Keywords: nanoindentation, metal foam, deconvolution, elastic properties, image analysis, homogenization.

This paper describes an identification of small-scale elastic properties on highly porous metal foam [1]. Commercially available aluminium foam 'Alporas', an Al-alloy with thickening (Ca) and blowing (TiH₂) agents, was studied. The resulting material is characterized by a closed pore system with very thin pore walls (~100 µm) and large pores (~2.6 mm). The overall porosity of the sample reaches 93%.

The task of finding intrinsic elastic properties of microscale components was solved by using statistical nanoindentation [2,3]. A large number of small indents (with penetration depth of approximately 250 nm) was performed over the micro-scale material level. It was confirmed that at least two mechanically distinct components need to be distinguished. They were, similarly to electron microscope and image analyses, denoted as Al- and Ca/Ti-rich areas. Intrinsic phase properties were assessed with the deconvolution algorithm that seeks for the best fit of the experimental histogram by the given number of Gaussian functions. The mean values of Young's moduli for the two phases were estimated as 61.883 GPa and 87.395 GPa together with respective volume fractions.

At first, an analytical homogenization (namely the Mori-Tanaka scheme [4]), was employed for the assessment of effective elastic properties at the micro-level. The method estimates effective properties based on individual phase properties and their volume fractions. For verification, advanced homogenization based on fast Fourier transformation (FFT) was used. The discretized problem of the weak formulation to the system of governing differential equations can be solved using the method based on FFT. The transformation leads to a nonsymmetric linear system equations that is solved, for example using the conjugate gradient method [5]. The linear system depends only on the stiffness coefficients at grid points that can be obtained using nanoindentation and thus the effective stiffness tensor can be calculated.

A comparison of the analytical and FFT schemes demonstrates very good agreement. It means that (i) the analytical scheme gives a very good estimate for this particular case and (ii) the material behaves as close-to-isotropic over the RVE (L~100 µm). The effective cell wall properties were found to be close to 70 GPa irrespective to the homogenization procedures used.

1

J. Banhart, "Manufacture, characterisation and application of cellular metals and metal foams", Progress in Materials Science 46, 559-632, 2001. doi:10.1016/S0079-6425(00)00002-5

2

G. Constantinides, K.R. Chandran, F.-J. Ulm, K.V. Vliet, "Grid indentation analysis of composite microstructure and mechanics: Principles and validation", Materials Science and Engineering: A, 430(1-2), 189-202, 2006. doi:10.1016/j.msea.2006.05.125

3

J. Nemecek, V. Šmilauer, L. Kopecký, "Nanoindentation characteristics of alkali-activated aluminosilicate materials", Cement and Concrete Composites, 33(2), 163-170, 2011. doi:10.1016/j.cemconcomp.2010.10.005

4

A. Zaoui, "Continuum Micromechanics: Survey", Journal of Engineering Mechanics, 128, 2002. doi:10.1061/(ASCE)0733-9399(2002)128:8(808)

5

J. Zeman, J. Vondrejc, J. Novák, I. Marek, "Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients", Journal of Computational Physics, 229(21), 8065-8071, 2010. doi:10.1016/j.jcp.2010.07.010

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