Modulus continued shear

The continuing search to determine the shear modulus and shear strength consists of a collection of tests. Several tests are discussed because each has faults, as will be seen, and because, to some extent, there is no universal agreement on the best way to measure the shear properties. 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

Equations 1.2 to 1.4 represent material functions under large deformations (e.g., continuous shear of a fluid). One may recall a simple experiment in an introductory physics course where a stress (a) is applied to a rod of length Z, in a tension mode and that results in a small deformation AL. The linear relationship between stress (ct) and strain (j/) (also relative deformation, y = AL/L) is used to define the Young s modulus of elasticity E (Pa) ... 

Experimental detection of the gel point is not always easy since the equilibrium shear modulus is technically zero at the gel point and any applied stress will eventually relax, but only at infinite time. From the classical theory, the attributes of the gel point are an infinite steady-shear viscosity and a zero equilibrium modulus at zero frequency limit (Figure 6-3) (Flory, 1953). These criteria have been widely employed to detect the gel point of chemical gels. However, because continuous shearing affects gel formation, accurate information from viscosity measurement is not possible in the close vicinity of the gel point. Further, information regarding the transition itself could only be obtained by extrapolation, thereby introducing uncertainties in the determination of the gelation moment. 

In rheology of condensed systems the shear modulus, G, is usually used as a characteristic of elasticity. In mechanics of continuous (isotropic) media it was shown that the modulus of shear for solid-like objects equals approximately 2/5 of the Young modulus, E... 

In these equations, the subscripts h and s refer to the hard continuous phase and the soft rubbery phase, respectively, E = Young s modulus, G = shear modulus, vh = Poisson s ratio (— 0.35 for polystyrene and most rigid polymers), and cj>8 and h = volume fractions of the components. 

Fig. 5. Regular temperature behavior for elastic-stiffness constants such as Young s modulus, the shear modulus, and the bulk modulus. The main features are continuous decrease with increasing temperature, zero slope at zero temperature, a i dependence at very low temperatures, relative flatness at low temperatures, and linear slope at high temperatures. A, the zero-temperature deviation from linear behavior, is a quantum effect due to zero-point oscillations.

We shall follow the same approach as the last section, starting with an examination of the predicted behavior of a Voigt model in a creep experiment. We should not be surprised to discover that the model oversimplifies the behavior of actual polymeric materials. We shall continue to use a shear experiment as the basis for discussion, although a creep experiment could be carried out in either a tension or shear mode. Again we begin by assuming that the Hookean spring in the model is characterized by a modulus G, and the Newtonian dash-pot by a viscosity 77. ... 

R. L. Hewitt and M. C. de Malherbe, An Approximation for the Longitudinal Shear Modulus of Continuous Fibre Composites, Journal of Composite Materials, April 1970, pp. 280-282. 

Shear modulus is also changed with the blending type (Fig. 11). Preheated blends provide higher modulus than the preblends. It is obvious that shear modulus either decreases or remains the same at the initial level of NBR for preblends, but beyond 45% of NBR there is a further rise in the G value irrespective of shear rates whereas at the lower shear rate, preheated blends show the continuous rise in the shear modulus for the entire composition range. 

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. 

Chain stretching is governed by the covalent bonds in the chain and is therefore considered a purely elastic deformation, whereas the intermolecular secondary bonds govern the shear deformation. Hence, the time or frequency dependency of the tensile properties of a polymer fibre can be represented by introducing the time- or frequency-dependent internal shear modulus g(t) or g(v). According to the continuous chain model the fibre modulus is given by the formula... 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

Figure 6. The log-log plot of the shear modulus G measured at f 0.015 Hz, versus (X - X )/X. The slopes of the continuous lines give the critical exponents t.

The rheology of the sol-gel transition was undertaken with special care in order to avoid gel disruption. A critical behaviour for the shear modulus with respect to the helix amount, is noticed. A simple relation between the rheological parameters and the degree of helix formation is pointed out in a limited range of helix amounts (X<15%). These experiments will continue on the fully matured gels. 

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