Shear Strain and Rate

Vrc y,y r 11 o - volume fraction of dispersed and matrix phase, respectively - volume fraction of the crosslinked monomer units - volume fraction of phase i at phase inversion - maximum packing volume fraction - percolation threshold - shear strain and rate of shearing, respectively - viscosity - zero-shear viscosity - hrst and second normal stress difference coefficient, respectively... 

Figure 2.15 Responses of shear strain /and rate of shear strain (dy/dt) to increasing shear stress for three nonlinear rheological models.

Shear deformations. In these tests the Interface Is deformed in shear to a certain extent or at a certain rate and the shear stress required is measured as a function of the shear strain and/or time. The interfacial shear modulus G or the Interfaclal shear viscosity t]° cem be calculated using [3.6.17] or [3.6.16], respectively. By analogy to the technique described above, stress relaxation experiments can also be carried out from these one Ccm obtain an interfacial shear stress relaxation modulus G ] ) = y(t)/(Ax/Ay). For solld-llke interfacial layers fracture can be studied as a function of time by deforming the interface at various shear rates and measuring the required shear stress as a function of the shear strain. 

In these cases the relative velocity of the shearing plates is not constant but varies in a sinusoidal manner so that the shear strain and the rate of shear strain are both cyclic, and the shear stress is also sinusoidal. For non-Newtonian fluids, the stress is out of phase with the rate of strain. In this situation a measured complex viscosity (rf) contains both the shear viscosity, or dynamic viscosity (t] ), related to the ordinary steady-state viscosity that measures the rate of energy dissipation, and an elastic component (the imaginary viscosity ij" that measures an elasticity or stored energy) ... 

Assuming that there is no slip at the solid/liquid interfaces, the velocity of movement, v, of an clement of fluid relative to the lower plate increases linearly from zero at z = 0 to V at z-= h, as shown in Figure 8.1(b). The ratio V/h is called the shear rate, or the rate of shear strain, and is denoted by D. The force needed to maintain the steady motion is proportional to the area, A, of the plates (ignoring edge effects), and the ratio F/A is called the shear... 

Newton s law of viscosity states that there is a linear relation between the shear stresses and rates of strain. Let us first examine this law for the case of simple shear where there is only one strain component. For explicitness consider the planar Couette problem of a steady shear flow generated by the parallel motion of one infinite plate at a constant speed U with respect to a second fixed infinite plate, the plates being separated by a small distance h with the pressure p constant throughout the fluid (Fig. 2.2.1). The role of boundary conditions in... 

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... 

The measurement of rheological properties for non-Newtonian, lipid-based food systems, such as dilatant, pseudoplastic, and plastic, as depicted in Figure 4.1, are much more difficult. There are several measurement methods that may involve the ratio of shear stress and rate of shear, and also the relationship of stress to time under constant strain (i.e., relaxation) and the relationship of strain to time under constant stress (i.e., creep). In relaxation measurements, a material, by principle, is subjected to a sudden deformation, which is held constant and in many food systems structure, the stress will decay with time. The point at which the stress has decayed to some percentage of the original value is called the relaxation time. When the strain is removed at time tg, the stress returns to zero (Figure 4.8). In creep experi-... 

In the capillary and Couette (as well as rotating parallel-plate) geometries, a gradient of shear exists in the radial direction. The cone and plate geometry has the advantage that constant shear strain and shear rate are applied at all radial distances. When the cone angle m is very small, tu- < 0.1 rad, analysis of the equations of motion indicates that the shear stress on the plate is... 

It is well known that the continuum theory in the Navier-Stokes equations only validates when the mean free path of the molecules is smaller than the characteristic length scale of the gas flow. Otherwise, the fluid will no longer be in thermodynamic equilibrium and the linear relationship between the shear stress and rate of shear strain cannot be applied. The commonly used... 

In the elastic response for a Hookian liquid the stress-strain relationship is a(t) = Gyif). For the Newtonian liquid it is shear strain and shear rate at time t, while a t) is the shear stress. 

Shear stress n. Force per imit area acting in the plane of the area to which the force is applied. In an elastic body, shear stress is equal to shear modulus times shear strain. In an inelastic fluid, shear stress is equal to viscosity times the shear rate. In viscoelastic materials, shear stress will be a function of both shear strain and shear rate. 

One sees in Eqs. (6.48) and (6.51) that the strain rate is linearly proportional to the stress and inversely proportional to the grain size. In Eq. (6.48), the expression is given in terms of shear strain and macroscopic shear stress. The above expressions explain why large-grained materials are preferential for creep applications at high temperatures. 

Experiments show that in steadily sheared foams and concentrated emulsions, the viscosity coefficient n depends on the rate of shear strain, and in most cases the Herschel-Bulkley equation [931] is applicable ... 

The term [dxi/dA 2] represents the deformation of the material and is defined as the shear strain y. Thus, the shear rate is the rate of deformation or the rate of shear strain and is expressed as reciprocal seconds (sec ). 

In these equations, y t) and y t) are the shear strain and shear rate at any time t, and a t) is the shear stress at the same time. A single constant completely defines the mechanical response in each case, the shear modulus G for the solid and the shear viscosity rj for the liquid. To reiterate, the current stress depends only on the current strain for the solid and only on the current strain rate for the liquid. The history of loading plays no part in either case. Hooke s law accurately describes the small-strain behavior of many solid materials, and Newton s law is broadly applicable to small-molecule liquids except near the glass transition. 

In most treatises,"- 3 the strain tensor is defined with all components smaller by a factor of 2 than inequation 3, so that 711 = dui/dxi and 721 = du2/bx + bui/bx ). However, such a definition makes discussion of shear or shear flow somewhat clumsy either a practical shear strain and practical shear rate must be introduced which are twice 721 and 721 respectively, or else a factor of 2 must be carried in the constitutive equations. Since most of the discussion in this book is concerned with shear deformations, we use the definition of equation 3 which follows Bird and his school" and Lodge. - This does cause a slight inconvenience in the discussion of compressive and tensile strain, where a practical measure of strain is subsequently introduced (Section F below). In older treatises on elasticity, strains are defined without the factor of 2 appearing in the diagonal components of equation 3, but with the other components the same. 

In an inelastic fluid, shear stress is equal to viscosity times the shear rate. In viscoelastic materials, shear stress will be a function of both shear strain and shear rate. 

Viscosity (/i) is a very important fluid property that is responsible for the resistance fluids offer to flow. When a fluid particle is subjected to a shear stress, it undergoes a change in shape as shown in Fig. 4.3 (b). The ratio (dx/dy) is a measure of this change in shape and is called shear strain. The rate of shear strain (R ) is... 

Since r, is nearly constant, it follows that the shear strain and shear rate will also be nearly constant. From B in spherical coordinates (Table 1.4.1 andeq. 1.4.13)... 

While we have considered only shearing deformations, the superposition principle applies to deformations having any kinematics. To generalize Eq. 4.3 to account for this, we need only replace the shear stress, shear strain, and shear rate by the corresponding tensorial quantities. The tensorial form of the Boltzmann superposition principle can then be used to determine... 

This behavior is typical of a thixotropic substance where the structure is a function of the shear strain and die shear strain rate. It is imperative therefore that cognizance be taken of any pre-shearing prior to an experimental observation. Furthermore evidence will be presented later confirming that shearing can alter the initial or equilibrium state. A technique of investigating the rate of recovery of the structure will also be discussed. 

By analogy with Eq. (3.1), we seek a description for the relationship between stress and strain. The former is the shearing force per unit area, which we symbolize as as in Chap. 2. For shear strain we use the symbol y it is the rate of change of 7 that is involved in the definition of viscosity in Eq. (2.2). As in the analysis of tensile deformation, we write the strain AL/L, but this time AL is in the direction of the force, while L is at right angles to it. These quantities are shown in Fig. 3.6. It is convenient to describe the sample deformation in terms of the angle 6, also shown in Fig. 3.6. For distortion which is independent of time we continue to consider only the equilibrium behavior-stress and strain are proportional with proportionality constant G ... 

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. 

If the maximum resolved shear stress r and the plastic shear strain rate y are defined according to (it is assumed that the Xj and Xj directions are equivalent)... 

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