Elastic-viscous analogy

For an isotropic, linearly viscous, incompressible mataial, the constitutive equation is easily obtained from Eq. (11.17) by invoking the elastic-viscous analogy the strain is replaced by the strain rate, E is replaced by the shear viscosity t], and V becomes 1/2 (for an incompressible material), giving (for the x direction)... 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

The Poisson-MaxweU theory that the viscous flow of a hquid is analogous to the yieldihg of a solid under forces exceeding the elastic limit ( 3.IX F) has received much attention recently. According to Maxwell, if P is the shearing force per unit area, i the time, 6 the deformation, n the shear modulus ( 4.IX F), then for a solid free from viscosity ... 

The viscous properties of a smectic A are characterized by the same five independent viscosities that characterize the nematic. As we shall see, however, the elastic properties of the smectic are very different from those of a nematic, and some flows permitted to the nematic are effectively blocked for the smectic. For smectic C, for which the director is tilted with respect to the layers, there are some 20 viscosities needed to characterize the viscous properties (Leslie 1993). Formulas for these, derived using a method analogous to that used for nematics by Kuzuu and Doi (1983, 1984) can be found in Osipov et al. (1995). The smectic phase for which rheological properties are most commonly measured is smectic A, however, and hereafter we will limit our discussion to it. 

The mechanical response of viscoelastic materials to mechanical excitation has traditionally been modeled in terms of elastic and viscous components such as springs and dashpots (1-3). The corresponding theory is analogous to the electric circuit theory, which is extensively described in engineering textbooks. In many respects the use of mechanical models plays a didactic role in interpreting the viscoelasticity of materials in the simplest cases. However, it must be emphasized that the representation of the viscoelastic behavior in terms of springs and dashpots does not imply that these elements reflect the molecular mechanisms causing the actual relaxation... 

Note that the simple Hooke s law behavior of the stress in a solid is analogous to Newton s law for the stress of a fluid. For a simple Newtonian fluid, the shear stress is proportional to the rate of strain, y (shear rate), whereas in a Hookian solid, it is proportional to the strain, y, itself. For a fluid that shares both viscous and elastic behavior, the equation for the shear stress must incorporate both of these laws— Newton s and Hooke s. A possible constitutive relationship between the stress in a fluid and the strain is described by the Maxwell model (Eq. 6.3), which assumes that a purely viscous damper described by Eq. 6.1 and a pure spring described by Eq. 6.2 are connected in series (i.e., the two y from Eqs. 6.1 and 6.2 are additive). 

Equation (6-94) has been found to be valid for a number of filled systems up to a value of f of about 0.3, whereas (6-95) and (6-96c) can be used at somewhat higher concentrations. These equations were first used to describe the viscosity of liquids with suspended solid particles. In fact equation (6-94) was derived using basic hydrodynamic principles. Equations of this type have been "borrowed" to be used for the elasticity of filled elastomers, based on the analogy between steady viscous flow and elastic deformation as described in equations (3-4) and (2-14), respectively. Certainly an additional justification... 

Previously introduced, the thermodynamic surface tension 7 represents the elastic resistance to surface dilation. Furthermore, two types of viscosities are defined within the interface, a dilational viscosity and a shear viscosity. For a surfactant monolayer, the surface shear viscosity rjS is analogous to the three-dimensional shear viscosity the rate of yielding of a layer of fluid due to an applied shear stress. The phenomenological coefficient s represents the surface dilational viscosity, and expresses the magnitude of the viscous forces during a rate expansion of a surface element. Figures 10a and 10b illustrate the difference between the two surface viscosities. 

Recent work showed [30] that in this case EC sets in with a continuous transition from the homogeneous state directly, similarly to the classical configuration of case A. It has been shown that the pattern forming mechanism, including the role of the elastic, dielectric, viscous and CH terms, is analogous to that of case A, and the standard model is applicable. The anal34ical one-mode neutral-curve expression for homeotropic initial alignment [31] is as follows ... 

The four-parameter model provides a crude quahtative representation of the phenomena generally observed with viscoelastie materials instantaneous elastie strain, retarded elastic strain, viscous flow, instantaneous elastie reeovery, retarded elastie reeovery, and plastic deformation (permanent set). Also, the model parameters ean be assoeiated with various molecular mechanisms responsible for the viscoelastic behavior of linear amorphous polymers under creep conditions. The analogies to the moleeular mechanism can be made as follows. 

These relationships are known as Newton s Law of viscous flow a is termed the fluidity and -q the dynamical shear viscosity. Newton s Law is analogous to Hooke s Law, except shear strain has been replaced by shear strain rate and the shear modulus by shear viscosity. As shown later, this analogy is often very important in solving viscoelastic problems. In uniaxial tension, the viscous equivalent to Hooke s Law would be a=7] ds/dt), where q is the uniaxial viscosity. As v=0.5 for many fluids, this equation can be re-written as <7-=3Tj(de/dO using t7=t /[2(1+v)], the latter equation being the equivalent of the interrelationship between three engineering elastic constants, (fi=E/[2il + v)]). 

As a third example, consider viscous flow between coaxial cylinders, as shown in Fig. 5.5. The outer cylinder is rotating with a linear velocity wb, while the inner cylinder is at rest. The particles in the fluid move in concentric circles around the axis of rotation. In polar coordinates, only the tangential component of fluid velocity, u, is non-zero. Using the elastic analogy, from Eq. (4.26) one can write... 

The simplest approach to analyzing viscous flow is to continue with the previous assumptions. The primary assumption is that the motion is slow, with negligible inertial forces. This allows terms involving acceleration, which lead to non-linearities in the equations, to be neglected and thereby preserves the analogy with static elasticity. 

In the above analysis of slow viscous flow, the materials were assumed to be incompressible. This is not, however, necessary to derive the flow equations. If the compressibility is included, two viscosity coefficients are needed. This is analogous to the need for two constants to describe the elastic behavior of isotropic materials. 

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