The parameter \(\varphi_{c}\) shows the maximum value of damage that is allowed to heal during unloading. For a material point which goes through the damage \(\varphi\geq\varphi_{c}\) the resulting stress-strain curves will be nonlinear, with the amount of departure from linearity dependent on the specific form of the degradation function.

3. Numerical solution

This section is applied to the two-dimensional implementation, which is performed under plane strain conditions in a finite element code within the Abaqus by means of UEL and UMAT subroutines. The purpose of this section is to introduce a straightforward implementation procedure. The eight-node quadrilateral finite elements are used.

Finite element approximation

In more compact form, the displacement field and phase-field are visualized inside an eight-node quadrilateral element by the following relations: