Four types of uniaxial compressive tests were performed in order to evaluate the mechanical properties of the foam. All tests were displacement-controlled and used an Instron 3369 testing machine with one extensometer. Tests were based upon standard ISO/DIS 13314(E) (“Compression Test for Porous and Cellular Metals”). The displacement rate was between 0.5 mm/min and 1.0 mm/min (0.3%/min and 1.3%/min) and platens were lubricated with automobile axle grease, and at least three tests of each type were performed:

Tests #2 and #3 showed that there is a significant difference between the apparent stiffnesses calculated from extensometer data and crosshead displacement data (3150 MPa vs. 700 MPa), suggesting that hollow sphere steel foam experiences strong localized strain near the platens. Engineering Poisson’s ratio (test type 1) increased from approximately 0.0 to about 0.2 in the inelastic range, where the Poisson’s ratio appeared to level off between 20% and 50% applied axial strain. Unloading moduli (test type 3) began around 2000 MPa in the elastic range, and rapidly increased to approximately 3150 MPa, a value from which there is minimal deviation with increasing strain (see Figure 2). Compressive stress-strain curves can be divided into three regimes: linear-elastic, yield plateau with approximately constant slope, and a final portion of steeply rising stress. The strain corresponding to the onset of the rapidly rising portion of the stress-strain curve is labeled as the densification strain [4].

No established definition exists for the onset of densification, and the following definition was adopted. Let E_t,0.05(ɛ) be the tangent modulus of the material determined by performing a linear regression on the stress-strain curve over the range (ɛ − 0.05, ɛ + 0.05), and define E_t,0.05(ɛ_proof) to be the value of this tangent modulus in the window immediately following the 0.01 proof stress (essentially a 0.01 offset version of the yield stress). Thus, the densification strain is defined as: (1)

In essence, the densification is assumed to begin when the tangent modulus exceeds for the first time the post-yield tangent modulus. Densification begins approximately at a strain of 65% for our hollow sphere foam, at a stress of 17 MPa (test type 4, see Figure 2). Full results are summarized in Table 1.

Tensile testing was based upon the ASTM E8-08 (“Tension Testing of Metallic Materials”) standard, and specifically the “plate-like” specimen described therein. Modified grips consisting of 25 mm slots cut into both ends of the sample with solid plates epoxied into these two slots were used. Tension was applied at 0.5 mm/min (1.0%/min), and an extensometer was attached to the test section of the specimen (see Figure 3). Three samples were tested in this manner, plus a fourth one that was tested with repeated unloadings.

Tension tests showed a wide variation in results (see Figure 4). Fracture occurred at strains between 1.5% and 4.0%, and ultimate stresses were between 4 and 6 MPa. In tests #1 and #2, the fracture occurred along two distinct cracks (see Figure 3), while in test #3, the entire cross-section fractured across a single dominant crack. On a mesostructural level, fracture surfaces were characterized by pullout of the sphere-to-sphere welds (see Figure 3). Unloading moduli were nearly identical to those observed in compression, at approximately 3100 MPa, and only once fracture began did the moduli begin to drop rapidly.

Shear tests were performed upon hollow sphere foam (Figure 5). Tests showed remarkable consistency, with elastic modulus and ultimate stress varying by less than ±10% (see Figure 6 and Table 2). Some ductility is evident in that the material yields before it reaches its ultimate strength. There are also two distinct slopes in the post-yield behavior. The second, smaller slope, beginning at about 0.07 shear strain, is likely where friction between the heterogeneous fracture surfaces begins.

Comparing to Friedl et al. [5] tests on 8% foams, a similar yield stress (4.6 MPa) but an elastic modulus (640 MPa), nearly 5 times greater than their tests, was observed in compression. In tension, similar values for yield stress (3.3 MPa), fracture strain (3.3%), and fracture stress (5.3 MPa) were observed. This similarity, despite higher relative density of our foam (14.5%), may be explained by Friedl’s use of a stronger base metal (316 L stainless).

Gibson and Ashby [6] developed mathematical models for predicting the effective properties of metal foams. Comparing all of their available “open-cell” equations to experimental results, the results are within the predicted range with the exception of Poisson’s ratio, which is predicted to be approximately 0.3, differing substantially from our experimental values given in Table 1.

The mesostructure of sintered hollow sphere foam consist of the spheres and the welds between those spheres. The spheres have been shown to be random close-packed (RCP) [7]. Wouterse and Philipse [8] tested five RCP algorithms, and showed that two different variations of the “Mechanical Contraction Method” resulted in RCP stackings that were most similar to an experimental stacking. The simpler of those two algorithms, the “Modified Mechanical Contraction Method”, was selected [8,9] for our study. After sphere locations were determined, the cylindrical welds were inserted to connect spheres, or overlapping walls were merged. Sphere size, wall thickness, weld diameter, and sphere location were allowed to vary randomly in order to capture variation in foam mesostructure (Figure 7).

Several simulations were performed in order to validate the full stress-strain curve of the simulations with those obtained experimentally. The study also investigated size effects within the simulations. One 7 mm cube, two 6.25 mm cubes, and several 5.5 mm cubes were simulated. All inputs were based upon microscopy studies when possible. The base metal yield stress 172 MPa was based upon the experimental test performed on compacted samples, and the elastic modulus was initially assumed to be the standard 200,000 MPa. However, based upon microscopy studies of intact foam samples, a microporosity (that is, porosity within the sphere walls themselves) of 20% was estimated and therefore the initial base elastic modulus was reduced by this value. The validation simulations showed increasing accuracy as model size increased from 5.5 to 7 mm (see Figure 8a,b).

Similarly to experimental tests by Andrews [10], smaller samples have lower apparent strengths and stiffnesses. No known, published research has attempted to study the effect of model size upon Poisson’s ratio. The simulated Poisson’s ration decreased with the sample size. The simulations of 7 mm models match experimental engineering Poisson’s ratio with reasonable accuracy. While the experimental data is noisy, Poisson’s ratio rises with compressive strains, which indicates a gradual increase in foam incompressibility.