Equations yield stress

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

If the sum of the mechanical allowances, c, is neglected, then it may be shown from equation 15 that the pressure given by equation 33 is half the coUapse pressure of a cylinder made of an elastic ideal plastic material which yields in accordance with the shear stress energy criterion at a constant value of shear yield stress = y -... 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

For high-cycle fatigue of uncracked components, where neither or. the yield stress, it is found empirically that the experimental data can be fitted to an equation of form... 

A composite material used for rock-drilling bits consists of an assemblage of tungsten carbide cubes (each 2 fcm in size) stuck together with a thin layer of cobalt. The material is required to withstand compressive stresses of 4000 MNm in service. Use the above equation to estimate an upper limit for the thickness of the cobalt layer. You may assume that the compressive yield stress of tungsten carbide is well above 4000 MN m , and that the cobalt yields in shear at k = 175 MN m . What assumptions made in the analysis are likely to make your estimate inaccurate ... 

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

Taking into account all the various stipulations, we shall still assume that yield stress has a certain physical meaning and it can be measured by a stationary method proceeding from the flow curve. However, to measure the points and achieve such a clear pattern as shown in Fig. 1 is not always convenient and it is rather labor-consuming. In practice, it is convenient to use a semi-analytical procedure. It is based on the utilization of an equation for flow curves taking into account the existence of yield stress. The most widespread equation of this kind is the Casson equation. It assumes that the x(y) dependence for filled polymers is expressed in the following way ... 

Fig. 4. Diagram illustrating the method for determining a yield stress by constructing a flow curve in the coordinates of the Casson equation. The content of the filler is more for curve 2 than for curve 1...

Since non-Newtonian flow is typical for polymer melts, the discussion of a filler s role must explicitly take into account this fundamental fact. Here, spoken above, the total flow curve includes the field of yield stress (the field of creeping flow at x < Y may not be taken into account in the majority of applications). Therefore the total equation for the dependence of efficient viscosity on concentration must take into account the indicated effects. 

According to the structure of this equation the quantity cp indicates the influence of the filler on yield stress, and t r on Newtonian (more exactly, quasi-Newtonian due to yield stress) viscosity. Both these dependences Y(cp) andr r(cp) were discussed above. Non-Newtonian behavior of the dispersion medium in (10) is reflected through characteristic time of relaxation X, i.e. in the absence of a filler the flow curve of a melt is described by the formula ... 

Another approach to determining the contribution being made by each of the possible com-pression/decompression mechanisms involves monitoring the degree and rate of relaxation in tablets immediately after the point of maximum applied force has been reached. Once a powder bed exceeds a certain yield stress, it behaves as a fluid and exhibits plastic flow [121,122], Certain investigators [122] have studied plastic flow in terms of viscous and elastic elements and have derived the following equation ... 

Because this material will not flow unless the shear stress exceeds the yield stress, these equations apply only when r > r0. For smaller values of the shear stress, the material behaves as a rigid solid, i.e.,... 

By equating the vertical component of the yield stress over the surface of the sphere to the weight of the particle, a critical value of = 0.17 is obtained (Chhabra, 1992). Experimentally, however, the results appear to fall into groups one for which F(i fa 0.2 and one for which F(i fa 0.04—0.08. There seems to be no consensus as to the correct value, and the difference may well be due to the fact that the yield stress is not an unambiguous empirical parameter, inasmuch as values determined from static measurements can differ significantly from the values determined from dynamic measurements. 

Equation (32) clearly illustrates the point that as rp becomes very small relative to c, i.e., for large cracks, the fracture stress may be orders of magnitude smaller than the true yield stress of the material. 

A poor flowout is generally related to yield stress behavior. A method of assess ing( l) this behavior of a coating is through Casson plots based on the following equation ... 

As for the derivation of Eqs. 122,123 and 124 only the transitions 1—>2 have been counted, these equations do not describe recovery processes, where the transitions 2 —>1 are important as well. These approximations have been made for convenience s sake, but neither imply a limitation for the model, nor are they essential to the results of the calculations. Equation 124 is the well-known formula for the relaxation time of an Eyring process. In Fig. 65 the relaxation time for this plastic shear transition has been plotted versus the stress for two temperature values. It can be observed from this figure that in the limit of low temperatures, the relaxation time changes very abruptly at the shear yield stress Ty = U0/Q.. Below this stress the relaxation time is very long, which corresponds with an approximation of elastic behaviour. 

This material is seen to be shear thinning. It is possible that it may exhibit a yield stress but confirmation of this would require measurements at lower shear rates. Note that the Rabinowitsch-Mooney equation is still valid when a non-zero yield stress occurs. 

When mixing a liquid exhibiting a yield stress, it is clear that material near the impeller will be fluid while that further away, where the shear stress has fallen below the yield stress ry, will be stagnant. Mixing therefore occurs only in a cavern around the impeller. The cavern diameter Dc for a flat blade single impeller can be calculated from the equation... 

Here the fitting parameters are the slope of the line (the plastic viscosity, rip) and the Bingham or dynamic yield stress (the intercept, constitutive equations will be introduced later in this volume as appropriate. 

In practice this grossly overestimates the yield stress, which may be a factor of 103 less than we would predict from this equation. The reason is that it is relatively easy for motion to occur across the end of the dislocation where there is a mismatch in the lattice planes. Of course the basic structure of the crystal is not changed and so when we pause the experiment and start again we find the same modulus. Figure 2.6 illustrates the process with a cubic lattice. 

So far we have restricted our discussion to Newtonian liquids, but the analysis will change somewhat if the systems are non-Newtonian. A useful illustration of the problems that arise is the case of a Bingham plastic. This gives us a linear response, as does a Newtonian liquid, but in this case there is an intercept or yield stress. The constitutive equation for a Bingham plastic is... 

The results of Equation (3.56) are plotted in Figure 3.14. It can be seen that shear thinning will become apparent experimentally at (p > 0.3 and that at values of q> > 0.5 no zero shear viscosity will be accessible. This means that solid-like behaviour should be observed with shear melting of the structure once the yield stress has been exceeded with a stress controlled instrument, or a critical strain if the instrumentation is a controlled strain rheometer. The most recent data24,25 on model systems of nearly hard spheres gives values of maximum packing close to those used in Equation (3.56). 

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