Issue 12

M. Angelillo et alii, Frattura ed Integrità Strutturale, 12 (2010) 63-78; DOI: 10.3221/IGF-ESIS.12.07 68 The numerical method we propose to get out of the “elastic pocket” is to adopt for the interface energy the relaxed form depicted in Fig. 2a. With such cohesive type energy the system will access a descent direction at u= L  °/E. The stress thresold is set again to infinity as soon as the system accesses the gradient flow from A to B; the fractured minimum is reached through steepest descent. In this example the mesh skeleton is fixed, therefore the crack can open up only at point 2. A way to make the crack move is to consider the mesh 1, 2, 3 variable in the sense that the coordinate x(2) of point 2 is taken as a further variable in the energy:               2 2 2 2 c 1 / 2 EA u / x 2 1 / 2 EA (u u) / L x 2 a = 0 E (u,a, x 2 ) 1 / 2 EA u / x 2 1 / 2 EA (u u a) / L x 2 G a 0 a 0                 and the minimum is searched with respect to u, a, x(2). Actually all the fractured minima are equivalent if the bar is homogeneous and there is no reason for the bar to move the crack from the midpoint. This can be easily verified by preminimizing  (u,a,x(2)) with respect to u (that gives u°=u x(2)/L) and substituting u° for u in the energy:   2 2 c 1 / 2 EA u / L a = 0 E (a) E (u u ,a, x 2 ) 1 / 2 EA / L (u a) + G a 0 a 0              that is actually idependent of x(2). To make the node movements worthwile we consider that the bar is weackened at x=0.366L (Fig. 5), that is the thoughness G c is not homogeneous and has a minimum at x=0.366L. The bar is discretized into 10 elements in such a way that there is no node at the weackest point. The energy is then a function of 27 variables: 9 node displacements, 9 displacement jumps, 9 node positions: {u(i), a(i), x(i)}, i=2,...,10. A value of the displacement u, large enough to have G c <1/2EA/L u 2 , is assigned and the minimum state is searched with a descent method using as starting state the displacement field depicted i n Fig. 6a. Figure 5 : Weackened 1d bar: FE discretization. (a) (b) (c) Figure 6 : Three snapshots of the evolution of the displacement u and of the node positions a(i), from starting point to convergence (diplacement is represented along the vertical axis).