Effective mechanical behavior of hyperelastic honeycombs and two-dimensional model foams at finite strain

Abstract

The aim of the present study is the analytical and numerical determination of the effective stress–strain behavior of solid foams made from hyperelastic materials in the finite strain regime. For the homogenization of the microstructure, a strain energy-based concept is proposed which assumes macroscopic mechanical equivalence of a representative volume element for the given microstructure with a similar homogeneous volume element if the strain energy of both volume elements is equivalent, provided that the volume average of the deformation gradient is equal for both volume elements. The concept is applied to an analysis of hyperelastic solid foams using a two-dimensional model. The effective stress–strain behavior is analyzed under uniaxial and biaxial loading conditions in the tensile and in the compressive range as well as under simple shear deformation. It is observed that the effective mechanical behavior of cellular solids at infinitesimal and finite deformation is essentially different on both, the quantitative and the qualitative level.

Introduction

Solid foams and other cellular materials gain increasing importance as core construction materials for structural sandwich panels in aerospace technology and other areas of lightweight construction. In comparison to homogeneous materials, the main advantage of cellular solids is their low specific weight which enables the construction of ultralight structures. Due to their low relative density with large pore volume fraction, foamed materials can easily experience large deformations since large deformations on the macroscopic level usually require much smaller deformations of the individual struts constituting the foam structure. For stiff metallic foams, the deformation in the finite strain regime is usually elastic–plastic while finite elastic strains may occur in foams made of polymer and rubber-like materials.

For reasons of numerical efficiency, the analysis of macroscopic structures made of foamed materials during the design process is preferably performed in terms of effective properties rather than by means of a direct model for the given microstructure. Since solid foams are materials with structural hierarchy [1], no detailed information regarding the deformation processes on the microscopic level is required, if only quantities on a much larger scale than the microstructural intrinsic length scale are to be determined. Nevertheless, even a pure macroscopic analysis requires a proper effective material model which is related to the microstructure.

Since the pioneering work on the mechanics of cellular solids by Gent and Thomas [2] as well as by Patel and Finnie [3] appeared, much work has been performed with respect to an appropriate modelling of the effective material behavior of solid foams. Most of the studies are related to infinitesimal deformation. Today, comprehensive treatises on infinitesimal deformation of solid foams are already found in textbooks such as the well-known work by Gibson and Ashby [4]. On the other hand, only few publications are available on the geometrically nonlinear effective deformation of foamed materials in the finite strain regime.

The effective geometrically nonlinear behavior of regular, elastic tetrakaidecahedral [5] foams was first considered by Warren and Kraynik [6]. Zhu et al. [7] re-analyzed this type of foam model under high-strain compression. The theoretical results in this study are found in good agreement with experimental data on polyurethane foams. An analysis on the microstructure evolution under large effective deformation is provided by Wang and Cuitiño [8] using a model similar to the model employed by Warren and Kraynik [6].

Alternatively to the cited approaches considering the effective stress–strain behavior of solid foams, other studies are directed exclusively to the buckling and postbuckling behavior of solid foams under compressive loads. In this context, Hutzler and Weaire [9] provided an analytical study on the buckling of two-dimensional honeycombs. The same configuration has been investigated by Papka and Kyriakides [10], [11], [12] using experimental and numerical methods based on a regular multiple cell model. This approach had been applied previously to two-dimensional cellular structures consisting of tubular base cells (see [13]). In their studies, elastic–plastic polycarbonate or metallic material behavior on the cell wall level is considered. The numerical homogenization is performed by a stress averaging technique under prescribed external displacements of a finite element model of the complete specimen according to the experimental set-up. A study on the effective failure surfaces of regular and irregular hexagonal honeycombs has been provided by Triantafyllidis and Schraad [14]. In this study, special interest is directed to the geometrically nonlinearity. Both, the failure due to cell wall buckling and the failure due to cell wall plasticity (as well as combinations) are analyzed. Other recent work on buckling and finite deformation of two-dimensional cellular solids includes the publications by Ohno et al. [15] and by Okumura et al. [16] on the effective mechanical behavior of honeycombs under compressive in-plane load.

The present study is concerned with the homogenization of cellular solids with general cellular geometry and topology. The approach assumes mechanical equivalence of a representative volume element for the given microstructure and a similar volume element consisting of the homogeneous, effective medium if the strain energy in both volume elements is equal, provided that both volume elements are subject to a macroscopically equivalent effective deformation. The deformation of both volume elements is defined to be macroscopically equivalent, if the volume average of the deformation gradient for both elements is equal. The effective stress tensor is determined as usual in the hyperelastic framework using the partial derivative of the effective strain energy density with respect to the corresponding complementary strain tensor. This homogenization approach extends the small strain concept presented previously by the present authors [17], [18] into the finite strain regime. Alternative strain energy based concepts have been used by Warren and Kraynik [6] in an analysis of isotropic tetrakaidecahedral foams as well as by Ponte Castañeda and Suquet [19] although the latter study is not directed specifically to cellular solids.

The derived homogenization concept is applied to the analysis of solid foams made from a hyperelastic material. Therefore, a hyperelastic response will be obtained also on the effective level of structural hierarchy. For reasons of numerical efficiency, a two-dimensional microstructural model is employed. Nevertheless, it can be assumed that all basic effects in the mechanical behavior of solid foams are recovered by this simplified model since the underlying microscopic deformation mechanisms are equivalent in both two and three dimensions. The stress–strain behavior of hyperelastic foams is analyzed under both tensile and compressive conditions as well as under biaxial and simple shear deformation. In parameter studies, the influence of the microstructural parameters on the effective stress–strain behavior is studied. Significant qualitative differences between the effective mechanical behavior of solid foams in the infinitesimal and in the finite strain regimes are observed since the effect of cell wall alignment is not adequately modelled in small strain approaches. Specific effects include more sophisticated effects of the relative density and deformation-induced anisotropy of initially isotropic foams.