where the subscriptions within the arrays indicate node numbers, and\(N_{I}\) express the element vector-field shape function correlated with node \(I\). \(\mathbf{u}_{I}\) and \(\varphi_{I}\) are the displacement and phase-field values at node \(I\), respectively. In this work the standard 8-node quadrilateral element employed. So, \(m=8\). The corresponding derivative quantities are given by