As can be seen from Modeling, the equations governing the development and relaxation of residual stresses and deformations are highly coupled and nonlinear. In addition, time integrations are also involved, which provide a time-dependent attribute to the whole system. Having said that, it is yet possible to derive differential expressions that make the numerical treatment significantly easier. For this, let us assume that all fields are known at time , and that we want to calculate their values at the next time .

Following Chambers [6], the stress tensor can be written as the following:

(26–11)

where and denote the spherical and deviatoric contributions of the stress tensor, respectively, at time . The expressions for these contributions are as follows:

(26–12)

(26–13)

In Equation 26–11, is the thermal strain given by

(26–14)

As can be seen, Equation 26–12 and Equation 26–13 involve the reduced time , which can be evaluated as

(26–15)

where is the integration variable. Two auxiliary functions are also found in Equation 26–12 and Equation 26–13, and are given by

(26–16)

(26–17)

It is interesting to note that the Equation 26–15 – Equation 26–17 involve an integration over the time interval , which can be performed via a simple sub-integration algorithm, over subintervals.

Finally, following Markovsky et al. [24], the fictive temperature can be evaluated with the following formula:

(26–18)

with

(26–19)

Once again, a sub-integration algorithm can be used over subintervals.

When considering the relationships in Equation 26–11 – Equation 26–19, we see that several auxiliary fields are calculated, although they are not explicitly required by the user. These fields include the individual modes for the bulk and shear stress and , their corresponding functions and , and the individual fictive temperature . All these quantities are involved in equations that are solved locally. Therefore, piecewise continuous interpolation can be selected, which is also the simplest and cheapest interpolation from the point of view of computation.

It is important to guarantee a stable behavior of the solver for the calculation of residual stresses and deformations for any bulk and shear relaxation spectrum. Therefore, a technique similar to the DEVSS [17] method is invoked. It consists of adding a term to the momentum equation that is expressed as a function of displacements gradient, and to remove its counterpart expressed as a function of the deformation tensor. The evaluation of the deformation tensor is included in the simulation.