Calculation of stress–strain curves from relaxation data in the rubbery flow region

Stress relaxation curves for polysulfone and Lexan polycarbonate are only time dependent at a constant temperature if strain is defined as ?_H = In (l/l₀) and the “true” cross-sectional area A = A₀/(1 + ?) is used. The strain-independent stress relaxation curves can be used to calculate stress–strain curves at different rates of strain according to the linear viscoelastic theory. The agreement between experimental and calculated stress–strain curves is good at least up to about 60% strain in the range of 0.01 to 0.2 in./min rate of extension if an average rate of strain defined by ?_H = 1/t ln(l/l₀) is used.     

Soft Matter in a Tight Spot: Nanorheology of Confined Liquids and Block Copolymers

The linear frequency-dependent shear rheology and force–distance profiles of molecularly-thin fluids of very different structure were contrasted: a globular molecule octamethylcyclotetrasiloxane (OMCTS), branched alkanes (3-methylundecane and squalane), and a polymer brush in near-theta solution (polystyrene-polyvinylpyridine). In each case the data suggest a prolongation of the longest relaxation time (τ₁) with increasing compression. At frequencies ω > 1/τ₁ the shear response was “solid-like”, but at ω < 1/τ₁ it was “liquid-like”. OMCTS under mild compression exhibited seeming power-law viscoelastic behavior with G′(ω) = G″(ω) over a wide frequency range. Of the branched-molecule fluids, 3-methylundecane exhibited oscillatory force–distance profiles; this confirms prior computer simulations. But squalane (6 pendant methyl groups in an alkane chain 24 carbons long) showed one sole broad attractive minimum. Polymer brushes in a near-theta solvent exhibited changes qualitatively similar to those OMCTS, in particular, a smooth progression of longest relaxation time, generating a transition from “liquid-like” to “solid-like” shear rheology with decreasing film thickness. The common trend of shear response in these systems, in spite of important differences in molecular structure and force–distance profiles, is emphasized.     

Effect of blending on the rheological properties of polystyrene

Experimental non-Newtonian viscosity, primary normal stress difference, complex viscosity, and shear stress relaxation were taken for highly fractionated polystyrene in Aroclor 1248 (a chlorinated biphenyl) as well as for blends differing in molecular weight and concentration. The data are described by three parameters: a zero-shear rate value, the slope of the log–log plots in the high shear rate region, and a time constant defined as the inverse of the shear rate at the intersection of the low and high shear rate asymptotes. For the functions measured, the low shear rate region is characterized by a dependence on M_w and the high shear rate region by a dependence on M_n. Implications to polymer processing are discussed.     

Stress relaxation of elongated strip of poly(vinylidene fluoride) in ethyl acetate vapor

Poly(vinylidene fluoride) films in ethyl acetate vapor were studied at 30°C for vapor pressures of p = 0, 12, 30, 40, 66, 85 torr and elongations ε = 4.5%, 7%, 9.5%, 19%, 29%, and 44%. A cyclic experiment was also performed at ε = 7% and p = 40 torr for three sorption/desorption cycles. Assuming, as a first approximation, that the stress relaxation of the “dry” and “wet” polymers is proportional to the elastic strain, ε_el, empirical calculations were performed and compared to experimental results. In general, the presence of a vapor or gas in a polymer matrix enhances the stress relaxation by softening or plasticizing the polymer and transforming a portion of the elastic strain, ε_el, into the plastic strain, ε_pl. As the transformation continues, the sorption and stress of the wetted elongated sample change simultaneously with time until an optical “overshoot” and a mechanical or stress “undershoot” is observed. This result seems to be the consequence of the differential change of the stress of the “dry” and “wetted” samples with sorption time, τ = t ? t_o(t = time; t_o = initial sorption time), which depends on the differential time dependencies of the transformations of the elastic and plastic strains.     

On the non‐newtonian viscous behavior of polymer melts in terms of temperature and pressure‐dependent hole fraction

A new theoretical non‐Newtonian viscosity model is developed by taking the fractional series expansion of Eyring's shearing strain rate. A broad range of experimental rheological data of various polymer melts including polyethylenes, polypropylene, polystyrene, poly (methyl methacrylate), and polycarbonate are fitted well using the proposed model. From the model; zero shear, constant shear‐stress and constant shear‐rate viscosities are derived as a linear function of viscosity related quantity, Y_h, called “thermo‐occupancy function” and their coefficients are discussed in detail. The thermo‐occupancy function is expressed in terms of temperature and structural vacancies such as hole fraction computed from the Simha‐Somcynsky Hole Theory (SS). In addition, the derivative of the logarithm of viscosities with respect to the hole fraction, named as viscoholibility, is observed decreases with the increasing hole fraction. © 2014 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2014 , 131, 40540.     

Shear modulus in closely packed gel suspensions

Rheological properties and swelling were examined in a series of concentrations of particles of crosslinked polyacrylate gels in water or salt solutions. The modulus during steady shear G_s = 2τ²/P₁₁-P₂₂ was determined from shear stress τ and primary normal force difference P₁₁-P₂₂ in a cone-and-plate rheometer. G_s was nearly constant with shear rate for the gel particles in the closely packed condition. The dynamic storage modulus G′ determined by ecentric rotating disc rheometry increased with increasing frequency for all concentrations. The apparent equilibrium shear modulus G_e determined by stress relaxation agreed closely at all concentrations and ionic strengths with the corresponding values of G_s, and hence G_s is considered a good estimate of equilibrium shear modulus for this gel material.     

Molecular weight distribution and rheological properties of polyisobutylene solutions

The molecular weight distribution of a series of polyisobutylenes was determined using osmotic pressure measurements, gel permeation chromatography, and intrinsic viscosity. All of the polymers except for one, a blend of the highest and lowest molecular weight constituents, had similar moderate molecular weight distributions. The “extended chain length” method of calibrating the gel permeation chromatograph for polyisobutylenes was found to be effective. Steady state and transient shear stresses and normal stresses were measured on 5% decalin solutions of these polymers. The zero shear viscosity increased with the 3.3 power of molecular weight, and the zero shear normal stress coefficient (σ₁₁ ? σ₂₂)/Γ² varied with the 7.5 power. Relative elastic memory as measured by (σ₁₁ ? σ₂₂)/σ₁₂ or stress relaxation increased with increasing molecular weight (and at constant number- or weight-average molecular weight) with breadth of distribution. Stress overshoot also correlated with this tendency.     

A Survey of Rheological Properties of One-Component Epoxy Adhesives

Flow characteristics of seven commercially available one-component epoxy adhesive pastes were measured using a controlled shear stress rheometer and a controlled shear rate rheometer over a temperature range from 5°C to 60°C. Combining data obtained from both controlled rate and controlled stress experiments over a wide range of shear rates, we observed Newtonian flow (shear stress proportional to shear rate) at very low shear rates, a plateau “shear thinning” region at intermediate shear rates, and a second region of linear dependence of shear stress on shear rate at high shear rates. The adhesive pastes exhibited a very broad range of rheological behavior. Two flow parameters important to adhesive application technology, the plastic viscosity and the apparent yield stress, were measured for each adhesive. The plastic viscosity ranged from 11.6 to 329.5 Pa. s; the apparent yield stress ranged from 56.2 to 413 Pa. The temperature dependence of the rheological parameters of the epoxy adhesive pastes was also determined. The results are reported as the activation energies, E_η and E_σ , of plastic viscosity and apparent yield stress, respectively. The apparent yield stress of each adhesive paste was much less sensitive to changes in temperature than was the plastic viscosity. This suggests that the processing characteristics are likely to show qualitative as well as quantitative changes with temperature.     

A model relating the elastic properties of high-density polyethylene melts to the molecular weight distribution

An earlier model relating the variation of the steady-shear melt viscosity of high-density polyethylene to the molecular weight distribution is applied toward predicting the steady-shear elastic compliance, the first normal stress difference, and relaxation spectrum as a function of shear rate from the molecular weight distribution. The model envisions the cutting off of longer relaxation times as the shear rate is raised such that at any shear rate ${\rm \dot \gamma }$ the molecular weights and their corresponding maximum relaxation times τ_m are partitioned into two classes; the relaxation times are partitioned into operative and inoperative states, depending on whether they are less than or greater than τ_c, the maximum relaxation time allowed at ${\rm \dot \gamma }$. Equations relating molecular weight and relaxation time to the steady-shear elastic compliance and viscosity are assumed valid at nonzero shear rates, except for the partitioning effect of shear rate. The shear rate dependence of the first normal stress difference and the steady-shear viscosity for polyethylene melts is successfully predicted over the range covered by the cone-and-plate viscometer. The assumed proportionality constant between τ_c and 1/${\rm \dot \gamma }$ was determined to be 1.7. Using this relation, the maximum relaxation time at 190°C for a polyethylene molecule of molecular weight M is given by τ_m = 1.4 × 10^?19 (M)^3.33. Reasonable agreement has been obtained between the experimentally determined relaxation spectrum of a polyethylene melt and that predicted from the molecular weight distribution. The agreement is best at the longest relaxation times.     

LOW-SHEAR-RATE RHEOMETRY AND POLYMER QUALITY CONTROL

Possibilities for improved quality control of polymerization reactions arise from the measurement of N1 (the first normal stress difference) as well as the viscosity η. At low shear rates in particular, N₁ and η are associated with different regions of the relaxation spectrum H(τ); and cannot, in general, be calculated from one another. Above a certain polymer concentration, N₁ and η give information about two very different averages of the molecular weight distribution, with N₁ being remarkably sensitive to small changes in the high molecular weight “tail” of the distribution. By considering usage of these and other rheological measurements, together with ranges of shear rate or frequency, a tentative qualitative classification of different levels of rheological discrimination is suggested. Choice of on-line rheological sensor could be influenced in the first instance by choice of level

One on-line sensor — a “Stressmeter” — is discussed in detail. Evidence is given to show that this discriminates well between viscous and normal stress properties, and that, for a number of liquids tested, actual values of normal stress differences can be obtained to a useful degree of approximation. Such data for a wide range of liquids are given, including what are thought to be the first published normal stress data on polyethylene solutions and gelatin solutions.     

Maxwell Fluid Model for Generation of Stress–Strain Curves of Viscoelastic Solid Rocket Propellants

Solid rocket propellants are modeled as Maxwell Fluid with single spring and single dashpot in series. Complete stress–strain curve is generated for case‐bonded composite propellant formulations by taking suitable values of spring constant and damping coefficient. Propellants from same lot are tested at different strain rate. It is observed that change in spring constant, representing elastic part is very small with strain rate but damping constant varies significantly with variation in strain rate. For a typical propellant formulation, when strain rate is varied from 0.00037 to 0.185 per second, spring constant (K) changed from 5.5 to 7.9 MPa, but damping coefficient (D) varied from 1400 to 4 MPas. For all strain rates, stress–strain curve is generated using developed Maxwell model and close matching with actual test curve is observed. This indicates validity of Maxwell fluid model for case‐bonded solid propellant formulations. It is observed that with increases in strain rate, spring constant increases but damping coefficient decreases representing solid rocket propellant as a true viscoelastic material. It is also established that at higher strain rate, damping coefficient becomes negligible as compared to spring constant. It is also observed that variation of spring constant is logarithmic with strain rate and that of damping coefficient follows a power law. The correlation coefficients are introduced to ascertain spring constants and damping coefficients at any strain rate from that at a reference strain rate. Correlation for spring constant needs a coefficient “H,” which is function of propellant formulation alone and not of test conditions and the equation developed is K₂=(K₁‐H)×{ln(dε₂/dt)/ln(dε₁/dt)}+H. Similarly for damping coefficient (D) also another constant “S” is introduced and prediction formula is given by D₂=D₁×{(dε₂/dt)/(dε₁/dt)}^S. Evaluating constants “H” and “S” at different strain rates validate this mathematical formulation for different propellant formulations. Close matching of test and predicted stress–strain curve indicates propellant behavior as viscoelastic Maxwell Fluid. Uniqueness of approach is to predict complete stress–strain curves, which are not attempted by any other researchers.     

Superposition of small strains on large deformations as a probe of nonlinear response in polymers

The incremental response ΔG(t) obtained by superposing a deformation, Δγ, on a large deformation, γ₁, has been determined in step shear experiments for a polyisobutylene solution and for a poly(methylmethacrylate) glass in torsion. For both systems ΔG(t) at γ₁ was found to be smaller than the linear viscoelastic modulus, G(t), at zero prestrain. ΔG(t) was found to increase with increasing time, t_e, after imposition of the large deformation. It was also observed that the “apparent relaxation spectrum” associated with ΔG(t) narrows and shifts to shorter times when compared to the spectrum associated with the linear viscoelastic modulus, ΔG(t). The results for the solution art-well described by the nonlinear constitutive equation of the BKZ elastic fluid theory. It is found that ΔG(t) for the glass falls between the behavior predicted by the BKZ theory and the linear viscoelastic behavior.     

Characterization of the aging behavior of raw epoxidized natural rubber with a rubber processing analyzer

Tests of the strain sweep, frequency sweep, and stress relaxation for raw epoxidized natural rubber were carried out with a rubber processing analyzer. The results showed that the complex viscosity, η*, decreased with the prolongation of the aging time in the region of Newtonian flow, but in the region of non‐Newtonian flow, the decrement of η* with a rising shear rate decreased with the prolongation of the aging time. The torque (S′) response from the strain sweep indicated that aging brought about an obvious decrease in the increment of S′ with rising strain in the linear viscoelastic region and a small increase in the slope of the plateau on the curve of the S′ response in the nonlinear viscoelastic region. The stress relaxation rate constants k and b, calculated according to the equations S_t = S₀e^?kt and S_t = S₁t^?b (where S_t, S₀, and S₁ are the stresses at relaxation time t, t = 0, and t = 1, respectively), increased, and the stress relaxation time obtained directly from the rubber processing analyzer shortened with the prolongation of the aging time. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 100: 1277–1281, 2006     

Stress–strain behavior of SAN/glass bead composites above the glass transition temperature

Relaxation and stress–strain behavior of SAN–glass bead composites are studied above the glass transition temperature. The strain imposed on the polymeric matrix of the composite is defined as ?_p = ?_c/(1 ? ?^?). Stress relaxation data for the filled polymer which is independent of strain can be calculated by multiplying the relaxation modulus (at a certain strain) by (1 + ?_p). Stress–strain curves at constant strain rate and for different concentrations of the filler can be shifted to form a master curve independent of filler content if the tensile stress is plotted versus ?^p. The relaxation modulus increases with increasing the filler concentration and can be predicted by a modified Kerner equation at 110°C.     

The onset of nonlinear viscoelasticity in multiaxial creep of glassy polymers: A constitutive model and its application to PMMA

A physically based, isostructural, constitutive model is described for simulating the onset of nonlinear viscoelasticity in multiaxial creep of glassy polymers, as needed in stress analyses of load-bearing components. In the linear viscoelastic limit, shear response reduces to that of a generalized Maxwell model, while hydrostatic response is Hookean. Nonlinearity enters through Eyring-type rate process kinetics. The equations of the model are solved numerically using a pseudo-linear approximation through each time step, leading to an incremental equation for stress that would be convenient for use in finite element analyses. The model and its assumptions were tested using tension, shear and combined tension/shear creep experiments on well-aged poly(methyl methacrylate) at 70°C. Reproducibility tests confirmed the assumption of constant glass structure for strains up to ～ 1.5 × 10^?2. Shear and pressure activation volumes were obtained by fitting the dependence of the shear compliance on stress invariants. The data showed unequivocally that shear activation volumes vary with log(relaxation time), and excellent agreement was obtained for a linear variation, consistent with the “compensation rule” of polymer thermo-viscoelasticity. The activation volumes are large (many monome units), indicating the cooperative nature of the elementary flow process. Interestingly, they are of the same order as those applying to yield and plastic flow. Although the model finds success in simulating creep, it fails to describe so accurately the strain recovery on unloading. Possible explanations are suggested.     

Mechanical degradation of high molecular weight polymers in dilute solution

The mechanical shear degradation of polydisperse polyisobutene and monodisperse polystrene in oils of different viscosities in the concentration range of 0.1% to 1% was studied using a high-shear concentric cylinder viscometer under laminar and uniform well-defined shear field conditions. Molecular weight distributions (MWDs) were measured by gel permeation chromatography (GPC). Degradation of polydisperse polyisobutene solutions narrows the distributions principally through the breaking down of large molecules. Degradation of monodisperse polystyrene broadens the distributions at lower shear stress. At higher shear stresses, the distributions do not broaden as much but are still broader than those of the original polymer. The final M_w/M_n ratios are considerably different from the value of 2 expected for random degradation. Hence, the degradation is likely a nonrandom process. It was found that the extent of degradation has a negative concentration dependence coefficient at relatively high molecular weight and a positive concentration dependence at lower molecular weight. Competing mechanisms of “stretching” and “entanglements” for degradation were postulated to explain the results. The degradation data indicate that the shear stress is the controlling parameter, not the shear rate. The shear degradation is independent of initial molecular weight and viscosity of the solvent.     

Tensile stress measurements on linear and branches low-density polyethylene melts. Interpretation of results with a generalized Maxwell model

Dynamic shear experiments in the linear range of deformation and extensional tests at constant strain rate have been carried out on a linear low-density polyethylene (LLDPE) melt and on two branched low-density polyethylene (LDPE) melts with different amounts of long-chain branching. Both the dynamic shear moduli and the tensile stress obey the time–temperature superposition principle. A simple model based on a nonaffine generalized Maxwell model with two relaxation times is proposed to describe the rheological behavior in elongation of these melts. Close agreement between the model and the experimental data can be obtained by adjusting the two relaxation times and the “slip parameter” of entanglements. The variations of these parameters with strain rate and their relationship with molecular structure are discussed.     

Temperature scanning stress relaxation behavior of water responsive and mechanically adaptive elastomer nanocomposites

The decrease of stress at constant strain, that is, the stress relaxation process as a function of temperature, is a central mechanical characteristics of elastomer nanocomposites for their potential applications. However, in the conventional stress relaxation test, the relaxation behavior is usually determined as a function of time at constant temperature. The present work reports the temperature scanning stress relaxation (TSSR) characteristics of a new kind of mechanically adaptive elastomer nanocomposite by monitoring the nonisothermal relaxation behavior as a function of temperature. This kind of adaptive elastomer nanocomposite was prepared by introducing calcium sulfate (CaSO₄), as the water-responsive phase into the hydrophilic elastomer matrix. The influence of water-induced structural changes on TSSR behavior was investigated. Water treatment had a strong effect on the shape of the relaxation spectrum of the nanocomposite. It was revealed that the in situ development of hydrated nano-rod crystal structures of CaSO₄ in the elastomer matrix was responsible for the changes in the mechanical relaxation behavior of the composites. Atomic force microscopy was used to verify this nano-rod crystal morphology in the elastomer matrix. The mechanism of water-induced mechanical reinforcement of the composite was explored from dynamic mechanical analysis of the material and correlated with its stress relaxation behavior. © 2019 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2020 , 137, 48344.     

Statistical entropy model for relaxation in stressed polymer glasses

The molecular kinetic theory near the glass transition, bused on the existence of free volume distribution, is extended to incorporate the effects of stress and stress rate. The fundamental equations for the volume relaxation and recovery in stressed amorphous polymers are derived in accordance with the balance of nonequilibrium statistical entropy. Using these kinetic equations, an earlier nonequilibrium criterion for the glass transition temperature, T_g, is generalized to include the effects of stress and stress rate. In contrast to the prevalent thinking toward free volume theories, an explicit expression between T_g and stress is developed and reveals that T_g does not continue to increase at all pressures but levels off to a “universal” asymptote at very high pressure (>10 K bars). The expression is applicable to any tension and compression stress conditions. A comparison between theory and experiment under constant stresses determines the activation volume tensor which reveals the molecular mechanism relating T_g and the plastic yield of glassy polymers.     

First normal stress difference in capillary extrusion flow of nanometer calcium carbonate‐filled PLLA biocomposites

The nanometer calcium carbonate (nano‐CaCO₃)‐filled poly‐L ‐lactide (PLLA) biocomposites were prepared using a twin‐screw extruder. The first normal stress difference of the composites were measured by means of a capillary rheometer under experimental conditions with temperatures ranging from 170 to 200°C and shear rates varying from 50 to 10³ s^?1. The first normal stress difference (N₁) increased roughly linearly with increasing shear stress (τ_w). The sensitivity of the N₁ to τ_w increased with an increase of the die length–‐diameter ratio, and the N₁ value varied slightly with the filler weight fraction (?_f) as test temperature was constant. When the shear stress was fixed, the N₁ reached a minimum value for ?_f = 1%. The values of the N₁ of the composite melts decreased roughly linearly with a rise of temperature when the shear rate was constant. POLYM. ENG. SCI., 2013. © 2012 Society of Plastics Engineers