Page 491 - Introduction to Continuum Mechanics

Non-Newtonian Fluids 475

Let

we can write Eq. (8.4.6) as

or

The above equation is the integral form of constitutive equation for the linear Maxwell fluid
written in terms of the infinitesimal strain tensor E (instead of the rate of deformation tensor
D). The function/(s) in this equation is known as the memory function. The relation between
the memory function and the relaxation function is given by Eq. (8.4.7).
The constitutive equation given in Eq. (8.4.8) can be viewed as the superposition of all the
stresses, weighted by the memory function/^), caused by the deformation of the fluid particle
(relative to the current time) at all the past time (t' = - » to the current time /)•
For the linear Maxwell fluid with one relaxation time, the memory function is given by

For the linear Maxwell fluid with discrete relaxation spectra, the memory function is:

and for the Maxwell fluid with a continuous spectrum