Stress Relaxation Modulus Research Articles

Stokes' first problem for linear viscoelastic fluids with finite memory

We consider Stokes' first problem for a viscoelastic fluid. The memory of the fluid is truncated to a finite time interval and discontinuities in the stress relaxation modulus or its derivatives are allowed at the point of truncation. We investigate secondary waves which are generated by the interaction of these singularities in the memory with earlier waves.

Functional form for the stress relaxation moduli of binary blends consisting of nearly monodisperse polymers and the effect of tube size enlargement. 1

Description detaillee de la forme fonctionnelle pour le module de relaxation de contrainte (theorie binaire) precedemment proposee brievement. Bon accord entre les predictions de la theorie avec les resultats de l'analyse. Discussion des relations entre la theorie monodispersee et la theorie binaire

Nonseparable Behavior in Rubber Viscoelasticity

Abstract New large-strain rubber viscoelasticity results for a filled and an unfilled IIR vulcanizate and previously published results for two NR gum vulcanizates have been discussed. The data show that the “mixed” response functions of large-strain stress relaxation, and the incremental storage and relaxation moduli do not demonstrate factorizability of time and strain effects. This is a consequence of the elastic and relaxation contributions in each of the mixed functions being different. The incremental dynamic data also show that the loss modulus for the filled IIR and unfilled NR vulcanizates (unavailable for the unfilled IIR) are separable functions of time and strain. This directly shows that the relaxation spectra for the filled IIR and unfilled NR vulcanizates are independent of strain for the deformations studied. In fact, it is argued that a necessary and sufficient condition for the relaxation spectrum to be independent of strain is that the loss modulus is a factorizable function of time and strain effects. The quantitative success of the Generalized Solid-Liquid (GSL) model in representing the viscoelastic behavior of the gum NR vulcanizate has been reviewed. Although the GSL model applies only to unfilled vulcanizates, it has also been successfully used to qualitatively interpret the results for the filled IIR compounds. Both successes are attributed to the physical assumptions intrinsic to the GSL model; more specifically, 1) the relaxation spectrum is independent of the state of strain, and 2) the deformational dependences of elastic and relaxation contributions to the overall response of the system need not be the same. Physical arguments justifying these assumptions have been covered. It has also been shown with the aid of the GSL model, that a material might exist which demonstrates factorizability in stress relaxation and incremental loss modulus behaviors but nonfactorizability in the incremental storage and relaxation moduli.

Melt Rheology of Two Engineering Thermoplastics: Poly(ether Imide) and Poly(2,6‐Dimethyl‐1,4‐phenylene Ether)

Material functions of two engineering plastics [a poly(phenylene ether) and a poly(ether imide)] were characterized, including the shear viscosity, first normal stress coefficient, storage and loss moduli, growth and relaxation of shear stress, and first normal stress coefficient and relaxation moduli. The oscillatory shear and relaxation moduli data were employed to determine the temperature‐dependent parameters of Wagner model. Various material functions, which were determined on the basis of this model in conjunction with the fitted parameters agreed reasonably well with the experimental results. The reported data and parameters should facilitate a better understanding of the processability characteristics of these two engineering plastics.

Chain entanglements. II. Numerical results

Numerical methods are described for the calculation of time correlation functions and the stress relaxation moduli for the model developed in the preceding paper [J. Chem. Phys. 89, 3892 (1988)]. Illustrative results are presented for the storage and loss modulii, the shear compliance, and the translational diffusion constant, with parameters appropriate to polybutadiene.

Relaxation spectra and viscoelastic properties of the binary linear flexible polymers

A binary blending law was proposed for a mixture which was composed of two narrowly distributed linear polymer fractions with different molecular weights and the high entanglement density. The relaxation spectrum was used to express the binary blending law and it was derived from the chain relaxation processes based on tube model theories as well as from the earlier observations on the viscoelasticity of linear polymers with bimodal molecular weight distribution. The proposed law showed good agreements with experimental data over a wide range of time or frequency for the viscoelastic properties such as shear stress relaxation modulus and dynamic moduli. It also showed good-fitted results for the compositional dependences of some viscoelastic constants such as zero-shear viscosity and recoverable compliance.

Characterization of polymer molecular weight distribution by transient viscoelasticity: Polytetrafluoroethylenes

AbstractThe relationship between the relaxation time spectrum H(τ) in the terminal zone and the volume‐fraction differential molecular‐weight‐distribution function P(M) is derived by considering binary chain contacts for stress transmission, equation image where β and λ are constants for a given chemical type. This is used to determine the molecular‐weight‐distribution curves from the stress relaxation modulus spectrum (above the crystal melting point) at 370°C for a number of commercial and experimental poly(tetrafluoroethylenes) (PTFEs). It is found that PTFEs typically have bimodal molecular‐weight distributions. The lower‐molecular‐weight peak conforms essentially to the “most‐probable” distribution, and the higher‐molecular‐weight peak to the binary coupling distribution. The entanglement molecular weight Me is 5490, and the number of main‐chain atoms between entanglement points is 110, consistent with a flexible chain. The zero‐shear melt viscosity at 370°C is η0 = 1.79 × 10−13 Mw2.94±0.13, where η0 is in Pa.s and Mw/Me = 2,000 to 12,000. The monomeric friction coefficient is also determined.

Physical Ageing and Annealing in Fibers and Textile Materials

The static correspondence principle for viscoelastic composites is used to study the ageing and viscoelastic properties of the wool keratin composite and wool textile as semblies. The equilibration process of ageing before or during stress relaxation or creep tests changes the mechanical properties of the materials. In a short-term test, the ageing that occurs during the test is negligible. Increasing the ageing time of the material before mechanical testing causes a horizontal shift of the stress relaxation or creep curves in the logarithmic time scale. The effect of physical ageing on the me chanical properties of the material can therefore be represented by a single time shift factor. In a long-term test, the ageing processes that occur during the test have a pronounced influence on the mechanical properties. Therefore, the effect of ageing both before and during testing needs to be taken into account. An effective time can be used to describe the change of mechanical properties due to the ageing process. The conclusion is independent of the geometrical configuration and the stress relaxation moduli of constitutent phases. The relationship between the various ageing processes in static and dynamic tests is also discussed.

Viscoelastic Relaxation in Glassy Polymers

AbstractThe nonequilibrium mechanism controlling the viscoelastic response in the glassy state is analyzed in accordance with our multiple hole energy model. The change in the nonequilibrium glassy state defines the Struik physical aging exponent and determines the motion of the relaxation time spectrum, the transition of the global shift factor and the slow decay of the stress relaxation modulus especially at longer times in the glassy state where the Kohlrausch-Williams-Watts function fails. Although the distribution of relaxation times has its time scales shifted with the aging time, the shape of the spectrum remains very much the same which supports the notion of thermo-aging-rheological simplicity in the glassy and transition states.

Stress Relaxation and Differential Dynamic Modulus of Polyisobutylene in Large Shearing Deformations

Stress relaxation in shear has been measured at 25°C for a polyisobutylene sample with viscosity average molecular weight 7.7×105, at shear strains γ from 0.2 to 0.5, over a time scale from 102 to 104.5 s. In this range, the relaxation modulus G(γ;t) could be expressed by the product G(0;t)h(γ); at γ=0.5, h=0.76. Simultaneous measurements of the differential storage and loss shear moduli, G′(ω,γ;t) and G″(ω,γ;t) were made throughout the relaxation process by intermittently superposing small oscillating deformations with a maximum additional strain of 0.01, at a frequency of about 0.33 Hz, which falls in the middle of the plateau zone of the polymer. For γ=0.2, G′ and G″ remained unchanged from their values at zero static strain, G′(ω,0) and G″(ω,0) respectively, confirming that the density of entanglements (or topological obstacles, or temporary network junctions), remains constant throughout the relaxation process at small strains. At higher strains, G″ was nearly constant, but G′ was somewhat smaller at the first measurement after imposition of static strain and slowly recovered to its original zero-strain value. The behavior could be qualitatively interpreted by the Doi theory which includes both an equilibration process characterized by a time τe and a disengagement process characterized by a time τd; the decrease in G′ would correspond to loss of entanglements by the contraction of stretched molecules during equilibration and the subsequent increase to reestablishment of new entanglements by reptation during disengagement.

The continuous‐time random walk description of the non‐equilibrium mechanical response of crosslinked elastomers

AbstractThe application of the continuous‐time random walk model to the description of the non‐equilibrium stress response of a crosslinked network is discussed. Different versions of the continuous‐time random walk model lead to the prediction of a stress relaxation function having either a fractional‐power law form or a fractional‐exponential law form. Both forms are tested against experimental data on the stress relaxation and loss modulus behaviours of crosslinked natural rubber.

Stress–relaxation and stress–strain behavior of poly(ethylene‐co‐vinylacetate) at varying crosslink density

AbstractThe stress–relaxation modulus of an unfilled ethylene vinyl acetate polymer at three different degrees of crosslinking was measured at 15 temperatures over a temperature span of about 160°C. At each temperature, the time response was measured for at least 3 decades of time. From these data it was possible to construct a master curve for each degree of crosslinking. The time–temperature shift factors, aT, were found to be related to temperature by the relation log aT = −A(T–273), where the parameter A has an average value of 0.234 for the three materials.

Stress Relaxation and Differential Dynamic Modulus of Carbon Black-Filled Styrene-Butadiene Rubber in Large Shearing Deformations

Abstract Combined measurements of shear stress relaxation and differential dynamic storage and loss shear moduli G′ and G″ have been made on styrene-butadiene rubber (type 1502) containing 50 phr N299 carbon black and 10 phr Sundex 790 oil, both cured and uncured, and compared with similar measurements on the cured and uncured gum rubber. The range of temperature was −22.5° to 63°C, and of static shear strain 0.01 to 0.40; the maximum oscillatory shear strain was about 0.005 and the frequency was usually 0.64 Hz. Dynamic measurements of G′ and G″ with no superposed static strain and stress relaxation measurements at small strains on the filled samples could be reduced by frequency-temperature superposition with αT shift factors that were somewhat larger than those that applied to the raw or crosslinked gum. At about 0.6 Hz, the filler increased the storage modulus by about a factor of 5 for the cured sample and about 10 for the uncured; the moduli for the cured and uncured filled samples were almost identical. The order of rates of relaxation was unfilled uncured > filled uncured > filled cured > unfilled cured. At small static strains, the stress relaxation of the filled samples (both uncured and cured) was substantial, but the differential dynamic moduli G′ and G″ remained nearly equal to their values with zero static strain throughout the relaxation process. With increasing static strain γ, the strain-dependent relaxation modulus G(γ;t) decreased by as much as a factor of 3 for the filled, cured sample and 4 for the filled, uncured sample. The modulus G(γ;t) could not be factored into a strain-dependent and a time-dependent function. During stress relaxation of the filled samples (both cured and uncured) at large strains, the differential storage modulus G′(ω,γ;t) experienced a drop followed by slow recovery toward its initial value at 25°C; the loss modulus G″(ω,γ;t) was unchanged. For the filled, cured sample, complete recovery to the initial value G′(ω,0) (where 0 refers to zero static strain) was accomplished by removing the shearing stress and heating for one day at 63° followed by return of the temperature to 25°C. The results are interpreted in terms of a structure which combines a crosslinked (or, in the case of uncured, an entangled) polymer network and a network of carbon black particles. At small strains, stress relaxation is thought to be accomplished primarily by rearrangements of carbon particles and/or polymer molecules bridging them, without structural damage in the sense that the rearranged structure has the same properties as the original. These rearrangements are impeded by crosslinking in the cured vis-a-vis the uncured filled rubber. The kinetics of rearrangement may be governed by configurational changes of the polymer molecules since the temperature shift factors do not differ greatly from those for the gum rubber. At large strains, the particle network can be damaged but can regain its structure by a healing process which is accelerated at higher temperatures. The conclusions apply to the particular compound studied here and not necessarily to other rubber-black compounds, which according to the literature show great diversity of properties.

Rheology of Fibrin Clots. VI. Stress Relaxation, Creep, and Differential Dynamic Modulus of Fine Clots in Large Shearing Deformations

A fine, unligated clot of human or bovine fibrin prepared from purified fibrinogen is subjected to a large torsional deformation (maximum shear strain γ up to 1.37) with superposed small oscillating deformations (Δγ ca. 0.03) at frequencies from 0.2 to 1 Hz. The “secant modulus” is defined as Gi(γ)=σi/γ, where σ is stress and the subscript i refers to an initial measurement about 25 s after imposition of strain. The storage modulus G′(ω,γ) refers to a differential oscillating measurement at frequency ω superposed on a static strain γ. For γ up to about 0.1, Gi(γ) was independent of γ and equal to G′(ω,0) measured at about 1 Hz and zero static strain. At large static deformations, the differential storage modulus G′(ω,γ) could be used to monitor changes in structure. Since there is very little time dependence of the relaxation modulus in the range from 1 to 60 s, and the loss tangent is very small, Gi′(ω,γ) could be considered simply as the differential modulus Gi(γ)+γdGi/dγ, and agreed with the latter exp...

Effects of phase separation on network and viscoelastic properties in SBS thermoplastic elastomers

Stress relaxation of poly(styrene-b-butadiene-b-styrene) thermoplastic elastomers is studied in dependence of molecular weight and degree of hydrogenation in the temperature range between −30° and +80 °C. The influence of these parameters on the structure of the physical network and the degree of partial mixing in the domain boundary is investigated by separating the stress-relaxation modulus into a viscoelastic term and an equilibrium network modulus calculated from the relaxation-time spectrum. The temperature dependence of the one-second relaxation modulus is quantitatively described by use of a modified Kerner model for the simulation of the viscoelastic term. The modification allows the estimation of the volume fraction of interfacial material and its correlation to the parameters which govern phase separation.

Studies on Mechanical properties of Soft Lining Materials

Viscoelastic properties of silicone material (code-named MDX-4-4210) and 4 commercial soft liners and related materials (Kurepate, Neo-Snugger, Super-Soft and Viscogel) were examined using a stress relaxation testing instrument. All measurements were preformed in water at a temperature of 37±0.5°C.In the case of other 4 materials, except Visco-gel, the decreasing rates of relative stress {f(t)/f(0)} with time were relatively small. Furthermore, the decreases of stress relaxation modulus {Er(t)} of these materials with period soaked in water at 37°C were also very small, with exception of Visco-gel. Code-named MDX-4-4210 showed the most superior elastic properties in materials used. The relative stress in Visco-gel diminished markedly with time.These results indicate that all materials, excluding Visco-gel, have possessed the favourable properties as soft liners and related materials since it is necessary for these materials to remain in oral cavity for long period of month.

Analysis of the Tensile Behavior of Wheat Gluten at Constant Strain Rates. The Effect of Secondary Bonding Modification

The effect of secondary bonding on the tensile behavior of gluten at constant strain rate was studied. Freeze‐dried gluten was reconstituted with water, deuterium oxide, and 1M, 3M, and 5M aqueous solutions of urea. The tensile properties were measured at 23±1°C at strain rates from 4.77×10−4 to 4.77×10−1 s−1. Separation of time and strain effects was achieved up to Cauchy strains ranging from 0.43 to 2.86, depending on the medium used to reconstitute gluten. Both the stress relaxation modulus, calculated from the constant strain rate modulus, and the rate of stress relaxation were higher for gluten reconstituted with deuterium oxide than for gluten reconstituted with water. The stress relaxation modulus of gluten reconstituted with 1M urea was 30% lower than that of gluten reconstituted with water but the rate of stress relaxation was the same in both materials. As the concentration of urea increased, the stress relaxation rate decreased progressively while the stress relaxation modulus increased. These results are discussed in terms of changes in conformation and interactions resulting from the partial disruption or strengthening of secondary bonds caused by urea or deuterium oxide, respectively.

Rheological studies of the polymerization of elastomeric impression materials. III. Dynamic stress relaxation modulus.

As part of a detailed study of the curing of a range of dental elastomeric impression materials, the 10-s stress relaxation modulus was measured as a function of the curing time. The influence of the catalyst/base ratio was also investigated and compared with the pregel viscosity previously studied. Using the theory of network formation and an understanding of the chemistry of these materials, the modulus data were interpreted in terms of the polymerization kinetics. The addition silicones exhibited the most abrupt transition between the gel and final set stage, which is close the ideal response. The clinical significant of the results is discussed.

Rheological studies of the polymerization of elastomeric impression materials. I. Network structure of the set state

As part of a detailed study of the curing of a range of dental elastomeric impression materials, their chemistry was investigated. This information, the dependence of the stress relaxation modulus of the set material on the catalyst/base ratios, and the level of diluent were used to elucidate the network structure. Such information is required before the rheological behavior during cure can be understood. The clinical significance of the recommended catalyst/base ratio is discussed.

A Direct Experimental Determination of the Elastic Contribution of Chain Entangling in a Tightly Crosslinked Elastomer

Abstract The experimental result, that the equilibrium force is nearly equal to the pseudoequilibrium force immediately prior to quenching and irradiation, allows the following conclusions: (1) Chain scission during crosslinking is not a serious problem. (2) The network of highly entangled linear chains is effectively at elastic equilibrium immediately prior to crosslinking in the strained state. This would not be the case if the entangled structure remained untrapped. (3) The effect of chain entangling in tightly crosslinked elastomers is large, also at elastic equilibrium. In fact, it is almost quantitatively equal to the pseudo-equilibrium stress relaxation modulus of the uncrosslinked linear polymer. This result is in agreement with the results from the Langley and the two-network methods.