Benefits of metal foams and developments in modelling techniques to assess their materials behaviour: A review

Abstract

A review assessing the relative benefits of metal foam core sandwich panels with respect to honeycomb, polymeric foam and truss cores has been conducted. It is noted that metal foams are able to combine low density with good bending stiffness and strength, making them attractive core materials for use in industrial applications (e.g. aircraft wing structures). The current modelling tools available for metal foams are also reviewed. These fall under three categories: analytical methods using dimensional analysis, finite element methods utilising a repeating unit cell, and finite element methods utilising the random Voronoi technique. It is noted that analytical methods do not take into account the effect of imperfections in the microstructure. Finite element methods utilising a repeating unit cell also fail to capture the natural variations in microstructure that are observed in most cellular materials. The effects of imperfections are discussed, and it is observed that these reduce the hydrostatic strength of foams by differing amounts. The Voronoi technique can produce a geometrically more realistic model of the foam structure than the other two methods. Finite element methods are also being developed that utilise a three-dimensional tomographic image of a real foam as the geometric description of the model and it is possibly here that the most exciting developments for the modelling of foams resides.

Keywords

Metal foam Sandwich panel Modelling Finite elements Review

Introduction

Metal foams are a relatively new class of materials that show good potential for lightweight structures, energy absorption, and thermal management.1^–3The first attempts to manufacture metal foams date back to 1943 when Benjamin Sosnick tried to foam aluminium with mercury.4In the 1950s, it was found that liquid metals could be foamed by pretreating them to enhance their viscosity, for example by oxidising the melt or adding oxide particles.5 In the early 1970s, the Ford Motor Co evaluated aluminium foam samples, but initial developments were unsuccessful, leading to a decline in R&D of metal foams post-1975. 6 6,7Research picked up again towards the end of the 1980s when the Shinko Wire Co. developed the Alporas foam manufacturing process.8 In 1990, Joachim Baumeister brought the compacted powder foaming process developed in the 1950s to a considerable level of sophistication.9

Today, several different manufacturing techniques exist to make metal foams, five of which are commercially established – these are reviewed by Banhart1 and by Ashby et al.,2 which present a comprehensive review of metal foams, their manufacturing methods, and their advantages in industry. Further details can be found in Refs. 10–17. The above review work concentrates mainly on the manufacturing technologies and experimental methods for foam materials and structures. However, significant efforts have also been made to develop analytical as well as finite element (FE) modelling techniques for assessing the material behaviour of foam materials.18^–26The aim of the present paper is to provide a comprehensive review of these numerical techniques.

For completeness, an overview of metal foams is provided in the section on ‘Overview of metal foams and their structures’ – noting notably their advantage as cores for lightweight sandwich panels. Alternative sandwich core materials are summarised, discussed and presented in the section on ‘Sandwich structure core types and their relative benefits’. On top of this, the current modelling tools available for metal foams are discussed. These fall under three categories:

analytical methods, utilising dimensional analysis that gives the dependence of the foam properties on the relative density but not the cell geometry

FE methods utilising a repeating unit cell

iii

FE methods utilising the random Voronoi technique. This approach gives a more accurate representation of the cell geometry of the foams.26

The detailed research results on the above numerical techniques are reviewed, analysed and presented in the sections on ‘Analytical modelling methods of metal foams’ and ‘Numerical modelling methods of metal foams’ respectively. The ‘Discussion’ section provides an assessment of the relative merits of each method, as well as discussing the future trends for modelling of metal foams. Conclusions are given at the end of the paper.

Overview of metal foams and their structures

There exist several different manufacturing methods to make metal foams. The different techniques are used for different subsets of metals to create porous materials with a limited range of relative densities and cell sizes. Some methods produce closed cell foams, others open cell. The cost of each process varies significantly – from $7 to $12 000/kg.27 Most commercially available metal foams are currently based on aluminium or nickel.28Characterisation and testing methods have been developed for metal foams, including uniaxial compression tests on cylindrical foam specimens, uniaxial tension tests on dogbone specimens, torsion tests and 4-pt bending for fatigue testing.29^–32 The foam structure can be examined using notably optical microscopy, scanning electron microscopy and X-ray tomography. 1 1,33

Metal foams display high values of E^1/3/ρ, where E is the Young's modulus of the foam and ρ the foam density. This material index is derived in Ref. 27 for a panel of specified length, width and stiffness with the objective of minimising the mass. The greater the value of E^1/3/ρ, the lighter a panel can be for a given stiffness. E^1/3/ρ therefore characterises the bending stiffness of lightweight panels and suggests the use of metal foams as light, stiff panels.27The weight of metal foam sandwich structures is comparable to waffle stiffened aluminium panels, but they have lower manufacturing costs.2

Furthermore, metal foams have high values of , where σ_y is the elastic limit of the foam. This material index is derived in Ref. 27 for a panel of specified length, width and strength with the objective of minimising the mass and hence characterises the bending strength of lightweight panels. The greater the value of , the stronger a panel will be for a given mass. Consequently, a metal foam panel is stronger than one of the same material which is solid. Strength limited foam core sandwich panels can also offer weight savings over traditional stinger stiffened panels.2

Metal foams are therefore able to combine low density with good bending stiffness and strength. This makes them attractive as cores of lightweight sandwich structures.27 Such sandwich structures show promise in the design of aircraft wing boxes, which are at present typically fabricated utilising thin panels that comprise of a skin stiffened by stringers. 34 34,35 The resultant panels are light and stiff, but relatively expensive to produce due to high machining costs and the inefficient use of material. They also display significant anisotropy in the bending plane (i.e. they are not equally stiff about all axes of bending).36 The stabilisation of the thin sheets can often be more efficiently achieved by the use of sandwich structures. The lightweight core serves to separate the skins, hence increasing the moment of inertia of the panel while keeping weight to a minimum. The result is a structure that is efficient in resisting both bending and buckling loads.18 Sandwich structures also provide a continuous stiffness distribution within the skin panel. This integral stiffening leads to a reduced parts count for assemblies and hence less logistics, parts manufacturing and assembly work.37

Sandwich structures were first used in England in the construction of the Mosquito night bomber of the Second World War. In 1943, the Vultee BT-15 fuselage was constructed from fibreglass reinforced polyester as the face material with both a glass fabric honeycomb and balsa core.38Today, composite sandwich structures are used extensively in aircraft design (for example on the Airbus A380).35Forty-six per cent of the external surface of the Boeing 757/767 is honeycomb sandwich and the Boeing 747 comprises of a fuselage cylindrical shell that is primarily Nomex honeycomb sandwich.38It should be noted, however, that the use of sandwich structures in commercial aviation is at present limited to secondary structures only. To be applicable to primary structures (that is, structural components that are necessary to sustain design ultimate flight and ground loads), sandwich structures must ensure that any damage during the service life of the component does not result in failure prior to the damage being detected – this would need to be demonstrated by appropriate tests and analyses, and the definition of maximum allowable damages and their visibility, and it is here that the ability to accurately model the behaviour of metal foams is of critical importance. Sandwich structures suffer the disadvantage that structural failures cannot always be detected by standard methods such as visual inspection and ultrasonic pulse echo. This may lead to the need to inspect the components using repeated non-destructive testing, while considering at the same time the economic requirements for in service life.37 Moreover, for sandwich structures to be used in primary structures, the time and ease of in service repairs must be considered. The key drivers for the use of a sandwich panel as a primary structure include its ability to fail safe, the toughness of the materials, and its capability to withstand impact damages.37

Metal foam sandwich panels also show promise in automotive applications due to their lightweight construction and energy absorbing capabilities. One example is German car maker Karmann's concept car called the Ghia roadster, which consists of aluminium foam sandwich panels for structural and energy absorbing parts that are half the weight and ten times stiffer than steel.40 In addition, Fiat and the Norwegian University of Science and Technology have conducted studies that show car crash boxes (assemblies located at the front of a car, designed to crumple in the event of a collision to protect passengers and minimise vehicle damage) that comprise of a tube filled with a metal foam display an improved axial energy absorption capability compared to that of empty tubes, and more than the sum of the individual energies of the tube and the foam.41There is also a marked improvement in the energy absorption in off-axis collisions of metal foam filled tubes due to the isotropic nature of the foams.42 Other potential applications exist in shipbuilding, the railway industry, the biomedical industry and civil engineering. 1 1,42

The various application fields for metal foams, including concepts that are still in the verification phase, have been extensively reviewed by Banhart.1 These comprise three overarching categories: structural, functional and decorative applications. Structural applications tend to involve closed cell foams – for instance, light weight construction has gained importance in ship building and as such light, stiff closed cell aluminium foam sandwich panels with excellent damping behaviour show promise in this area. Functional applications tend to involve open cell foams. Highly conductive open cell foams based on copper or aluminium can be used as heat exchangers – the open cell structure allows gases or liquids to flow through it, allowing heat to be added or removed. Decorative applications for metal foams also exist – for instance, aluminium foams have been used to build designer furniture. Foams based on gold or silver have potential to be a new material for jewellery.1

However, despite these potential applications, mass production of metal foams has been limited primarily due to an insufficient ability to produce high quality materials with good reproducibility of their properties, due to a lack of control over the structure and morphology of the foams, and high manufacturing costs.1,2,10^–17 In addition, for most manufacturing processes, there is currently no applicable analytical or numerical model that can predict the effect of parameter changes – at present, improvements are typically made by trial and error.1 However, with improvements in the manufacturing processes of metal foams, there is a potential for future industrial mass market applications.1

The current manufacturing methods for producing sandwich panels that combine a metal foam core with metal face sheets can be classed under two distinct categories: ex situ bonding is the process of bonding the face sheets directly onto a sheet of metal foam; and in situ bonding is the process of combining the metal foam manufacture with bonding to the face sheets.39 The ultimate objective of the bonding process is to achieve a bond between the face sheets and foam core that is above the strength of the foam.43 Technologies are emerging for creating syntactic metal foam structures (i.e. foams with an integrally shaped skin). This would allow cheap, lightweight structures to be moulded in a single operation and it is perhaps here that current metal foam technology holds the greatest promise. 2 2,13

Sandwich structure core types and their relative benefits

The two most common sandwich core types in industrial applications are honeycomb and foam (polymeric or metallic) cores.28An additional core type comprises of truss structures.

Honeycomb cores consist of ‘any array of identical prismatic cells which nest together to fill a plane’.18The cells are typically hexagonal in section, though they can also be triangular, square, rhombic or circular.44 Polymer and metal honeycombs are used as sandwich panel cores in aerospace components, though honeycombs can also be made from ceramics and paper.18 Figure 1ashows an aramid fibre (Nomex) reinforced honeycomb structure.45 The structure is effectively two-dimensional (2D) and regular, making honeycombs easier to analyse than foams which have three-dimensional (3D) cell arrangements.46

Figure 1

a aramid fibre (Nomex) reinforced honeycomb45 and b definitions of parameters for honeycomb cell46

The truss core sandwich panel includes a corrugated sheet or a truss core disposed between two face sheets.

Honeycomb cores

It can be seen from Fig. 1athat a typical honeycomb is made up of a set of hexagonal cells.45 The dimensions of the cell are defined by the cell wall lengths l and c, the angle between the two cell walls θ, and the thickness of the cell walls h, as shown in Fig. 1b.46

Honeycomb properties are anisotropic; that is, the in plane stiffnesses and strengths are different to the out of plane ones. When a honeycomb is compressed in plane (i.e. the stress acts orthogonal to the axis of the cells – the plane X₁X₂ in Fig. 1b), the cell walls initially bend, and the deformation is linear elastic. Depending on the material of the cells, they then collapse beyond a critical strain by either elastic buckling, plastic yielding, creep, or brittle fracture. Eventually, adjacent cell walls will touch, the cells will close up and the structure densifies, resulting in a sharp increase in stiffness. In tension, the cell walls initially bend and, depending on the material, either yield plastically or fracture in a brittle manner.18 When loading occurs in the out of plane direction (the X₃ direction in Fig. 1b), the cell walls either extend or compress and the moduli and collapse stresses are much higher, i.e. honeycomb structures are much stiffer and stronger in the out of plane direction.18

Figure 2a and b shows compressive stress–strain curves for a honeycomb structure loaded in plane direction and the out of plane direction respectively.18 It can be seen from Fig. 2 that there are three distinct stages in the compressive stress–strain curves: an initial linear elastic region, followed by a Plateau region, and finally a densification stage where the stress rises rapidly with strain. Increasing the relative density of a honeycomb structure alters the stress–strain curves (see Fig. 2).

Figure 2

a in plane direction honeycomb stress–strain curves and b out of plane direction honeycomb stress–strain curves46

Analytical models have been developed to evaluate the moduli and collapse stresses of honeycombs in both uniaxial and biaxial loading for both in plane and out of plane loading scenarios. These are presented by Ashby and Gibson18 and Lu and Yu.46These show that one of the governing criterion which determines the performance of a honeycomb structure is its relative density, defined as the overall density of the cellular material (ρ*) divided by the density of the solid of which the cellular material is made (ρ_s). Finite element methods have also been developed – these and the analytical methods are discussed respectively in the sections on ‘Analytical modelling methods of metal foams’ and ‘Numerical modelling methods of metal foams’.

Sandwich structures that comprise of honeycomb cores can display high strength and light weight. However, as mentioned previously, they are also highly anisotropic. They are also difficult to form into complex curved shapes because of induced anticlastic curvature. Durability issues have been linked to moisture intrusion into the panels resulting in internal corrosion and facesheet debonding. 36 36,47

Metal foam cores

As with honeycombs, the properties of a foam are largely controlled by its relative density. In addition, the material the foam is made from and its cell type (open or closed) also dictate the properties.46 Foam properties are further affected by anisotropy and defects, i.e. wiggly, buckled, or broken cell walls and cells of significant size.27 The response of foams and their analysis mirrors that of honeycombs.

Metal foams can, unlike honeycombs, display isotropy in mechanical properties. They can be made with integral skins, which presents the possibility to make composite structures without adhesive bonding, and can be readily formed into curved shapes.47 They display a densification stage when subjected to a compressive stress, where the stress rises rapidly with strain as the foam cells crush – this has the implication that the integrity of a metal foam core sandwich panel is not necessarily compromised when subjected to impacts. Furthermore, open cell foams do not trap moisture (i.e. they are less susceptible to corrosion than honeycombs).34 Open cell cores could provide a dual function, and potentially be used for the storage or drainage of fuel in aircraft wing structures. Honeycomb cores or traditional stringer stiffened panels do not offer this advantage.48

Metal foam cores can exhibit values of E^1/3/ρ in the range 2–5 GPa^0·33/(mg m⁻³) whereas steels are typically around 0·7 GPa^0·33/(mg m⁻³) and aluminium around 1·5 GPa^0·33/(mg m⁻³). Metal foam cores can also exhibit values of in the range 2–10 MPa^0·5/(mg m⁻³), whereas steels are typically around 1·8 MPa^0·5/(mg m⁻³) and aluminium around 3·7 MPa^0·5/(mg m⁻³).² As noted earlier, this suggests their use as the cores of lightweight sandwich structures.

Polymeric foam cores

The analytical models to evaluate the moduli and collapse stresses of open cell and closed cell polymeric foams in both compression and tension are identical to those for metal foams, which are discussed in the section on ‘Analytical modelling methods of metal foams’.18

Figure 3 shows compressive stress–strain curves for a polyurethane foam.46It can be seen that the curves exhibit the same trend as per metal foams and honeycombs.

Figure 3

Stress–strain curves for closed cell rigid polyurethane foams of various densities46

Polymeric foams tend to be cheaper than their metallic counterparts.36 As with open cell metal foams, open cell polymeric foams do not trap moisture.36Closed cell polymer foam cores give increased thermal insulation at moderate weight, but creep even at ambient temperatures.47

The structure of polymeric foams is similar to that of metallic foams, but they do not exhibit metallic characteristics such as electrical conductivity, etc. Unless protected, polymer structures suffer more damage to lightning strikes than metallic ones and allow significant proportions of lightning current to flow into onboard systems (e.g. electrical wiring).28

Truss cores

Work has been conducted to investigate the properties of miniature truss core sandwich panels.45 These miniature truss core sandwich structures are similar in design to large engineering structures such as bridges and skyscraper frames. Figure 4a shows a typical miniature truss core sandwich structure.49

Figure 4

a LM22 truss core sandwich panel with solid face sheets,50 b Kagomé truss core panel,49 c tetrahedral triad unit with leg members of rectangular cross-section;45 relative property comparisons for honeycomb and tetrahedral truss core systems: d shear modulus and e shear strength45

Deshpande and Fleck50 have analysed tetragonal and pyramidal shaped trusses. They find that both cores display significant anisotropy and are susceptible to plastic buckling, resulting in bending asymmetry. Work has also been conducted to investigate the properties of 3D Kagomé truss core topologies, as shown in Fig.4b.49^–52It is found in Ref. 49 that while tetragonal truss cores do not display isotropy after yielding, Kagomé cores do. The Kagomé core also has the greater load capacity and appears to be the superior core choice for ultra light panels.

Truss core sandwich panels tend to be advantageous for some applications since they may be fabricated with face sheets having a heavier gauge than those of honeycomb structures.

In the United States, ∼40 of bridges are not able to handle present demands and require replacement. Approximately half the cost of bridge replacement comes from rerouting traffic during the construction process. Sandwich structures with truss cores offer a potential to provide premade deck panels that can be installed within days as opposed to the weeks required by traditional construction methods.47

Work by Wicks and Hutchinson48 as well as Deshpande and Fleck50 indicates that sandwich structures with periodic open cell truss cores can be as stiff, strong and light as hexagonal honeycomb core panels. Also, the open nature of truss cores means they do not trap moisture and could provide a dual function (e.g. they could potentially be used for the storage or drainage of fuel).48 Open cell cores based upon tetrahedral truss concepts can allow fluids to easily flow through, making them less susceptible to internal corrosion and depressurisation induced delamination.45 However, truss core sandwich panels also tend to be expensive and difficult to manufacture, generally requiring batch type processing. Automated manufacture of miniature truss cores remains at present expensive.45

Analytical modelling methods of metal foams

The mechanical behaviour of metal foams is set by the cell structure and mechanical properties of the solid material. In an attempt to understand the mechanical response of foam materials under loading, analytical modelling tools have been developed for metal foams.53

The important length scale in metal foams is cell size, which is significantly large compared to the grain size that dictates properties in dense metals.54 The unit cell of cellular/lattice materials is in the order of millimetres or micrometres, which allows them to be treated both as structures and materials. The lattices can be studied using traditional methods of mechanics; however, one must also treat the lattice as a ‘material’ in its own right, with its own set of effective properties that allows a direct comparison with fully dense materials.55

Analytical methods to determine the basic properties of metal foams (e.g. the moduli and collapse stresses) are presented by Ashby and Gibson.18 This analysis was extended to include size effects by Onck et al.,19in which the FE method was used – Onck et al.'s work19 is discussed in the section on ‘Finite element methods utilising repeating unit cell’. Chen et al.20 examined the effects of periodic defects (i.e. cell waviness and non-uniform wall thickness) in altering the shape and size of the yield surface analytically, using a unit cell model for periodic hexagonal honeycombs. Ashby and Gibson's18 as well as Chen et al.'s20 findings are summarised in Table 1.

Table 1

Unit cell structures and associated equations used in analytical modelling of open cell metal foams 16 16,18

For isotropic, open cell foams Ashby and Gibson18considered the cubic model shown in Table 1. It consists of solid struts surrounding a void space and connected at joints. Cellular solids are characterised by their relative density, which for the structure shown in Table 1 (with t<<L) is given by (1) where is the density of the foam, ρ_s is the density of the solid of which it is made, L is the cell size and t is the thickness of the cell edges. Three possible collapse mechanisms exist under compressive loading: plastic bending of the cell edges, elastic buckling of the cell edges or successive fracturing of the cell edges. The one that requires the lowest stress dominates. The analysis by Ashby and Gibson18 produced the equations shown in Table 1 when the foam is loaded in compression. In equations (2)–(4), is the Plateau stress, σ_y,s is the yield strength of which the solid is made, is the stress that causes the foam to collapse by elastic buckling, E_s is the modulus of the solid of which the strut is made, is the crushing stress and σ_MOR is the modulus of rupture of the solid strut. The constants of proportionality can then be determined by fitting experimental data. Ashby and Gibson18find that experimental data for open cell foams that collapse plastically are adequately described by equation (2) when the constant of proportionality is ∼0·3. Experimental data for elastomeric open cell foams are adequately described by equation (3) when the constant of proportionality is ∼0·05, and that for brittle foams are adequately described by equation (4) when the constant of proportionality is ∼0·2.

Most closed cell foams also follow the above scaling laws, which is unexpected as the cell faces must carry membrane stresses when the foam is loaded. However, the cell faces are very thin and rupture or buckle at such low stresses that their contribution to stiffness and strength is small, leaving the cell edges to carry most of the load.55 Ashby and Gibson's analysis18 suggests that the hydrostatic strength of an isotropic foam is governed by cell wall stretching and scales with , while the uniaxial strength is controlled by cell wall bending and scales with . These predictions neglect the effect of imperfections in the microstructure (waviness of cells, non-uniform cell wall thickness, etc.).

The key equations used in Chen et al.'s20 analysis of the effect of cell wall waviness and non-uniform cell wall thickness in altering the shape and size of the yield surface are presented in Table 1. Equation (5) describes the initial transverse deflection w(x) for a wavy imperfection along each cell. w_o is the amplitude of the waviness, n is the number of ripples in a length l/2 (assuming each beam is symmetrical, see Table 1), and x is the distance from the midjoint O along the cell edge. By considering a wavy beam of length l/2 and thickness t which is clamped at one end and subjected to a transverse force Q and axial force P at the other end, Chen et al.20 produced equation (6) for the yield locus in (P,Q) space. In equation (6), it is assumed that plastic collapse is by the formation of a plastic hinge at a distance x_c from the fixed end O. w_c and ψ_c denote respectively the initial transverse deflection and associated slope at x_c. The effect of wavy imperfections on the yield strengths of perfect honeycombs is shown in Fig. 4a – σ_U is the uniaxial yield strength and σ_H is the hydrostatic yield strength (hydrostatic loading causes yielding in honeycomb and foam structures in conftrast to the assumption for homogeneous solids) of a honeycomb with cell wall waviness, and and are the corresponding yield strengths of a perfect honeycomb.20

As can be seen from Fig. 5a, Chen et al.20 found that cell wall waviness reduces the hydrostatic yield strength of the regular honeycomb structure significantly. This is because cell wall waviness changes the deformation mechanism from cell wall stretching to cell wall bending under hydrostatic loading. The uniaxial yield strength is only slightly reduced as the deformation mechanism is cell wall bending for both a perfect and wavy honeycomb when loaded uniaxially.

Figure 5

Effects of a waviness and b non-uniform wall thickness on yield strengths of perfect honeycombs20

Chen et al.20 considered the effect of non-uniform wall thickness on the size and shape of the yield surface using the geometric model shown in Table 1. They assumed that cell wall thickness decreases linearly from the joint O to the midpoint of the cell edge and they used simple beam theory such that the predictions are limited to small variations in cell wall thickness. They determined the yield surface by analysing the plastic collapse of a clamped beam whose thickness varies linearly with length. The critical load for collapse of such a beam is given by equation (7), where x_c is the distance of the plastic hinge from the built-in end. The thickness t_c at x_c is given by equations (8)–(10). The effect of non-uniform wall thickness on the yield strengths of perfect honeycombs is shown in Fig. 5b.20

As can be seen from Fig. 5b, Chen et al.20 found that non-uniform wall thickness slightly reduces the hydrostatic yield strength – they explained this by noting that under hydrostatic loading the deformation mechanism is cell wall stretching, but the yield strength is reduced due to the thinning of the cell walls towards the midpoints of the struts. The uniaxial yield strength increases slightly with non-uniform wall thickness as in this instance yield is due to the formation of plastic hinges in the vicinity of the joint, such that a redistribution of the cell wall material towards the joint will increase the plastic collapse moment and hence the yield strength.

Chen et al.20 also studied the effects of random defects (i.e. cell size variations, fractured cell walls, cell wall misalignments and missing cells) using the FE method. This approach and their findings are discussed in the sections on ‘Finite element methods utilising repeating unit cell’ and ‘Finite element methods utilising random Voronoi technique’.

Finally, Zhu et al.56analytically determined the elastic constants (Young's modulus, shear modulus and Poisson's ratio) of an open cell foam loaded in uniaxial tension, using tetrakaidecahedral cells on a bcc lattice, as shown in Fig. 6, by considering the bending, twisting and extension of the cell edges.

Figure 6

Three tetrakaidecahedral cells in bcc lattice, with lattice repeat vector shown: typical members under load, when tensile stress is applied along z axis, are shown in bold56

It is assumed in Ref. 56 that the material of the cell edges is an isotropic elastic solid with Poisson's ratio of 0·5. It is found in Ref. 56 that the Young's modulus, shear modulus and Poisson's ratio can be expressed as functions of relative density, for an equilateral triangle cross-section, as follows (11) (12) (13) where R is the relative density of the open cell foam, E₁₀₀ is the Young's modulus of the foam in the 100 lattice vector direction (note that for the foam analysed, E₀₀₁ = E₀₁₀ = E₁₀₀), G₁₂ is the shear modulus of the foam in lattice axes and ν₁₂ is the Poisson's ratio of the foam in lattice axes. In Ref. 56, it is noted that the relationship between the relative density and the width of cell edges depends on the cross-sectional shape; indeed, the Young's modulus is found to be 38 higher, for a given relative density, if the edge cross-sections are Plateau borders rather than equilateral triangles.

Numerical modelling methods of metal foams

In an attempt to understand the mechanical response of foam materials under loading, numerical modelling tools have been developed for metal foams.53 These can be separated under two distinct techniques:

FE methods utilising a repeating unit cell such as a tetrakaidecahedron

FE methods utilising the random Voronoi technique. This approach gives a more accurate representation of the cell geometry of the foams.26

These methods are now reviewed in turn.

Finite element methods utilising repeating unit cell

The second approach to studying the mechanical behaviour of metal foams is to analyse a repeating unit cell, such as a tetrakaidecahedron, using FEs. Simone and Gibson21 used FE analysis of idealised 2D (hexagonal honeycomb) and 3D (closed cell tetrakaidecahedral foam) cellular materials to consider the effect of the distribution of solid between cell faces and edges on the mechanical properties. Specifically, they used FE analysis to estimate the relative elastic modulus and relative plastic collapse strength as a function of relative density and the distribution of solid material. For the 3D case, they utilised the Kelvin tetrakaidecahedron unit cell as it is the lowest energy unit cell known consisting of a single polyhedron.

The tetrakaidecahedron cell is defined by six planar square faces and eight hexagonal faces that are non-planar, but have zero mean curvature. The analysis in Ref. 21 makes the simplification that all cell faces are planar so as to eliminate non-linearities caused by wall curvature and isolate the effects of the solid distribution. An aggregate of the tetrakaidecahedral cells used to model the closed cell foam is shown in Fig. 7a.

Figure 7

a idealised tetrakaidecahedral foam structure and b normalised modulus and normalised peak stress versus fraction of solid in Plateau borders for tetrakaidecahedral foams21

It is found in Ref. 21 that shifting material away from the cell faces of a closed cell foam into Plateau borders along the edges has minimal effect on the elastic modulus and causes a reduction in the peak stress, which is shown in Fig. 7b. This is accounted for in21 by noting that closed cell foams deform primarily by the in plane stretching of the cell faces, and shifting material towards the cell edges reduces the net cell face area with respect to in plane axial deformation, thus leading to a reduction in stress.

The foam behaves more like an open cell foam as the volume fraction of the solid shifted to the edges increases to the point where almost all of the solid material is in the cell edges as opposed to the faces.

Onck et al.19 modelled the effect of the size of a rigid indenter on the indentation strength of a regular hexagonal honeycomb of unit depth using the FE analysis program ABAQUS. A honeycomb of sufficient size to eliminate any influence of the boundaries was selected. Each cell wall was modelled using beam elements and the solid cell wall material was assumed to be elastic perfectly plastic with E_s = 70 GPa, ν_s = 0·3 and σ_ys = 300 MPa. The relative density was taken to be 0·09. The indenter was displaced uniformly into the honeycomb while the opposite edge of the honeycomb was fixed in the direction of indenter displacement and free to translate in the normal direction.

Figure 8 shows Onck et al.'s results19 as a plot of indentation strength normalised with respect to compressive strength versus the ratio of indenter width to cell size.

Figure 8

Normalised indentation strength plotted against ratio of indenter width to cell size17

It can be ascertained from Fig. 8 that the indentation strength decreases as the ratio of indenter size to the cell size increases. Onck et al.19 explain that this trend may be understood by noting that the total load on the indenter is equal to the sum of that required to crush the honeycomb beneath the indenter and that required to fully yield the cell walls at the perimeter of the indenter, resulting in the following equation (14) where σ_ind is the indenter strength, is plastic collapse stress of the honeycomb, is the plastic shear strength and S/w is the ratio of cell size to indenter width. Equation (14) has been plotted in Fig. 8, and it can be seen that there is good agreement with the FE results when C₂ = 7·23.

Moreover, in Ref. 57 Onck et al. compared equation (14) developed in Ref. 19 to experimental results for axisymmetric indentation tests carried out on a closed cell aluminium foam, and found that the indentation data are well described by equation (14).

Hodge and Dunand22 developed a 3D FE model to predict the creep properties of nickel rich NiAl foams. They used a repeating unit cell consisting of three orthogonal sets of four parallel hollow or solid struts with square cross-section, connecting at joints arranged on a square lattice (see Fig. 9a). They considered two relative densities for both types of struts (hollow and solid). It should be noted that the actual architecture of the NiAl foam is much less regular than the simple cubic lattice shown in Fig. 9a. While the model is a simplified representation of the general geometry of the NiAl foam, it does capture important features such as hollow struts and 3D periodicity.

Figure 9

a geometric FE model for foam with hollow struts (relative density = 5; in solid strut model, struts have smaller width but same cross-sectional area) and b compressive strain rate versus stress curves at 900°C as measure experimentally on 20 pores per linear inch foams and as calculated by FE for hollow struts22

Creep of the NiAl material in Ref. 22 within the struts and joints is assumed to take place according to the power law equation (15) where A is the Dorn constant, n is the stress exponent, Q is the activation energy, R is the gas constant, T is the temperature, is the steady state strain rate and σ is the uniaxial applied stress.

It is found in Ref. 22 that the 3D FE model predicts creep rates in reasonable agreement with experimental data from creep tests between 800 and 1100°C with compressive stresses between 0·1 and 1·5 MPa of NiAl foams consisting of open cells with hollow struts (see Fig. 9b). Based on the numerical results from the FE model, a simplified analytical model is proposed in Ref. 22 whereby struts parallel to the applied stress deform by creep in a purely compressive mode, while perpendicular struts prevent buckling but provide no directional load bearing capacity. It is found that the analytical model produces results that are very similar to the predictions of the numerical model and in good agreement with the experimental data.

A significant limitation of the above unit cell modelling approach is that it does not capture the natural variations in microstructure that are observed in most cellular materials, for instance, cell size variations. Models of FE that allow for these to be accounted for are discussed in the section on ‘Finite element methods utilising random Voronoi technique’.

Chen et al.20 considered the effect of cell wall misalignments using the FE method by displacing in random directions the joints of a perfect hexagonal honeycomb by a constant distance αl, where l is the length of each side of the honeycomb and the fraction α gives the magnitude of the imperfection. The displaced cellular structure for the case α = 0·2 is shown in Fig. 10a. The uniaxial yield strength σ_U and the hydrostatic yield strength σ_H of a honeycomb with cell wall misalignments, normalised by the corresponding yield strengths and of a perfect honeycomb, are plotted as functions of the imperfection measure α in Fig. 10dfor a honeycomb of relative density 0·10. For completeness, the ratio of uniaxial to hydrostatic yield strength σ_U/σ_H of the imperfect honeycomb is included in Fig. 10d. It is concluded from Fig. 10d that the cell wall misalignments lead to a large reduction in hydrostatic strength. Chen et al.20 also noted that by varying the bending to stretching strength ratio of a beam element, it is found that under hydrostatic stressing the deformation mechanism of cell wall bending dominates over cell wall stretching as α increases.

Figure 10

Typical FE mesh for honeycombs with a cell wall misalignment (α = 0·2); b fractured cell walls (number fraction = 1); c 1 cell missing. Effect on uniaxial and hydrostatic yield strengths of 2D foams with relative density 0·10 of d cell wall misalignments; e fractured cell walls and f missing cells20

Chen et al.20 also considered the effect of fractured cell walls on the strength of foams by randomly removing 1 of the cell edges of a perfect honeycomb. Figure 10bshows the FE mesh of a perfect honeycomb with 1 of its cell edges removed randomly. Chen et al.'s20 results are shown in Fig. 10e, which plots the uniaxial and hydrostatic yield strengths of a honeycomb with fractured cell walls, normalised by the corresponding strengths for a perfect honeycomb, as a function the percentage of fractured cell walls for an initially perfect honeycomb of relative density 0·10. For completeness, the ratio of uniaxial to hydrostatic yield strength σ_U/σ_H of the imperfect honeycomb is included in Fig. 10e. It can be seen from Fig. 10e that the removal of cell walls results in pronounced weakening. Chen et al.20noted that by varying the bending to stretching strength ratio of a beam element, the bending of cell walls is found to be the dominant deformation mechanism under hydrostatic loading for perfect honeycombs with fractured cell walls. This differs from the hydrostatic compressive behaviour of honeycombs without fractured cell walls, which is dominated by cell wall stretching.

Finally, Chen et al.20analysed the effect of missing cells by removing a cluster of adjacent cells within a perfect hexagonal honeycomb structure (see Fig. 10c). Their results are shown in Fig. 10f, where it can be deduced that the presence of a single hole reduces the hydrostatic strength significantly – the presence of a hole induces bending of the cell walls for hydrostatic loading, which produces the reduction in strength.

Finite element methods utilising random Voronoi technique

The third approach to studying the mechanical behaviour of metal foams is to model the real foam structure (which can be represented as a stacking of randomly distributed cells of various shapes and sizes which fill the space completely) by the random Voronoi technique. This technique can produce a geometrically more realistic model of the foam structure.23

The objective is to create FE models in a manner that is similar to the way metal foams are produced in reality, and to make the models large enough to provide reliable input for generation of homogenised engineering properties.21 This is achieved by generating distributions of cell nuclei (or points) in space numerically, and simulating cell growth around the nuclei through generation of Voronoi tessellations. Voronoi tessellations are a form of space decomposition; given a set of N points in a plane, a Voronoi tessellation divides the domain in a set of polygonal regions, the boundaries of which are the perpendicular bisectors of the lines joining the points. Each polygon contains only one of the N points. The resulting structure is a Voronoi foam. If the nuclei in the model are distributed periodically, the cellular foam microstructure is then regular. A random distribution of nuclei is modelled with a random space tessellation. The randomly distributed points then become the centres of the foam cells; flat cell boundary faces appear where two neighbouring cells come into contact. Open cell foams can be modelled by locating the struts where three cell faces meet and subsequently removing the cell faces. The initial distribution of the nuclei completely determines the final geometry of the Voronoi tessellation and hence foam microstructure.23

Zhu et al.26 investigated how cell irregularities affect the elastic properties of open cell foams using 3D FE analysis. They constructed 3D random structures by first defining a cubic control volume (or representative volume element). In order to achieve a periodically repetitive unit volume, the representative volume element is considered to be surrounded by 26 equivalent boxes. A distribution of virtual cell nuclei is then placed in the representative volume element by generating x, y and z coordinates independently from the pseudorandom numbers between zero and one, starting at one corner of the cube. Once the first point is specified, each following random point is accepted only if it is greater than a minimum allowable distance δ from any existing point, until n nuclei are seeded in the cube. The cell nuclei are defined as the centre points of solid spheres that are packed in the space. 26 26,58 The regularity of a 3D Voronoi tessellation can be measured by equation (16) (16) where d₀ is the minimum distance between any two adjacent nuclei in a regular lattice with n identical cells. To construct a random tessellation, the maximum δ should be less than d₀, else it is impossible to obtain n cells. For a regular lattice, δ = d₀. For a totally random tessellation, δ = α = 0.26

All struts in the foam are represented mechanically by beams rigidly connected in vertices. It is assumed for simplicity in Ref. 26 that all the struts have the same and constant cross-section, and thus the analyses was limited to models having low relative density. The results of Ref. 26 are given in Fig. 11 and suggest that, for low density foams, highly irregular foams have a greater Young's modulus and shear modulus, and smaller bulk modulus than a perfect foam. It was also found that the Poisson's ratio does not change with cell regularity, but does reduce gradually with increasing relative density (see Figs. 11 and 12).

Figure 11

Effects of cell regularity on reduced a Young's modulus, b bulk modulus, c shear modulus and d Poisson's ratio of random Voronoi foams having constant relative density of 0·01: diamond points represent theoretical results, and bars represent computational results26

Figure 12

Effects of relative density on Poisson's ratio of random Voronoi foams with varying degrees of regularity parameter α (Ref. 26)

Huang and Gibson24 created an open cell Voronoi structure and inputted it into ABAQUS to analyse the steady state creep response of a foam. The creep response of foams is of interest when they are used at high temperatures relative to their melting point. For instance, metal foams are attractive for heat transfer devices and lightweight structural sandwich panels, both of which may require them to be used at high temperatures. It was found that the creep response of a damaged Voronoi foam with struts randomly removed increases rapidly as the fraction of struts removed increases (see Fig. 13).

Figure 13

Creep rate of damaged Voronoi foam, normalised by that of intact foam at same nominal relative density, plotted against fraction of struts removed24

Chen et al.20 studied the effects of cell size variations using the FE method. They considered the multiaxial yield response of 2D honeycombs with both Γ-Voronoi and δ-Voronoi distributions of cell size (for the former, the minimum distance separating two adjacent generation points is unconstrained, while for the latter it must be larger than a minimum prescribed value). Their findings are shown in Fig. 14 – it can be seen that the hydrostatic strengths of these two structures are less than that of a perfect honeycomb by a factor of 2 or 3, though the microstructures are not sufficiently dispersed in cell size to switch the deformation response from cell wall stretching to cell wall bending under hydrostatic loading. Furthermore, the found that the uniaxial and plastic properties of random Voronoi models are well described by those of a perfect honeycomb, regardless of whether the cells are distributed according to the Γ- or δ-law.20 Comparable results for the elastic properties of a δ-Voronoi distributed honeycomb are presented by Silva et al.25 for 2D foams, and by Grenestedt and Tanaka59for 3D foams.

Figure 14

Effect of size variations and cell wall misalignments on hydrostatic yield strength of 2D cellular foams20

Chen et al.20 went on to consider the effect of fractured cell walls in Γ-Voronoi structures, and it was found that the yield behaviour of a Γ-Voronoi structure with relative density 0·10 is essentially the same as that shown in Fig. 10e for a perfect honeycomb. This can be explained by the fact that fractured cell walls have a much stronger effect than variations in cell size on reducing the yield strength of 2D foams.

Discussion

The ability to accurately model metal foams is of prime importance to their application in industry. For example, as noted in the section on ‘Overview of metal foams and their structures’, the use of sandwich structures in commercial aviation is at present limited to secondary structures only. To be applicable to primary structures, sandwich structures must ensure that any damage during the service life of the component does not result in failure prior to the damage being detected – this would need to be demonstrated by appropriate tests and analyses, and it is here that the capability to accurately model the behaviour of metal foams is of value.

Models of FE that utilise repeating unit cells are unable to capture the natural variations in the microstructure, however they do still capture some of the important features of foams such as 3D periodicity and the cross-sectional shape of the cell walls; such models have been shown to provide good agreement with experimental data and theoretical predictions. 19 19,22 Models of FE utilising the random Voronoi technique can produce a geometrically more realistic model (and are hence more representative) of the foam structure than analytical methods or FE methods that utilise a repeating unit cell. 24 24,26 However, they can require longer model construction and running times and hence cost.

Analytical methods do not take into account the effect of imperfections in the microstructure, and are simplified representations of foam structures. However, they do provide a means to quickly assess the mechanical properties of a foam before proceeding to more complicated numerical methods, and can provide a benchmark for validation of a new FE model. Moreover, theoretical models of foam mechanics permit the identification of the deformation mechanisms that control mechanical behaviour – in FE analysis, there may be no physical understanding of the dominant deformation mechanisms.56

Methods of FE are also being developed that utilise a 3D tomographic image (a non-destructive visualisation of a foam at the scale of its cellular microstructure obtained by X-ray tomography) of a real foam as the geometric description of the model (see Fig. 15). 60 60,61Such techniques could model both open cell as well as closed cell foams and could prove to be useful in predicting the mechanical response of cellular materials. It is possibly here that the most exciting developments for the modelling of foams resides.

Figure 15

Three-dimensional tomography image of closed cell foam60

Conclusions

The available sandwich core types found in industrial applications have been reviewed. It has been ascertained that sandwich structures that comprise of metal foam cores can display isotropy in mechanical properties, can be made with integral skins, can be readily formed into curved shapes, open cell foams do not trap moisture and combine low density with good bending stiffness and strength.

The current microstructural modelling tools available for metal foams have been reviewed. These fall under three categories: analytical methods using dimensional analysis, FE methods utilising a repeating unit cell, and FE methods utilising the random Voronoi technique. It has been noted that analytical methods do not take into account the effect of imperfections in the microstructure, and FE methods utilising a repeating unit cell also fail to capture the natural variations in microstructure that are observed in most cellular materials. It has been observed that imperfections reduce the hydrostatic strength of foams by differing amounts. The Voronoi technique can produce a geometrically more realistic model of the foam structure than the other two methods.

Finite element methods are also being developed that utilise a 3D tomographic image of a real foam as the geometric description of the model and it is possibly here that the most exciting developments for the modelling of foams resides.

Footnotes

Acknowledgements

I would like to thank Dr Daniel Balint, Professor Jianguo Lin and Professor Tony Atkins for providing valuable advice and recommendations with regards to the preparation of this review.

This review was the commended review of the 2011 Materials Literature Review Prize of the Institute of Materials, Minerals and Mining, which is administered by the Editorial Board of Materials Science and Technology.

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