The yield stress equation

Why the large dielectric loss tangent is ncccssar for the ER cfTcct Elao proposed a qualitative model on the assumption that the particle turning process and particle polarization process arc both important to the ER cfTcct, and the interfacial polarization would be responsible for the ER effect [26], 

Tlic above-equations mean that one can easily know (he second slep internal energy and entropy change if the total internal energy and entropy change, as well as the first step internal energy and entropy change, are determined. 

Maxwell equation is thus still used for computing ff and in Hao s work 

For the step 1, the interfacial polarization was assumed to be inactive, which is only physically likely under the condition of = c. In such a condition, 

The AU2 should be less than zero, as the inter-particle force in the ER crystalline lattice is attractive. Thus Fq. (53) should be expressed 

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An interesting approach for deriving the relationship between the yield stress and particle volume fraction was proposed by lV[e/./asalnia [111] on the basis of Helmholtz free energy change under an electric field [109], The yield stress equation was derived as ... 

Since p is a very important parameter in the yield stress equation (69), it will be interesting to further explore which physical parameters would greatly affect p value and then the yield stress. According to the definition. 

A quantitative comparison between the prediction derived from the yield stress equation and the experimental results was made by Hao for the zeolite/silicone oil system [29J. The static dielectric constant of the pure zeolite material and the yield stress of zeolite/silicone oil suspension of the particle volume fraction 0.23 were experimentally measured at different temperatures. The calculated yield stress values from Eq. (69) vs. temperature is shown in Figure 17 as a solid line. For comparison, the experimentally measured data are also shown in Figure 17 as black points. As wc can sec, the predicted values agree very well with the experimental ones, indicating that Eq. (69) is able to predict the yield stress of ER fluids, indeed. 

Note that the yield stress equation contains an important parameter p. If p is positive, the yield stress will increase with which is consistent with the prediction given by the polarization model [35,39]. However, if p is negative, then the yield stress will decrease with which can not be explained by the polarization model. The parameter p only becomes positive when the dielectric loss tangent of the dispersed solid material is larger than 0.1 [64,73,78]. 

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) ... 

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... 

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). 

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. 

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... 

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. 

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). 

Fluids that show viscosity variations with shear rates are called non-Newtonian fluids. Depending on how the shear stress varies with the shear rate, they are categorized into pseudoplastic, dilatant, and Bingham plastic fluids (Figure 2.2). The viscosity of pseudoplastic fluids decreases with increasing shear rate, whereas dilatant fluids show an increase in viscosity with shear rate. Bingham plastic fluids do not flow until a threshold stress called the yield stress is applied, after which the shear stress increases linearly with the shear rate. In general, the shear stress r can be represented by Equation 2.6 ... 

In a similar way, the yield stress increases under applied hydrostatic pressure, P, leading to a modified Eyring equation ... 

Converting penetration depth to hardness has the advantage of normalizing consistency values so that they are less dependent on the penetration load. This is the rationale behind hardness testing in metallurgy. In these cases, the contact pressure as defined by hardness in Equation 2 is used to deduce the yield stress of a material (Tabor, 1996). However, the yield stress is the resistance to an applied shear stress, but it is not the only resistance to a penetrating body. The elastic properties of a fat, and the coefficient of friction between the cone and the fat sample will also impede the penetration of the cone (Tabor, 1948). Kruisher et al. (1938) tried to eliminate friction effects and advocated the use of a flat circular penetrometer with concave sides. 

Much more effort has gone into relating hardness value to the yield stress of fats than to their elastic properties. For example, the International Dairy Federation proposed (Walstra, 1980) that penetration depth be converted to an apparent yield stress (AYS) for sharp-ended cones according to the equation ... 

Spreadability is another important parameter of butter texture. A spreadability index (S) can be calculated from the yield stress value obtained for butter before and after working using a constant-weight penetrometer, as shown in Equation (5), where fu and /w are the yield stress values before and after working respectively (Haighton, 1965). 

The yield stress varies with temperature in accordance with the equation [1, 28]... 

Princen has considered the behaviour of foams and emulsions [10] subjected to shear stress, employing a two-dimensional hexagonal package model and accounting for the foam expansion ratio and the contact angles. He derived equations for the shear modulus and the yield stress... 

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