Issue 51

F. Frabbrocino et alii, Frattura ed Integrità Strutturale, 51 (2020) 410-422; DOI: 10.3221/IGF-ESIS.51.30 412 The crack growth is predicted by the use of a moving mesh methodology based on an arbitrary Lagrangian-Eulerian (ALE) formulation. In particular, two coordinate systems are introduced, known as Referential (R) and Moving (M) ones (Fig. 1). A one to one relationship between the R and the M is defined by the following mapping operator   :   M R X X ,t        1 R M X X ,t       (1) where R X  and M X  identify the positions on the computational points in R and M configurations, respectively. More details on the derivation of the governing equations are reported in [21]. According to ALE formulation, the time and spatial derivatives of a generic physical field, in referential and material configurations can be related by the following relationships:   1 1 X R X R f J f f f X f X,t f X J f                                with   R r d X X ,t dt        (2) where X   represents the relative velocity of the grid points in the material reference system. Analogously, starting from Eqn.(2), the second time derivative is evaluated recursively as :   2 X X X X X X f f f X fX f X X f X X                                        (3) where   is the gradient operator function. Figure 1 : Relationship between R and M coordinate systems. The governing equations in the material configuration can be written by means of the principle of d’Alembert, taking into account virtual works of inertial, external and internal forces: V V V u dV u u dV t u ndA f u dV                           (4) where, n  is the unit normal vector,  is the mass density,   is the Cauchy stress tensor, t  is the traction forces vector on the free surface, f  is the volume forces vector dV and dA are the volume and the loaded area in the material configuration. Substituting Eqn. (2)-(3), consistently to ALE formulation, the governing equations given by Eqns. (4)-(5) should be reformulated to take into account the transformation rule between the Lagrangian and referential coordinate system (Fig.4):