III. RESULTS AND DISCUSSION

Figure 1 shows the CT-scan images of pure Al foams having ρ_rel of (a) 0.32 and (b) 0.35 and those of 7075-T6 alloy foams with ρ_rel of (c) 0.32 and (d) 0.35, respectively. As can be observed, all the foam samples have homogeneous distribution of interconnected open-cells. Typical C _s of 0.75–1.0 mm can be observed in both sets of samples. Figure 2(a) presents the stress–strain curves of pure Al foams having ρ_rel = 0.25, 0.28, 0.32, 0.35, 0.40, 0.45, 0.47, and 0.50 compressed at έ = 0.002 s⁻¹ (quasi-static), and Fig. 2(b) shows the zoom-in view of the elastic region of the curves. Figure 3(a) shows the stress–strain curves of pure Al foams having ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50 compressed at έ = 2.0 s⁻¹ (static loading), while Fig. 3(b) depicts the zoom-in view of the elastic region of the curves. As reported elsewhere,^{Reference Paul and Ramamurty7,Reference Ruan, Lu, Chen and Siores30,Reference Cady, Gray, Liu, Lovato and Mukai31} the deformation process initiates in a single band at the weakest portion of the foam, mostly at the contacts with the loading surface and proceeds one layer after the other. In most of the samples in the present work, it was found that the first deformation band (DB) (formed by contributions of multiple mechanisms of bending, shearing, and crushing) appeared mostly at the top layer of the samples touching the upper plate of the universal testing machine. This ensures that the foams developed by the present process have nearly homogeneous cell/density distribution throughout the foam matrix. The compressive response of the pure Al foams shows a typical deformation process that can be classified into three regions, viz, linear-elastic, plateau, and densification regions. In the linear-elastic region where the stress increases rapidly with a small increase of compressive strain, cells of the top layers progressively experience compression. A significant fall in the slope of the stress–strain curve is usually observed after the initial collapse (onset of the plateau region). With further increase in the strain, weak layers of the undeformed region collapse in a layer-wise manner through the propagation of the DB. The above deformation process repeats till ε_d is achieved in the foam where majority of the cell walls come into self-contact (onset of the densification region), followed by a small increase in strain and the rapid rise of the corresponding stress. The nonserrated stress–strain curves also confirm that the pure Al foams possess nearly uniform distribution of pores, strut thickness, and ρ_rel. The typical values of Y _s observed in the MFs having different ρ_rel (compressed at έ = 0.002 and 2.0 s⁻¹) are summarized in Table I. It can be noticed that the Y _s of the MFs shows a monotonically increasing trend with ρ_rel. With the increase in ρ_rel, the moment of inertia to deform/bend the struts both in the elastic and plastic regions increases, which leads to the enhancement of the compressive properties (Y _s and E) of the MFs.^{Reference Bafti and Habibolahzadeh10} In all the samples, the tangential modulus (E _pl) in the plateau region (starting point of the plastic region till the onset of the densification region) is much smaller than in the elastic region. As can be observed in Fig. 2(a), σ_pl increased with increasing ρ_rel while an opposite trend was observed for ε_d (Table I). In addition, the developed foams exhibited έ sensitivity. The typical Y _s values (Table I) estimated from the plots in Fig. 2(a) for ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50 are 4.75 MPa, 7.31 MPa, 7.7 MPa, 8.32 MPa, and 13.72 Mpa, respectively, when compressed at έ = 0.002 s⁻¹. The corresponding Y _s values of the same set of samples when compressed at έ = 2.0 s⁻¹ [as estimated from the plots in Fig. 3(a)] are 5.1 MPa, 9.88 MPa, 17 MPa, 22.29 MPa, and 26.58 MPa, respectively. With the increase in έ by 1000 times, the strength of the foams doubled in the samples with higher ρ_rel, viz, 0.40, 0.45, and 0.50. However, the enhancement was marginally low for ρ_rel = 0.32 and 0.35. Since the surface skin of the samples are very thin (open-cell structure), hence the skin contribution does not seem to have major contribution in enhancing the Y _s and other mechanical properties.^{Reference Youn and Kang32,Reference Peroni, Solomos and Pizzinato33} The έ sensitivity of the mechanical properties of MFs are attributed to the έ sensitivity of the parent material, micro-inertial effect, compression and flow of gas in the cells, shape, and size of the cells, and microstructural properties (e.g., defects), which may cause different deformation modes of the material at different έ.^{Reference Wang, Shen, Lu and Zhao27,Reference Calladine and English34–Reference Campana and Pilone39} The present samples have open-cell morphology; therefore, the influence of air/gas on the mechanical properties can be ruled out (there is no chance of entrapped gas in the foams). As mentioned before, the foams have nearly homogenous C _s distributions and show nonserrated stress–strain curves. Therefore, the contributions from the defects affecting the mechanical properties are not so prominent. According to the micro-inertial hardening theory proposed by Klintworth and Stronge,^{Reference Klintworth and Stronge40} at low έ, micro-inertia has a nominal effect, and the dominating mode of deformation is the buckling of struts.^{Reference Paul and Ramamurty7} When MFs are subjected to low έ, the cells get longer time to respond to the applied load (in achieving equilibrium) by releasing the stress through crushing/bending of cells and hence can transmit the impact stress properly after the deformation of the cells. In the case of compression under high έ, the inertia of the struts opposes the buckling process and forces it to occur at a delayed time. Therefore, the threshold buckling force is higher than the quasi-static compression, resulting in enhanced buckle loads at high έ.^{Reference Calladine and English34} The observed έ effect is much prominent in the high ρ_rel samples. The Y _s values increased significantly at έ = 2.0 s⁻¹ compared to έ = 0.002 s⁻¹ (Table I). Although such a phenomenal increase in Y _s does not look plausible, repeated experiments confirmed the same type of behavior with minute variation in strength. Apart from ρ_rel and έ sensitivity of the specimen, the strut thickness can also be a decisive factor in enhancing the Y _s values. In the developed MFs, the strut thickness was controlled by scheming the size of the salt beads used as preforms and the infiltration pressure. The samples with ρ_rel > 0.35 were formed with salt beads of 1.5–2 mm size while the rest were formed with 0.75–1 mm size beads. No such dominating effect of salt bead size (or C _s) or strut thickness on Y _s was observed for έ = 0.002 s⁻¹ with increasing ρ_rel. However, for έ = 2.0 s⁻¹, a prominent increase in Y _s values were observed only for larger C _s (1.5–2 mm, ρ_rel > 0.35) and not in the samples with smaller C _s (0.75–1 mm, ρ_rel < 0.40). It is to be noted that έ = 2.0 s⁻¹ is a static rate rather than the dynamic rate where the micro-inertial effects dominate. Therefore, it can be concluded that C _s and strut thickness are influencing the strength of the foam samples and also contributing toward their έ sensitivity.

FIG. 1.

CT-scan images of pure Al foams having ρ_rel values of (a) 0.32 and (b) 0.35, and 7075-T6 alloy foams with ρ_rel values of (c) 0.32, and (d) 0.35, respectively.

FIG. 2.

(a) Stress–strain curves of pure Al foams of ρ_rel = 0.25, 0.28, 0.32, 0.35, 0.40, 0.45, 0.47, and 0.50 compressed at έ = 0.002 s⁻¹ and (b) the zoom-in view of the elastic region of the curves.

FIG. 3.

(a) Stress–strain curves of pure Al foams having ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50 compressed at έ = 2.0 s⁻¹ and (b) the zoom-in view of the elastic region of the curves.

TABLE I.

Mechanical properties of pure Al foams derived at two different strain rates (έ = 0.002/s and 2.0/s).

^a The bracketed terms denote the values at ε_d.

The values after the ± sign represent the standard deviation.

Figure 4(a) shows the compressive stress–strain curves of 7075-T6 alloy foams having ρ_rel = 0.25, 0.28, 0.32, 0.35, 0.40, 0.45, 0.47, and 0.50, when compressed at a quasi-static loading rate of 0.002 s⁻¹. The zoom-in view of the elastic region of the curves is shown in Fig. 4(b). As can be noticed, the alloy foams exhibited quite different stress–strain curves, compared to the curves of pure Al in Fig. 2(a) for the same έ. The samples under deformation showed elastic and plateau region but not the densification region as the samples could not withstand the load till the densification of the foams. The curves are jagged in nature with continuous ups and downs and possessed very small plateau region. The foams revealed their brittle nature with the fracture initiated soon after reaching the maximum strength, i.e., Y _s. Cracks were generated in the samples when compressed beyond Y _s and gradually the solid content started falling out of the boundary (in parts) indicating the local brittle fracture. By contrast, the foams showed complete universal compressive characteristics when subjected to a έ of 2.0 s⁻¹. Figure 5(a) shows the stress–strain curves of 7075-T6 alloy foams having ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50 when compressed at έ = 2.0 s⁻¹ (static loading), and Fig. 5(b) presents the zoom-in view of the elastic region of the curves. The possible reasons behind such a wide variation in deformation behavior for two different έ could be the microstructural features of the struts and the phenomenon by which the DB progressed throughout the sample. As mentioned earlier, at quasi-static έ, the cells get enough time to respond to the applied load and the stress is released by means of fracture in the cells. However, at static and dynamic processes, due to fast compression, the propagation of DB to the next layers takes place with partial rupturing of the cells. This is reflected in the stress–strain curves as a flat plateau region [Fig. 5(a)]. The typical Y _s values observed in the alloy foams of ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50 are 9.60 MPa, 13.39 MPa, 19.32 MPa, 23.00 MPa, and 26.72 MPa when compressed at έ = 0.002 s⁻¹. The values increased to 21.63 MPa, 25.41 MPa, 26.20 MPa, 29.49 MPa, and 36.35 MPa in the corresponding samples when compressed at έ = 2.0 s⁻¹ (Table II). Similar to the pure Al foams, a monotonic increasing trend of Y _s with increasing ρ_rel was observed. It is quite rational that the strength of the foams would improve with the increase in ρ_rel due to the increased moment of inertia to bend the struts. Among the pure Al and 7075-T6 foams of almost similar ρ_rel and processed under same experimental conditions, the later yielded higher Y _s at both strain rates (0.002 and 2.0 s⁻¹). This can be attributed to the properties of the parent materials. The bulk 7075-T6 alloy has higher strength than pure Al. It is also apparent that the two set of samples have some difference in the microstructure of the struts (e.g., grain size, defects, etc.).

FIG. 4.

(a) Stress–strain curves of 7075-T6 alloy foams having ρ_rel = 0.25, 0.28, 0.32, 0.35, 0.40, 0.45, 0.47, and 0.50 compressed at έ = 0.002 s⁻¹ and (b) the zoom-in view of the elastic region of the curves.

FIG. 5.

(a) Stress–strain curves of 7075-T6 alloy foams having ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50 and compressed at έ = 2.0 s⁻¹ and (b) the zoom-in view of the elastic region of the curves. (c) Stress–strain curves of 7075-T6 alloy foams having ρ_rel = 0.35 for two different C _s values when compressed at έ = 0.002 s⁻¹.

TABLE II.

Mechanical properties of 7075-T6 alloy foams derived at two different strain rates (έ = 0.002/s and 2.0/s).

^a The bracketed terms denote the values at ε_d.

The values after the ± sign represent the standard deviation.

As can be noticed in Table II, the simultaneous increase in both the C _s (0.75–1.0 to 1.5–2.0 mm) and ρ_rel (<0.35 to >0.40) result in significant rise in the strength values during quasi-static loading, while no such inference can be made for static loading. Apart from sample dimension and ρ_rel of the foams, the dimension of the cells and strut thickness also play crucial roles in determining the overall mechanical properties of the foams. Reports on the effects of C _s on the mechanical properties of foams reveal a complex behavior.^{Reference Kanahashi, Mukai, Nieh, Aizawa and Higashi2,Reference Nieh, Higashi and Wadsworth4,Reference Cao, Wang, Ma, Zhao and Yang8,Reference Jiang, Wang and Zhao38} In the present work, foams with ρ_rel > 0.35 were formed using salt particles of size 1.5–2 mm, while the rest were fabricated with particles of 0.75–1 mm size (Table II). To investigate the effects of C _s on the strength of the foams, compressive tests at quasi-stating loading (έ = 0.002 s⁻¹) were conducted on few sets of samples of same ρ_rel but different C _s values (0.75–1.0 and 1.5–2.0 mm). Figure 5(c) shows the stress–strain curves of 7075-T6 alloy foams having ρ_rel = 0.35 for two different C _s values when compressed at έ = 0.002 s⁻¹. The sample with larger C _s showed higher Y _s with slight deviation in E. It suggests that at quasi-static έ, the strength of the 7075-T6 foams is enhanced with the increase in C _s (larger the C _s, higher is the strength). With the simultaneous increase of both C _s (0.75–1 to 1.5–2 mm) and ρ_rel (<0.35 to >0.40) at quasi-static έ, a large rise in the strength substantiated that C _s plays a vital role (along with ρ_rel) in controlling the strength of the foams. The observed Y _s values at έ = 2.0 s⁻¹ are much higher compared to the values observed at έ = 0.002 s⁻¹. This indicates that the strength of the foams is sensitive to έ. However, the percentage increase in the Y _s values between the quasi-static and static έ for ρ_rel = 0.32 is 125%, which became as low as 28% with the increase in ρ_rel up to 0.45. This points out that under static έ and with simultaneous increase of ρ_rel and C _s, the rate of enhancement in Y _s decreases. At έ = 2.0 s⁻¹, there is a monotonous increase in the strength of the foams with increasing ρ_rel and samples seem to be sensitive to the έ. With smaller C _s, the sample is more sensitive to έ while larger C _s proves to be detrimental for Y _s with increasing ρ_rel at static έ. The percentage increase in Y _s between έ = 0.002 s⁻¹ and έ = 2.0 s⁻¹ with ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50 is approximately 125, 123, 36, 28, and 36%, i.e., the change decreased with increasing ρ_rel. However, as mentioned in Table II, the samples with ρ_rel = 0.32 and 0.35 were composed of C _s = 0.75–1 mm and the samples with ρ_rel = 0.40, 0.45, and 0.50 had C _s = 1.5–2 mm, i.e., for smaller C _s, the samples are more sensitive to έ.

Figure 6 shows the normalized compressive Y _s versus ρ_rel plots of pure Al at (a) έ = 0.002 s⁻¹ and (b) έ = 2.0 s⁻¹, and 7075-T6 foams at (c) έ = 0.002 s⁻¹ and (d) έ = 2.0 s⁻¹, respectively. For open-cell foams, the scaling law for normalized Y _s can be taken as,^{Reference Ashby, Evans, Fleck, Gibson, Hutchinson and Wadley11,Reference Gibson and Ashby22}

(2)$${{{Y_{\rm{s}}}} \over {{\sigma _{\rm{y}}}}} = C{\left( {{\rho _{{\rm{rel}}}}} \right)^{1.5}}\quad ,$$

where C is a parameter related to the topology of the foam and varies between 0.1 and 1, and σ_y is estimated from the microhardness data (∼HV/3).^{Reference Bafti and Habibolahzadeh10,Reference Ashby, Evans, Fleck, Gibson, Hutchinson and Wadley11,Reference Murr, Amato, Li, Tian, Cheng, Gaytan, Martineza, Shindo, Medina and Wicker25,Reference Mahmutyazicioglu, Albayrak, Ipekoglu and Altintas26} The average HV values for pure Al and 7075-T6 alloy foams were obtained as 85, 79, 70, 85, 88, 87, 80, and 87 and 129, 127, 132, 135, 139, 123, 117, and 120 for ρ_rel = 0.25, 0.28, 0.32, 0.35, 0.40, 0.45, 0.47, and 0.50, respectively. Although the scaling law for open-cell foams indicates that the normalized Y _s is proportional to (ρ_rel)^1.5, a better fitting of the experimental data was obtained with (ρ_rel)² except for the 7075-T6 Al alloy foams, when compressed at έ = 2.0 s⁻¹ [Fig. 6(d)].

FIG. 6.

Normalized Y _s − ρ_rel plots of Al foams compressed at (a) έ = 0.002 s⁻¹ and (b) έ = 2.0 s⁻¹, and 7075-T6 alloy foams compressed at (c) έ = 0.002 s⁻¹ and (d) έ = 2.0 s⁻¹, respectively.

Structural stiffness is another important property that defines the characteristics of MFs and helps in classifying them for different applications. The E value defines the extent up to which any material can resist deformation on application of external load. As mentioned before, the E of the foams depend on ρ_rel, pore geometry, and cell topology. The structural flaw induced stress concentrations in the foams may lead to a slight plastic bending at a very low stress level. Consequently, the E value obtained from the loading stress–strain curve may not be the real representation of the stiffness. By contrast, the unloading slope remains roughly constant unless the foam was loaded up to its plastic collapse stress.^{Reference Wang and Zhou41} Hence, stiffness values are generally preferred from the unloading data. However, in the literature, many authors reported E values derived from the loading data to understand the compressive properties of the foams. Song et al.^{Reference Song, Chen, Yanagita and Frew42} and Wouterson et al.^{Reference Wouterson, Boey, Hu and Wong43} reported E values estimated from the initial portions of the loading data. Esen and Bor^{Reference Esen and Bor44} calculated E from curve fitting the linear elastic region of the stress–strain curves prior to the onset of the elastic–plastic region. Saha et al.^{Reference Saha, Kabir and Jeelani45} and Michailidis et al.^{Reference Michailidis, Stergioudi and Tsouknidas46} were also measured E from the slope of the initial linear part of the stress–strain curve. Chang et al.^{Reference Chang, Huang, Yang, Kuo and Lee47} defined E as the slope of the linear region of the stress–strain curve in the 5–15% of the strain range. Recently, Koohbor et al.^{Reference Koohbor, Mallon, Kidane and Lu48} also reported the same method of deriving the effective E by best line fitting of the stress–strain plots prior to the elastic limits. In the present work, we also reported the E values estimated from the loading curve. In MFs, some imperfections, such as cell edge curvature, materials concentrated at cell nodes rather than cell edges, corrugations and curvature of cell walls, and nonuniform ρ*, contribute significantly in degrading the stiffness values.

It was also observed that E follows an increasing trend with the decrease in Ø, similar to Y _s for both the samples. The moment of inertia to bend/deform the struts both in elastic and plastic regions increases with ρ_rel, which lead to the enhancement of the compressive properties (Y _s and E) of the MFs. The typical values of E of the pure Al foams when compressed at έ = 0.002 s⁻¹ are 65 MPa, 117 MPa, 309 MPa, 354 MPa, 385 MPa, 390 MPa, 490 MPa, 734 MPa for ρ_rel = 0.25, 0.28, 0.32, 0.35, 0.40, 0.45, 0.47, and 0.50, respectively. The trend of E under static compression (έ = 2.0 s⁻¹) was also found to be similar to the quasi-static compression but the magnitude of E enhanced significantly with the increase in ρ_rel. The mass distribution in the foam matrix also plays a vital role in controlling the strength and stiffness of the foams.^{Reference Simone and Gibson35} As described earlier, the present foams possess uniformity in the pore and density distribution. Therefore, we conclude that the stiffness will not be affected by the morphology of the present foams. The typical E values of the foams at έ = 2.0 s⁻¹ are 112 MPa, 376 MPa, 498 MPa, 1008 MPa, and 3212 MPa for ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50, respectively. The changes are higher as compared with quasi-static compression. It is expected that the έ sensitivity of E and Y _s should quantitatively behave in a similar manner. However, in the present set of samples, it can be noticed that E is more sensitive to έ. The possible reason could be the microplasticity in the samples.^{Reference Zhou, Shrotriya and Soboyejo49} In general, when the foam is compressed at high έ, the inertia of the struts opposes the buckling process, which causes the rupturing or deformation of the cells at a delayed time. As a result, a rise in E is observed with ρ_rel. Some areas/domains in the MFs may possess plastic property or microdefects (impurity phase, cracks) trapped in the struts which under compression releases the localized stress (in the elastic region), resulting in the reduction of strength. Since the effect of local microplasticity in ductile metals is observed well before the onset of bulk yielding, the onset of local microplasticity may not lead to significant shape changes in the individual struts and the foam block, but their presence may lower down the Y _s.^{Reference Zhou, Shrotriya and Soboyejo49}

Similar behavior of E was observed in the 7075-T6 alloy foams. The E values observed at έ = 0.002 s⁻¹ for ρ_rel = 0.25, 0.28, 0.32, 0.35, 0.40, 0.45, 0.47, and 0.50 are 95 MPa, 256 MPa, 342 MPa, 387 MPa, 462 MPa, 545 MPa, 674 MPa, and 916 MPa, respectively. With the increase in έ by 1000 times (έ = 2.0 s⁻¹), the E values were enhanced by 105%, 90%, 120%, 137%, and 188% (701 MPa, 736 MPa, 1018 MPa, 1292 MPa, and 2637 MPa) for ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50, respectively. In the present study on pure Al and 7075-T6 alloy foams, dissimilar trends of E with έ are observed, even though the processing parameters and cell parameters are similar. Therefore, it can be concluded that the mechanical properties of the MFs also depend on the parent material, apart from the ρ_rel, cell morphology, and έ. A large variation in the C _s as well as a large population of morphological defects are common features of cellular metals/alloys and hence a significant variation in the foam properties can also be expected.^{Reference Paul and Ramamurty50} However, in the present work, the foams possess a very narrow C _s distribution and hence we conclude that the observed behavior of the foams describes their inherent response to the applied load (and έ).

The σ_pl and ε_d primarily depend on ρ_rel, foam microstructure, and the applied έ. Raj et al.^{Reference Raj, Parameswaran and Daniel51} reported σ_pl of Al foams to increase with the increase in both ρ_rel and έ. Wang et al.^{Reference Wang, Shen, Lu and Zhao27} also observed σ_pl to increase from low to medium έ for closed-cell Al alloy foams. However, in Alulight and Duocell foams, the σ_pl is almost insensitive to έ (up to 5000 s⁻¹).^{Reference Campana and Pilone39} In the present work, the σ_pl values of the pure Al foams obtained from energy absorption efficiency-based approach [Eq. (1)] at έ = 0.002 s⁻¹ are 12 MPa, 18 MPa, 19 MPa, 26 MPa, 32 MPa, 34 MPa, 39 MPa, and 40 MPa for ρ_rel = 0.25, 0.28, 0.32, 0.35, 0.40, 0.45, 0.47, and 0.50, respectively. The σ_pl values at έ = 2.0 s⁻¹ are 24 MPa, 32 MPa, 56 MPa, 59 MPa, and 77 MPa corresponding to ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50. To study the έ sensitivity of σ_pl, we have adopted the reported criterion of 20% increase in the value.^{Reference Wang, Shen, Lu and Zhao27} With the increase in έ by 1000 times, σ_pl nearly doubled the values as compared with έ = 0.002 s⁻¹ for ρ_rel = 0.50. Wang et al.^{Reference Wang, Shen, Lu and Zhao27} reported a similar kind of increase in σ_pl values for ρ_rel = 0.09 when έ was increased by 5 orders of magnitude. The samples with lower ρ_rel, viz, 0.32, 0.35, 0.40, and 0.45 showed an enhancement in σ_pl of 24.19%, 37.23%, 77.21%, and 76.82%, respectively. The possible reason for such high rise could be the large strut thickness which requires higher deformation load. With the increase in έ, the plastic deformation is often confined to the initial cell layer where impact occurs. Mondal et al.^{Reference Mondal, Goyal and Das52} reported that σ_pl highly depends on ρ_rel and not on the έ for very light Al foams. Since the present foams possess larger strut thickness, the inertial effect at high έ along with ρ_rel shall contribute toward the enhancement of σ_pl. An increasing pattern of σ_pl with ρ_rel was also observed in the 7075-T6 alloy foams (although the increment is much smaller than the pure Al foams). The typical σ_pl values at έ = 2.0 s⁻¹ are 15.79 Mpa, 24.44 Mpa, 28 Mpa, 31.55 Mpa, and 19.23 Mpa for ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50, respectively.

Similarly, the ε_d is also a vital parameter to characterize the mechanical properties of foams and is mainly influenced by the έ, morphology, and strut microstructure of the foams.^{Reference Wang, Shen, Lu and Zhao27,Reference Schüler, Fischer, Bührig-Polaczek and Fleck28} As can be observed in Figs. 2(a), 3(a), and 5(a), both Al and 7075-T6 alloy foams hardly exhibit any abrupt switching between different deformation regions and hence an accurate identification of ε_d is difficult. As mentioned before, we have followed an energy efficiency-based approach to identify the most accurate marking of the onset of the densification mechanism in the present samples. The obtained values of ε_d are given in Tables I and II for pure Al and 7075-T6 alloy foams, respectively. The ε_d values of the pure Al foams at έ = 0.002 s⁻¹ decreased with the increase in ρ_rel, as expected due to the increase in solid content in the foam. The typical values of ε_d are 0.584, 0.535, 0.538, 0.52, 0.53, 0.485, 0.48, and 0.464 corresponding to ρ_rel = 0.25, 0.28, 0.32, 0.35, 0.40, 0.45, 0.47, and 0.50, respectively. It can also be observed that with the increase in έ, the decreasing trend of ε_d is not so prominent and no dependency of ε_d on the C _s for both the έ is revealed. For έ = 2.0 s⁻¹, the typical values of ε_d are 0.54, 0.54, 0.55, 0.51, and 0.53 corresponding to ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50, respectively. The ε_d decreased nearly by an amount of 0.02 or more (∼2.5–8%) for most of the samples at quasi-static έ, whereas for static έ, both increase and decrease in ε_d were observed with increasing ρ_rel. The 7075-T6 alloy foams showed an opposite behavior. When compressed at έ = 0.002 s⁻¹, the samples did not show the densification rather it collapsed soon after achieving the Y _s at very low strain [Fig. 4(a)]. The foams with low ρ_rel (∼0.25–0.35) collapsed even before reaching ε = 0.3. For higher ρ_rel, the foams collapsed completely above ε = 0.4. It can also be observed that above the Y _s value, there is a continuous decrease in the stress. It indicates the fracture of the cells taking place in the foams. By contrast, for έ = 2.0 s⁻¹, full stress–strain curves were obtained. The maximum value of ε_d of 0.61 was obtained for ρ_rel = 0.40, while a minimum of 0.54 was obtained for ρ_rel = 0.35. No particular trend of the ε_d values with ρ_rel can be derived. Few reports are available in the literature highlighting the effects of ρ_rel and έ sensitivity on ε_d. Wang et al.^{Reference Wang, Shen, Lu and Zhao27} reported negligible dependence of ε_d on έ for Al foams with closed-cell morphology. Mondal et al.^{Reference Mondal, Goyal and Das52} showed that the ε_d of closed-cell Al-fly ash composite foams is almost invariant to both the ρ_rel and έ. Yi et al.^{Reference Yi, Zhu, Zu, Hu and Yi53} reported that the compaction strain for the Al alloy decreases with the increase in έ. Han et al.^{Reference Han, Cheng, Li and Wang37} mentioned that the ε_d decreases with έ in open-cell pure Al foams processed by infiltration method. Bafti et al.^{Reference Bafti and Habibolahzadeh10} reported that ε_d decreases with the increase in ρ_rel of Al foams produced by powder-carbamide spacer route. As can be noticed in the present set of pure Al and 7075-T6 foams, the ε_d did not reveal any particular trend with respect to ρ_rel at έ = 2.0 s⁻¹.

In MFs, the applied compressive load is absorbed during the deformation process. In this regard, the W and I _m are two important parameters of foams, defined as follows^{Reference Kanahashi, Mukai, Nieh, Aizawa and Higashi2,Reference Paul and Ramamurty7} :

(3)$$W = \int_{\rm{0}}^{{\varepsilon _{\rm{m}}}} {\sigma \left( \varepsilon \right){\rm{d}}\varepsilon } \quad ,$$

(4)$${I_{\rm{m}}} = {1 \over {{\sigma _{\rm{m}}}{\varepsilon _{\rm{m}}}}}\int_{\rm{0}}^{{\varepsilon _{\rm{m}}}} {\sigma \left( \varepsilon \right){\rm{d}}\varepsilon } \quad ,$$

where σ(ε) is the compressive stress (function of ε), ε_m is the limit of plateau strain up to the onset of densification, and σ_m is the corresponding compressive stress. The W increases with ρ_rel (Y _s and σ_pl also increase with ρ_rel) since the volume of the solid content in the foams increases and so do the area under the stress–strain curve. Larger area under the stress–strain curve reflects higher energy absorption. Therefore, for excellent absorption capacity of the foams, it is good to have large elastic region and also the deformation in the plateau region should progress with nearly constant or slowly increasing stress with strain. As shown in Table I, the values of W (for Al at έ = 0.002 s⁻¹) in both the elastic and plateau regions increase with ρ_rel, with the maximum value of 17.29 MJ/m³ for Ø = 50%. To assess the effects of έ on the absorption energy of the foams with different ρ_rel, comparative studies were performed between samples with same ρ_rel and foam morphology. At higher έ, the energy absorption increased and depicted a higher rate of increase above ρ_rel = 0.35. The percentage increase in the energy absorption at higher έ for ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50 are 26%, 37%, 76%, 91%, and 117%, respectively. The large C _s in the foams plays a contributing role in this regard. The foams with ρ_rel > 0.35 have C _s in the range 1.5–2 mm (Table I). As mentioned before, the C _s also contributed positively to enhance the compressive properties at elevated έ, thereby increasing the area under the stress–strain curve. By contrast, I _m at both the έ did not follow any specific trend and the sample with ρ_rel = 0.45 was found most efficient in terms of energy absorption till the ε_d. In addition, the observed I _m values mostly show positive έ sensitivity.

The trend of W of 7075-T6 alloy foams was also found to be similar with those of pure Al foams. Since the foams fractured at small strain when compressed at quasi-static loading (έ = 0.002 s⁻¹), the absorption energy in the elastic region (till ε = 0.1) is only considered for the analysis. As expected, the W values in the elastic region mostly increased with ρ_rel, excluding the ρ_rel = 0.50 sample. With the increase in the έ from 0.002 to 2.0 s⁻¹, the observed change (%) in the W values in the elastic region were 626%, 444%, 518%, 498%, and 493% for ρ_rel = 0.32, 0.35, 0.40, 0.45, and 0.50, respectively. This reflects that the energy absorption is έ sensitive. The absorption efficiency at static compression also increased with ρ_rel and the foam with ρ_rel = 0.45 was found to be the most efficient with I _m = 92%.