A fractal study on meso structures and size effect of metallic foams

Meso structure of metallic foam, such as morphologies ofcells, cell walls and pores, is very important to its mechanical properties.Unlike honeycombs and some other cellular solids which have sequential andperiodical cell structures, metallic foams are inherently disordered in thecell level: non-equally sized and various shaped cells, non-equallydimensioned and curvy cell walls, random pores and cracks. It is impracticalto extract a representative cell and build ordered models, and thesimplified homogenous models can not reveal the importance of cellstructures to the global mechanical properties. A common parameter inanalytical models is relative density, which is a macroscopic value thatcannot reveal the meso structure of metallic foam. In this paper, a fractalapproach is introduced to bridge the morphological parameters of cellstructures and the mechanical properties of metallic foams. Cell morphologyof Al foams is proved to be a fractal geometry of self-similar in a certainscale, using the slit island method proposed by Mandelbrot. A series of Alfoams with different relative densities and meso structures were examined bythe box-counting dimension method, the information dimension method and theslit island method. It is found that the fractal dimension is in directproportion to the ratio of characteristic wall thickness to mean celldiameter. By mapping the cell morphology to the generalized Sierpinskicarpet, the mechanical properties of Al foams are expressed as the functionof fractal dimension. In addition, the fractal model is combined with thesize effect model proposed by Onck et al, and the revised size effect modelfor metallic foams incorporates stochastic characteristic of mesostructures. The results show that the fractal-based model can not onlyreveal the variation of yield strength with specimen size, but also bridgethe meso structures and mechanical properties of Al foams directly.