Issue 47

M.F. Funari et alii, Frattura ed Integrità Strutturale, 47 (2019) 277-293; DOI: 10.3221/IGF-ESIS.47.21 282 former is basically fixed or at least re-meshed in those cases in which large distortions occur. The mapping between the two configurations is governed by the following expressions:   M R X X        1 R M X X       (7) where : R M C C    with   X Y      is assumed to be invertible with continuous inverse,   , M R X X   are material and referential positional vector functions with   T M M M X X Y   and   T R R R X X Y   . In addition, the transformation between the RC and the MC is also expressed by gradient operator defined as a function of the Jacobian matrix, on the basis of Eq.(7):     1 M R J        X Y R R X Y R R X Y J X Y                       (8) According to the ALE methodology, rezoning or regularization equations are required to modify the mesh position, reducing mesh distortions. In addition, on the external boundaries, boundary and initial conditions should be enforced to constraint the mesh motion to do not exceed the structural domain. Therefore, according to a mesh regularization method based on Laplace’s method, the following relationships hold: 2 2 0 0 0, X Y U X on S             (9) where 2   is the nabla operator and S U is the external boundary of the core system. Additional equations are required to modify the mesh positions of moving boundaries, which describe preexisting internal cracks. However, in order to quantify the crack advance in terms of displacement and angle of propagation, a crack growth criterion should be introduced. Similarly to what proposed to simulate the core-to-skin interfaces debonding, a fracture function based on local-Griffith approach criterion defined in terms of the ERR is assumed. Moreover, the Maximum Energy release rate criterion is used [24] for the crack angle prediction. In particular, the crack tip boundary conditions are defined in terms of incremental displacements by means of the following equations:     2 1 cos , sin , with tan and T T T T J X Y arc J                         (10) where  is the angle of crack propagation   , T T X Y     are the crack tip components along ( X R , Y R ) coordinate system, (J 1 , J 2 ) are the components of the J integral along normal (e 1 ) and tangential (e 2 ) directions. Moreover, T   is the incremental displacement of the tip, which is determined by solving the following constrained optimization problem: 0, 0 0 T F T F T F f f f            (11) N UMERICAL I MPLEMENTATION he model is implemented numerically by using a finite element approximation. In particular, the structural formulation of the skins follows a Timoshenko beam model, whereas the core is based on a 2D plane-stress description. Skins and core are connected through internal forces arising from the TSL of the moving interface elements. It is worth noting that the governing equations of the skins are not affected by the ALE interface formulation, since it contributes as weak contribution in the structural formulation. Contrarily, in the case of the core region, governing equations should be modified by changing the variables and their derivatives from the MC to the RC, by using T