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5.5.1 Reduced Relaxation Functions
Reduced relaxation functions are monotonically decreasing functions of time that satisfy and . Many different forms of reduced relaxation functions are available in FEBio, given in the FEBio User's Manual. The simplest and most commonly used relaxation function is the exponential function , where is the relaxation constant. In viscoelasticity theory it is common to use a combination of relaxation functions with distinct relaxation constants , described as a Prony series of the form The coefficients are normalized by to enforce . Alternatively, we could have written This type of relaxation function is said to have a discrete spectrum of coefficients corresponding to each .
It is also possible to define a continuous relaxation spectrum such that the reduced relaxation function is given by To satisfy the continous relaxation spectrum must satisfy
For example, Fung  proposed a relaxation spectrum of the form When substituted into eq.(5.5.1-3) it produces where is the incomplete gamma function and is the exponential integral function, which satisfy . An alternative model proposed later by Fung  is which produces
A generalization of Fung's earlier continuous relaxation spectrum may be derived from the work of Malkin  who proposed to use a function proportional to . If we constrain this spectrum to the range it takes the form When substituted into (5.5.1-3) this continuous relaxation spectrum produces the reduced relaxation function In the limit as , the expression of eq.(5.5.1-10) reduces to eq.(5.5.1-6).
Another example for a continuous relaxation spectrum is the exponential spectrum which produces where is the modified Bessel function of the second kind, of order 1.