RHEOLOGICAL PARAMETERS FOR VISCOELASTIC MATERIALS 
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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 TIMEVARYING FLOWSinusoidal timevarying 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 timevarying 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 2. Sinusoidal deformation of a cubical volume of fluid. The sinusoidal timevarying 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 stressstrainshear 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 Qfactor 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 Qfactor 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 springdashpot 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 cmgramsecond(cgs) units and the meterkilogramsecond (mks) or standard international (SI) units. The following is a table of units for the viscoelastic parameters.
Conversion factors:

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