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474 Generalized Linear Maxwell Fluid with a Continuous Relaxation Spectrum.

8.4 Generalized Linear Maxwell Fluid with a Continuous Relaxation Spectrum.

The linear Maxwell fluid with a continuous relaxation spectrum is defined by the constitu-
tive equation:

where the relaxation function 0(f) is given by

The function //(A)/A is the relaxation spectrum. Eq. (8.4.2a) can also be written

As we shall see later that the linear Maxwell models considered so far are physically
acceptable models only if the motion is such that the components of the relative deformation
gradient (i.e., deformation gradient measured from the configuration at the current time t, see
Section 8.5 ) are small. When this is the case, the components of rate of deformation tensor
D are also small so that [see Eq. (v), Example 5.2.1]

where E is the infinitesimal strain measured with respect to the current configuration.
Substituting the above approximation in Eq. (8.4.1) and integrating the right hand side by parts,
we obtain

The first term in the right hand side is zero because 0(«) = 0 for a fluid and E(0=0 because
the deformation is measured relative to the configuration at time t. Thus,

Or, letting t—t' = s, we can write the above equation as