Stress Relaxation Modulus Research Articles

A note on viscoelastic bodies whose material properties depend on the density

In this note, we study the response of a viscoelastic body whose stress relaxation modulus and creep compliance depend on the density of the body in such a manner that the stress and strain appear linearly in the constitutive equation. Such models would be useful to study the response of porous viscoelastic bodies undergoing small deformations, as the moduli depend on the porosity, and hence the density. We study the problem of tension–torsion of cylinders of arbitrary cross-section within the context of this constitutive relation.

Analysis of linear viscoelasticity of aging soft glasses

Aging soft glassy materials do not follow time-translational invariance and violate the principles of linear viscoelasticity, such as the relation between the dynamic moduli in the frequency domain and the stress relaxation modulus in the time domain. Using an aqueous suspension of hectorite clay, a model aging soft glassy material, we account for time-dependent behavior by transforming the experimentally obtained stress relaxation and dynamic moduli from the real-time domain to the effective-time domain by normalizing the real-time by a time-dependent relaxation time. We find that the two sets of experiments probe different aging and deformation timescales, and simultaneous analysis provides insights into material behavior over a wide range of timescales. Careful analysis of the dynamic moduli reveals contamination of (slow) aging α modes by (fast) nonaging β modes, which is not clearly evident in the stress relaxation modulus. On the other hand, the stress relaxation measurements probe timescales long enough to characterize the terminal relaxation of the suspension. A toy Maxwell model is used to elucidate the interaction of the slow aging and fast nonaging modes, and their effect of measurements. The synergistic relationship between the two sets of experiments offers guidelines for characterizing time-dependent materials by generalizing the principles of linear viscoelasticity.

Using fractional derivatives for improved viscoelastic modeling of textile composites. Part II: Fabric under different temperatures

An improved modelling of the viscoelastic behavior of fabric composites at different temperatures is presented by a new means of fractional calculus. To this end, a four-parameter fractional derivative Zener model is first used to describe the dynamic behavior of the fabric, where the constitutive relation’s response to a unit step of strain (for determination of the time dependent stress relaxation modulus) can be readily found in the literature. This response function is then fitted to the experimental data of samples of a typical prepreg composite made of E-glass fibers comingled with polypropylene fibers, both at the dry and consolidated conditions. It is shown that the Zener model can better capture the experimental data when compared to the conventional approaches such as integer order modelling and Prony series. The results, particularly, showed that the viscoelastic behavior of the material at lower temperature regimes is far more accurate using the fractional derivatives approach. Finally, the variation of storage and loss moduli of the dry fabric with external cyclic loading were scrutinized, to determine the rubbery and glassy regions of the frequency response.

Transverse relaxation characteristic and stress relaxation model considering molecular chains of HTPB coating based on pre-strained thermal aging

In order to accurately describe the transverse relaxation characteristic and stress relaxation modulus of HTPB coating during pre-strain thermal aging process, a one month thermal aging test was carried out at 70 °C with pre-strain of 0%, 3%, 6% and 9%, respectively. The low-field 1H NMR and stress relaxation modulus tests were carried out for HTPB coating at different aging stages. The stress relaxation model considering the molecular chains was proposed according to the changes of crosslinking chain and dangling chain of HTPB coating during pre-strain aging. The results showed that with the increase of aging time, the decay rate of transverse relaxation curve became faster, the transverse relaxation time decreased, the value of combined parameter qMrl increased, the proportion of crosslinking chain decreased, while the proportion of dangling chain increased. Moreover, the stress relaxation modulus increased, the crosslinking network structure of HTPB coating became denser and the degree of crosslinking increased. At the initial aging stage, the pre-strain will destroy the crosslinking network structure of HTPB coating to a certain extent. With the increase of aging time, the effect of pre-strain will gradually weaken and the influence of aging on materials will gradually increase. The correlations between the stress relaxation model considering the molecular chains and the test results were more than 0.9950, which can accurately describe the stress relaxation modulus of HTPB coating during the pre-strain thermal aging process.

Application of fractional calculus methods to viscoelastic behaviours of solid propellants.

A three-branch viscoelastic model based on fractional derivatives is proposed for the viscoelastic behaviours of solid propellants. The simulation results show a satisfactory agreement with the stress relaxation modulus and complex modulus of solid propellants. As a comparison, the static modulus is also characterized by traditional viscoelastic model with integer-order derivatives. Results show that the application of the fractional derivatives to the viscoelastic constitutive model can effectively reduce the number of the required parameters while giving an accurate prediction of viscoelastic behaviours of solid propellants. Moreover, a simple and effective direct search method based on simulated annealing and Powell's mothed is proposed for the data fitting. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Using fractional derivatives for improved viscoelastic modeling of textile composites. Part I: Fabric yarns

In this article, the viscoelastic behavior of fabric yarns is modeled by a new means of fractional calculus. First, the constitutive relation of the fractional “Poynting-Thomson” model is developed to investigate the behavior of yarns. The proposed material constitutive relation is a differential equation of fractional order, where the response to a unit step of stress (for determination of creep compliance), or a unit step of strain (for stress relaxation modulus) can be readily found in the literature. Here, focusing on the stress relaxation, the response function is fitted to the experimental data of yarns in a typical woven fabric prepreg, at both dry and partially consolidated conditions. The yarns were made of E-glass fibers comingled with polypropylene fibers. The results showed a significant agreement with experimental data along with improved predictions of the new fractional modeling approach when compared to other approaches such as the integer order model and Prony series. For early relaxation times, especially for [Formula: see text] a considerable discrepancy was observed between the values of relaxation modulus obtained by the experiments and that by the integer-order derivative model. However, the results extracted via the fractional derivatives were in close agreement with experimental results at all relaxation times. Using the fractional properties of the yarns, the variation of storage and loss moduli of the yarns with external frequency loading was also predicted, capturing both the rubbery and glassy regions of the material frequency response.

Stress relaxation in symmetric ring-linear polymer blends at low ring fractions.

We combine linear viscoelastic measurements and modelling in order to explore the dynamics of blends of the same-molecular-weight ring and linear polymers in the regime of the low volume fraction (0.3 or lower) of the ring component. The stress relaxation modulus is affected by the constraint release (CR) of both rings and linear components due to the motion of linear chains. We develop a CR-based model of ring-linear blends that predicts the stress relaxation function in the low fraction regime of ring component in excellent agreement with experiments. Rings trapped by their entanglements with linear chains can only relax by linear-chain-induced constraint release, resulting in much slower relaxation of rings than of linear chains. The relative viscosity of the blend with respect to the linear melt viscosity η L at ring overlap volume fraction increases proportionally to the square root of ring molecular weight . Our experimental results clearly demonstrate that it is possible to enhance the viscosity and simultaneously the structural relaxation time of linear polymer melts by adding a small fraction of ring polymers. These results not only provide fundamental insights into the physics of the CR process but also suggest ways to fine-tune the flow properties of linear polymers by means of adding rings.

Structure Evolution of Epoxidized Natural Rubber (ENR) in the Melt State by Time-Resolved Mechanical Spectroscopy.

In this work, time-resolved mechanical spectroscopy (TRMS) was used to accurately characterize the rheological behavior of an epoxidized natural rubber (ENR) containing 25 mol% of epoxy groups. Conventional rheological tests are not suitable to characterize with accuracy the frequency-dependent linear viscoelastic behavior of materials, such as ENR, in a transient configurational state. For this reason, TRMS was used to determine the true rheological behavior of ENR, as well as to gain some insights into the changes of its macromolecular architecture under the dynamic conditions experienced during the measurements. The constructed master curves for the moduli revealed a gradual transition of the ENR rheological state from liquid-like to solid-like through the formation of an “elastic gel” throughout the bulk of the polymer. Furthermore, the evolution of the stress relaxation modulus revealed a slow relaxation mechanism, resulting from thermally activated reactions in the molten state attributed to the formation of crosslinks. Finally, the crosslink density evolution was estimated from the TRMS data and compared with results derived from equilibrium solvent-swelling measurements. These demonstrated the accuracy of the TRMS data in the prediction of the structural changes that can take place in polymers during processing.

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.

Physical and viscoelastic properties of carrots during drying

It is essential to understand the physical and mechanical properties of a product since these properties affect the structure, texture, and ultimately consumer acceptance. The effect of drying conditions on dynamic viscoelastic properties, stress relaxation function and creep compliance, and physical properties, such as moisture distribution, color parameters, and shrinkage, was studied. An increase in drying temperature and duration resulted in a decrease in moisture content and volume, which were highly correlated (R = .988). Water evaporation followed the falling rate period, demonstrating that the water transport was limited by internal resistances. The decomposition of carotenoids led to a decrease in magnitude of color parameters (L, a, and b), between 30.1% and 51.6% with 4 hr drying. It was observed that the material shrinkage and moisture content highly affected the mechanical properties; increased stress relaxation modulus and decreased creep compliance values of the sample. The creep behavior, expressed with Burger's model (R2 ≥ .986), was highly dependent on moisture content. The linear viscoelastic region of carrots was found to be at strains lower than 3%. The three-element Maxwell model well fitted to describe the viscoelastic behavior of carrots (R2 ≥ .999, RMSE ≤ 2.08 × 10-4 ). The storage moduli (G') were higher than loss moduli (G″), indicating that samples presented solid-like behavior. The findings can be used to improve the textural attributes of carrots and carrot-based products.

Dynamics and Viscoelasticity of Disordered Melts of Symmetric Diblock Copolymers

Simulations of coarse-grained models are used to study relationships among chain motion, composition fluctuations, and stress relaxation in unentangled melts of symmetric diblock copolymers. Measurements of the dynamic structure factor S(q,t) are reported as a function of wavenumber q, time t, and χN, where χ is the Flory–Huggins interaction parameter and N is degree of polymerization. The function S(q,t) is found to be a nearly exponential function of time, S(q,t) ∝ e–t/τ(q), for wavenumbers similar to or less than the wavenumber q* at which the static structure factor S(q) ≡ S(q,t=0) is maximum. The relationship between the decay time τ(q) and S(q) is used to define an effective wavenumber-dependent diffusivity D(q) for fluctuations of wavenumber q. The function D(q) is shown to change very little with changes in χN and to be a monotonically decreasing function of the nondimensional wavenumber qRg, where Rg is polymer radius of gyration. The linear shear stress relaxation modulus G(t) is inferred from measurements of the shear stress autocorrelation function. At low values of χN, far from the order–disorder transition (ODT), the modulus G(t) agrees with predictions of the Rouse model. Near the ODT, G(t) develops an additional slowly decaying feature arising from slow decay of composition fluctuations with q ∼ q*. The behavior of G(t) near the ODT is predicted nearly quantitatively by a modified version of the model of Fredrickson and Larson (FL), in which the prediction of the FL theory for the slowly decaying component is added to the prediction of the Rouse theory for contributions arising from single-chain relaxation, using the independently measured behavior of S(q,t) as an input to the theory.

Acidic-Thermal Ageing Effect on Compression Stress Relaxation of Silicone Rubber

The compression properties of silicone rubbers used as gaskets in PEM fuel cells are studied. The specimens are aged under different test conditions, viz, high temperature, humid air, and acidic solutions, prepared to match real PEM fuel cell operation conditions. The compression and stress relaxation tests are conducted. Temperature, humid air and acidic solution exert a serious effects on the mechanical performance of silicone rubbers. All the three factors can cause an increase in the stress relaxation modulus and permanent compression deformation. The effects of high temperature and an acidic solution are more pronounced. This can accelerate the deterioration of mechanical properties and decrease the sealing efficiency of gaskets, which would influence the durability of PEM fuel cells. It creates the basis for life prediction of silicone rubbers under appropriate accelerated durability test conditions.

Experimental determination of the viscoelastic parameters of K-BKZ model and the influence of temperature field on the thickness distribution of ABS thermoforming

This study investigated an influence of the temperature field on thickness distribution of thermoformed products using complex and high-aspect-ratio mold. The optimum temperature field was obtained to achieve a more uniform thickness distribution in the thermoformed products by using finite element simulation. The material properties of acrylonitrile-butadiene-styrene (ABS) polymer sheet were obtained by two rheological measurement tests. The linear viscoelastic properties, such as the storage modulus and loss modulus, were measured by a small amplitude oscillatory shear (SAOS) test for wide ranges of frequency and temperature. The discrete relaxation time and discrete relaxation modulus were obtained by nonlinear regression. The fitting parameters C1 and C2 for the WLF model were obtained by curve fitting. The nonlinear viscoelastic property, such as stress relaxation modulus, was measured by a step strain test. The damping function and fitting parameter α of Wagner-Demarmels (WD) model were determined by curve fitting. Then, the Kaye–Bernstein-Kearsley-Zapas (K-BKZ) constitutive equation was utilized to the thermoforming simulation in order to investigate the material behavior of the polymer sheet. The numerical results showed that a more uniform thickness distribution could be achieved with the optimum temperature field of the sheet. The thinnest part of the products was improved by more than 30%.

Rheological Tuning of Entangled Polymer Networks by Transient Cross-links.

The remarkable functionalities of transiently cross-linked, biopolymer networks are increasingly becoming translated into synthetic materials for biomedical and materials science applications. Various computational and theoretical models, representing different transient cross-linking mechanisms, have been proposed to mimic biological and synthetic polymer networks, and to interpret experimental measurements of rheological, transport, and self-repair properties. Herein, we employ molecular dynamics simulations of a baseline entangled polymer melt coupled with parametrized affinities for binding and unbinding of transient cross-links (CLs). From these assumptions alone, we determine the emergent CL mean density and fluctuations, and the induced rheology, across the 2-parameter space of binding and unbinding affinities for a moderately long chain, entangled the polymer melt. For sufficiently weak (short-lived) CLs, nonmonotonicity emerges with respect to the affinity to form CLs: the stress relaxation, viscous, and elastic moduli all shift above the baseline if CLs form rapidly, reverse below the baseline as CLs form slowly, and reverse again, recovering the baseline as CLs form very slowly. For sufficiently strong (long-lived) CLs and sufficiently fast CL formation, a dramatic rise emerges in the viscous and elastic moduli at all frequencies, more prominently in the elastic moduli at medium to high frequencies, inducing a sol-gel transition. These results are placed in context with the experimental and theoretical literature on transient polymer networks.

Brownian dynamics investigation of the Boltzmann superposition principle for orthogonal superposition rheology

The most general linear equation describing the stress response at time t to a time-dependent shearing perturbation may be written as the integral over the past history t' of a time dependent relaxation modulus, depending on t - t', multiplied by the perturbing shear rate at time t'. This is in agreement with the Boltzmann superposition principle, which says that the stress response of a system to a time dependent shearing deformation may be written as the sum of responses to a sequence of step-strain perturbations in the past. In equilibrium rheology, the Boltzmann superposition principle gives rise to the equality of the shear relaxation modulus, obtained from oscillatory experiments, and the stress relaxation modulus measured after a step-strain perturbation. In this paper, we describe the results of Brownian dynamics simulations of a simple soft matter system showing that the same conclusion does not hold when the system is steadily sheared in a direction perpendicular to the probing flows, and with a gradient parallel to that of the probing deformations, as in orthogonal superposition rheology. In fact, we find that the oscillatory relaxation modulus differs from the step-strain modulus even for the smallest orthogonal shear flows that we could simulate. We do find, however, that the initial or plateau levels of both methods agree and provide an equation relating the plateau value to the perturbation of the pair-function.

Creep and stress relaxation behavior for natural cellulose crystal of wood cell wall under uniaxial tensile stress in the fiber direction

We investigated the temporal changes in creep and stress relaxation behavior in both microscopic crystalline cellulose and macroscopic strain of wood specimen using Japanese cypress (Chamaecyparis obtusa Endl.) to understand the viscoelastic properties of wood cell walls. Specimens 600 µm in thickness were observed by the X-ray diffraction and submitted to tensile load. The crystal lattice strain of (004) plane and macroscopic strain of specimen were continuously detected during creep and stress relaxation tests. It was found that the creep compliance based on macroscopic strain showed a gradual increase after instantaneous deformation due to loading and then the parts of creep deformation remained as permanent strain after unloading. On the other hand, crystal lattice strain showed a different behavior for macroscopic strain; it kept a constant value after instantaneous deformation due to loading and then increased gradually after a certain period of time. These differences between macroscopic and microscopic levels were never found in the stress relaxation tests in this study. Relaxation modulus at the macroscopic level only showed a decreasing trend throughout the relaxation process. However crystal lattice strain kept a constant value during the macroscopic relaxation process. In addition, the microfibril angle (MFA) of wood cell wall has a role of mechanical behavior at microscopic level; crystal lattice strains were smaller with increasing MFA at both creep and relaxation processes. Creep compliance and stress relaxation modulus at the macroscopic level decreased and increased with increasing MFA, respectively. Our results on the viscoelastic behavior at microscopic level evidenced its dependency on MFA.

Slip Spring-Based Mesoscopic Simulations of Polymer Networks: Methodology and the Corresponding Computational Code.

In previous work by the authors, a new methodology was developed for Brownian dynamics/kinetic Monte Carlo (BD/kMC) simulations of polymer melts. In this study, this methodology is extended for dynamical simulations of crosslinked polymer networks in a coarse-grained representation, wherein chains are modeled as sequences of beads, each bead encompassing a few Kuhn segments. In addition, the C++ code embodying these simulations, entitled Engine for Mesoscopic Simulations for Polymer Networks (EMSIPON) is described in detail. A crosslinked network of cis-1,4-polyisoprene is chosen as a test system. From the thermodynamic point of view, the system is fully described by a Helmholtz energy consisting of three explicit contributions: entropic springs, slip springs and non-bonded interactions. Entanglements between subchains in the network are represented by slip springs. The ends of the slip springs undergo thermally activated hops between adjacent beads along the chain backbones, which are tracked by kinetic Monte Carlo simulation. In addition, creation/destruction processes are included for the slip springs at dangling subchain ends. The Helmholtz energy of non-bonded interactions is derived from the Sanchez–Lacombe equation of state. The isothermal compressibility of the polymer network is predicted from equilibrium density fluctuations in very good agreement with the underlying equation of state and with experiment. Moreover, the methodology and the corresponding C++ code are applied to simulate elongational deformations of polymer rubbers. The shear stress relaxation modulus is predicted from equilibrium simulations of several microseconds of physical time in the undeformed state, as well as from stress-strain curves of the crosslinked polymer networks under deformation.

Expression of normal stress difference and relaxation modulus for ternary nanocomposites containing biodegradable polymers and carbon nanotubes by storage and loss modulus data

In this paper, the first normal stress differences (N1) and the relaxation modulus (G (t)) are predicted for prepared poly (lactic acid) (PLA)/poly (ethylene oxide) (PEO)/carbon nanotubes (CNT) nanocomposites using the experimental results of storage and loss moduli. N1 and G (t) are calculated for these samples as a function of shear rate. Moreover, the effects of various parameters on N1 and G (t) are revealed to validate the equations. N1 increases by shear rate, but it shows a plateau at low shear rates. The addition of nanoparticles to the blends increases N1 demonstrating that the nanoparticles enhance the elasticity. G (t) decreases upon increasing in the shear rate in all samples and the addition of CNT to polymer blends causes a high G (t). A high storage modulus and small loss modulus enhance N1 and G (t), whereas poor storage modulus lowers N1 and G (t). Additionally, G (t) improves significantly at small strain and high N1. This work presents useful guidelines for calculation of N1 and G (t) and understanding the origin of these terms in polymer systems.

Fractional viscoelastic models with non-singular kernels

New definitions of fractional derivative with non-singular kernels are proposed recently. In the present paper, we apply the classical and new definitions of fractional derivative to four fractional viscoelastic models, namely, fractional Maxwell model, fractional Kelvin–Voigt model, fractional Zener model and fractional Poynting–Thomson model. For each fractional viscoelastic model, the stress relaxation modulus, creep compliance and dynamic modulus are derived analytically under the classical and new fractional derivative definitions. The performance of these models under different fractional derivative definitions is further compared. The results show that the fractional Zener model and fractional Poynting–Thomson model are equivalent in all conditions. Compared with the classical fractional derivative definition with a power function kernel, the fractional derivative definition with a logarithmic function kernel can be used to describe the ultraslow creep and relaxation behaviors. However, the performance of fractional Maxwell model with the exponential function kernel is close to that of integer-order Maxwell model. Fractional Maxwell model and fractional Zener model with the Mittag–Leffler function kernel do not provide accurate descriptions of the stress relaxation modulus at shortest time and the storage modulus at highest frequency. Thus, specific modification is needed when applying the new definitions of fractional derivative to viscoelasticity.

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.