Yield stress prediction

Corn stover, a well-known example of lignocellulosic biomass, is a potential renewable feed for bioethanol production. Dilute sulfuric acid pretreatment removes hemicellulose and makes the cellulose more susceptible to bacterial digestion. The rheologic properties of corn stover pretreated in such a manner were studied. The Power Law parameters were sensitive to corn stover suspension concentration becoming more non-Newtonian with slope n, ranging from 0.92 to 0.05 between 5 and 30% solids. The Casson and the Power Law models described the experimental data with correlation coefficients ranging from 0.90 to 0.99 and 0.85 to 0.99, respectively. The yield stress predicted by direct data extrapolation and by the Herschel-Bulkley model was similar for each concentration of corn stover tested. 

The yield stress values given in Table 3 demonstrate that the yield stresses determined with the Herschel-Bulkley model were lower than the yield stresses determined with all the other methods at equal concentrations. The yield stress predicted by direct data extrapolation and by the Herschel-Bulkley model was similar for each concentration of corn stover. 

Figure 7.27. Yield stress prediction for different particle sizes. [Adapted, by permission, from Pukanszky B, Voros G, Polym.Composites 17, No.3, 1996, 384-92.]...

Yield stresses can also be obtained by extrapolation of shear rate-shear stress data to zero shear rate according to one of several flow models. The application of several models was studied by Rao et al. (AS.) and Rao and Cooley (Al) The logarithm of the yield stresses predicted by each model and the total solids (TS) of the concentrates were related by quadratic equations. The equations for the yield stresses predicted by the Herschel-Bulkley model (Equation 4) which described very well the flow data of Nova and New Yorker tomato cultivars were ... 

Of the models Hsted in Table 1, the Newtonian is the simplest. It fits water, solvents, and many polymer solutions over a wide strain rate range. The plastic or Bingham body model predicts constant plastic viscosity above a yield stress. This model works for a number of dispersions, including some pigment pastes. Yield stress, Tq, and plastic (Bingham) viscosity, = (t — Tq )/7, may be determined from the intercept and the slope beyond the intercept, respectively, of a shear stress vs shear rate plot. 

For most practical purposes, the onset of plastic deformation constitutes failure. In an axially loaded part, the yield point is known from testing (see Tables 2-15 through 2-18), and failure prediction is no problem. However, it is often necessary to use uniaxial tensile data to predict yielding due to a multidimensional state of stress. Many failure theories have been developed for this purpose. For elastoplastic materials (steel, aluminum, brass, etc.), the maximum distortion energy theory or von Mises theory is in general application. With this theory the components of stress are combined into a single effective stress, denoted as uniaxial yielding. Tlie ratio of the measure yield stress to the effective stress is known as the factor of safety. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

You want to predict how fast a glacier that is 200 ft thick will flow down a slope inclined 25° to the horizontal. Assume that the glacier ice can be described by the Bingham plastic model with a yield stress of 50 psi, a limiting viscosity of 840 poise, and an SG of 0.98. The following materials are available to you in the lab, which also may be described by the Bingham plastic model ... 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

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 influence of fillers has been studied mostly at hl volume fractions (40-42). However, in addition, it is instructive to study low volume fractions in order to test conformity with theoretical predictions that certain mechanical properties should increase monotonlcally as the volume fraction of filler is Increased (43). For example, Einstein s treatment of fluids predicts a linear increase in viscosity with an increasing volume fraction of rigid spheres. For glassy materials related comparisons can be made by reference to properties which depend mainly on plastic deformation, such as yield stress or, more conveniently, indentation hardness. Measurements of Vickers hardness number were made after photopolymerization of the BIS-GMA recipe, detailed above, containing varying amounts of a sllanted silicate filler with particles of tens of microns. Contrary to expectation, a minimum value was obtained (44.45). for a volume fraction of 0.03-0.05 (Fig. 4). Subsequently, similar results (46) were obtained with all 5 other fillers tested (Table 1). 

The creep stress was assumed to be shared between the polymer structure yield stress and the cell gas pressure. A finite difference model was used to model the gas loss rate, and thereby predict the creep curves. In this model the gas diffusion direction was assumed to be perpendicular to the line of action of the compressive stress, as the strain is uniform through the thickness, but the gas pressure varies from the side to the centre of the foam block. In a later variant of the model, the diffusion direction was taken to be parallel to the compressive stress axis. Figure 10 compares experimental creep curves with those predicted for an EVA foam of density 270 kg m used in nmning shoes (90), using the parameters ... 

Harden s (27) market survey of the growth of polyolefin foams production and sales shows that 114 x 10 kg of PE was used to make PE foam in 2001. The growth rate for the next 6 years was predicted as 5-6% per year, due to recovery in the US economy and to penetration of the automotive sector. In North America, 50% of the demand was for uncrosslinked foam, 24% for crosslinked PE foams, 15% for EPP, 6% for PP foams, 3% for EVA foams and 2% for polyethylene bead (EPE) foam. As protective packaging is the largest PE foam use sector, PE foam competes with a number of other packaging materials. Substitution of bead foam products (EPP, EPE, ARCEL copolymer) by extruded non-crosslinked PE foams, produced by the metallocene process was expected on the grounds of reduced costs. Compared with EPS foams the polyolefin foams have a lower yield stress for a given density. Compared with PU foams, the upper use temperature of polyolefin foams tends to be lower. Eor both these reasons, these foams are likely to coexist. 

Substitution of the stresses from Eqs. (5.86) to (5.87) into Eq. (5.84) and division by ei yields a prediction for the axial composite tensile modulus ... 

Princen [64] discovered that the yield stress, xQ, was strongly dependent on <(>, increasing sharply with increasing phase volume. x was also found to depend linearly on the surface tension. Variation with the mean droplet radius, however, did not match theoretical predictions this was reportedly due to the presence of a finite film thickness between adjacent droplets. 

Both criteria predict the same value of the yield stress for uniaxial tensile and compressive states, although it was well established that for... 

For many years, several authors have tried to explain and predict the yield stress of polymers (crosslinked or not), as a function of the experimental test parameters (T, e) and/or structural parameters (chain stiffness, crosslinking density). These models would be very useful to extrapolate yield stress values in different test conditions and to determine the ductile-brittle transition. 

Eyring s equation may be regarded as a good phenomenological description of yield stress as a function of test parameters (T, e), but it cannot be related to physical processes at the molecular scale. The equation can be used at high e for impact properties and for the prediction of the ductile brittle transition temperature. Eyring s equation can be modified with two sets of parameters if two relaxations are involved in the range of temperatures and strain rates (Bauwens-Crowet et al., 1972). 

No significant difference is found for the predictions of the yield stresses from these three criteria. However, Tresca s criterion is more widely used than the other two because of its simplicity. When two solid spherical particles are in contact, the principal stresses along the normal axis through the contact point can be obtained from the Hertzian elastic... 

Since wet foams contain approximately spherical bubbles, their viscosities can be estimated by the same means that are used to predict emulsion viscosities. In this case the foam viscosity is described in terms of the viscosity of the continuous liquid phase (tjo) and the amount of dispersed gas (4>). In dry foams, where the internal phase has a high volume fraction the foam viscosity increases strongly due to bubble crowding, or structural viscosity, becomes non-Newtonian, and frequently exhibits a yield stress. As is the case for emulsions, the maximum volume fraction possible for an internal phase made up of uniform, incompressible spheres is 74%, but since the gas bubbles are very deformable and compressible, foams with an internal vol-... 

Figures 2-4 show that no experimental data were recorded at low impeller shear rates. Experimental data began at y = 8.53 s4 for 21% solids, 5.15 s 1 for 23% solids, and 3.43 s 1 for 25% solids. The reason for the missing data is that the helical impeller viscometer has limitations. Owing to possible viscometer error, data were not recorded until the impeller torque was >10% of the full-scale torque. Therefore, no experimental data were recorded at low impeller rotational speeds. The lack of experimental data at low shear rates made comparison of rheologic models at low shear rates and the prediction of yield stress impossible.

Experimental rheologic data were fit to the power law, Herschel-Bulkley, and Casson models. The power law model does not predict yield stress. Yield stress for 21% grain slurries predicted by the Herschel-Bulkley model was a negative value, as shown in Table 6. Yield stress values predicted by the Herschel Bulkley model for 23 and 25% solids were 8.31 and 56.3 dyn/cm2, respectively. Predicted yield stress values from the Casson model were 9.47 dyn/cm2 for 21% solids, 28.5 dyn/cm2 for 23% solids, and 44.0 dyn/cm2 for 25% solids. 

Fig. 22. Comparison of predicted (solid lines [28]) and measured (circles [42, 43]) temperature dependence of yield stress...

Fig. 26. Comparison of the predicted (solid lines [46]) and measured (points [45]) strain rate dependence of the compressive yield stress of silica filled epoxy at different filler concentrations...

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