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**

=2*f* is the angular frequency

*t* is time

**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 (

**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).