Simple shear definition

In the semi-dilute regime, the rate of shear degradation was found to decrease with the polymer concentration [132, 170]. By extrapolation to the dilute regime, it is frequently argued that chain scission should be nonexistent in the absence of entanglements under laminar conditions. No definite proof for this statement has been reported yet and the problem of isolated polymer chain degradation in simple shear flow remains open to further investigation. 

This definition gives y = G when applied to a simple shear flow (i.e., K = 0). 

Note 1 The relationships between the Xi andX, / = 1, 2, 3, in simple shear are a particular case of the general relationships given in Definition 1.3. 

Simple shear flow causes rotation, and also the liquid inside a drop subjected to shear flow does rotate. The drop is also deformed, and the relative deformation (i.e., the strain) is defined as D = (L — B)/ L + B) see Figure 11.7a for definition of L and B. For small Weber numbers, we simply... 

Here the absolute value of the velocity gradient is called the shear rate. For a newtonian fluid it is known that in this simple shear flow only the shear stress zxy is nonzero. However, it is possible that all six independent components of the stress tensor may be nonzero for a non-newtonian fluid according to its definition. For simple shearing flow of an isotropic fluid it can be proven [6] that the total stress tensor can have the general form... 

A final point to consider in our definition of strain is that the normal and shear components were summed to obtain a displacement in a given direction (Eqs. (2.9)-(2.11)). For large or finite strains, one would expect the displacements produced by one component to influence the others. For example, if a body were given a simple shear and then, with d , held constant, one superimposed a large normal strain the angle of shear y would change. If, however, both these... 

There are three types of melt behavior in a simple shear flow dilatant (D) (shear thickening) Newtonian (N), and pseudoplastic (P) (shear thinning). Similarly, in an extensional flow, the liquids may be stress hardening (SH), Troutonian (T), or stress softening (SS). By definition, the response considered here is taken at sufficiently... 

Rheology studies the relationship between force and deformation in a material. To investigate this phenomenon we must be able to measure both force and deformation quantitatively. Steady simple shear is the simplest mode of deforming a fluid. It allows simple definitions of stress, strain, and strain rate, and a simple measurement of viscosity. With this as a basis, we will then examine the pressure flow used in capillary rheometers. 

We see that the Coulomb yield criterion therefore defines both the stress condition required for yielding to occur and the directions in which the material will deform. Where a deformation band forms, its direction is one that is neither rotated nor distorted by the plastic deformation, because its orientation marks the direction that establishes material continuity between the deformed material in the deformation band and the undistorted material in the rest of the specimen. If volume is conserved, the band direction denotes the direction of shear in a simple shear (by the definition of a shear strain). Thus for a Coulomb yield criterion the band direction is defined by Equation (11.6). 

Development of an understanding of turbulence requires consideration of the details of turbulent motion. Much of our intuitive sense of fluid flow is based on what we can observe with the naked eye, and much of this intuitive sense can be applied to an understanding of turbulence, if we proceed with some care. We begin with the classical definition of simple shear flow, as shown in Figure 2-6. In this figure a Newtonian fluid is placed between two flat plates. The top plate moves with velocity Vx, requiring a force per unit area of plate surface (F/A) to maintain the motion. The force required is in proportion to the fluid viscosity. 

If we now consider the state of stress in an isotropic material, by definition the material has no preferred directions. In simple shear flow, we have... 

In defining the material functions that describe responses to simple-shear deformations, a standard frame of reference has been adopted. This is shown in Fig. 10.4. The shear stress <7is the component < i (equal to <7i2 because of the symmetry of the stress tensor), and the three normal stresses are <7u, in the direction of flow (xj), Gjj in the direction of the gradient and <733, in the neutral (x ) direction. As this is by definition a two-dimensional flow, there is no velocity and no velocity gradient in the Xj direction. However, in describing shear flow behavior, we will follow the conventional practice of referring to the shear stress as <7, and the shear strain as y, where neither symbol is in bold or has subscripts. 

The above definition is appropriate for mildly deformed droplets. However, for highly deformed droplets (Fig. 6.18b) the appropriate measure of deformation is D = UR, where L is now the half length and R is the radius of the undeformed droplet. The steady simple shear solution presented by Cox (1969), which in the limit of small deformation includes the theory of Taylor (1934), is... 

The situation is not so simple when these various parameters are time dependent. In the latter case, the moduli, designated by E(t)and G(t), are evaluated by examining the (time dependent) value of o needed to maintain a constant strain 7o- By constrast, the time-dependent compliances D(t) and J(t)are determined by measuring the time-dependent strain associated with a constant stress Oq. Thus whether the deformation mode is tension or shear, the modulus is a measure of the stress required to produce a unit strain. Likewise, the compliance is a measure of the strain associated with a unit stress. As required by these definitions, the units of compliance are the reciprocals of the units of the moduli m in the SI system. 

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