Determination of Spectra from Viscoelastic Functions Using First-Order Approximations

7 DETERMINATION OF SPECTRA FROM VISCOELASTIC FUNCTIONS USING FIRST-ORDER APPROXIMATIONS 

In the previous sections we have seen that the compliance and relaxation viscoelastic functions can be expressed in terms of the retardation and relaxation spectra, respectively. However, the spectra cannot be determined beforehand they can only be calculated from viscoelastic functions. For example, N s) and N s) in Eqs. (9.5) and (9.11) can be obtained by using the expressions 

Because the relaxation spectra are similar for transient and dynamic relaxation viscoelastic functions, H t) can also be obtained from the storage relaxation modulus. The plot of the kernel of the integral of Eq. (9.8), x /(l + (o x ), versus logcax is a sigmoidal curve that intercepts the ordinate axis at 0.5 and reaches the value of 1 in the limit cox oo (see Fig. 9.5). The kernel can be approximated by the step function 

Therefore the storage relaxation modulus can be written approximately as 

The value of — In co in the lower limit of the integration arises from the fact that here x = 1 /co. Then the relaxation spectrum is approximately given by 

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