Boltzmann Integral and Excitation-response Theory

The viscoelastic response is commonly described by using a form of Boltzmann s hereditary integral, referred to as the excitation-response theory (Gurtin and Sternberg 1962 Yamamoto 1972 Christensen 2003). 

If the input x t) is given by a collection of step functions as shown in Fig. 2.13b, the response is written as 

Then for a general form of the input function x(t), we have 

2 Stress Relaxation and the Relaxation Spectra Generalized Maxwell Model 

First we consider a simple uniaxial response. If a strain s is given by 

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