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### Material Functions Derived from Oscillatory Tests

In a typical sinusoidal oscillation experiment, the applied stress and resulting strain wave forms can be described as follows:

= 0 cos t

= 0 cos (t -

where 0 is the stress amplitude

0 is the strain amplitude

=2f is the angular frequency

t is time

is the phase lag (loss angle)

Figure 3. Sinusoidal wave forms for stress and strain functions.

The phase lag and amplitude ratio ( 0/ 0) will generally vary with frequency, but are considered material properties under linear viscoelastic conditions. For an ideal solid, = 0º, and the response is purely elastic, whereas for a Newtonian fluid yielding a purely viscous response, = 90º.

The material functions can be described in terms of complex variables having both real and imaginary parts. Thus, using the relationship:

Then the stress and strain can be expressed as follows:

where is termed the complex strain amplitude. The shear storage modulus [or storage modulus, for short], which represents the in-phase (elastic) component of oscillatory flow, is defined as:

The out-of-phase (viscous) component is termed the shear loss modulus [or loss modulus, for short]:

The complex shear modulus, G*, is then defined as follows:

so that:

 Figure 4. Vectorial representation of moduli.

The function measures the relative importance of viscous to elastic contributions for a material at a given frequency.

Additionally, a complex viscosity, *, can be defined using the complex strain rate, = j, such that:

or alternatively

where ' is termed the dynamic viscosity, and is equivalent to the ratio of the stress in phase with the rate of strain (0 sin ) to the amplitude of the rate of strain (0). The term '' is referred to as the out-of-phase viscosity, and is equivalent to the ratio of the stress 90º out of phase with the rate of strain ( 0 cos ) to the amplitude of the rate of strain ( 0) in the forced oscillation.

Finally, an absolute shear modulus is defined as the ratio of the amplitude of the stress to the amplitude of the strain in forced oscillation ( simple shear), or:

Alternatively, forced oscillation experiments can be equivalently described in terms of compliance, as opposed to the derivation above based on the modulus. Similar arguments lead to the following analogous terms:

complex shear compliance, J* The ratio of the complex strain ( *) to complex stress ( *) in forced oscillation (simple shear).

shear storage compliance, J' The ratio of the amplitude of the strain in phase with the stress (0 cos ) to the amplitude of the stress ( 0) in forced oscillation (simple shear).

shear loss compliance, J'' The ratio of the amplitude of the strain 90º out of phase with the stress (0 sin ) to the amplitude of the stress (0) in forced oscillation (simple shear).

absolute shear compliance, |J*| The ratio of the amplitude of the strain (0) to the amplitude of the stress (0) in forced oscillation (simple shear).

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