Plastics models

Note that different perfectly plastic models for three dimensional case are considered in (Mosolov, Myasnikov, 1971). 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

All chemists use models. Beginning chemistry students use plastic models to help them understand and visualize the structures of molecules. Recently, both students and experienced researchers have begun to use chemical drawing programs for the same purpose. 

Bond density surfaces are also superior to conventional models when it comes te describing chemical reactions. Chemical reactions can involve many changes in chemica bonding, and conventional formulas are not sufficiently flexible to describe what happen (conventional plastic models are even worse). For example, heating ethyl fonnate t( high temperatures causes this molecule to fragment into two new molecules, foraii( acid and ethene. A conventional formula can show which bonds are affected by ths reaction, but it cannot tell us if these changes occur all at once, sequentially, or in soms other fashion. 

Make a sketch of each decalin isomer, and label the orientation of the bridgehead hydrogens with respect to each ring (equatorial or axial). Build a plastic model of each isomer and determine its conformational flexibility (a flexible molecule can undergo a ring flip, but a locked molecule cannot). Is flexibility responsible for stabihty ... 

Examine the structure of cyclodecane, a molecule which contains the same number of carbons as decalin, but only has one ring (a model of the most stable conformation is provided). Compare it to cis and trans decalin. Make a plastic model of cyclodecane. Is it flexible or locked What conformational properties of cyclodecane can be anticipated from the properties of decalins What properties cannot be anticipated How do you account for this ... 

Electronic marketplace/E-commerce In addition to the many databases available and person-to-person contacts, E-commerce in plastics has been conducted through suppliers web sites or the dot-commerce independent web sites that link material buyers with sellers in transactions or auction formats. During the year 2000 five plastic producers/suppliers and various elastomer producers/suppliers created a new and important business model of a joint-venture web site. It provides multiple companies to join forces to do business. This is a strategy some observers call competition and others regard as just another form of selling in. an electronic format. Regardless of how it is perceived, the model will help propel e-commerce into the mainstream of processor procurement due to the size and wealth of the companies involved. The plastic model example is the largest online business-to-business site todate. 

The rheological properties of a particular suspension may be approximated reasonably well by either a power-law or a Bingham-plastic model over the shear rate range of 10 to 50 s. If the consistency coefficient k is 10 N s, /m-2 and the flow behaviour index n is 0.2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham-plastic model ... 

What will be the pressure drop, when the suspension is flowing under laminar conditions in a pipe 200 m long and 40 mm diameter, when the centre line velocity is 1 m/s, according to the power-law model Calculate the centre-line velocity for this pressure drop for the Bingham-plastic model. 

R. B. Jackson and M. M. Caldwell, Integrating resource heterogeneity and plant plasticity modelling nitrate and phosphate uptake in a patchy soil environment. J. Ecol. 84 891 (1996). 

If the value of p is set equal to 1/2 in the Sisko model, the result is equivalent to the Bingham plastic model ... 

The Bingham plastic model can describe acrylic latex paint, with a yield stress of 112 dyn/cm2, a limiting viscosity of 80 cP, and a density of 0.95 g/cm3. What is the maximum thickness of this paint that can be applied to a vertical wall without running ... 

You must determine the horsepower required to pump a coal slurry through an 18 in. diameter pipeline, 300 mi long, at a rate of 5 million tons/yr. The slurry can be described by the Bingham plastic model, with a yield stress of 75 dyn/cm2, a limiting viscosity of 40 cP, and a density of 1.4 g/cm3. For non-Newtonian fluids, the flow is not sensitive to the wall roughness. 

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 ... 

The Bingham plastic model usually provides a good representation for the viscosity of concentrated slurries, suspensions, emulsions, foams, etc. Such materials often exhibit a yield stress that must be exceeded before the material will flow at a significant rate. Other examples include paint, shaving cream, and mayonnaise. There are also many fluids, such as blood, that may have a yield stress that is not as pronounced. 

A pipeline is installed to transport a red mud slurry from an open tank in an alumina plant to a disposal pond. The line is 5 in. sch 80 commercial steel, 12,000 ft long, and is designed to transport the slurry at a rate of 300 gpm. The slurry properties can be described by the Bingham plastic model, with a yield stress of 15 dyn/cm2, a limiting viscosity of 20 cP, and an SG of 1.3. You may neglect any fittings in this pipeline. 

A procedure analogous to the one followed can be used for non-Newtonian fluids that follow the power law or Bingham plastic models (Darby and Melson, 1982). 

Determine the size of the smallest sphere of SG = 3 that will settle in applesauce with properties given in Problem 19, assuming that it is best described by the Bingham plastic model [Eq. (11-49)]. Find the terminal velocity of the sphere that has a diameter twice this size. 

For greater concentrations of fine particles the suspension is more likely to be non-Newtonian, in which case the viscous properties can probably be adequately described by the power law or Bingham plastic models. The pressure drop-flow relationship for pipe flow under these conditions can be determined by the methods presented in Chapters 6 and 7. 

It is well known that enhanced deposition in the first few airways occurs due to the turbulence produced. Turbulent diffusion is accounted for by using factors (ratio of observed deposition to calculated diffusion deposition) to correct the diffusion deposition. These had formerly been measured by Martin and Jacobi (1972) in a dichotomous plastic model of the upper airways. The data used here are from measurements performed by Cohen (1986) using hollow casts of the upper bronchial tree which included a larynx. This cast was tested using cyclic flow with deposition measured for 0.03, 0.15 and 0.20 urn diameter particles. Her turbulent diffusion factors are used in the calculation here (14 for generation 0, and 2 for generations 1 to 6). 

Figure 9. A plastic model of a nasal cavity showing the different positions (numbers 1-5) of the sensors. Reprinted with permission from ref. 14. Copyright 2003 American Chemical Society.

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