Complex rate of strain

Ratio of complex stress (a ) to complex rate of strain (y ) in the forced oscillation of a material... 

Viscosity is the ratio of a stress to a strain rate [Eq. (I l-9)j. Since the complex modulus G has the units of stress, it is possible to define a complex viscosity t] as the ratio of G to a complex rate of strain ... 

The complex viscosity, which is the complex stress to complex rate of strain ratio ... 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

Experience with applying the Reynolds-stress model (RSM) to complex flows has shown that the most critical term in (4.52) to model precisely is the anisotropic rate-of-strain tensor 7 .--1 (Pope 2000). Relatively simple models are thus usually employed for the other unclosed terms. For example, the dissipation term is often assumed to be isotropic ... 

In an important class of materials, called Newtonian, this relationship is linear and one parameter—the viscosity—specifies the constitution of the material. Water, low-viscosity fluids, and gases are Newtonian fluids. However, most polymeric melts are non-Newtonian and require more complex constitutive equations to describe the relationship between the stress and the rate of strain. These are discussed in Chapter 3. 

The complex relationship between the configurational distortion produced by a perturbation field in polymers and the Brownian motion that relaxes that distortion make it difficult to establish stress-strain relationships. In fact, the stress at a point in the system depends not only on the actual deformation at that point but also on the previous history of deformation of the material. As a consequence the relaxation between the stress and strain or rate of strain cannot be expressed by material constants such as G or /, as occurs in ideal elastic materials, but rather by time-dependent material functions, G t) and J t). It has been argued that the dynamics of incompressible liquids may be characterized by a function of the evolution of the strain tensor from the beginning up to the present time. According to this criterion, the stress tensor would be given by (3,4)... 

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) ... 

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... 

In terms of the complex viscosity 7, the phase difference between the stress and the rate-of-strain can be expressed. Generally the stress induced by the rate-of-strain given by Eq. (4.31) can be written as... 

Solutions of polymers exhibit a number of unusual effects in flows. Complex mechanical behavior of such liquids is governed by qualitatively different response of the medium to applied forces than low-molecular fluids. In hydrodynamics of polymers this response is described by rheological equation that relates the stress tensor, a, to the velocity field. The latter is described by the rate-of-strain tensor, e... 

The viscosity of a liquid is a parameter that measures the resistance of that liquid to flow. For example, water has a very low viscosity, while honey has a much larger or thicker viscosity. Newtonian fluids have constant values of viscosity, which means that the stress in a flowing liquid is proportional to the rate of strain of the flow. Non-Newtonian liquids do not have constant viscosity, but rather have viscosities that can be functions of the rate of strain, the total amount of strain, and other flow characteristics. Huids are usually non-Newtonian as a result of microscopic additives such as polymers or particles. These additives alter the viscosity of a liquid and impart nonlinear flow behavior, such as viscoelasticity. The non-Newtonian behavior of many complex liquids is described thoroughly in several texts, for example [1]. In this entry we focus on behavior and applications of polymer solutions in microfluidic devices. For example, DNA is a biopolymer that is common in microfluidics applications such as gene sequencing and amplification. 

The ratio of the imaginary part of to P is often called snrface viscosity, although in this case the complex natnre of E arises naturally from the diffusion problem and is umelated to any relationship between interfacial stress and the rate of strain. The reason for this terminology is that the analysis of wave motion in Chapter 5 is carried out without any explicit consideration of snrfactants, but... 

More complex models can be formed by connecting a number of such elements in series. However, even these are only linear visco-elastic models in which the rate of straining is directly proportional to the stress. For polymers at typical structural levels of load, the stress-strain rate is often highly non-linear. A molecular interpretation of this can be found in a thermally activated rate process model involving motion of chain segments... 

In more generalised flows, both the stress and the rate of deformation (strain rate) are tensor quantities, and the constitutive relationship between these may be very complex (Schowalter, 1978 Bird et a/., 1987a). The relationship between stress and shear rate frequently depends on the shear rate (flow rate), as is the case in a simple shear thinning fluid. However, for some fluids the stress may depend on the strain itself as well as on the rate of strain, and such fluids show some elasticity or memory behaviour, in that their stress at a given time depends on the recent strain history such fluids are... 

It may be shown that any combination of linear elements must be linear, so any models based on these linear elements, no matter how complex, can represent only Unear responses. Just how reaUstic is a linear response Its most conspicuous shortcoming is that it permits only Newtonian behavior (constant viscosity) in equilibrium viscous flow. For most polymers at strains greater than a few percent or so (or rates of strain greater than 0.1/s), a linear response is not a good quantitative description. Moreover, even within the... 

Memory effects are revealed by experiments in which a complex fluid is subject to a time-dependent shear rate. Included under this rubric are measurements of the startup stress when a constant shear rate is suddenly imposed on an initially stationary system, and the stress when a system subject to some nonzero constant rate of strain suddenly has the rate of strain increased, decreased, or reversed. A prominent feature in measurements of stress on sudden imposition of a large rate of strain is stress overshoot, in which the stress first increases to a value much larger than its steady-state value, and then relaxes back to its steady-state value. Contrariwise, if the shear rate applied to a polymer fluid is held constant for a long time and then suddenly reduced, the stress may show undershoot the stress declines to a value well below its steady-state value and then increases back to its steady-state value. Related features have been seen for N. Bird, et al. also note measurements on responses to superposed flows, in particular the combination of a constant rate of shear flow with an oscillatory shear parallel or perpendicular to the constant shear(7). Bird, et al. further assert that multiple oscillations around the steady-state stress are sometimes observed before the steady state is attained. Recent studies involving step strains or oscillatory shear superposed on steady shear are reported by Li and Wang(8). 

Studies of stress overshoot on sudden imposition of constant rate of strain include Osaki, etal. 0) and Inoue, etal. ). Osaki, etal. also report the time dependence of N. Inoue, et al. note a potential artifact perturbing stress measurements, namely shear-induced phase separation. Representative experiments on double strain rates are presented by Oberhauser, et al. 2) and by Wang and Wang(13). The observed stress has a complex time dependence including overshoot and undershoot ... 

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