Shear stresses Plate

The lubrication mechanism of the bulk and thin film forms of boric acid is similar to that of the other known lamellar solids. Specifically, under shear stresses, plate-like crystallites of H3BO3 can align themselves parallel to the direction of relative motion and then slide over one another with relative ease to provide low friction, as shown in Figures 6.9 and 6.10 [24]. 

Normal Stress (Weissenberg Effect). Many viscoelastic fluids flow in a direction normal (perpendicular) to the direction of shear stress in steady-state shear (21,90). Examples of the effect include flour dough climbing up a beater, polymer solutions climbing up the inner cylinder in a concentric cylinder viscometer, and paints forcing apart the cone and plate of a cone—plate viscometer. The normal stress effect has been put to practical use in certain screwless extmders designed in a cone—plate or plate—plate configuration, where the polymer enters at the periphery and exits at the axis. 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

A sliding plate rheometer (simple shear) can be used to study the response of polymeric Hquids to extension-like deformations involving larger strains and strain rates than can be employed in most uniaxial extensional measurements (56,200—204). The technique requires knowledge of both shear stress and the first normal stress difference, N- (7), but has considerable potential for characteri2ing extensional behavior under conditions closely related to those in industrial processes. 

Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 6-1 illustrates this concept. A fluid is bounded by two large paraU plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U. The fluid continues to deform as long as the force is applied, unlike a sohd, which would undergo only a finite deformation. 

The force is direcdly proportional to the area of the plate the shear stress is T = F/A. Within the fluid, a linear velocity profile u = Uy/H is estabhshed due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient y = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. 

In spite of these problems, polymer melts have been sufficiently studied for a number of useful generalisations to be made. However, before discussing these it is necessary to define some terms. This is best accomplished by reference to Figure 8.2, which schematically illustrates two parallel plates of very large area A separated by a distance r with the space in between filled with a liquid. The lower plate is fixed and a shear force F applied to the top plate so that there is a shear stress (t = F/A) which causes the plate to move at a uniform velocity u in a direction parallel to the plane of the plate. 

It is assumed that the liquid wets the plates and that the molecular layer of liquid adjacent to the top plate moves at the same velocity as the plate whilst the layer adjacent to the stationary plate is also stationary. Intermediate layers of liquid move at intermediate velocities as indicated by the arrows in the diagram. The term shear rate is defined as the rate of change of velocity with cross-section (viz. d /dr) and is commonly given the symbol ("y). It is not altogether surprising that with many simple liquids if the shear stresses are doubled then the shear rates are doubled so that a linear relationship of the form... 

The layers in the plate-like structure of talc are Joined by very weak van der Waals forces, and therefore delamination at low shear stress is produced. The plate-like structure provides high resistivity, and low gas permeability to talc-filled polymers. Furthermore, talc has several other structure-related unique properties low abrasiveness, lubricating effect, and hydrophobic character. Hydrophobicity can be increased by surface coating with zinc stearate. 

Assuming that the melt viscosity is a power law function of the rate of shear, calculate the percentage difference in the shear stresses given by the two methods of measurement at the rate of shear obtained in the cone and plate experiment. 

Study of transverse shearing stress effects is divided in two parts. First, some exact elasticity solutions for composite laminates in cylindrical bending are examined. These solutions are limited in their applicability to practical problems but are extremely useful as checl oints for more broadly applicable approximate theories. Second, various approximations for treatment of transverse shearing stresses in plate theory are discussed. 

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

Shear-stress-shear-strain curves typical of fiber-reinforced epoxy resins are quite nonlinear, but all other stress-strain curves are essentially linear. Hahn and Tsai [6-48] analyzed lamina behavior with this nonlinear deformation behavior. Hahn [6-49] extended the analysis to laminate behavior. Inelastic effects in micromechanics analyses were examined by Adams [6-50]. Jones and Morgan [6-51] developed an approach to treat nonlinearities in all stress-strain curves for a lamina of a metal-matrix or carbon-carbon composite material. Morgan and Jones extended the lamina analysis to laminate deformation analysis [6-52] and then to buckling of laminated plates [6-53]. 

Now recognize an apparent contradiction in classical plate theory. First, from force equilibrium in the z-direction, we saw transverse shear forces and Qy must exist to equilibrate the lateral pressure, p. However, these shear forces can only be the resultant of certain transverse shearing stresses, i.e.. 

However, these transverse shearing stresses were neglected implicitly when we adopted the Kirchhoff hypothesis of lines that were normal to the undeformed middle surface remaining normal after deformation in Section 4.2.2 on classical lamination theory. That hypothesis is interpreted to mean that transverse shearing strains are zero, and, hence, by the stress-strain relations, the transverse shearing stresses are zero. The Kirchhoff hypothesis was also adopted as part of classical plate theory in Section 5.2.1. 

Accordingly, we find it difficult to determine the distribution of the transverse shearing stress in a beam, much less in a plate. Procedures for determining the approximate transverse shear stress distribution in plates are described in Section 6.5.2. 

Area of plate = A Figure 3.4. Shear stress and velocity gradient in a fluid... 

When a mbber block of rectangular cross-section, bonded between two rigid parallel plates, is deformed by a displacement of one of the bonded plates in the length direction, the rubber is placed in a state of simple shear (Figure 1.1). To maintain such a deformation throughout the block, compressive and shear stresses would be needed on the end surfaces, as well as on the bonded plates [1,2]. However, the end surfaces are generally stress-free, and therefore the stress system necessary... 

Model reactors, e.g. viscosimeters (cylinder or cone-plate systems), channel currents and jets (see Table 2) have been used very often to test the shear stress... 

Fig. 25. Total and viable cell concentrations of TB/C3 hybridomas versus duration of shear in a cone and plate viscometer (shear stress 208 Nm ). The error bars indicate the 95% confidence intervals [62]...

Flow chambers are based on the theory of parallel plates. They should provide a defined two-dimensional laminar flow of medium over a monolayer of cells. Based on this theory Levesque et al.[9] described an equation for the calculation of the shear stress. Shear stress i is then given as... 

In the Couette cell the shear stress varies signficantly with radial position across the gap as r2. Should a more uniform stress environment be required then the cone-and-plate geometry may be used [17]. An example apparatus is shown in Figure 2.8.7. 

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

Fig. 4.3.3 (a) Shear flow of a Newtonian fluid defined as the ratio of the shear stress and trapped between the two plates (each with a shear rate, (b) A polymeric material is being large area of A). The shear stress (a) is defined stretched at both ends at a speed of v. The as F/A, while the shear rate (y) is the velocity material has an initial length of L0 and an gradient, dvx/dy. The shear viscosity (r s) is (instantaneous) cross-sectional area of A. 

Design procedures for tube-plates are given in BS PD 5500, and in the TEMA heat exchanger standards (see Chapter 12). The tube-plate must be thick enough to resist the bending and shear stresses caused by the pressure load and any differential expansion of the shell and tubes. The minimum plate thickness to resist bending can be estimated using an equation of similar form to that for plate end closures (Section 13.5.3). 

The shear stress in the tube-plate can be calculated by equating the pressure force on the plate to the shear force in the material at the plate periphery. The minimum plate thickness to resist shear is given by ... 

For turbulent flow, we shall use the Chilton-Colburn analogy [12] to derive an expression for mass transfer to the spherical surface. This analogy is based on an investigation of heat and mass transfer to a flat plate situated in a uniform flow stream. At high Schmidt numbers, the local mass transfer rate is related to the local wall shear stress by... 

To better understand the way that shear stresses affect a system, we can look at an idealized system. In the example shown in Fig. 6.4 we will sandwich a layer of liquid between two metal plates. We hold the bottom plate stationary, while the top one can slide parallel to the bottom plate at some velocity, v, while maintaining continual contact with the liquid. Between the plates, the top layer of the fluid that is in direct contact with the top plate moves with it. The... 

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