Equilibrium stress-relaxation

The method (27) can best be explained with reference to Figure 2. After stretching to 10, the force f is measured as a function of time. The strain is kept constant throughout the entire experiment. At a certain time, the sample is quenched to a temperature well below the glass-transition temperature, Tg, and cross-linked. Then the temperature is raised to the relaxation temperature, and the equilibrium force is determined. A direct comparison of the equilibrium force to the non-equilibrium stress-relaxation force can then be made. The experimental set-up is shown in Figure 4. 

A new stress-relaxation two-network method is used for a more direct measurement of the equilibrium elastic contribution of chain entangling in highly cross-linked 1,2-polybutadiene. The new method shows clearly, without the need of any theory, that the equilibrium contribution is equal to the non-equilibrium stress-relaxation modulus of the uncross-linked polymer immediately prior to cross-linking. The new method also directly confirms six of the eight assumptions required for the original two-network method. 

Stresses cannot relax completely any more. The upper curves in Fig. 2 show this curving to the right, where at long times an equilibrium stress level will eventually be reached. More data at longer times would be required in order to clearly identify the value of Ge. However, Ge can be estimated from the curves,... 

Equilibrium stress-strain dependences were determined in extension using a stress relaxation arrangement described earlier (21). Dry non-extracted samples were measured at 150 C in nitrogen atmosphere and extracted samples swollen in dimethylformamide were measured at 25 C. The equilibrium value of stress 6 e was reached within 2-4 min except of a few dry samples with the lowest tig, for which the equilibrium stress was determined using an extrapolation procedure described earlier (21). 

A8. The Helmholtz elastic free energy relation of the composite network contains a separate term for each of the two networks as in eq. 5. However, the precise mathematical form of the strain dependence is not critical at small deformations. Although all the assumptions seem to be reasonably fulfilled, a simpler method, which would require fewer assumptions, would obviously be desirable. A simpler method can be used if we just want to compare the equilibrium contribution from chain engangling in the cross-linked polymer to the stress-relaxation modulus of the uncross-linked polymer. The new method is described in Part 3. 

The choice of exposure times is complicated when more than one reaction is taking place as the rates will be different. Generally, it is best if equilibrium absorption is reached relatively quickly in comparison to chemical changes and its effects treated separately from the subsequent chemical changes. This is not unlike the situation for physical and chemical stress relaxation. 

An equilibrium model may not be representative of the true situation commonly faced in the laboratory. The relaxation behaviour of the samples becomes progressively longer with increasing volume fraction. It is quite reasonable to suppose that, at high particle concentrations and low electrolyte concentrations, the relaxation times become so long that it is impractical to allow all the stresses and strains to relax from the sample prior to measurement. Stress relaxation studies for a range of particles that show nearly complete relaxation is shown in Figure 5.16.21... 

To the best of our knowledge, the supercoil conformation of the monoden-dron jacketed polystyrene is one of the first observations of a defined tertiary structure in synthetic polymers. The plectoneme conformation could be caused by underwinding or overwinding of a backbone from its equilibrium state [168]. Quick evaporation of the solvent certainly can cause a residual torsion in the molecule as it contracted in itself. Unlike macroconformations of biomolecules, where the tertiary structures are often stabilized by specific interactions between side groups, the supercoil of the monodendron jacketed polymers is metastable. Eventually, annealing offered a path for the stress relaxation and allowed the structural defects to heal [86]. 

The rheology of many of the systems displayed gel-like viscoelastic features, especially for the long-range attractive interaction potentials, which manifested a non-zero plateau in the shear stress relaxation function, C/t), the so-called equilibrium modulus, which has been considered to be a useful indicator of the presence of a gel. The infinite frequency shear rigidity modulus, was extremely sensitive to the form of the potential. Despite being the most short-... 

A good diagnostic for creep and stress relaxation tests is to plot them on the same scales as a function of either compliance (J) or modulus (G), respectively. If the curves superimpose, then all the data collected is in the linear region. As the sample is overtaxed, the curves will no longer superimpose and some flow is said to have occurred. These data can still be useful as a part of equilibrium flow. The viscosity data from the steady-state part of the response are calculated and used to build the complete flow curve (see equilibrium flow test in unit hi.2). 

The adsorption and desorption kinetics of surfactants, such as food emulsifiers, can be measured by the stress relaxation method [4]. In this, a "clean" interface, devoid of surfactants, is first formed by rapidly expanding a new drop to the desired size and, then, this size is maintained and the capillary pressure is monitored. Figure 2 shows experimental relaxation data for a dodecane/ aq. Brij 58 surfactant solution interface, at a concentration below the CMC. An initial rapid relaxation process is followed by a slower relaxation prior to achieving the equilibrium IFT. Initially, the IFT is high, - close to the IFT between the pure solvents. Then, the tension decreases because surfactants diffuse to the interface and adsorb, eventually reaching the equilibrium value. The data provide key information about the diffusion and adsorption kinetics of the surfactants, such as emulsifiers or proteins. 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

Figure 7.3. Determination of elastic and viscous components. Incremental stress-strain curve constructed by stretching a specimen in strain increments of 2 to 5% and allowing the specimen to relax to an equilibrium stress before an additional strain increment is added. The elastic fraction is defined as the equilibrium stress divided by the initial stress. (Adapted from Silver, 1987.)...

Measurements. The morphology of the blends was studied by optical microscopy (Leitz Dialux Pol), transmission electron microscopy (Jeol 100 U), and scanning electron microscopy (Cambridge MK II). Ultramicrotome sections were made with an LKB Ultratome III. Samples for scanning electron microscopy were obtained by fracturing sheets at low temperature. The fracture surfaces were etched with a 30% potassium hydroxide solution to hydrolyse the polycarbonate phase. Stress-relaxation and tensile stress-strain experiments were performed with an Instron testing machine equipped with a thermostatic chamber. Relaxation measurements were carried out in flexion (E > 108 dyn/cm2) or in traction (E < 108 dyn/cm2). Prior to each experiment, the samples were annealed to obtain volumetric equilibrium. 

The origin of the phenomenon of ageing lies in the fact that glasses are not in thermodynamic equilibrium their volume and entropy are too large, hence there is a tendency to volume reduction (volume retardation). Decreases of rates of stress relaxation and creep are consequences of this phenomenon. 

At equilibrium and at a particular RVP, the amount of adsorbed water held by a cellulose generally will be greater if it has been obtained following desorption from a higher RVP and not by adsorption from a lower RVP. The cause of this hysteresis is not fully established [303]. One explanation is based on the internal forces generated when dry cellulose swells, limiting the amount of moisture adsorbed whereas when swollen cellulose shrinks, stress relaxation occurs since the cellulose is plastic and permits a higher uptake of moisture. 

The equilibrium 20 is important not only in the synthesis of linear polysiloxanes but also in their applications. The effects of water vapor on inducing chain cleavage at high temperature are not only reduced molecular weights but also a dramatic increase in the rates of chemically induced stress relaxation at 250 °C in cross-linked poly(dimethylsiloxane) networks under load (70). Slow hydrolytic bond cleavage in cross-linked networks is seen even in studies of stress relaxation in air at room temperature, and appreciable rates of stress relaxation in the loaded networks are measured at temperatures as low as 70 (7i). The stress relaxation is greatly accelerated... 

On the other hand the nature of the retractive forces in the yield and post-yield regions has been the subject of much controversy. Bull (1945), Woods (1946a,b), Astbury (1947), Elod and Zahn (1949a), and Breuer (1962) have concluded from the effects of temperature on the retractive forces that entropy contributes very little to retractive forces at strains up to 30 %. Meyer and Haselbach (1949) and Meyer et al. (1952), however, consider that the fibers must reach an equilibrium condition before measurements are made and conclude that the forces are entirely entropic. There can be no doubt that after stress relaxation at high temperatures the residual force is largely entropic (Feughelman and Mitchell, 1959), but this force is only a fraction of the initial force. 

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