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The mechanical properties of closed-cell aluminium foams are governed by their geometrical and topological evolution during impact. Here we non-destructively investigate the deformation mechanisms of a closed-cell aluminium foam sample at the cell scale. The sample has been compressed with 21 interrupted drop-weight impacts at a nominal strain rate of 40 s−1 and the post-impacted sample has been imaged at four pertinent strain states with high-resolution X-ray micro-computed tomography (XCT). Moreover, a number of qualitative and quantitative structural analyses are carried out using advanced 3D image analyses to understand the effect of foam geometry/topology on its mechanical response. Our results show that the deforming microstructure creates strong correlations across a range of geometrical, topological and shape characteristics. Quantitative image analyses of the sample at four different strain states reveal that the regions with large void fractions predominantly undergo collapse and subsequently reduces the structural heterogeneity. Further, we show that the deformation mechanism leaves topological signatures in the evolving microstructure, which can be used to better understand the mechanical response of the sample at various stages of impact-deformation. We demonstrate here, for the first time, how the topological quantities can be used to explore the foam deformation mechanisms and to correlate the deformation with mechanical response during impact.

Crystalline frameworks in 3D Euclidean space can be constructed by projecting tilings of 2D hyperbolic space onto three-periodic minimal surfaces, giving surface reticulations. The technique involves Delaney–Dress tiling theory, group theory, differential and non-Euclidean geometry. Preliminary results of this approach, found at http://epinet.anu.edu.au , are discussed and compared with other approaches.

We describe a systematic approach to generate nets that arise from decorations of periodic minimal surfaces. Such surfaces are covered by the hyperbolic plane, in the same way that the euclidean plane covers a cylinder. Thus, a symmetric hyperbolic network can be wrapped onto an appropriate minimal surface to obtain a 3D periodic net. This requires symmetries of the hyperbolic net to match the symmetries of the minimal surface. Here, we tabulate all such symmetry groups that are compatible with the H minimal surface. c

We consider finite point-set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set. We use the minimal spanning tree of the finite set of points to compute the number and size of its -connected components. By extrapolating the limiting behavior of these quantities as ! 0 we can say whether the underlying set appears to be connected, totally disconnected, or perfect. We demonstrate the effectiveness of our techniques for a number of examples, including a family of fractals related to the Sierpinski triangle, Cantor subsets of the plane, the Henon attractor, and cantori from four-dimensional symplectic sawtooth maps. For zero-measure Cantor sets, we conjecture that the growth rate of the number of -components as ! 0 is equivalent to the box-counting dimension. © 2000 Elsevier Science B.V. All rights reserved.