Law with Effective Time

Recall from Chapter 6 that the constitutive law describing the stress and strain relation of a viscoelastic material can be written as 

one can consider the modulus at temperature T and time t to be the same as the modulus at a reference temperature To at a reduced time At a small time increment later, dt, the modulus at temperature T has changed to a new value correspondingly later in reduced time at the reference temperature 

Physically, this expression represents that all relaxation times in a time increment at temperature T are 1/a times slower/faster than those occurring in the reduced time increment at the reference temperature. Integrating one obtains an expression for reduced time, or ejfective time as it is often called, as 

Note that the shift factor is a function of time according to the temperature history as the temperature changes, so does the shift factor. To account for a temperature history, the constitutive law can thus be written in effective time space as 

Mapping this constitutive law to the real time domain results in 

---