The small dimensionless parameter \(\eta\) models an artificial residual stiffness of a totally broken phase, \(\varphi=1\), and is essentially needed to prevent numerical difficulties. For numerical reasons (stability) \(\eta\) may not be chosen too small. However, too large values for\(\ \eta\) overestimate the bulk energy in fractured zones.
By applying variational principles, the minimization problem, Eq.(6) can be reformulated as the system of the stress equilibrium equation, div\(\mathbf{\sigma}\left(\mathbf{u},\varphi\right)=0\).
The second-order Cauchy stress tensor, \(\mathbf{\sigma}\),