Power Law Viscoelastic Functions

On a log-log plot, many viscoelastic functions are approximately linear over significant time ranges. This suggests the use of a relaxation function of the form 

Observe that U(0, given by (1.2.33 a) is not defined in this case. However, the complex modulus can be defined by (1.5.2c), giving 

Putting z = ico and rotating it to a point x on the positive real axis, the integral can be evaluated in terms of the Gamma function (Al.4.1). Rotating back, we find the 

The fact that 0 is frequency-independent is an interesting feature of the model. Experimentally, this is found to be a good approximation for nearly all high polymers over a broad range of frequencies [Kolsky (1956), Pipkin (1972)]. From (1.5.34) we obtain 

Ar( +p)r( -p) Apn which is most easily shown by going backwards, by means of [see (1.5.29b)] 

---