Shear modulus macroscopic

Using the fluctuation-dissipation theorem [361, which relates microscopic fluctuations at equilibrium to macroscopic behaviour in the limit of linear responses, the time-dependent shear modulus can be evaluated [371 ... 

Use the bounding techniques of elasticity to determine upper and lower bounds on the shear modulus, G, of a dispersion-stiffened composite materietl. Express the results In terms of the shear moduli of the constituents (G for the matrix and G for the dispersed particles) and their respective volume fractions (V and V,j). The representative volume element of the composite material should be subjected to a macroscopically uniform shear stress t which results in a macroscopically uniform shear strain y. 

Contrary to the phase separation curve, the sol/gel transition is very sensitive to the temperature more cations are required to get a gel phase when the temperature increases and thus the extension of the gel phase decreases [8]. The sol/gel transition as determined above is well reproducible but overestimates the real amount of cation at the transition. Gelation is a transition from liquid to solid during which the polymeric systems suffers dramatic modifications on their macroscopic viscoelastic behavior. The whole phenomenon can be thus followed by the evolution of the mechanical properties through dynamic experiments. The behaviour of the complex shear modulus G (o)) reflects the distribution of the relaxation time of the growing clusters. At the gel point the broad distribution of... 

In the composite, there is a distribution of orientation and of separation distances of the interaggregate bonds with respect to direction of the applied strain. As a result, the composite can be considered as a collection of those elementary models, each having a different yb. By introducing the weighting function N(yh) d h> which gives the number of links that break when the polymer is stretched from yfe to yb + dy, the complex excess shear modulus of the macroscopic filler system is given by... 

The shear moduli are Pf for the fluid shear viscosity, p, for the solid shear modulus, and Pm for the macroscopic (matrix) shear modulus. Under the conditions of porosity and pressure variations, the effective densities of the liquid phase are ... 

These systems are typically of very high viscosity and may even possess a yield stress [146]. They exhibit elastic properties, and their shear modulus is typically in the range of lO -lO Pa [56,146-149]. It should be noted here again that these cubic phases are formally no longer microemulsions, but they are structurally very closely related. Their microstructural units are almost identical to those in the corresponding microemulsion, but their macroscopic appearance is considerably different i.e., whereas the microemulsion is a liquid of low viscosity, the cubic phase is a solidlike material. 

These parameters are related to one another and can be determined theoretically or through the use of a variety of experimental techniques. Usually they are calculated from typical macroscopic network properties, like swelling degree Q or Young s modulus E (for tensile or compression strain) or shear modulus G (for shearing strain), which can be determined by physical methods as will be shown in Sections 4.S.2-4.3.4. 

When a polymer exhibits a maximum in the imaginary part of the dielectric permittivity (the loss permittivity, e") at frequencies less than 200 Hz, it becomes possible to make comparisons with the frequency dependence of shear moduli and most specifically with the loss shear modulus, G". This has been done for polypropylene diol, also called poly(oxypropy-lene), where there is reported a near perfect superposition of the frequency dependence of the normalized loss shear modulus with that of the normalized loss permittivity as reproduced in Figure 3. The acoustic absorption frequency range of interest here is 100 Hz to 10 kHz, yet present macroscopic loss shear modulus data can be determined at most up to a few hundred Hz. Nonetheless, for X -(GVGIP)32o there is a maximum in loss permittivity, e", near 3 kHz that develops on raising the temperature through the temperature range of the inverse temperature transition. With the width of the loss permittivity curve a distinct set of curves as a function of temperature become... 

Fig. 2. Macroscopic development of rheological properties (eg, / o = zero-shear viscosity) and mechanical properties (eg, = equilibrium shear modulus) during network formation,...

It was observed empirically by Hooke that, for many materials under low strain, stress is proportional to strain. Young s modulus may then be defined as the ratio of stress to strain for a material under uniaxial tension or compression, but it should be noted that not all materials (and this includes polymers) obey Hooke s law rigorously. This is particularly so at high values of strain but this section only considers the linear portion of the stress-strain curve. Clearly, reality is more complicated than described previously because the application of stress in one direction on a body results in a strain, not only in that direction, but in the two orthogonal directions also. Thus, a sample subjected to uniaxial tension increases in length, but it also becomes narrower and thinner. This quickly leads the student into tensors and is beyond the scope of this chapter. The subject is discussed elsewhere [21-23]. There are four elastic constants usually used to describe a macroscopically isotropic material. These are Young s modulus, E, shear modulus, G, bulk modulus, K, and Poisson s ratio, v. They are defined in Figure 9.2 and they are related by Equations 9.1-9.3. 

Voigt has shown how, starting with the monocrystalline elastic stiffiiess moduli, Cy, it is possible to derive expressions for tne macroscopic shear modulus, Gy, and the bulk modulus, Ky. For a mac-roscopically isotropic aggregate of cubic monocrystais, the VoiGT approach yields ... 

We propose a way to obtain averaged macroscopic quantities like density, momentum flux, stress, and strain from "microscopic" numerical simulations of particles in a two-dimensional ring-shear-cell. In the steady-state, a shear zone is found, about six particle diameters wide, in the vicinity of the inner, moving wall. The velocity decays exponentially in the shear zone, particle rotations are observed, and the stress and strain-tensors are highly anisotropic and, even worse, not co-linear. From combinations of the tensorial quantities, one can obtain, for example, the bulk-stifftiess of the granulate and its shear modulus. 

Equation 1.16 can be used for the description of the temperature dependence of displaying one principal difference from the filled polymers. As the G value for amorphous and semi-crystalline polymers, the macroscopic shear modulus should be used instead of its value for a loosely packed matrix. This is explained so that, contrary to the value, the G value for the polymer amorphous state is determined by the structure of both quasi-phases [35]. The truth is that the application of the G value in Equation 1.16 only for a loosely packed matrix would mean determination of the cluster property (x ) from properties of another structure component only, which is physically meaningless. 

Because the material is assumed to be macroscopically isotropic it is necessary to determine only two material constants. The most convenient constants are the bulk modulus K and shear modulus G, since from (1.13) we have ... 

The shear modulus relates stress and strain. These are macroscopic quantities used in coarse-grained or continuum descriptions of materials. The bubble model takes a mesoscopic perspective, viewing the material as a collection of individual bubbles the shear modulus is related to the bubbles displacements via Equation 12.15. To develop this relationship, the elastic energy stored by the deformation. 

This expression relates the macroscopic shear modulus to the microscopic (relative) motion of the bubbles. We now drop subscripts and prefactors and speak only of the typical normal and tangential motions, Am and Am- -. When the packing is sheared, it responds so as to minimize the elastic energy AU. Assuming the two terms in the energy expansion are of similar order [9], it then follows that (in three dimensions)... 

In one series of measurements, the crosslinker concentration was varied in gels based on 15 % acrylic acid. The pH in these systems is around 2.3, hence they are essentially nonionic with only around 0.25 % of the carboxylic acid groups being dissociated. Figure 1 gives an overview on the macroscopic properties of the gels. The shear modulus, G, measured by small uniaxial compression of the gel cylinders as reported elsewhere (7) is plotted versus the theoretical network density, Vth. Vth is twice as large as the molar concentration of... 

Finally, substituting Equations 8.23 and 8.24 into Equation 8.26 gives the desired equation relating the macroscopic shear modulus Glt to the constituent shear moduli as a function of fiber volume fraction ... 

---