Equilibrium stress

The variations in forces and moments during vibration are given by Equations (5.16) and (5.17). T ie membrane prestress state (equilibrium stress state) is specified by Nj<, Ny, and N. ... 

Direct measurements of equilibrium stress-strain isotherms of SAH are complicated by the gel softness. Nevertheless, a number of experiments on compression and tension of the gels has been reported (see, for example, Refs. [18, 21, 42]). The method of dynamic light scattering is free from such inconveniences... 

These experimental results show conclusively that the deformation factor occurring in the theoretical equation of state offers only a crude approximation to the form of the actual equilibrium stress-strain curve. The reasons behind the observed deviation are not known. It does appear, however, from observations on other rubberlike systems that the type of deviation observed is general. Similar deviations are indicated in TutyP rubber (essentially a cross-linked polyisobutylene) and even in polyamides having network structures and exhibiting rubberlike behavior at high temperatures (see Sec. 4b). 

Similar results have been obtained for natural rubber vulcanized in like manner. Here also the observed equilibrium stress tends... 

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,... 

The equilibrium stress-strain isotherms in elongation, and the swelling ratios in benzene, were measured at 25°C for these networks. Network chain densities calculated from these measurements exceeded the values predicted from stoichiometry. 

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). 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

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. 

Some interesting aspects of the interface kinetics appear only when temperature and latent heat are included into the model, if the process of heat conductivity is governed by a classical Fourier law, the entropy balance equation takes the form Ts,= + x w where s = - df dr. Suppose for simplicity that equilibrium stress is cubic in strain and linear in temperature and assume that specific heat at fixed strain is constant. Then in nondimensional variables the system of equations takes the form (see Ngan and Truskinovsky, 1996a)... 

In Eq. (1), a is the equilibrium stress (Nm 2) supported by the swollen specimen a is the stretched specimen length divided by the unstretched length (extension ratio) v2 is the volume fraction of dry protein and p is the density of dry protein. In the common case of tetrafunctional crosslinks, the concentration of network chains n (mol network chains/g polymer) is exactly one-half the concentration of crosslinks, so that n = 2c. The hypothesis that a specimen behaves as if it were an ideal rubber can be confirmed by observing a linear relation with zero intercept between a and the strain function (a — 1/a2) and by establishing a direct proportionality between a and the absolute temperature at constant value of the extension ratio, as stipulated by Eq. (1). 

The cross-link density can be determined by equilibrium swelling or from equilibrium stress-strain measurements at low strain rate, elevated temperature, and sometimes in the swollen state3 °... 

When using equilibrium stress-strain measurements, the cross-link density is determined from the Mooney-Rivlin equation ... 

During this same period, the equilibrium stress-strain properties of well characterized cross-linked networks were being studied intensively. More complex responses than the neo-Hookean behavior predicted by kinetic theory were observed. Among other possibilities it was speculated that, in some unspecified way, chain entanglements might be a contributing factor. 

Smith,T.L., Frederick, . E. Ultimate tensile properties of elastomers. IV. Dependence of the failure envelope, maximum extensibility, and equilibrium stress-strain curve on network characteristics. J. Appl. Phys. 36,2996-3005 (1965). 

In the simplest study of this type, Al-ghamdi and Mark [138] studied reinforcement of PDMS by two zeolites of different pore sizes. The zeolites were a zeolite 3A (pore diameter 3 A) and a zeolite 13X (pore diameter 10 A), both with a cubic crystalline structure. They were simply blended into hydroxyl-terminated chains of PDMS which were subsequently end-linked with tetraethoxysilane to form an elastomeric network. These elastomers were studied by equilibrium stress-strain measurements in elongation at 25°C. Both zeolites increased the modulus and related mechanical properties of the elastomer, but the effect was larger for the zeolite with the larger pore size. 

Figure 2.46 Typical non-equilibrium stress-strain curve in elongation. After F. W. Billmeyer, Jr., Textbook of Polymer Science, 2nd Ed., Wiley-Interscience, New York, 1971. Reproduced by permission of John Wiley and Sons.

Therein, D-1 is the positive definite, isotropic, fourth order viscous compliance, where r/ s and (s are the macroscopic viscosity parameters, D, , is the inelastic solid deformation rate and = F, 1 rfEQ F 71 is the corresponding non-equilibrium stress tensor. Furthermore, the superscript ( ) indicates the belonging to the intermediate configuration. 

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.)...

Weakly crosslinked epoxy-amine networks above their Tg exhibit rubbery behaviour like vulcanized rubbers and the theory of rubber elasticity can be applied to their mechanical behaviour. The equilibrium stress-strain data can be correlated with the concentration of elastically active network chains (EANC) and other statistical characteristics of the gel. This correlation is important not only for verification of the theory but also for application of crosslinked epoxies above their Tg. 

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