Isotropic constitutive models for metallic foams

Abstract

The yield behaviour of two aluminium alloy foams (Alporas and Duocel) has been investigated for a range of axisymmetric compressive stress states. The initial yield surface has been measured, and the evolution of the yield surface has been explored for uniaxial and hydrostatic stress paths. It is found that the hydrostatic yield strength is of similar magnitude to the uniaxial yield strength. The yield surfaces are of quadratic shape in the stress space of mean stress versus effective stress, and evolve without corner formation. Two phenomenological isotropic constitutive models for the plastic behaviour are proposed. The first is based on a geometrically self-similar yield surface while the second is more complex and allows for a change in shape of the yield surface due to differential hardening along the hydrostatic and deviatoric axes. Good agreement is observed between the experimentally measured stress versus strain responses and the predictions of the models.

Introduction

Over the past few years, low cost aluminium foams have been produced for a wide range of potential applications such as the cores of sandwich panels and various automotive parts. A typical aim is to develop lightweight structures which are adequately stiff and strong yet absorb large amounts of energy. The successful implementation of metallic foams requires the development of design methods based on engineering constitutive laws. A major aim of the current study is to provide a simple but reliable constitutive description of the yield behaviour of metallic foams.

Early theoretical studies (Gibson and Ashby, 1997) suggest that the hydrostatic strength of an isotropic foam is governed by cell wall stretching and scales with the relative density $\text{ρ}\text{̄}$, whereas the uniaxial strength is governed by cell wall bending and scales with $\text{ρ}\text{̄}{}^{\mathrm{3/2}}$. Thus, for a relative density of $\text{ρ}\text{̄}\text{=0.1}$, the hydrostatic strength of the perfect structure is about three times the uniaxial strength. These theoretical predictions neglect the effect of imperfections in the microstructure. Recently, the effect of morphological defects on the elastic and plastic properties of foams has been addressed by various authors (e.g. Silva et al., 1995, Kraynik et al., 1997, Grenestedt, 1998). The main findings of these and other studies have been reviewed by Chen et al. (1999). Chen et al. (1999) comprehensively studied the effect of various geometrical imperfections on the in-plane yielding behaviour of 2D cellular materials using a combination of analytical and finite element methods. They found that cell wall waviness significantly reduces the hydrostatic strength of the regular honeycomb structure. Similarly, random imperfections in the form of cell wall misalignments and fractured cell walls reduce the hydrostatic strength to the same level as the uniaxial strength. The effect of a random dispersion of cell size in a Γ- or δ-Voronoi structure is relatively small and the yield surfaces of these structures have approximately the same size and shape as those of ideal hexagonal honeycombs. Chen et al. (1999) predict nearly circular yield surfaces in the stress space of mean stress versus effective stress for honeycombs with either fractured cell walls or cell wall misalignments. Kraynik et al. (1997) found a similar sensitivity of hydrostatic strength to the presence of cell wall misalignments in open cell 3D elastic foams. These analyses demonstrate that a small degree of imperfections suffices to induce bending deformation in the cell walls under all macroscopic stress states.

Experimental data for the multi-axial yield of foamed metals are limited. The main contributions are those of Triantafillou et al., 1989, Gioux et al., 2000. Triantafillou et al. (1989) conducted axisymmetric tests on an open-cell aluminium foam under combined axial tension and radial compression. Gioux et al. (2000) reported yield data for closed and open cell aluminium foams under a variety of biaxial, shear and axisymmetric loadings. The presence of experimental scatter in these studies has made it difficult to establish the shape of the yield surfaces. Moreover, only the initial yield surface has been addressed. In many design situations, for example in energy absorbing devices, an understanding of the post-yield behaviour is essential.

Miller (2000) has recently proposed a continuum plasticity framework for metallic foams. He modified the Drucker–Prager yield criterion and introduced three adjustable parameters to fit the yield surface to the then available experimental data; these data are the uniaxial tensile and compressive yield strengths, and the ratio of radial to axial plastic strain rate in an axisymmetric test, i.e. the “plastic Poisson’s ratio”. He assumed an associated flow rule and based the hardening law upon the uniaxial compressive behaviour to give a complete constitutive representation of the plastic behaviour. In contrast, the constitutive model of Zhang et al., 1997, Zhang et al., 1998 for polymer foams adopts a non-associated flow rule to account for the low observed values of plastic Poisson’s ratio. Zhang et al. use a hardening rule based upon the plastic volumetric strain. For the case of metallic foams micromechanical arguments suggest that associated flow on the microscopic scale translates to the macroscopic scale, (Hill, 1967, Gurson, 1977) therefore, we anticipate that plastic normality is satisfied.

In this paper the yield surfaces of an open cell and a closed cell aluminium foam are measured for axisymmetric compressive stress states. The evolution of the yield surfaces under uniaxial and hydrostatic compression is explored. Two phenomenological isotropic constitutive models with different levels of complexity are then developed to model the observed behaviour.