Issue 12

M. Angelillo et alii, Frattura ed Integrità Strutturale, 12 (2010) 63-78 ; DOI: 10.3221/IGF-ESIS.12.07 66        \ ]]u([[ )( )u, (E ds da    (4) under the boundary conditions described above. Space discretization To discretize the problem we split the domain into triangles, as shown in Fig. 1, and identify  with the part of the skeleton of the triangulation that is not at the boundary of  . In order to duplicate nodes and edges of the skeleton of the mesh, we introduce special interface elements with zero thickness placed along the edges of the continuous elements. A topological view of these elements is depicted i n Fig. 13a for a plane strip. Figure 1 : Typical BVP: strip with a straight crack in “mixed” mode. In order to give to the skeleton the ability to choose optimal crack paths, the positions of the nodes of the triangulation in the reference configuration  , are taken as further variables and can be moved to minimize the total energy. Approximate interface (surface) energy density In the numerical applications we approximate the equilibrium trajectory of the system by considering crack propagation as based on critical points of the energy. To get out of possible small energy wells, either numerical (due to the finite element mesh) or physical (due to fracture initiation) on introducing the limit tensile stress  °, the limit shear stress  ° the thoughness  =G c , the shear stiffness k and the functions (Fig.2) 1 2 2 2 2 ( ) 2arctan( ) 2 2 ( ) arctan( ) log (1 ( ) ) 2 2 a a a e e k b k b b k b k                                             (5) where  = 1.90086 , we adopt the following approximate relaxed form of the surface energy density: )( )( ),( 2 1 b a ba     (6) (a) (b) Figure 2 : Comparison between exact and relaxed interface energies.