Fig. 1

a Comparison between the discrete discontinuity of the LEFM theory (left) with the smeared discontinuity of the PF model (right); b 1D approximation function which smear out the discontinuity, the damage \({\mathfrak {d}}\) follows the exponential based function \({\mathfrak {d}} = \text {e}^{-\mid x \mid / l}\). c Phase field approach for thin-walled structures adopting the solid shell approach