Prediction of Young's modulus and Poisson ratio for foamed metals using octahedron model

Simplified treatment of model

For the convenience of calculation, the strut in the abovementioned unit octahedron can be taken as a small cylinder. In addition, let the porosity be θ. The side length of cube containing unit octahedron is a, so the length of strut L and the radius of strut r for the octahedron will be obtained from Fig. 1 as follows21 (1) (2)

Deduction of relation

When the foamed body is subjected to tensile load, the included angle between the strut and the axis of the unit octahedron (see that in Fig. 1) will decrease or has a tendency to decrease. Therefore, the strut can be taken as a cantilever of which node A is fixed, and node B is subjected to load (see Fig. 2). In this figure, α₀ is the original included angle between the strut and axis of the unit octahedron and α₀ = arccos(3^1/2/3), f is the applied load to the strut, and f₁ and f₂ are the parallel and vertical forces to the axis of the strut resulting from f respectively.

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Figure 2

Analytical schematic for elongation of unit octahedron

Based on the relationship between maximum stress σ_max and bending moment M for the cantilever from materials mechanics, we have (3) where Z is the bending modulus of the cantilever. The following analysis can be made.

As a cantilever, it will have a greater tendency to deflect under larger bending moment for the strut and will have a stronger antideflection ability with higher bending modulus. Therefore, when the stress applied to strut reached the proportional limit for corresponding dense material, the deflection angle α will increase with increasing bending moment M_p and decrease with increasing bending modulus Z. From the above, we know (4) where α_p is the included angle between the strut and its axis when the proportional limit for corresponding dense material is reached within the strut, and α₀ is the original included angle between the strut and axis of the unit octahedron, with 0<α_p<α₀. From equation (3) we also have (5) where σ₀ is the tensile strength of corresponding dense material.

Thus, the deflection angle α can be approximately regarded to be proportional to M_p and inversely proportional to bending modulus Z22 (6) where γ is a coefficient determined by the material (prepared by the same process). From the above equation, it can be seen that α_p is the feature included angle for the proportional limit decided by the material species itself.

The factual tensile force applied to the foamed metal will increase from zero when the porous body undergoes tensile loads; the included angle between the strut and the tensile direction (see Fig. 1) will reduce, that is, the unit octahedron will be gradually elongated. When the tensile force applied to the foamed material increases, the maximum stress within the strut will reach to the proportional limit for the dense material, and the strut will deflect to reach to the position where non-linear elastic deflection will happen.

A sectional profile for the node was produced from Fig. 2, as shown in Fig. 3.

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Figure 3

Node sectional diagram of unit octahedron for analysing elastic strain limit from Fig. 1

In the unit octahedron, combining equations (1) and (2) and the geometric relationships in Fig. 3, the length of strut excluding the node size will be (7) Therefore, the original height of pore in the unit octahedron will be (see Figs. 2 and 3) (8) Correspondingly, for the pore, the maximum size in the direction perpendicular to the height is (9) As to the high porosity foamed metal, the deformation of which mainly results from the deflection of strut by bending moment, the elongation of the strut is negligible, and the strain is primarily dependent on the deflection of struts.

When the included angle between the strut and the tensile direction in the unit octahedron turns to be α_p, the height of pore will become (10) Correspondingly, for the pore, the maximum size in the direction perpendicular to the height will become (11) Therefore, the apparent linear elastic strain limits ϵ_limit1 in the tensile direction and ϵ_limit2 in the direction perpendicular to the tensile force will be respectively (12) (13) From the above two equations, the apparent Poisson ratio for the porous body will be directly derived as (14) Since the α_p is dependent on the material species itself, the right side of equation (14) is a material constant.

Let we have (15) where K_v is a constant that depends on the material making of the porous body; therefore, the apparent Poisson ratio for the high porosity reticulated foamed metal will also be a material constant and independent of the porosity. It is consistent with the results from the study of Gibson and Ashby.1 However, it is simpler and more direct with more obvious consistency and logicality for the present deduction.

In addition, following the methods of Liu,15 the apparent proportional limit for the foamed metal will be derived as (16) where K_p is a material constant, σ_p is the proportional limit for dense material, and m is the plastic brittle index for dense material with its value between 1 and 1·5.15,26

The apparent Young's modulus for the foamed metal can be directly obtained from equations (12) and (16) (17) Let we can obtain (18) where K_E is also a material constant that depends on the material species.

Simplification of model

The actual pore structure in the foamed material is complicated with different sizes and shapes. Neither the cubic model proposed by Gibson and Ashby1 nor the tetrakaidecahedral model by Kelvin or octahedral model in the present paper can comprehensively represent the actual pore structures in the foamed materials. Even so, it is not necessary to find a model that can represent the actual pore structure accurately in general cases, since any theoretical model is of the ideal condition for the simplified analysis to understand the features and properties of foamed materials. As to exactly understand the micro-image within porous bodies, the new methods27,28 of field emission microscopy and material point models of foams can be utilised. Basically, the applicability for the model developed is dependent on whether it can correlate the analysed results well with the measured properties.

Analysis for octahedron model

Similar to the Gibson–Ashby and Kelvin models, the octahedron model is also an analytical structure–property model, and it is not an absolutely structural simplified model.1 The octahedron model is characterised by the 3D isotropy, close packed pore units, high symmetry of unit cells and equivalent struts; it can represent the basic structural features for isotropic 3D reticulated foamed materials. Therefore, the octahedron model is a comprehensive model combining the reality and abstract, structure and property. The unit cell was treated through an octahedron model in order to analyse the property practically while not against the actual structure. It can reach acceptable results in a simple way, and it is also quite independent of the actual different structures in the foamed materials.

Bending problem for struts

From the perspective of mechanics, a pole should be taken as a slim pole when a ratio of 10 for length to radius L/r was reached, and the bending deformation should be considered. In fact, the bending deformation of struts may not be thought as negligible in the octahedron model. The rupture will happen when the strut deflects to some extent. Based on the analysis for the octahedron model, to what extent the strut will rupture can be solved by using the material constant, and there would be no change in the mathematical–physical expression form.

Plastic brittle index m

The index value of m is in the range of 1–1·5,15,26 and m = 1 only appears in the extremely soft condition of struts like foamed lead. In addition, the ‘octahedral analytic model’ proposed by the present authors is an abstractly simplified model for foamed metals with high porosity, and its structural form is the simplification for the real cell structure of foamed metals. As an approximation, the ‘octahedral analytic model’ can be applied to the description of the low density porous materials like the Gibson–Ashby and Kelvin models. The octahedron model demonstrates its rationality theoretically, and it can reflect the deformation mechanisms for tension/compression and bend simultaneously even though it is not the real cell model. It is still applicable to low density porous materials, and it also should be noted that more work need to be conducted to predict the Poisson ratio for foamed materials.