Strain dependent viscosity

As the director rotates, the shearing stress shows periodic maxima and minima. After solving Eq. (10-29) for the strain-dependent director angle 9 y), the strain-dependent viscosity afy is given by (Gu and Jamieson 1994)... 

Dependencies of longitudinal viscosity upon time at different extension velocities coincide up to the start of deviation from linear behavior. The higher the extension velocity, the earlier deviation is observed. An interesting fact is that in all cases the deviation from linearity is observed at one and the same critical strain independent of the molecular weight of the polymer and temperature. The value of critical strain depends only upon the nature of the polymer and lies within the limits of 0.4 — 1. 

Since pressure driven viscometers employ non-homogeneous flows, they can only measure steady shear functions such as viscosity, 77(7). However, they are widely used because they are relatively inexpensive to build and simple to operate. Despite their simplicity, long capillary viscometers give the most accurate viscosity data available. Another major advantage is that the capillary rheometer has no free surfaces in the test region, unlike other types of rheometers such as the cone and plate rheometers, which we will discuss in the next section. When the strain rate dependent viscosity of polymer melts is measured, capillary rheometers may provide the only satisfactory method of obtaining such data at shear rates... 

The big difference between normal isotropic liquids and nematic liquids is the effect of anisotropy on the viscous and elastic properties of the material. Liquid crystals of low molecular weight can be Newtonian anisotropic fluids, whereas liquid crystalline polymers can be rate and strain dependent anisotropic non-Newtonian fluids. The anisotropy gives rise to 5 viscosities and 3 elastic constants. In addition, the effective flow properties are determined by the flow dependent and history dependent texture. This all makes the rheology of LCPs extremely complicated. 

Problem 10.3(b) (Worked Example) Derive Eq. (10-31), the time- or strain-dependent shear viscosity in the absence of Frank elastic stresses. 

Velocity maps were measured with How-compensated spin-echo sequences [Rof l], The expefimental data were analysed in terms of a power law for the shear stress a[2 which solves the constitutive equation (T 2 = f)(dy/d/)" for the shear-rate-dependent viscosity rj and the applied shear strain y. The azimuthal velocity at radius r and rotation speed wr of the inner cylinder can be expressed as... 

The complex viscosity can be related to the steady-shear viscosity rf) via the empirical Cox-Merz rule, which notes the equivalence of steady-shear and dynamic-shear viscosities at given shearing rates ri y) = rj (co). The Cox-Merz rule has been confirmed to apply at low rates by Sundstrom and Burkett (1981) for a diallyl phthalate resin and by Pahl and Hesekamp (1993) for a filled epoxy resin. Malkin and Kulichikin (1991) state that for highly filled polymer systems the validity of the Cox-Merz rule is doubtful due to the strain dependence at very low strains and that the material may partially fracture. However, Doraiswamy et al. (1991) discussed a modified Cox-Merz rule for suspensions and yield-stress fluids that equates the steady viscosity with the complex viscosity at a modified shear rate dependent on the strain, ri(y) = rj yrap3), where y i is the maximum strain. This equation has been utilised by Nguyen (1993) and Peters et al. (1993) for the chemorheology of highly filled epoxy-resin systems. 

Most pure liquids and solutions of small molecules show a simple proportionality between sheer stress and sheer strain rate. Such liquids are known as Newtonian, after their discoverer. Solutions of polysaccharides usually show Newtonian behaviour only below a certain critical concentration, c, at which the chains start to interact. Above this concentration, except at very low sheer strain rates, viscosity decreases rapidly with sheer strain rate until at very high values of y it becomes constant. This is not a small effect with xanthan gum (see Section 4.6.10.3.1), the fall in viscosity is lO -fold. The dependence of viscosity on sheer strain rate is often given by... 

Dynamic mechanical strain-controlled measurements for both concentrated fabric softeners are shown in Figure 4.26. There are significant differences between the two products as regards the magnitude of the complex viscosity and complex modulus components and their strain dependence. Product B exhibits a higher viscosity and markedly longer linear region. The zero shear viscosity of product B is approximately 95 mPa s whereas that of product A is approximately half of this value at 50 mPa s. 

An important distinction between polymeric liquids and suspensions arises from their different microstructures and is evidenced by the elastic recoil phenomena that polymers exhibit but suspensions do not. The polymeric or macromolecular system when deformed under stress will recover from very large strains because like an elastic material the restoring force increases with the deformation. With a suspension, however, the forces between the particles decrease with increasing separation so that there is limited mechanism for recovery. There are, however, a variety of rheological properties common to polymeric liquids that suspensions will exhibit including shear rate dependent viscosity and time-dependent behavior. We shall discuss these differences in more detail in the following section. 

The heuristic extension of the rheological state equation from Bingham to Casson materials is straightforward. It can be derived directly from Eqn. (7) by introducing a strain rate dependant viscosity 77(72) instead of Tjg... 

The viscosity of molten glass displays a broad region of relatively strain-independent viscosity behavior albeit with a high degree of temperature dependence. A number of enqnrical formula are available to fit the experimentally measured data. In this per, the Vogel-Fidcher-Tammann VFT) equation is used... 

Strain dependence. Furthermore, in steady shear a strong gel will rupture completely whereas a weak gel will flow, albeit without obeying the Cox-Merz superposition principal. In this case the complex viscosity (r] ) is higher than the flow viscosity ( //). This indicates the presence of weak interactions between the molecules that contribute to the viscosity measured by the non-destructive oscillatory technique but not to the viscosity measured by the destructive flow technique. 

For a viscoelastic liquid in shear, the ratio oz t)/y in equations 56 to 58 is sometimes treated as a time-dependent viscosity J (r) which increases monoton-ically (provided the viscoelasticity is linear) to approach the steady-state viscosity Meissner has pointed out that, in an experiment involving deformation at constant strain rate followed by stress relaxation at constant deformation, viscoelastic information can be obtained without imposing the restriction in equation 11 of Chapter 1 that the loading interval is small compared with the time elapsed at the first experimental stress measurement. ... 

The time dependence of stress under conditions of constant strain rate has been discussed for the case of linear viscoelasticity in Section FI of Chapter 3. For uniaxial extension at constant strain rate ei = (l/ ) d /dt), the time-dependent tensile stress ffriO is often expressed in terms of a time-dependent viscosity = An example of the stress growth in such elongational flow from... 

If a constant shear rate 7 is imposed on a viscoelastic liquid, the stress rises as described by cr(t) or the time-dependent viscosity function Tj+Ct) = o (t)/7, and these quantities approach their steady-state values as steady flow is reached. If the steady-state flow is terminated, the subsequent stress relaxation is described by or the time-dependent viscosity function rj t)y, discussed in Sections C2 and C3 of Chapter 1. If the viscoelastic behavior is linear, these functions are monotonic and related to the relaxation modulus G(t) by equations 13 and 14 of Chapter 1. The functions and are independent of 7 and their sum is simply jjo- At large strain rates, however, both and T (t) become strongly dependent on 7. 

Elastomers are typically viscoelastic and exhibit both viscous as well as elastic characteristics when rmdergoing deformations. The viscous component (o = t], where, a is the stress, t] is the coefficient of viscosity and is the change of strain as a function of time) takes care of the energy dissipated as heat after a strain/stress is applied and followed by its removal while the elastic component (o = Ee) brings back the material towards original dimension, or, in other words, strain depends on stress applied to the materials and time. The overall strain is then governed by the equation where the two terms are separable, as in Eq. (2) is called as linear viscoelasticity and usually it is applicable only for small deformations, where, t is... 

Study of strain dependence of tanS shows that, for all compounds tan8 increases with increasing in strain. The trend of tanS can be divided in two zones i.e. elastic zone (tanS < 1) and viscose zone (tanS > 1). It is indicated that the elastic zone of reclaimed is longer than NR, and in NR/reclaimed rubber blends with increasing reclaimed rubber content elastic zone becomes longer. This is due to the fillers left in reclaimed rubber. Non-homogeneity of phases and non-uniformed dispersion of fillers in NR/reclaimed rubber 50/100 and 75/50 cause shorter elastic zones in these blends than virgin natural rubber [90]. 

The class of non-Newtonian fluids discussed above (i.e. fluids showing a shear-dependent viscosity) is simply one subset of the types of behaviour observed in polymeric fluids. The shear-dependent fluids considered above are assumed to be inelastic, although some polymer solutions show some degree of elasticity. When elastic materials are deformed through a small displacement they tend to return to their original configuration. If a shear stress is applied to an ideal solid, then for small displacements the displacement, which is the strain, y, is proportional to the applied stress and Hooke s... 

In terms of the strain-hardening modulus, this has been developed by the use of Kuhn and Griin models and Kratky models to relate the development of molecular orientation and meehanical anisotropy (see Section 8.6.3). With regard to the strain rate sensitivity the strain rate-dependent viscosity has been developed by studies of creep and yield behaviour (see Sections 11.3 and 12.5.1). 

Fig. 7.19. Time dependent viscosities for shear and extensional flow, and as predicted by Lodge s equation of state. Calculations are performed for different Hencky strain rates en, assuming a single exponential relaxation modulus G(t) exp — t/r...

Wagner [92,93] too provided a method for the prediction of normal stress difference from shear viscosity data using a strain-dependent single integral constitutive equation of Berstein et al. [46] type as foUows ... 

Shenoy and Saini [32] have provided a simplified approach to the prediction of primary stress difference in polymer melts as discussed in Giapter 5. This is based on the relationship between the unified normal stress difference function and the unified viscosity function curves through a strain-dependent single-... 

In addition to the trace, another quantity, often referred to simply as the strain rate, is of interest. The strain rate, taken from the modulus of the tensor, is a positive-definite representation of all possible components of the strain rate tensor. It is used to determine the viscosity in strain-dependent non-Newtonian fluids and is also helpful as a reporting tool for mixing applications. In particular, regions with a high strain rate play an important role in liquid dispersion. 

Figure 9.21 shows the time-dependent viscosities derived from Eqs. (9.178) and (9.189) for both simple shear and extensional flow. For simplicity a single exponential relaxation with a relaxation time t is assumed for G t"). The dotted line represents the time-dependent viscosity for simple shear, which is independent of 7. A qualitatively different result is found for the extensional flow. As we can see, the time-dependent extensional viscosity ff t) increases with ch and for en > 0.5t a strain hardening arises. 

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