Dependence of moduli

Fig. 25. Frequency dependence of moduli growth rate at the LST of crosslinking PDMS. Data from Scanlan and Winter [58]...

To calculate temperature dependencies of G and G", the experimental dependencies of real and imaginary parts of the modulus were taken for cured epoxy resin.The properties of the interphase layer have been chosen assuming that the curves of the temperature dependence of moduli have the same shape but are shifted along the temperature axis to a lower temperature ( soft interphase) or higher temperature ( rigid interphase). 

Strengths and moduli of most polymers increase as the temperature decreases (155). This behavior of the polymer phase carried over into the properties of polymer foams and similar dependence of the compressive modulus of polyurethane foams on temperature has been shown (151). 

The effect of temperature on PSF tensile stress—strain behavior is depicted in Figure 4. The resin continues to exhibit useful mechanical properties at temperatures up to 160°C under prolonged or repeated thermal exposure. PES and PPSF extend this temperature limit to about 180°C. The dependence of flexural moduli on temperature for polysulfones is shown in Figure 5 with comparison to other engineering thermoplastics. 

Corresponding referential, current spatial, and unrotated spatial inelastic constitutive equations are equivalent, and identical results are obtained from their use, if corresponding moduli and elastic limit functions are used. In the current spatial constitutive equations, if the dependence of the spatial moduli... 

The dependence of the spatial moduli c and b on f has been emphasized in writing the stress rate relation (5.154). This dependence implies that the these quantities are varying as the deformation proceeds, quite apart from their dependence on e and k. If CC and are assumed to be constant, independent of E and K, then, in component form, (5.155) becomes... 

It is the dependence of the spatial constitutive functions on the changing current configuration through F that renders the spatial constitutive equations objective. It is also this dependence that makes their construction relatively more difficult than that of their referential counterparts. If this dependence is omitted, then the spatial moduli and elastic limit functions must be isotropic to satisfy objectivity, and the spatial constitutive equations reduce to those of hypoinelasticity. Of course, there are other possible formulations for the spatial constitutive functions which are objective without requiring isotropy. One of these will be considered in the next section. 

Upon comparison of Eqs. 29 and 36, it is readily apparent that both theories predict the same power law dependence of the contact radius on particle radius and elastic moduli. However, the actual value of the contact radius predicted by the JKR theory is that predicted by the DMT model. This implies that, for a given contact radius, the work of adhesion would have to be six times as great in the DMT theory than in the JKR model. It should be apparent that it is both necessary and important to establish which theory correctly describes a system. 

The mechanical properties of polymers are of interest in all applications where they are used as structural materials. The analysis of the mechanical behavior involves the deformation of a material under the influence of applied forces, and the most important and characteristic mechanical property is the modulus. A modulus is the ratio between the applied stress and the corresponding deformation, the nature of the modulus depending on that of the deformation. Polymers are viscoelastic materials and the high frequencies of most adiabatic techniques do not allow equilibrium to be reached in viscoelastic materials. Therefore, values of moduli obtained by different techniques do not always agree in the literature. 

Analysis of the relationships between the moduli and bond strength between particles [222] has shown that for Vf = 0.1 — 0.15 the concentration dependence of the modulus corresponds to the lower curve in the Hashin-Shtrikman equation [223] (hard inclusion in elastic matrix), and for Vf — 0.34 to the upper boundary (elastic inclusion in a hard matrix). The 0.1 to 0.34 range is the phase inversion region. 

Above a critical hller concentration, the percolation threshold, the properties of the reinforced rubber material change drastically, because a hller-hUer network is estabhshed. This results, for example, in an overproportional increase of electrical conductivity of a carbon black-hUed compound. The continuous disruption and restorahon of this hller network upon deformation is well visible in the so-called Payne effect [20,21], as represented in Figure 29.5. It illustrates the strain-dependence of the modulus and the strain-independent contributions to the complex shear or tensUe moduli for carbon black-hlled compounds and sUica-hUed compounds. 

This equation calls attention to the well-established inverse relationship between the degree of equilibrium swelling of a series of rubber vulcanizates in a given solvent and the forces of retraction, or moduli, which they exhibit on stretching. The indicated approximate dependence of qm on the inverse three-fifths power of the modulus has been confirmed. 

The results of stress-strain measurements can be summarized as follows (1) the reduced stress S (A- A ) (Ais the extension ratio) is practically independent of strain so that the Mooney-Rivlin constant C2 is practically zero for dry as well as swollen samples (C2/C1=0 0.05) (2) the values of G are practically the same whether obtained on dry or swollen samples (3) assuming that Gee=0, the data are compatible with the chemical contribution and A 1 (4) the difference between the phantom network dependence with the value of A given by Eq.(4) and the experimental moduli fits well the theoretical dependence of G e in Eq.(2) or (3). The proportionality constant in G for series of networks with s equal to 0, 0.2, 0.33, and 0. Ewas practically the same -(8.2, 6.3, 8.8, and 8.5)x10-4 mol/cm with the average value 7.95x10 mol/cm. Results (1) and (2) suggest that phantom network behavior has been reached, but the result(3) is contrary to that. Either the constraints do survive also in the swollen and stressed states, or we have to consider an extra contribution due to the incrossability of "phantom" chains. The latter explanation is somewhat supported by the constancy of in Eq.(2) for a series of samples of different composition. 

The temperature dependences of the isothermal elastic moduli of aluminium are given in Figure 5.2 [10]. Here the dashed lines represent extrapolations for T> 7fus. Tallon and Wolfenden found that the shear modulus of A1 would vanish at T = 1.677fus and interpreted this as the upper limit for the onset of instability of metastable superheated aluminium [10]. Experimental observations of the extent of superheating typically give 1.1 Tfus as the maximum temperature where a crystalline metallic element can be retained as a metastable state [11], This is considerably lower than the instability limits predicted from the thermodynamic arguments above. 

Now these expressions describe the frequency dependence of the stress with respect to the strain. It is normal to represent these as two moduli which determine the component of stress in phase with the applied strain (storage modulus) and the component out of phase by 90°. The functions have some identifying features. As the frequency increases, the loss modulus at first increases from zero to G/2 and then reduces to zero giving the bell-shaped curve in Figure 4.7. The maximum in the curve and crossover point between storage and loss moduli occurs at im. 

Eigure 4.2. The E dependence of the storage G (solid symbols) and loss G" (open symbols) moduli of a mono-disperse silicon oil-in-water emulsion stabilized with SDS, with radius a = 0.53 jam, for three volume fractions from top to bottom (j> = 77%, 60%, and 57%. The frequency is 1 rad/s the lines are visual guides. (Adapted from [10].)... 

The frequency dependence of the moduli was measured by Mason et al. [7,8] and is shown in Fig. 4.3 for several values of <. In all cases, there is a plateau in G (, this extends over the full four decades of explored frequency, while for the lower 4>, the plateau is no longer strictly independent on co but reduces to an inflection point at G. In contrast, for all 4>, G" co) exhibits a shallow minimum, G". Mason et al. used G to characterize the elasticity and G" to characterize the viscous loss. 

Fig. 14. Temperature dependence of the dynamic storage (E ) and loss (E") moduli at 10 Hz as a function of composition...

Effect of Frequency-Dependence of Elastic Moduli on Dynamic Light Scattering. . 102... 

Here pg and p f are the mass densities of the gel and the solvent, respectively, K is a bulk modulus, c0 is the speed of sound, and i s is the solvent shear viscosity. The solvent bulk viscosity has been neglected. The terms proportional to / arise from an elastic coupling in the free energy between the density deviation of gel and that of solvent The p in Eq. (6.1) coincides with the shear modulus of gels treated so far. We neglect the frequency-dependence of the elastic moduli. It can be important in dynamic light scattering, however, as will be discussed in the next section. 

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