Issue 12

M. Angelillo et alii, Frattura ed Integrità Strutturale, 12 (2010) 63-78; DOI: 10.3221/IGF-ESIS.12.07 67 In (6)  ° represents the slope at the inflection point of the curve depicted in Fig. 2a ,  ° the slope of the dotted lines in Fig. 2b. Notice that while  and k are constants,  °,  °, are actually mesh dependent, that is their values must be tuned depending on the current mesh size h and should tend to infinity as h tends to zero, linearly with1/ h . Although the density (13) has a shape close to that of the approximate surface energy density defined in [2], the main difference between them is that all parameters in the function (6) have a clear physical meaning and no shape parameter is needed. F RACTURE NUCLEATION IN 1 D he study of a simple 1d example can clarify the main ingredients of our approach. Consider the long strip of brittle material depicted i n Fig. 3, modeled as a 1d continuous bar, fixed at the left end and subject to a given displacement u at the other end. Figure 3 : 1d bar: FE discretization. The bar has been discretized into two finite elements of length L/2. Inside the elements the material is linearly elastic and characterized by the elastic energy density  =1/2 EA  2 . The spirit of our approximation is to consider that fractures, that is discontinuities in the displacements, may occur at the interface between elements (the skeleton of the mesh), that is at point 2. To allow for the interface energy at node 2 we must duplicate node 2 into two spatially coincident nodes in the reference configuration  : 2’, 2”. Therefore we are reduced to a two degrees of freedom structure. The two degrees of freedom we choose are {u,a} and are defined in terms of the displacements u(2’), u(2”) as follows: u= u(2’), a = u(2”)- u(2’). Then the total energy  is 2 2 2 2 c 1 / 2 EA 2 u / L 1 / 2 EA 2 (u u) / L a = 0 E (u,a) 1 / 2 EA 2 u / L 1 / 2 EA 2 (u u a) / L G a 0 a 0                The graph of the energy  (u, a ) for a particular value of the given displacement u is depicted in Fig. 4. In the figure are visible two minima: one is ½ EA u 2 L, the “elastic” minimum for a =0 and u=u/2 (point A), that is the one for which there is no discontinuity; the other one is G c , the minimum for a =u and u=0 (point B), that is the bar is completely fractured. For u small the “elastic” minimum is smaller than the “completly fractured” minimum. As the displacement u is gradually increased from zero, with the system sitting in the “elastic” minimum, the energy minimum rises and becomes larger than G c . As it is evident fro m Fig. 4, wathever large be the elastic minimum with respect to G c , there is always an energy barrier to sormount to go from the local to the global minimum, that is there is no descent direction from A to B. Figure 4 : 3d plot of the energy  for u large. T