Keywords: yield surface, hardening, open-cell foams, Kelvin foam, homogenization, micromechanics.

The present paper is concerned with the numerical determination of the macroscopic yield behaviour of open-celled metal foams which are highly porous cellular solids (see Gibson and Ashby [1]). The outstanding properties of open-celled metal foams stem from the sponge-like microstructure. Since the porosity of the material can amount to 98%, metal foams offer a promising lightweight potential. In contrast to conventional materials, cellular solids are characterized by large plastic deformations on the macro level which occur even at hydrostatic loading.

In order to establish open-celled foams for engineering application, detailed knowledge about the mechanical behaviour is required. Phenomenological material models are used in structural analysis. Though, before such material models can be developed, information about the yield surface and the change of the elastic limit due to plastic strains are the prerequisites. In this context, micromechanical approaches and homogenization methods serve in a two-fold way. On the one hand, homogenization delivers the effective properties that can be used in structural analysis. On the other hand, micromechanical approaches reveal the underlying physics that is responsible for the yield behaviour on the macroscopic scale.

For the homogenization of the cellular microstructure, a strain-energy based procedure is applied (Hohe and Becker [2], Demiray et al. [3]). This procedure defines the macroscopic mechanical equivalence of a representative volume element for the given microstructure and a similar volume element of the homogeneous "effective" medium by the condition that the average strain energy density in both volume elements is equal provided that both volume elements are subject to deformation states which are macroscopically equivalent. The effective strain state is defined in terms of the macroscopic Green-Lagrange strain tensor which is determined using its definition in terms of the effective deformation gradient. The effective stress is defined by the second Piola-Kirchhoff stress tensor which is obtained as the partial derivative of the average strain energy density of the representative volume element for a prescribed effective strain state. Using the definition of the macroscopic strain gradient, the corresponding displacement loads on the micro level are evaluated. Periodic boundary condition are applied which assure that two opposite faces of the RVE deform in the same manner.

As a surrogate model for open-celled foams, the simplified Kelvin foam is considered. This microstructure consists of a body-cubic centred lattice of tetrakaidecahedral cells. Lord Kelvin [4] proposed this polyhedron for packing cells with equal volume in an efficient way. Hereby, the tetrakaidecahedron is a convex polyhedron which consists of six square faces and eight hexagons.

Numerical experiments are conducted to generate the macroscopic yield surfaces. One locus of the yield surface is obtained by scanning a radial load path in the strain space for a given offset plastic strain value. The numerical examples show that the initial yield surfaces in strain space exhibit a piece-wise planar surface with sharp corners. It is shown that this effect is caused by plastic hinges that develop at different planes of the model foam. The stress yield surface is a prolate ellipse which is aligned in direction of the hydrostatic axis. Due to the induced bending in the struts under all macroscopic stress states, the stress yield surfaces of the randomized model foams shrink significantly.

In the main part of this study, the further evolution of the yield surfaces is recorded for different loading situations. The numerical results show that a combination of hardening mechanisms develops. It is observed that the subsequent yield surfaces in strain space are mainly shifted. However, the yield surfaces exhibit also isotropic and distorsional hardening at relatively large deformations. Since an elastic/perfectly-plastic material is used for the struts, it can be concluded that the change of the cellular microstructure has a significant impact on the evolution of the yield surfaces. In detail, the subsequent yield surfaces may exhibit a nose for uniaxial tensile and shear deformation. As compared to the loading case of biaxial compression, for uniaxial compression the stress yield surface can rotate due to the alignment of the struts in the loading direction.

1

L.J. Gibson, M.F. Ashby, "Cellular Structures and Properties", Cambridge University Press, 1997

2

J. Hohe, W. Becker, "Effective mechanical behavior of hyperelastic honeycombs and two-dimensional model foams at finite strain", International Journal of Mechanical Sciences, 45, 891-913, 2003. doi:10.1016/S0020-7403(03)00114-0

3

S. Demiray, W. Becker, J. Hohe, "Strain-energy based homogenisation of two- and three-dimensional hyperelastic solid foams", Journal of Materials Science, 40, 5839-5844, 2004. doi:10.1007/s10853-005-5017-6

4

W. Thomson (Lord Kelvin), "On the Division of Space with Minimum Partitional Area", Philosophical Magazine, 24, 503, 1887. doi:10.1007/BF02612322

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