Issue 41

M.F. Funari et alii, Frattura ed Integrità Strutturale, 41 (2017) 524-535; DOI: 10.3221/IGF-ESIS.41.63 528 The interfaces moving mesh strategy Until the crack onset condition is not satisfied the ALE equations are inactive and the position of the computational mesh points is expressed in the Fixed Material Frame FM  identified by the x 1 -  x 2 coordinates (Fig. 2a), which coincides, at this stage, with the Moving (M) Frame M  ( X 1 -X 2 ) . It worth noting that the difference between FM  and M  is mainly in the mesh element number involved in the numeric model. In particular, in FM  a coarse mesh is required to identify the of the crack onset position, whose accuracy is verified introducing more finite elements by using a remeshing procedure in the M  configuration (Fig. 2b). This procedure leads to the evaluation of the crack initiation position 1 i X which sets to zero Eq. 3. The mathematical description of the moving mesh modeling is defined by a mapping operator Φ , which relies a particle in a Fixed Referential Frame R  and the one in current moving coordinate system (Fig. 2c). The mesh motion in terms of displacement field, at the i -th interface, is described as the difference between material 1 i X and the referential coordinates ( ) x : ( ) ( ) ( ) ( ) 1 1 , i i i i i i X X t t t t on x x x D = - =F - W (4) where i i B h W = ´ represents the region in which the debonding mechanisms are produced. In order to reduce mesh distortions, produced by the mesh movements, rezoning or smoothing equations are introduced to simulate the grid motion consistently to Laplace based equation which is, in the case of one dimensional domain for both Static (S) and Dynamic (D) cases, defined on the basis of the following relationships [15]:   2 i i , i ,t X         (S),   3 2 i i , ,t X t           (D) (5) Eqs.(5) should be completed by means of boundary equations to reproduce the crack tip motion on the basis of the assumed crack growth criterion, namely i f g . In particular, for a fixed position in which the crack initiation occurs, different boundary conditions should be introduced to enforce internal or external debonding mechanisms. In particular, once the position of the crack onset is determined, i.e.at 1 1 i X X  , a geometrical debonding with length equal to 2  is introduced in the numerical model, producing two potential finite crack tips in which debonding phenomena can be triggered (Fig. 2b). In order to described the evolution of debonding phenomena the following boundary conditions should be introduced, which are completed by the Kuhn-Tucker optimality conditions:     1 1 0 0 0 0 0 0 i i i i i i i i T T f T f T f i i i i i i i i T T f T f T f X ,g with X g X g , at X X X ,g with X g X g , at X X                                         (6) where i T X  indicates the displacement of the crack tip front of the k -th debonded interface, i f g is the fracture function defined in Eq,(3) and   /    represents the value of the    variable evaluated at left (-) or right (+) crack tip position. However, additional relationships are required for the ALE formulation to reproduce the crack tip motion and boundary conditions. In particular, the displacements of the computational nodes are assumed to be zero at the boundaries of the structure, whereas small portions, close to the crack tip for the left and right debonding mechanisms, namely i   and i   , are assumed to be moved rigidly, enforcing the computational nodes at the extremities to have the same displacements. This choice ensures that the NLs involved in the debonding mechanisms are constrained to a small portion containing the process zone, reducing the total complexities of the model. Consequently, the following boundary conditions should be considered in the analysis (Fig. 2b):