Issue 51

F. Fabbrocino et alii, Frattura ed Integrità Strutturale, 51 (2020) 410-422; DOI: 10.3221/IGF-ESIS.51.30 413               1 1 1 1 1 1 1 1 2 2 r r r r R R r R R V V R R R R r r r V C uJ uJ det J dV [ u u J X u J X u J J X X u J X J X ] u det J dV t u det( J )ndA f u det( J )dV                                                                                 (5) where r V and r  are the volume and the loaded area in the referential configuration, C  is the elastic moduli matrix collecting stiffness coefficients, det( J ) is the determinant of a scalar metric representing the ratio of differential areas. It is worth noting that Eqn. (5) requires the definition of a proper advancing scheme to enforce the crack tip displacements and a rezoning procedure to move the current positions of the crack tip front, keeping the computational mesh undistorted during the whole calculation. In the proposed modeling, the crack tip motion is consistent to the crack growth criterion based on the instantaneous crack tip speed [22]. A fracture function depending from fracture tip variables, such as for instance ERR or SIF is required. According to ALE approach, the region enclosed into the  contour is able to describe the crack motion by introducing the following boundary conditions (Fig. 2): 0 0 T F T F X cos , Y sin ,              (6) where 0  is the crack propagation angle, whereas F   is the incremental scalar quantity computed at the current iteration step, by adopting a proper dynamic crack growth criterion. Angle prediction and crack tip displacements are evaluated by solving the following equations:   0 0 0 G f G, ,G     0 F g G   G F F f G        with 0 F    and 0 F G G G f f       (7) where F   is the fracture multiplier, G represents the fracture variable, i.e. total ERR or SIF, 0  is the angle of propagation depending from the crack kinking function F g , G f is the fracture function. Eqn. (6)-(7) are completed by additional boundary conditions applied to the geometry contour lines and time initial conditions (Fig.2): 1 2 0 0 0 0 0 on X , Y , X Y , X Y t                   (8) The elements of the remaining regions of the structures are stretched due to a Laplace based regularization method leading to a consistent transition of the mesh during the crack growth: 2 0 X X ,     2 0 X Y ,     (9) The variational form of the governing equations of the ALE problem is derived starting from Eqn. (8)-(9), introducing weight functions     1 2 w X ,Y ,w X ,Y and then integrating by part. Internal boundary conditions regarding the prescribed crack tip speed, defined on the basis of Eqn. (7) are taken into account for by means of non-ideal weak constraints based on the Lagrangian Multiplier Method (LMM). Therefore, the resulting equations for ALE formulation are:   0 R F F F V S S X : X dV Xn dA X dA                                (10) where X  is the vector containing the horizontal and vertical nodal point positions,   is the LMM vector. Governing equations, given by Eqn. (5), (7) and Eqn. (10), introduce a nonlinear set of equations, which are solved numerically, using