Rate of shear deformation

In a Newtonian material the rate of shear deformation is proportional to the shear stress except at very low stresses this is not true of elastomers which are accordingly termed non-Newtonian. 

Fluid Dynamics is the study of the relationship between forces resulting motions of a medium that is continuously deformable by shear forces. The fluid medium is called a Newtonian fluid if the shear stress is proportional to the rate of shearing deformation. 

The occurrence of velocity differences across the flow direction is called shearing, and the difference in velocity per unit normal distance (velocity gradient) is called shear rate or rate of shear deformation (dy/dt, or y see below). 

Formula, 239 Rate of heat release, 482 Rate of shear deformation, 618 Rate of straining of the outer surface, 236 Reclaimed composite profiles, 617 Reclaimed resin, 51 Recycled resin, 48, 87 Nylon, 48... 

It has also been demonstrated experimentally that for most fluids the results of this experiment can be shown most conveniently on a plot r versus dVIdy (see Fig. 1.5). As shown here, dV/dy is simply a velocity divided by a distance. In more complex geometriejs, it is the limiting value of such a ratio at a point. It is commonly called the shear rate, rate of strain, and rate of shear deformation, which all mean exactly the same thing. Four different kinds of curve are shown as experimental results in the figure. All four are observed in nature. The behavior most commori in nature is that represented by the straight line through the origin. This line is called newtonian because it is described by Newton s law of viscosity... 

Combining the Eqs. (82) and (83), we will obtained the expression for the rate of shear deformation ... 

Below the yield stress 0 the rate of shear deformation (strain) (d /dt) is zero in all three nonlinear models. This is also true of the shear strain /for the Saint-Venant and Bingham models, whereas in the Prandtl-Reuss model /increases slowly with the shear stress t from zero to the yield stress 0. At this point, the value of /increases limitless as a step function, as it does in the two other models (Figure 2.15). 

Here, the exponent n characterizes the dependency of the apparent viscosity 77 on the shear stress t. For = 1, the system follows the ideal Newton equation, and rj = rj. However, for n < 1 the viscosity increases with increasing shear stress (dilatancy), while for n > 1 the viscosity decreases with increasing shear stress (structural viscosity) both are of great significance for many day-water suspensions (Hawlader et al, 2003). The dependence of apparent viscosity rj and rate of shear deformation dy/dt on shear stress r are shown in Figure 2.16a and b, respectively. 

Figure2.16 Dependency of apparent viscosity tj (a) and rate of shear deformation D = (dv/dt) = (d ydt) (b) on shear stress rfor viscoelastic liquids and suspensions, respectively.

If /3 < 1, the materials are termed pseudoplastic, and their viscosity decreases with the rate of shear deformation as the structure of the flowing material gradually becomes more ordered. If /3 > 1, the material displays so-called Bingham behavior (see Section 2.4.1.3), and the viscosity increases with increasing shear deformation rate. Such systems are also characterized by a yield stress 6 below which there is no deformation (shear deformation rate D = 0).lf P=l, ideal Newtonian behavior is observed, with a= rj. 

Extensional Viscosity. In addition to the shear viscosity Tj, two other rheological constants can be defined for fluids the bulk viscosity, iC, and the extensional or elongational viscosity, Tj (34,49,100—107). The bulk viscosity relates the hydrostatic pressure to the rate of deformation of volume, whereas the extensional viscosity relates the tensile stress to the rate of extensional deformation of the fluid. Extensional viscosity is important in a number of industrial processes and problems (34,100,108—110). Shear properties alone are insufficient for the characterization of many fluids, particularly polymer melts (101,107,111,112). 

In contrast to the behaviour of a solid, for a normal fluid the shear stress is independent of the magnitude of the deformation but depends on the rate of change of the deformation. Gases and many liquids exhibit a simple linear relationship between the shear stress r and the rate of shearing ... 

During the course of the experiment, pad surface roughness was measured by contact profilometry. Removal rate was found to be directly proportional to pad roughness (Fig. 11). Microscopic examination of the pad surface clearly showed the progressive smoothing away of the upper pad asperities (Figs. 12a-12d). No evidence of shear deformation of the asperities was... 

The completely general case is that of a fluid for which the relationship between shear stress and rate of shear is not linear and may also depend on both the duration of the shear and the extent of the deformation produced. Thus it is possible to divide non-Newtonian systems into three broad categories ... 

Table 4.1 presents typical values of viscosities of some common materials. One can see from this table that viscosity varies over several orders of magnitude. One may also note that, although the table lists a viscosity for glasses, the magnitude of the viscosity clearly suggests that the deformation of glasses at room temperatures will be extremely small so glasses are best treated as solids under normal conditions. (Recall the discussion in Section 4.1c of time scales and their relation to whether a substance is defined as a liquid or a solid.) Typical rates of shear for some familiar processes are shown in Table 4.2. 

DILATANCY. The property of certain colloidal solutions of becoming solid, or setting, under pressure. Also known as "inverse plasticity since there is an increase in the resistance to deformation with increase in the rate of shear. 

Fig. 29 Typical example of shear deformation zone. Thin film of PMMA at 53 °C and a strain rate of 2 x 10-3 s 1...

In the case of a fluid, the deformation y itself is not important since a fluid (by definition) will flow to take the shape of the vessel that contains it. Only the time derivative of y is important, and this quantity, also called the rate of flow or the rate of shear, is represented by the symbol y... 

At time t + At the rectangular fluid element is translated in the x direction and deformed into a parallelogram. We define the rate of shear as -dS/dt, where <5 is the angle shown in the figure. 

At 1-atm pressure in the surroundings, polysaccharide deformation and flow are normally initiated either by gravity or an applied shear rate (y) solvent (water) only flows under temperature (T) and concentration (c,) gradients. When T)i is constant or independent of the rate of shear (y in s 1) or stress (t), the flow is Newtonian. Very dilute polysaccharide dispersions are characterized mostly by Newtonian flow. At moderate concentrations, ti, may decrease (shear-thinning synonymous with pseudoplastic) or increase (shear-thickening synonymous with dilatant) nonlinearly with y for these dispersions, is replaced with (the apparent viscosity). Low DP and uniform distribution of substituents are conducive to tH high DP and nonuniform distribution are conducive to. A high T a is believed to elicit the human oral sensation of thickness. ... 

First, the rate of shear, which is not linear with the shearing stress due to the non-Newtonian behaviour, varies with the different types of polymer. The processability of different polymers with an equal value of the MI may therefore differ widely. An illustration of this behaviour is given in Fig. 15.14. Furthermore the standard temperature (190 °C) was chosen for polyethylenes for other thermoplastics it is often less suitable. Finally, the deformation of the polymer melt under the given stress is also dependent on time, and in the measurements of the melt index no corrections are allowed for entrance and exit abnormalities in the flow behaviour. The corrections would be expected to vary for polymers of different flow characteristics. The length-diameter ratio of the melt indexer is too small to obtain a uniform flow pattern. 

Consider a liquid contained between two parallel plates, each of area A cm2 (Figure 8-4). The plates are h cm apart and a force of P dynes is applied on the upper plate. This shearing stress causes it to move with respect to the lower plate with a velocity of v cm s-1. The shearing stress x acts throughout the liquid contained between the plates and can be defined as the shearing force P divided by the area A, or PI A dynes/cm2. The deformation can be expressed as the mean rate of shear y or velocity gradient and is equal to the velocity difference divided by the distance between the plates y = v/h, expressed in units of s-1. 

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