Approximate Relationships Between Viscoelastic Functions

Once a relaxation (retardation) spectrum is obtained from a relaxation (creep compliance) viscoelastic function, any other function can be obtained. Alternatively, approximate methods have been developed to calculate viscoelastic functions from one another (10). By taking into account 

Calculate the relaxation time spectrum from the equation 

Assume that is continuous, non-negative, and such that 

Let us assume further that the function to be found (continuous, nonnegative, and of bounded variation in the neighbourhood of each point) is in L (0, oo), i.e., it is modulus-integrable, and 

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The exact formal relationships between the various viscoelastic functions are conveniently expressed using Fourier or Laplace transform methods (cf. Section 5.4.2). However, it is often adequate to use simple approximations due to Alfrey in which the exponential term for a single Kelvin or Maxwell unit is replaced by a step function, as shown schematically in Figure 5.18. 

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