RHEOLOGICAL PARAMETERS FOR VISCOELASTIC MATERIALS

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In large spaces where the dimensions of the microstructure of a liquid are very small compared to those of the confinement space, the liquid can be treated as a continuum. In this case, the fundamental parameters, shear stress, shear strain, and shear rate (time rate of shear strain) are continuous throughout the liquid. STEADY VS. TIME-VARYING FLOW   European Beads Consider a small cubical volume of material. Under the action of forces that produces shear stress, the shape shifts to a parallelogram. Figure 1 shows such a volume at rest and immediately following application of force. The change in shape has two components, elastic deformation E and slippage S. The elastic deformation is accompanied by storage of elastic energy within the structure of the material, while the slippage is associated with a continuous input of viscous energy. When the force is removed, the deformed material undergoes a partial recovery of shape as the elastic energy is recovered; the shape change due to slippage is permanent. Thus, in steady flow the displacement component S continues to increase, and measurements of the nontime-varying force and velocity provide no information about the elastic energy component. In a time-varying flow, however, the elastic energy component also varies with time and may be either increasing or decreasing, while the viscous energy is always increasing. Consequently, the relation between the time-varying force and velocity reflects both the elastic and viscous properties of the material.FIGURE 1. Diagram showing a small cubical volume in shear. The displacement D due to deformation is composed of two parts: an elastic part E and a sliding part S.  With a constant force F, E remains constant but S continues to increase. When the force is removed, E diminishes to zero while S remains unchanged.

Figure 1 can be used for defining the shear stress, shear strain and shear rate (1, 2 and 3). H is the height of the cubicle volume in Figure 1.  
SINUSOIDAL TIME-VARYING FLOWSinusoidal time-varying flow provides a basis for clear differentiation of the elastic and viscous properties of the material, and for understanding the role of viscoelasticity in more complex time-varying flow, such as pulsatile flow. Figure 2 shows the cubical volume of fluid in oscillatory shear with accompanying sinusoidal functions descriptive of the shear rate and the shear stress.
FIGURE 2Sinusoidal deformation of a cubical volume of fluid. The sinusoidal time-varying shear rate and shear stress differ in phase by the angle phi as shown.  
The sinusoidal time variations in the stress (tau) and shear rate (gamma dot) are as shown in the Figure 2. The phase angle (phi)= 0° if the fluid is purely viscous, (phi)=90° if it is purely elastic, while (phi) lies between 0° and 90° if it is viscoelastic. With the sinusoidal time variation proportional to , the size and phase relation between the stress, strain, and shear rate are described using complex numbers (4, 5 and 6).
The components of the complex shear stress can be written as:  

 

Using these terms the complex coefficient of viscosity is defined by equation (8). Similarly complex rigidity modulus G* (9) can be obtained by taking the complex ratio of the shear stress to the shear strain as given in equations 4 and 5. In equation (9), G’ is the storage modulus and G” is the loss modulus. The complex coefficient of viscosity is related to the complex rigidity modulus by equation (10). Other viscoelastic constants can be found that are descriptive of the viscoelastic stress-strain-shear rate relations. These include the complex compliance (11) and the complex fluidity (12).
In view of the number of descriptive constants for viscoelastic materials and the simple and direct interrelations, one may find mixed usage, in which one constant is selected that is descriptive of energy loss and the second of energy storage. One of the more common mixtures (in earlier publications) is to use the viscosity and the storage modulus. In this mixture,. In some earlier work the complex viscosity is referred to as the complex “dynamic” viscosity. Use of the word “dynamic” has largely been dropped from current usage.
The instantaneous viscous energy loss and elastic energy storage vary with time at twice the frequency of the flow. Integrating over a complete cycle of the flow gives the energy dissipated per unit volume per cycle (13). Consequently, the average power dissipated per unit volume is given by equation (14). The elastic energy stored during the cycle builds to a maximum followed by recovery. This maximum is given by equation (15).
The components of the shear stress can be described in terms of these energies: the viscous stress is the rate of energy dissipation per unit volume, per unit shear rate. The elastic stress is the maximum energy stored during the cycle per unit volume, per unit strain.
   A Q-factor that is descriptive of the energy flow per cycle within a unit volume of material is defined by equation (16), using equations (11) and (13). The Q-factor can also be expressed in terms of the storage and loss moduli (17).
RELAXATION TIMEA simple Maxwell model element, consisting of an ideal elastic spring attached to an ideal dashpot, can serve to represent the viscoelastic behavior of energy storage and loss in a fluid for a specific measurement condition (frequency, shear rate, etc.). Analysis of this model gives the complex viscosity in terms of the dashpot constant and the spring constant.FIGURE 3. The spring-dashpot Maxwell model.  
The relaxation time is a measure of the time required for the energy stored in the spring to shift to the dashpot and dissipate. Equation (19) is used to determine the “apparent” relaxation time for any pair of viscosity and elasticity values for a liquid. If the single Maxwell model is an exact analog for the liquid, then Tr will be the same value for all measurement conditions. Otherwise, Tr will change as the frequency or flow amplitude is changed.
 UNITS FOR VISCOELASTICITYThe two commonly used units for describing viscoelastic parameters are the cm-gram-second(cgs) units and the meter-kilogram-second (mks) or standard international (SI) units. The following is a table of units for the viscoelastic parameters.

Conversion factors:

1 Pascal = 10 dynes/cm^2
1 Pascal-second (Pa-sec) = 10 Poise

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