Issue 36

J. Kováčik et alii, Frattura ed Integrità Strutturale, 36 (2016) 55-62; DOI: 10.3221/IGF-ESIS.36.06 58 0.01 0.1 1 0.1 1 10 100 1000  CS [MPa] volume fraction of metal [-] 0.01 0.1 1 0.01 0.1 1 10 100 1000  CS MPa] volume fraction of metal [-] Figure 2 : Scaling of the compression strength for AlSi11Mg0.6 foam without surface skin  20 x {10, 20, 30, 40} mm 2 . Figure 3 : Scaling of the compression strength for AlMg1Si0.6 foam without surface skin  20 x {10, 20, 30, 40} mm 2 . The observed values are also in contradiction with the prediction of the compression yield stress dependence on volume fraction of metal for closed cell foams according to Ashby et al. [2]:                       0 1 2/3 0 1 0 ys CS 'C CC '       . (3) Basically this equation is in general a combination of cell face stretching and cell edge bending. It is evident that the obtained dependence is almost the straight line in log-log space. For that reason, let us assume that the cell edge bending part can be neglected. Therefore the cell face stretching will prevail and the results ought to scale with the exponent of 3/2, but it is significantly lower value as was experimentally observed. Why obtained values of T f are low in comparison with the theoretical value 2.64 ± 0.3 and higher than 1.5 derived by Ashby et al. [2]? One significant solution can be the fact, that with the increasing porosity, the pore size in metallic foams becomes comparable with sample geometry and the size effect takes place. Nevertheless, after enlarging the volume of samples by 30% no increase of T f was observed (see Al 99.96 foams in Tab. 1 and Fig. 5). It is evident, that the question why the compression strength of metallic foams scales near the percolation threshold with T f of 1.89 – 2 cannot be simply explained by the sample size and the distance from the percolation threshold. It is necessary to look for another explanation. A possible solution can be found in the experimental work of Daoud and Coniglio [17]. They proposed that the free energy F of sol-gel transition scales as: f d .B .A F      , c c p p p    (4) where A , B are numerical constants, p is the volume fraction of sol, p c is the percolation threshold,  is the critical exponent for the correlation length, d is dimension of the problem and f is the critical exponent for the modulus of elasticity. The critical exponent f possesses two different universality classes in 3D [16]: f = 2.1 for central-force model when stretching forces dominate, and f = 3.76 for bond-bending model when bending forces dominate. The first term in Eq. (4) represents the contribution of the finite gel clusters below and above the percolation threshold. The second term to the free energy is the contribution of an infinite gel cluster. In 3D  = 0.88 [9] thus giving  .d = 2.64, which coincide with the theoretical prediction for T f = 2.64 ± 0.3 by Sahimi and Arbabi and also with Bergman’s bounds on T f .