Storage Modulus Master Curves Research Articles

Study of the matrix–particle interactions of polymer nanocomposites in the low‐frequency regime

AbstractIn this study, the dynamic moduli, in a wide frequency scale, of two series of polymer nanocomposites were qualitatively discussed. The dynamic mechanical analysis experimental results of polylactic acid (PLA)/silica nanocomposites, as well as PLA/graphene oxide‐carbon nanotubes hybrid nanocomposites, were employed. The study has been focused on the low‐frequency regime, estimating the terminal incline of the storage and loss modulus. In addition, the relative position of the two dynamic moduli was examined and results about the nanofiller's dispersion quality, as well as their interaction with the matrix were extracted. The consequence of nanofiller kind, loading, and distribution quality on the mechanical enhancement was thoroughly studied. In addition, the storage modulus master curves could be simulated by a model from the literature, considering the topological constraints of an entangled polymer reinforced by nanoparticles. It was shown that by adopting the linear contribution of the free and adsorbed chains to the relaxation function, the details of the master curves could be captured in the entire frequency range, quite effectively. The estimated model parameter values confirmed the analysis of the matrix–particle interactions.Highlights The SiO2 size effect on the terminal zone slope of the dynamic moduli. Successful description of the dynamic moduli by the tube reptation concept. Analysis of the physical meaning of the model parameter values. Relation between SiO2 dispersion quality and molecular motion restriction. Matrix‐particle interactions is the governing mechanism of enhancement.

Investigation of the time-domain linear viscoelastic response of warm mix asphalt mixture via discrete and continuous spectrum

Asphalt mixture is a temperature-sensitive material, and its viscoelastic properties are essential for pavement design and performance evaluation. This paper presents a method for investigating the time domain linear viscoelastic parameter response of asphalt mixture after adding different warm mix agents. The approach utilizes the generalized Sigmoidal function to construct master curves of storage modulus and loss modulus in the frequency domain, and both discrete and continuous spectrum analyses are used to analyze the viscoelastic behavior of the asphalt mixture. The effects of varying Kelvin and Maxwell element numbers on the discrete spectrum are compared to the spectral intensity and master curve obtained from the continuous spectrum. Results show that the generalized Maxwell model with 10−3 or 10−2 as the center point and the generalized Kelvin model with 103 or 104 can obtain more accurate results. When used as input for simulation software, the number of 11 elements can balance computational efficiency and accuracy. The study evaluates the impact of different warm mix agents on the relaxation modulus and creep compliance of asphalt mixture and discusses the practical implications of these findings for engineering applications. Adding Foam warm mix agent significantly lowers the asphalt mixture's relaxation modulus by approximately 44%, while Sasobit and Evotherm marginally raise it by 14% and 22%, respectively. The Foam warm mix agent also increases the equilibrium modulus of creep compliance to 0.091 MPa, 80% higher than that of HMA. The findings of this paper provide guidance for selecting the appropriate warm mix agent to improve pavement performance.

Evaluating fatigue resistance based on viscoelastic properties of asphalt mixtures

ABSTRACT In this study, the fatigue resistance of different asphalt mixture types subjected to the same aging condition was evaluated using linear viscoelastic properties of asphalt mixtures. The relaxation spectra were determined from the dynamic modulus test based on the linear viscoelastic theory, and the relationship between the parameters of relaxation spectra and the parameters of the storage modulus master-curves were developed. The Texas overlay test was conducted to evaluate the fatigue resistance with the number of load cycle to failure used as the measure of fatigue resistance. From the corresponding results, it is shown that the inflection point frequency of the dynamic modulus master-curves could be used to assess the fatigue resistance compared to other parameters. The Glover-Rowe parameter also exhibited a fair relationship with the fatigue resistance of the asphalt mixtures. The effect of the shape parameter (γ) of the dynamic modulus master-curves was quantified but seemed not very sensitive to fatigue resistance for the asphalt mixtures evaluated. In the study, a new parameter (VECIndex) was developed to evaluate the fatigue resistance, which considered the inflection point frequency and shape of the dynamic modulus master-curves, both of which influence the linear viscoelastic properties and fatigue resistance of asphalt mixtures.

Enhanced thermal stability and long-term mechanical durability at elevated temperatures of thermotropic liquid crystal polyester/glass fiber composites

We herein report the influences of glass fibers (GFs) on the thermal stability and long-term mechanical performance at elevated temperatures of thermotropic liquid crystal polyester (TLCP) composites for advanced applications in the industrial sectors of automotive, electric/electronics, aerospace, and construction. For this purpose, TLCP-based composites with 10–40 wt% GF loadings are fabricated by facile melt-mixing and injection-molding. The scanning electron microscopy (SEM) images and X-ray diffraction patterns demonstrate that the GFs are dispersed in the TLCP matrix with a microfibrillar structure. The specific interactions between TLCP and GFs in the composites are identified by ATR FT-IR spectroscopic analyses. The thermal degradation temperatures of the composites increase slightly with the increment of GFs with high thermal stability and heat capacity. The long-term mechanical durability of TLCP/GF composites at 150 °C is analyzed by obtaining time-dependent storage modulus master curves based on the stepped isothermal method and time–temperature superposition principle. The dynamic mechanical moduli, activation energy, and long-term mechanical durability are found to be highly enhanced for TLCP composites with higher GF loadings, which is owing to the efficient stress transfer from the TLCP matrix to reinforcing GF fillers via specific interactions at the composite interface.

Mechanical response of two different molecular weight polycarbonates at varying rates and temperatures

Polymers are widely used for lightweight design in industrial applications, such as helmets and car bumpers, where the most common causes of failure or damage are dynamic impact events. It is well known that the mechanical response of most polymers is highly dependent on the loading rate and temperature, and that it is not sufficient to use properties measured under static loads in the analysis of dynamic events. However, the time-temperature equivalence phenomenon offers the chance to predict high-rate performance using low-rate data. In this study, information about the constitutive behaviour of two different molecular weight polycarbonates, is obtained in low-rate experiments and then compared with the high-rate response. A master curve of storage modulus constructed from Dynamic Mechanical Analysis data is employed to understand the viscoelastic response under small-strain loading at various frequencies and temperatures. For the large-strain constitutive response, experiments at strain rates from 0.001 s-1 to 3000 s-1 are performed using a conventional crosshead device, hydraulic device, and split-Hopkinson pressure bar. Moreover, experiments at strain rates of 0.01 s-1 and temperatures from -60 to 120 °C are also performed, and the results are compared. This approach can distinguish ’constitutive’ rate dependence from the effects of specimen self-heating due to adiabatic heating under high-rate deformation. Meanwhile, the molecular weight effects on the mechanical response at varying rates and temperatures are also noted.

Predict the Phase Angle Master Curve and Study the Viscoelastic Properties of Warm Mix Crumb Rubber-Modified Asphalt Mixture.

To identify the most accurate approach for constructing of the dynamic modulus master curves for warm mix crumb rubber modified asphalt mixtures and assess the feasibility of predicting the phase angle master curves from the dynamic modulus ones. The SM (Sigmoidal model) and GSM (generalized sigmoidal model) were utilized to construct the dynamic modulus master curve, respectively. Subsequently, the master curve of phase angle could be predicted from the master curve of dynamic modulus in term of the K-K (Kramers–Kronig) relations. The results show that both SM and GSM can predict the dynamic modulus very well, except that the GSM shows a slightly higher correlation coefficient than SM. Therefore, it is recommended to construct the dynamic modulus master curve using GSM and obtain the corresponding phase angle master curve in term of the K-K relations. The Black space diagram and Wicket diagram were utilized to verify the predictions were consistent with the LVE (linear viscoelastic) theory. Then the master curve of storage modulus and loss modulus were also obtained. Finally, the creep compliance and relaxation modulus can be used to represent the creep and relaxation properties of warm-mix crumb rubber-modified asphalt mixtures.

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.

Effect of Temperature and Strain Rate on the Flexural Behavior of Wood-Polypropylene Composites.

The mechanical properties of wood-polypropylene composites exhibit typical viscoelasticity. However, there is little information on the mechanical properties of wood-polypropylene composites related to temperature and time, which limits the use of wood-polypropylene composites as structural components. Here, the effect of time (strain rate) and temperature on the flexural properties and the master curve of the storage modulus used to predict the long-term performance of wood-polypropylene composites were investigated. The results showed that the flexural strength and modulus increased linearly with the increase of wood contend, which can increase by 134% and 257% respectively when the mass fraction of wood powder reached 45%. Moreover, there was a positive linear relationship between flexural strength and ln strain rate, while the flexural strength and modulus decreased as temperature elevated. The storage modulus as a function of frequency (time) and temperature confirmed this trend. To evaluate the long-term performance, the storage modulus master curve was constructed and the respective activation energy was calculated, which revealed that the long-term performance of the samples depended on the matrix and the addition of an appropriate amount of wood powder was beneficial to improve their durability.

Loading rate and temperature dependence of flexural behavior in injection-molded glass fiber reinforced polypropylene composites

This paper is concerned with the influence of loading rate/temperature and frequency/temperature on static flexural behavior and dynamic mechanical properties of injection molded glass fiber reinforced polypropylene (GFPP) composites, respectively. Furthermore, the mechanism of long-term durability of PP and GFPP was investigated by the master curve of storage modulus constructed based on the Time-Temperature-Superposition (TTSP) method. Three-point bending tests at various loading rates/temperatures and dynamic mechanical analysis (DMA) at various frequencies/temperatures are performed. The time-temperature dependence of static flexural properties (modulus, E, and strength, σ) and dynamic properties (storage modulus, E’) in PP and GFPP is discussed. The results show that both static flexural behavior and dynamic mechanical behavior of PP and GFPP have significant time-temperature dependent phenomenon, which conforms to the time-temperature equivalence principle. The long-term durability of GFPP depends on the matrix resin rather than the GF reinforcement and its interface at temperatures above the glass transition temperature (Tg).

Experimental analysis and prediction of viscoelastic creep properties of PP/EVA/LDH nanocomposites using master curves based on time–temperature superposition

ABSTRACTPrediction of viscoelastic behavior of polymers over a long‐term period is of vital importance for engineering applications. An attempt was made to uncover the interplay between the morphology and viscoelastic behavior of compatibilized polypropylene/ethylene vinyl acetate (EVA) copolymer blends in the presence of layered double hydroxide (LDH) nanoplatelets. The time–temperature superposition (TTS) principle and WLF equations were merged to obtain master curves of storage modulus at defined reference temperatures enabling prediction of storage modulus at high frequency ranges which are not experimentally measureable. Moreover, the creep compliance master curves were acquired for different reference temperatures to predict the creep compliance of nanocomposites over long period of times. It was found that the presence of LDH decreases the creep compliance at long period of times while it decreases the unrecoverable deformation of EVA domains. A simple mechanism was proposed to explain the creep and recovery behavior of samples blend at different temperatures. © 2018 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018, 135, 46725.

Effect of aging level on viscoelastic properties of asphalt binder containing waste polyethylene

Due to inherent advantages, waste polyethylene (generated from domestic sources) has been used as asphalt modifier. This article discusses viscoelastic properties of the polyethylene modified asphalt binder. Several asphalt concrete mixtures were designed by varying polyethylene and asphalt content. Shredded polyethylene incorporated into asphalt concrete by dry mixing process. Using optimised asphalt content (at several polyethylene percentages), mixtures were prepared and compacted. These compacted specimens were aged for different duration in convection oven. Asphalt binder (from aged specimens) was extracted and tested for its viscoelastic properties using dynamic shear rheometer. Comparison of mastercurves indicated increase in complex modulus and decrease in phase angle values with addition of polyethylene at all reduced frequencies. However, the extent of change was highly dependent on frequency, aging duration and polyethylene content. This indicates that the addition of polyethylene provides more resilience to asphalt binder especially with less aging time. Storage modulus mastercurves (at lower frequencies) and relaxation modulus values (at longer time) indicate polyethylene provides additional stiffening to binder. Further, changes in viscous modulus with addition of polyethylene were marginal. Overall results indicate that waste polyethylene improves the properties of asphalt binder over extended loading period when heated for less time during mixing.

Reinforcement of epoxy resin/carbon fiber composites by carboxylated carbon nanotubes: a dynamic mechanical study

The composites of epoxy resins reinforced with unidirectional carbon fibers were prepared via hot pressing of carbon fiber prepregs. Multiwall carbon nanotubes (MWCNT) derivatives [parent CNTs, low carboxylated CNTs (LCCNTs) and higher carboxylated CNTs (HCCNTs)] were incorporated into the epoxy resin/carbon fiber (E/C) composite. Interestingly, remarkable modifications as consequences of homogeneous dispersion of CNTs in E/C matrix and prevention of void formation within the composite wereachieved by using carboxylated CNTs. The dynamic mechanical analysis (DMA) showe dimproved glass transition temperature (Tg) and storage modulus of the resulting E/C/HCCNT composite. The master curves of storage modulus were constructed for the samples using the Vogel-Fulcher-Tammann (VFT) and Williams-Landel-Ferry (WLF) equations as well as the time-temperature superposition principle (TTS). The apparent activation energies of molecular rearrangements were calculated at different temperatures based on the numerical derivatives and WLF equations. The experimental results indicate that COOH content of CNTs can have a remarkable effect on the ultimate properties and performance of the E/C/CNTs nanocomposites. Based on possessing light weight and good modulus (of 31 GPa and 20.6 GPa at -90 °C and 80 °C, respectively), E/C/HCCNT can be consideredas an interesting composite for structural applications by polymer engineers.

Predicting the high strain rate response of plasticised poly(vinyl chloride) using a fractional derivative model

Polymers are frequently used in fields as diverse as aerospace, biomedicine, automotive and in-dustrial vibration damping, where they are often subjected to high strain rate or impact loading. Poly(vinyl chloride) (PVC), and its plasticised variants (PPVC), are just two examples of this broad category of materi-als. Since many polymers exhibit strong rate and temperature dependence, including a low temperature brittle transition, it is extremely important to understand their mechanical responses over a wide range of loading con-ditions.PVC with 60 wt% plasticiser is used in this study, as its highly rubbery nature lends itself well to being used in various load mitigation and energy absorption applications. It is challenging to obtain high strain rate data on rubbery materials using conventional techniques such as the split-Hopkinson (Kolsky) bar. Therefore, alternative approaches are required. Based on previous work developing a framework to predict high rate re-sponseusing a fractional derivative model, Dynamic Mechanical Analysis (DMA) experiments are conducted on the PPVC to construct a master curve of storage modulus. These data are used to part-calibrate a modified Mulliken-Boyce model which also takes into account specimen heating to derive stress-strain relationships at strain rates varying from 0.001 s_1 to 13 500 s_1. This model is further calibrated against experiments conducted in a previous study and shown to provide an excellent description of the behaviour at these rates.

Evaluation of classic and fractional models as constitutive relations for carbon black–filled rubber

Many constitutive relationships have been derived to model the viscoelastic behaviors of materials, but limited works are done to point out which model is the most suitable one in certain conditions. In this article, we present the detailed comparisons of the classic rheological and fractional derivative models to make up for the deficiency. First, creep and stress relaxation tests of carbon black (CB)–filled rubber are carried out at the room temperature for 12 h, and frequency-sweep tests are conducted at a temperature ranging from Tg − 10°C to Tg + 100°C under tension loading. The master curve of storage modulus over a wide range of frequency covering about 21 decades is constructed according to the time/frequency–temperature superposition (TTS/fTS). And then the classic models (Maxwell, Kelvin, and their generalized models) and fractional derivative models (4, 5, 6, and 10 parameters Zener models) are used to fit the test data. The fitting results show that the classic generalized models and fractional Zener models are quantitatively equivalent. The classic generalized models with several parameters are much simpler and are able to predict the viscoelastic behaviors for short-term or narrow frequency conditions, while the fractional derivative models are suitable for wide frequency conditions.

Application of fractional calculus methods to asymmetric dynamical response of CB-Filled rubber

Frequency-sweep tests at various temperatures were conducted to study the asymmetric dynamic response of carbon black (CB) filled rubber. The master curves of storage modulus were constructed by use of time-temperature superposition (TTS), and the dynamic response of the material over a wide range of frequencies covering about 20 decades exhibits asymmetry. Based on the experimental results, the fractional Zener model and Prony series are verified to be unsuitable to reproduce the dynamicviscoelastic behavior. Therefore, a modified model by adding a spring-pot into the fractional Zener model is presented. The applicability of the modified model in describing the dynamic behavior of the CB-filled rubber is validated by the experimental results.

Thermomechanical viscoelastic analysis of woven-reinforced thermoplastic-matrix composites

The behaviour of four different woven-reinforced thermoplastic-matrix laminates was analysed using dynamic mechanical thermal analysis. The investigation was focussed on a temperature range relevant for applications in the automotive industry, namely between −30°C and 80°C. Storage modulus master curves were built for each material in three different orientations. Dependence of the tensile and shear moduli on frequency was described using Prony series. This description was then introduced in a phenomenological transversely isotropic viscoelastic material model. Shift factors determined in the frequency domain were converted to time domain and used in the proposed constitutive model. The model was implemented in Abaqus/Standard using a user material subroutine (UMAT) and validated by comparing relaxation and creep predictions with experimental data.

Viscoelastic properties of PS–POSS hybrid materials prepared by reactive processing

The viscoelastic properties of hybrid nanocomposites of polystyrene and polyhedral oligomeric silsesquioxane (PS–POSS) are determined in the molten state by small-amplitude oscillatory shear rheometry (SAOS) and in the solid state by dynamical-mechanical thermal analysis (DMA). PS–POSS samples synthesized to form various molecular structures and phase morphologies are investigated. Overall, both viscoelastic data sets correlate well with the POSS dispersion level and with the degree of grafting of the PS–POSS samples. From the SAOS data, enhanced POSS dispersion and degrees of grafting are characterized by deviations of the zero-frequency viscosity ratios of the Einstein and Einstein–Batchelor models and by linearity in the Han plots. PS–POSS samples with enhanced POSS dispersion and degrees of grafting are characterized as those with higher free-volume fractions and lower flow activation energies, as determined by storage modulus master curves that stem from applying the time-temperature superposition (TTS) principle to the DMA data.

Closed-form solution for horizontal and vertical shiftings of viscoelastic material functions in frequency domain

The closed-form shifting (CFS) algorithm is a simple mathematical methodology which determines the unique solution in the process of constructing master curves at selected reference temperature and pressure conditions. In a previous paper, the CFS algorithm has been fully described for monotonically increasing or monotonically decreasing functions only. This paper presents detailed steps of the generalized CFS methodology for non-monotonic functions, like the loss tangent. Performing shifting on the loss tangent, which does not require vertical shifting, is particularly important for materials which require vertical adjustment of dynamic viscoelastic functions, i.e., loss and storage moduli. Thus, based on horizontal shifting of the loss tangent, the CFS-based procedure of consecutive horizontal-vertical superposition for the storage modulus is proposed and analyzed. The analysis is done on the example of two synthetically generated non-monotonic tan delta segments and corresponding storage modulus segments in respect to different experimental parameters. It has been shown that the error brought by the shifting method into non-monotonic loss tangent and storage modulus master curves is twice smaller than the corresponding experimental noise level.

Influence of Chemically Treated Palygorskite Over the Rheological Behavior of Polypropylene Nanocomposites

Melt compounding was used to prepare thermoplastic composites reinforced with palygorskite; the chemical modification of the palygorskite surface and the control of extrusion parameters were used to promote good dispersion and a good interface with the matrix, affecting the rheological properties of the mixture, since the melt viscoelastic behavior is sensitive to changes in the molecular structures at nanoscale and mesoscale. The palygorskite nanoclay (diameter 30nm) was extracted from a site in the southeast of the state of Yucatan, Mexico; the palygorskite was subjected to purification and then superficially modified with 3-aminopropyltrimetoxysilane. These processes generated three types of specimens: unpurified, purified and purified-silanized palygorskite; which were subsequently incorporated (0.5 w%) to a polypropylene matrix by twin extrusion. Frequency sweeps were used to evaluate the interaction between the matrix and the nano-reinforcement. The master curves of storage modulus (G’), loss modulus (G”) and complex viscosity (η*) vs angular frequency (ωx) show a change of the interfacial interaction polypropylene/palygorskite when the palygorskite was chemically modified.

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.