Integral Formulation of Viscoelastic Problems

Among the equations that govern a viscoelastic problem, only the constitutive equations differ formally from those corresponding to elastic relationships. In the context of an infinitesimal theory, we are interested in the formulation of adequate stress-strain relationships from some conveniently formalized experimental facts. These relationships are assumed to be linear, and field equations must be equally linear. The most convenient way to formulate the viscoelastic constitutive equations is to follow the lines of Coleman and Noll (1), who introduced the term memory by stating that the current value of the stress tensor depends upon the past history 

Another approach to formulating the constitutive stress-strain relationship is through a simple variables change, % = t — x. After integration by parts, we obtain 

Since G t) is zero for negative values of t, we can set the upper limit of integration equal to infinity, that is. 

This integral representation has the advantage that its formulation does not require an appeal to models based on dashpots and springs, and it is purely formal. Alternatively, the roles of stress and strain could be reversed, giving 

The requirement of isotropy permits the representation of the fourth-order tensor in terms of two material functions in such a way that the stress-strain relationship becomes 

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