Constitutive models are integrated using explicit adaptive integration scheme with local substepping.
The constitutive model forms an ordinary differential equation of the form
The equation is for finite time step size solved using the Runge-Kutta method. Solutions that correspond to the
second- and third- order accuracy of Taylor series expansion are given by
where
The accuracy of the solution is estimated following Fehlberg as the difference between the second- and third- order solutions. The time step size
is accepted, if
where is a prescribed error tolerance. If the step-size is accepted,
is considered as a solution for the given
time step and the new time step size
is estimated according to Hull
If the step-size is not accepted, the step is re-computed with new time step size
In the case the prescribed minimum time step size or the prescribed maximum number of time substeps is reached, the finite element program is asked
to reject the current step and to decrease the size of the global time step.