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For a viscoelastic material the second Piola Kirchhoff stress can be written as follows : where is the elastic stress and is the relaxation function. Here we consider the special case where the relaxation function is given by With this function chosen for the relaxation function, we can write the total stress as Introducing the internal variables, we can rewrite (5.4-3) as follows, In FEBio, , so is the long-term elastic response of the material.
The question now remains how to evaluate the internal variables. From equation (5.4-4) it appears that we have to integrate over the entire time domain. However, we can find a recurrence relationship that will allow us to evaluate the internal variables at a time given the values at time . The last term can now be simplified using the midpoint rule to approximate the derivate. In that case we find the recurrence relation: The following procedure can now be applied to calculate the new stress. Given and corresponding to time , find and corresponding to time :
- calculate elastic stress:
- evaluate internal variables:
- find the total stress: