s, about 4.7 years) stress relaxation behavior of the blends at room temperature while the correlation coefficient of the shift factor fitting is 0.987. © 2020 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2020, 137, 48930.

Effect of filler content on the stress relaxation behavior of fly ash/polyurea composites

In this study, fly ash/polyurea (FA/PU) composites of various fly ash volume fractions were fabricated. The time-domain stress relaxation behaviors of pure polyurea and the FA/PU composites were measured by dynamic mechanical analysis (DMA) in tension mode at various temperatures. Both temperature and volume fraction of fly ash can affect the segmental motion of polyurea as well as the interaction between fly ash hollow spheres and polyurea matrix, which constitutes the intrinsic mechanism of stress relaxation. Master curves of relaxation modulus were constructed and compared for PU and FA/PU composites. A model was proposed to relate the relaxation modulus and volume fraction of fillers based on three-parameter fractional derivative model and Mori-Tanaka model. The numerical predictions obtained from the model are found to be in good agreement with the experimental results.

Effect of stress relaxation on accelerated physical aging of hydrogenated nitrile butadiene rubber using time-temperature-strain superposition principle

Abstract While the stress relaxation is a common phenomenon for polymers, its effect on the stress/strain accelerated aging of the hydrogenated nitrile butadiene rubber (HNBR) cannot be ignored. In this paper, the aging tests of HNBR at various temperature and strain levels were performed. To consider the influence of relaxation behavior during the accelerated aging process under external strains, a modified time-temperature-strain superposition principle considering stress relaxation (TTSSP-R) was proposed. With the general Maxwell model and the constructed master curve of relaxation modulus, the effect of stress relaxation on accelerated aging can be described. It was demonstrated that the proposed TTSSP-R approach provides an accurate evaluation of physical aging of HNBR with the consideration of stress relaxation. It could help the material designer to better understand the long-term performance and reliability of rubber components.

Creep, recovery, and stress relaxation behavior of nanostructured bioactive calcium phosphate glass-POSS/polymer composites for bone implants studied under simulated physiological conditions.

The creep and recovery and the stress relaxation behaviors of poly(butylene adipate-co-terephthalate) (PBAT) and polyhydroxyalkanoates (PHA) binary blends incorporating 30 wt % of a mixture of trisilanolisobutyl polyhedral oligomeric silsesquioxanes (POSS) and calcium phosphate glass (CaP-g) were investigated under simulated physiological and human body temperature conditions. The synergistic effect of PHA and CaP-g/POSS filler remarkably improved the creep behavior of the PBAT matrix and decreased its residual strain, consequently enhancing its elastic recovery. A considerable increase of the relaxation modulus of the hybrid materials was also observed upon incorporation of PHA and CaP-g/POSS. The relaxation modulus of the neat PBAT sample increased from ~60 MPa to ~1600 MPa after addition of 30 wt % CaP-g/POSS and 70 wt % PHA. However, after exposure of the composites to the simulated human body conditions for 14 days, a drop of dynamic mechanical properties of the studied material systems was observed along with formation of a desirable calcium phosphate phase on the material surface. The long-term (i.e., up to 7 × 105 s) viscoelastic behavior of the studied materials was successfully predicted using the time-temperature superposition principle and the obtained creep strain and the relaxation modulus master curves were satisfactorily fitted to the Findley power law equation and the generalized Maxwell model, respectively. This study demonstrates a facile method for tailoring CaP-g/POSS bioactive glasses composition for bone-like apatite formation on biopolymer surfaces. © 2019 Wiley Periodicals, Inc. J Biomed Mater Res Part B: Appl Biomater 107B: 2419-2432, 2019.

Long-term stress relaxation behavior of Polyaniline-EPDM blends using the time-temperature-strain superposition method

Long-term nonlinear viscoelastic stress response of polyaniline filled EPDM blends at various temperatures and strain levels were measured in short-term tests to check the applicability of time-temperature-strain superposition principle. A unified relaxation modulus master curve was constructed from the short-term tests. The unified master curve establishes the relaxation modulus over 18 months, which is 4.6 decades longer than the test duration. It was demonstrated that the time-temperature-strain superposition principle offers an accelerated testing methodology for determining the material’s long-term stress relaxation response at a reference strain and temperature.

Quantification of Uncertainty in the Master Curves of Viscoelastic Properties of Asphalt Concrete

Abstract The prediction of asphalt concrete (AC) behavior using a continuum damage mechanics approach requires viscoelastic properties like creep compliance (CC) and relaxation modulus (RM) values. Because of practical limitations, dynamic modulus (DM) and phase angle (PA) measurements are used to construct CC and RM master curves. Numerous approximate interconversion techniques have been used for this purpose. Because of issues during testing, fabrication processes, and interconversion approximations, significant scatter can be found in CC- and RM-constructed master curves. The present study proposes and compares quantification methods to address scatter found in CC and RM master curves. The scatter found in CC and RM master curves at different reduced time values is quantified using uncertainty quantification (UQ) techniques. For this purpose, several AC specimens with identical volumetric properties were prepared and tested for DM and PA values. Furthermore, CC and RM master curves were obtained using DM and PA through approximate interconversion techniques. Finally, CC and RM values obtained (at a fixed time using particular simulation techniques) with all specimens were combined. Using these values, scatter in CC and RM values (at a specific reduced time) was evaluated using UQ techniques. The results indicate that the choice of simulation technique affects the statistical parameters associated with probability density function (PDF) to a large extent. This, in turn, significantly affects the shape of the PDF at a chosen reduced time of interest. In other words, uncertainty found in CC and RM values are dependent on the choice of interconversion technique and time of interest.

Application of TTSP to non-linear deformation in composite propellant

A series of constant-rate tensile tests and relaxation tests were conducted to assess the mechanical properties of the hydroxy-terminated polybutadiene (HTPB) propellant. The relaxation modulus master curve was acquired using relaxation tests at the linear viscoelastic segment, and stress-reduced time master curves at different strain levels were established to determine whether the time–temperature superposition principle (TTSP) could be used to assess the non-linear mechanical property of the HTPB propellant. Results showed that the mechanical properties of HTPB propellant strongly depend on temperature; the extent of this dependence is enhanced with decrease in temperature, and the TTSP can be used in non-linear segments. With the application of the TTSP, the ultimate strength and the stress–strain curves of constant-rate tensile tests can be predicted at any temperature and strain rate, and the predicted σ–ε responses matched the test results closely, but led to some errors at high temperatures, this is because of some errors in the time–temperature shift factors.

Model and prediction of stress relaxation of polyurethane fiber

In this study, the effect of small strain (less than 10%) on hydrogen bond (H-bond) and crystallinity of dry-spun polyurethane fiber was investigated with fourier transform infrared spectroscopy and x-ray diffractometer, respectively. The results showed that the H-bond of hard segments hardly broke and its degree of crystallinity scarcely varied below strain of 10%. The fiber stress relaxation behavior at 25 °C under small strain was researched using dynamic mechanical analyzer. The stress relaxation modulus constitutive equation was obtained by transforming the non-linear relationship between stress and time into the linear relationship between stress and strain. The stress relaxation modulus master curve at 25 °C was established in terms of short-term stress relaxation tests at elevated temperatures (35 °C, 45 °C, 65 °C and 75 °C) according to time-temperature superposition principle (TTS) to predict long-term behavior within 353 year.

Establishing continuous relaxation spectrum based on complex modulus tests to construct relaxation modulus master curves in compliance with linear viscoelastic theory

A number of methods have been developed to determine the relaxation modulus, which is a fundamental material property that characterizes the rheological behavior of viscoelastic materials. However, the relaxation modulus master curves constructed using the current methods are likely to be accompanied by evident deficiencies. To correct these deficiencies, this study developed a method of using complex modulus tests to construct the master curve model of the relaxation modulus with a continuous relaxation spectrum that complied with the linear viscoelastic theory and allowed possible asymmetry in the relaxation spectrum curve. Using the data of the complex modulus tests, master curve models of the storage modulus and loss modulus were developed according to an approximate Kramers-Kronig relation. Based on the relationship between the relaxation modulus and the complex modulus, a specific model form of the continuous relaxation spectrum was established in terms of the same model parameters as those of the master curve models of the storage modulus and loss modulus. The established spectrum was found to be asymmetric and was ensured to be compliant with the linear viscoelastic theory. With the established relaxation spectrum, the master curve model of the relaxation modulus was formulated, and its numerical solution was determined as a function of the loading time; the master curves of the relaxation moduli were therefore constructed. The accuracy of the constructed master curves of the relaxation moduli were evaluated in a fairly wide time range and at the boundaries. The evaluation results demonstrated that the numerical method used to obtain the numerical solution of the relaxation modulus model produced negligible errors and that the approximate Kramers-Kronig relation provided a satisfactory representation of the mathematical interrelation of the viscoelastic parameters.The method developed in this study can be utilized to construct an accurate master curve of the relaxation modulus of any linear viscoelastic material. The constructed master curve of the relaxation modulus complies with the linear viscoelastic theory; specifically, it satisfies the approximate Kramers-Kronig relation and addresses the complete linear viscoelastic information delivered by both storage modulus and loss modulus.

A fractional model with parallel fractional Maxwell elements for amorphous thermoplastics

We develop a fractional model to describe the thermomechanical behavior of amorphous thermoplastics. The fractional model is composed of two parallel fractional Maxwell elements. The first fractional Maxwell model is used to describe the glass transition, while the second component is aimed at describing the viscous flow. We further derive the analytical solutions for the stress relaxation modulus and complex modulus through Laplace transform. We then demonstrate the model is able to describe the master curves of the stress relaxation modulus, storage modulus and loss modulus, which all show two distinct transition regions. The obtained parameters show that the modulus of the two fractional Maxwell elements differs in 2–3 orders of magnitude, while the relaxation time differs in 7–9 orders of magnitude. Finally, we apply the model to describe the stress response of constant strain rate tests. The model, together with the parameters obtained from fitting the master curve of stress relaxation modulus, can accurately predict the temperature and strain rate dependent stress response.

Prediction of fiber-directional flexural strength of carbon fiber-reinforced polypropylene based on time–temperature superposition principle

This study verified that the time–temperature superposition principle for fiber-directional flexural strength can be applied to thermoplastic composites undergoing instantaneous fast phenomena such as impact failure and long-term phenomena such as creep failure, by constructing the time- and temperature-dependent master curve of relaxation modulus of thermoplastic resin. The master curve could be transformed to another master curve that predicts fiber-directional flexural strength of carbon fiber-reinforced thermoplastic composites based on the micro-buckling failure theory expressed mainly by the resin’s elastic modulus. The experimental results obtained from high-speed bending test, static bending test at various temperatures, and creep bending test demonstrated that kink band failure occurred on the compressive surface of the specimen at every test condition. This validation and verification related to thermoplastic composites made it possible to predict static and dynamic flexural strengths at arbitrary temperature and creep flexural strength.

Effect of HDPEH polymer on viscoelastic properties of SBS modified asphalt

Styrene-Butadiene-Styrene (SBS) is one of the polymer that has been used for enhancing asphalt binder properties. The present work discusses the combined effect of High Density Polyethylene Homo Polymer (HDPEHP) and SBS on conventional and rheological properties of viscosity grade 30 asphalt commonly used in India. Six blends of binders were prepared using varying percentages of SBS and HDPEHP. Various rheological tests were performed on unaged, short-term aged and long-term aged binders to evaluate the effect of both polymers. Use of both polymers increased the viscosity of polymer modified blends marginally when compared to unmodified binder. Comparison of complex modulus mastercurves indicated that at lower reduced frequencies, complex modulus values were higher for polymer modified blends in relation to unmodified binder. Higher elasticity was provided by addition of HDPEHP when compared to other blends. Analysis of relaxation modulus and creep compliance mastercurves indicated combination of SBS and HDPEHP offered higher resistance under creep load and strain under long term conditions. Thus, it can be concluded that addition of HDPEHP enhances viscoelastic properties of SBS modified binder.

The Stress-Relaxation Behavior of Rice as a Function of Time, Moisture and Temperature

Abstract: Stress-relaxation behavior of single rice kernel was studied using a dynamic mechanical analyzer (DMA) in compression mode. The relaxation modulus was measured in a moisture content range of 12–30 % on dry basis (d.b.) and a temperature range of 25–80°C. A constant stain value of 1 % (within the linear viscoelastic range) was selected during the stress-relaxation tests. The relaxation modulus was found to decrease as the temperature and moisture increased. A master curve of relaxation modulus as a function of temperature and moisture content was generated using the time–moisture–temperature superposition principle. Results showed that the generalized Maxwell model satisfactorily fitted the experimental data of the stress-relaxation behavior and the master curve of relaxation modulus (R 2> 0.997). By shifting the temperature curves horizontally, the activation energy of the stress relaxation was obtained which significantly decreased with increase in the moisture content.

Estimation of low-temperature performance of recycled asphalt mixtures through relaxation modulus analysis

In cold climates, pavements undergo extreme thermal conditions which cause premature cracking and failure. The inclusion of Reclaimed Asphalt Pavement (RAP) could worsen this tendency due to the reclaimed bitumen stiffening effects. However, nowadays strict economic and environmental sustainability requirements strongly encourage the use of high RAP content in asphalt mixes.In this context, this paper discusses the low temperature performance of mixtures designed with 40% RAP according to the Bailey method. Different bitumen contents and polymer modification levels were employed. A mixture with 25% of unfractioned RAP was employed as the reference mixture.Laboratory tests were performed through Asphalt Thermal Cracking Analyzer (ATCA) by applying thermal loadings on restrained and unrestrained asphalt beams. Based on experimental data the relaxation modulus was evaluated to rank the materials. Its accurate determination is fundamental for designing long lasting pavements with proper performance at low temperature. The analysis was performed through a new analytical methodology for obtaining the relaxation modulus master curve by measuring thermally induced stress and strain. The solution, based on Boltzmann's equation and pseudo-variables concepts, accounts for time and temperature dependency of bituminous materials, avoiding complex integral transformations. The proposed solution successfully integrates the current ATCA analysis providing reliable estimations of relaxation properties fundamental in modelling of pavement behavior. Mixtures containing 40% RAP demonstrated enhanced relaxation capabilities at low temperature showing significant improved behaviours than the reference mixture. Although high amount of reclaimed material, proper selection of RAP and type and quantity of virgin bitumen can improve low temperature performance.