Issue 24

G. Cricrì, Frattura ed Integrità Strutturale, 24 (2013) 161-174 ; DOI: 10.3221/IGF-ESIS.24.17 166 2024A-T351 All Three Directions Equivalent Diameter (  m) Volume Fraction (%) Standard Deviation Nearest Neighbour (  m) Nearest Neighbour Standard Deviation (  m) Minimum Separation Distance (  m) Minimum Separation Distance Standard Deviation (  m) Average Diam of particles in size category (  m) All Sizes 0.76 0.33 10.61 8.01 8.55 8.09 1.75 1:2 0.13 0.06 16.14 12.11 14.65 12.39 1.41 2:3 0.10 0.05 35.18 26.48 32.82 27.30 2.41 3:4 0.07 0.04 62.38 46.78 59.61 48.71 3.41 4:6 0.12 0.07 58.90 40.68 53.76 42.22 4.87 6:8 0.11 0.08 93.54 64.45 86.12 66.44 6.88 8:10 0.09 0.09 130.27 96.68 122.14 101.91 8.86 10+ 0.14 0.21 126.95 97.90 111.94 99.72 13.23 Tabella 1 : Defects size distribution. As shown above, the cell dimension, and in particular the cell height, is a fundamental parameter because, when the strain localization occurs, the cell height coincides with the localized strip of material. So, from this point, the most of the energy released is proportional to this parameter. On the other side, in the present model the energy released is also influenced by the coalescence parameters and by the shape of the stress-strain curve (that depends on the nucleation and Tvergaard parameters). Then, the correct cell height should be chosen taking into account the influence of all the above parameters in the fracture process. If we consider the cell as a representative volume element (as it is usually done) it should contain a sufficiently representative voids distribution. Of course, a greater RVE contains more micro-structural information, but it is less useful in the present model because the strain would localize in a strip with fixed height. In conclusion, the cell height can’t be precisely defined without a critical consideration of all the constitutive law parameters influence. As an approximated evaluation, we can consider the bigger defects above the last 20% of the total volume fraction as it were the driving defects of the localization process. In this way, with reference to the Tab.1, an RVE, which dimension is coincident with the cell height, will contain one defect with 10+  m diameter and the appropriate number of minor defects as deduced from the size distribution (Tab. 2). Category Volume fraction Average diameter (  m) Spherical average diameter (  m) Volume (  m 3 ) Density (defects/ mm 3 ) Defects in the RVE 1:2 0.0013 1.41 1.79 3.00 433000 904 2:3 0.0010 2.41 3.07 15.1 66200 138 3:4 0.0007 3.41 4.34 42.8 16300 34 4:6 0.0012 4.87 6.20 125 8810 18 6:8 0.0011 6.88 8.50 322 3410 7 8:10 0.0009 8.86 11.3 752 1200 2 10+ 0.0014 13.23 17.7 2920 479 1 Tabella 2 : Defects distribution in the RVE (from Tab. 1 data rearranging). The cell height and the initial void volume fraction can be finally estimated: D = (10+ category density) 1/3 ≈ 130  m; f 0 = 0.0014 The D value is in good agreement with the literature values that range between 80÷200  m.