Issue 51

F. Frabbrocino et alii, Frattura ed Integrità Strutturale, 51 (2020) 410-422; DOI: 10.3221/IGF-ESIS.51.30 414 a customized finite element program based on COMSOL Multiphysics [23]. In particular, proper script files are carried out to manage the steps involved in the procedure, regarding the geometry variation due to the crack propagation and the mesh enrichment in the process zone. Moreover, the resulting algebraic equations are solved by using an implicit time integration scheme based on a variable step-size backward differentiation formula (BDF), in which a coupled approach, in which no splitting operators in the solving procedure for plane stress and ALE formulation are considered. More details on the implementation procedure are reported in [20]. Figure 2 : ALE formulation: boundary and initial conditions. E VALUATION OF THE ERR AND CRACK GROWTH CRITERION n the present section, the main formulas regarding the computation of the ERR and crack growth criterion are summarized. It is worth noting that the present formulation is quite general to be utilized in conjunction with other existing formulas and procedures typically developed in Fracture Mechanics to predict crack growth. In order to compute the ERR components, a path independent J integral formulation developed in [24] has been considered into the proposed numerical scheme by means of the following expression:         0 0 dS = dS dV c k  k i i ,k k i i ,k i i ,k i i ,k V V J lim W K n t u lim W K n t u u f u u u                                         (11) where   is a contour enclosing the crack tip, W and K are the strain and the kinetic energy densities and k n is the outward normal, , and i i i u u u   are the displacement velocity and acceleration of the material point, respectively,  is the material density,  is an arbitrary contour going around the crack tip. From the numerical point of view, case it is convenient to consider the following expression taking the limit of 0 V   , i.e 0    :     dS dV c k k i i ,k i i ,k i i ,k V J W K n t u u f u u u                          (12) The ERR components, evaluated with reference to the global coordinates system, can be projected on the local tip coordinates by using the following coordinate transformation rule (Fig.3): 0 0 0 0 x X y Y J cos sin J J sin cos J                        (13) with 0 0 0 0 x X Y y X Y J cos J sin J J sin J cos J          (14) I