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Studies on the dynamic compressive properties of open-cell aluminum alloy foams
Scripta Materialia 54 (2006) 83–87 www.actamat-journals.com
Studies on the dynamic compressive properties of open-cell aluminum alloy foams Wang Zhihua a
a,*
, Ma Hongwei
a,b
, Zhao Longmao a, Yang Guitong
a
Institute of Applied Mechanics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China b College of Science and Engineering, Jinan University, Guangzhou 510632, China Received 25 May 2005; received in revised form 20 August 2005; accepted 1 September 2005 Available online 6 October 2005
Abstract The compressive deformation behavior of open-cell aluminum foams with different densities and morphologies is assessed under quasi-static and dynamic loading conditions. The experimental results show that density, in contrast with cell size, is the primary variable characterizing the elastic modulus and yield strength of foams. Yield strength and energy absorption of foams are almost insensitive to strain rate, over a wide range of strain rates. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Aluminum alloy foams; Strain rate sensitivity; Compressive behavior; Energy absorption
1. Introduction Metal foams, as a new class of engineering materials, have potential for absorbing energy for they have an extended stress plateau in the compressive stress–strain curve. Recent improvements, as well as the development of production methods for metallic foams, have offered a variety of applications in fields such as the automobile, railway and aerospace industries. In these applications, the foam is subject to high-velocity deformations. Designing for these applications therefore demands a full characterization of their mechanical properties under a wide range of strain rates. The quasi-static mechanical properties of aluminum alloy foams, such as compressive strength and elastic modulus, have been extensively studied and reviewed [1]. However, related studies under dynamic conditions have been relatively limited due to the difficulty of characterizing the high strain rate behavior of aluminum alloy foams. Kenny [2] reported that the specific energy absorption of Alcan foam (open-cell) was independent of applied
*
Corresponding author. Tel.: +86 351 601 0560. E-mail address: [email protected] (W. Zhihua).
strain rate in the range 103–103 s1. No measurements were made however, of the stress–strain curve under dynamic and quasi-static loadings. In line with the findings of Kenny, Deshpande and Fleck [3] found that the strain rate dependence of mechanical strength was negligible for the Doucel (open-cell) and Alulight (closed-cell) foams. Lankford and Dannemann [4] also showed that open-cell aluminum foam did not exhibit the strain rate dependence of plateau stress. On the other hand, Mukai et al. reported that open-cell SG91A AL [5], closed-cell Alporas [6], and open-cell AZ91Mg [7], all had a high strain rate sensitivity of the plateau stress. Dannemann and Lankford [8] also demonstrated the strain rate dependence of plateau stress for Alporas (closed-cell) for high strain rates ranging from 4 · 102 to 2.5 · 103 s1. Thus, it is noted that, despite the fact that metallic foams are attractive materials for energy absorption, only limited data are available for dynamic strain rates. The deformation behavior of metallic foams can vary significantly by changing the intrinsic cell structures, such as wavy distortions of cell walls, microstructure of cell edge materials and cell morphology. For example, Lehmhus and Banhart [9] reported the effect of nine different heat treatments on the mechanical property of aluminum alloy
1359-6462/$ - see front matter 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2005.09.008
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foams at a static strain rate. They found that foams that fully heat-treated performed best with an increase of strength by 60–75% over the untreated samples. Kanahashi et al. [10,11] demonstrated that deformation behavior of open-cell alloy foams at both quasi-static and dynamic strain rates was strongly affected by heat treatment. These results suggest that heat treatment can improve the microstructure of the solid materials and, further, the dynamic response of metallic foams. Some studies have also been carried out to evaluate the effect of cell morphology on the mechanical properties of metallic foams. Nieh et al. [12] observed the effect of cell size and shape on the compressive behavior of open-cell AA6101-T6 aluminum foams with different relative density and showed that the former, in contrast to the latter, appeared to have a negligible effect on the strength of foam. It is noted that these studies were all carried out at static strain rates. The effect of cell morphology on the deformation behavior of metallic foams at dynamic strain rates has not been investigated. In the present paper, open-cell aluminum alloy foams with different cell sizes and relative densities were examined under quasi-static and dynamic conditions in order to estimate the effect of cell size on the compressive mechanical property and the energy absorption capability. 2. Experimental The open-cell aluminum alloy foams produced by the infiltration process were used in the present study. The composition of the cell wall material is Al–3 wt.%Mg– 8 wt.%Si–1.2 wt.%Fe. The relative density (defined as the density of the foam divided by the density of the cell wall material) of the foam ranged from 0.25 to 0.30 and the average cell sizes were 0.9 mm and 1.6 mm, respectively. Relevant properties of this material are listed in Table 1. Dynamic compression tests at strain rates in the range 102–104 s1 were performed at room temperature using the split Hopkinson pressure bar (SHPB) technique. We have assumed that the cellular material can be treated as a continuum: that is, that the sample dimensions should be as large as possible within the limitations of the Hopkinson pressure bar in order to obtain the actual properties of foam materials, However, to obtain the stress–strain relations with a large strain, including linear elasticity, collapse plateau and densification, the sample height needs to be minimized. AndrewsÕ study [13] showed that the height must be at least seven times the cell size, this being the minimum specimen size required to avoid edge effects, and the
experiments of Mukai [6] confirmed that the specific yield strength of a specimen with a height over 6 mm is almost constant. Therefore, the selected specimens are circular cylinders of diameter 35 mm and length 10 mm. With this choice of specimen dimensions, the specimens have at least 6–10 cells in all directions. A brief description of the experiment set-up is given below, with complete details given by Sathiamoorthy [14]. The striker, incident pressure and transmitter bars consisted of 37 mm diameter aluminum bars and their lengths were 800, 2000 and 2000 mm, respectively. The end surfaces were lubricated to reduce the frictional restraint. The compressive pulse is generated by axial impact on the incident pressure bar by the striker bar. When the compressive pulse reaches the specimen, a portion of the pulse is reflected from the interface, while the remainder is transmitted through to the transmitter bar. The incident pulse and reflected waves in the incident bar are recorded by the resistance strain gauge attached at the incident bar. The transmitted wave is also recorded by the semiconductor strain gauge attached at the transmitter bar. Typical incident, reflected and transmitted waves are shown in Fig. 1. Dynamic compressive stress–strain curves could be obtained from the measured results. Compressive tests were also performed at a quasi-static strain rate of 103 s1 using a servo-hydraulic test machine and specimens of 35 mm in diameter and 30 mm in height. Detailed experimental results are given in the following. 3. Results and discussion 3.1. Experimental results Fig. 2 shows the stress–strain curves for different cell sizes with the same relative densities under quasi-static and dynamic compression, respectively. The quasi-static deformation of open-cell aluminum alloy foams is uniform, which is different from the closed-cell foams with band
Table 1 Properties of open-cell aluminum alloy foam Property
Value
Composition of foam material Relative density (q*/qs) Average cell size Average wall thickness
Al–3Mg–8Si–1.2Fe (wt.%) 0.25–0.30 0.9 mm, 1.6 mm 0.6 mm
Fig. 1. Typical waves recorded from the incident and transmitter bars.
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tions of Nieh et al. [12], who reported a cell size effect at quasi-static strain rates in Doucel foams. Fig. 3 shows quasi-static and dynamic stress–strain curves for various densities of foams. The higher the foamsÕ relative densities are, the shorter the plateau region is, but higher densities also imply higher yield stresses. The relationship between the relative stress and the relative density for open-cell materials is given by Gibson and Ashby [1] as rpl 3=2 ¼ Cðq =qs Þ ; rys
ð1Þ
where rpl ; q are the plastic-collapse stress and the density of the cellular material, respectively, and rys, qs are the yield stress and the density of the cell wall material, respectively. Using this equation, the relationship between the relative stress and relative density for the presently studied aluminum alloy foams is plotted in Fig. 4, also included in the figure are the data from other aluminum foams [6]. The experimental data of the open-cell aluminum alloy foams,
Fig. 2. Stress–strain curves of aluminum alloy foams at different strain rates.
formation. At overall macroscopic strains greater than about 30% the whole specimen has crushed and uniform additional strain occurring as observed by Deshpande [3]. The compressive stress–strain curve of aluminum alloy foams, under either quasi-static or dynamic compression, exhibits three universal deformation characteristics: an initial linear-elastic region; an extended plateau region where the stress increases slowly as the cells deform plastically; and a final densification as collapsed cells are compacted together. These deformation characteristics of the aluminum alloy foams are similar to those of other metal foams [3,4]. The only significant difference between the dynamic and static stress versus strain curves is that, while the static curves are smooth, oscillations can be seen in the dynamic curves; the reason may be uninterrupted destabilization of aluminum alloy foams with the plastic collapse in cell walls. It can be seen from Fig. 2 that linear elasticity only appears at very low strain (smaller than about 0.05) and seems to be independent of the strain rate. It also can be readily observed in this figure that the cell size appears to have an insignificant effect on plastic collapse, even at the dynamic strain rates. This is very similar to the observa-
Fig. 3. Quasi-static and dynamic compression stress–strain curves of aluminum alloy foams of different densities and at different strain rates.
Fig. 4. Relation between the relative stress and relative density for different aluminum foams.
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in the present investigation are shown to be in reasonable agreement with the values predicted for the case of C = 0.3. This further indicates that relative density is the most important variable determining the mechanical properties of metallic foams. 3.2. Strain rate sensitivity It is of interest to note in Fig. 2 that, despite of six orders of magnitude difference in strain rate, the curves at both the dynamic and quasi-static strain rates are practically the same. It appears that there is no strain rate effect on stress for the open-cell foam in the present study. Note that we adopt the criterion of a 20% elevation in strength to define strain rate sensitivity. This is consistent with the fact that metallic foams are highly heterogeneous imperfect materials with dispersion strengths of the order of 20%. The specific yield stress (yield stress per (relative density)3/2) of the open-cell foams as a function of strain rate is plotted in Fig. 5. As can be seen in this figure, the strengths for the presented foams at the dynamic strain rates are consistent, and essentially the same as that observed at the quasi-static strain rate. This result indicates that the yield stress exhibits a slight strain rate dependence. In contrast, the Alporas foam shows a remarkable dependence. Its yield strength nearly doubles with a six orders of magnitude increase in strain rate. Two reasons may contribute to the strain rate sensitivity of a cellular material: cell morphology and strain rate sensitivity of the cell wall material. To characterize the stress dependence on the strain rate of the cell materials, Lindholm [15] have performed dynamic tests on 6000-series aluminum alloys, with compositions close to those of the present foams. They found that the strength increases by less than 15% when the strain rate is increased from 104 to 103. This suggests the absence of a strain rate effect in the studied aluminum alloy foams is associated with the cellular structure not the microstructure of the cell edge material. A simplified theo-
retical analysis presented by Deshpande [3] also demonstrates that the effect of strain rate on the dynamic strength of the cell wall material in aluminium alloy foams is less than the scatter band of the strength of the foams. Apparently, the intrinsic properties of the cell wall material are overwhelmed by the extrinsic properties of the cell structure during dynamic compression. Only limited studies on the strain rate effect on densification strain are available for aluminium alloy foams. If the densification strain is significantly reduced at a high strain rate, the foams may become mildly beneficial in absorption of impact energy, so it is important to investigate the variety of densification strain at high strain rates. From Fig. 3, one can see that the densification strain is sensitive to the relative density and the densification strain decreases with an increase of relative density. This is due to the fact that the lower density foam affords more opportunity for cell walls to collapse and deform. It can be seen from Fig. 2 that the densification strain exhibits a tendency to slightly decrease with increasing strain rate. 4. Energy absorption The increasing demands for active and passive safety of vehicles, especially in the automotive industry, lead to a specific interest in the energy absorption capacity of metallic foam materials. One significant advantage for aluminum foams is the isotropy of energy absorption, which is important in many crash situations. The absorption energy per unit volume WV at the specific strain can be evaluated by integrating the area under the stress–strain curve, namely: Z e WV ¼ r de ð2Þ 0
Correspondingly, the absorption energy per unit mass, WM, can be obtained as follows: Z e W M ¼ W V =q ¼ r de=q ð3Þ 0
Fig. 5. Strain rate dependence of yield stress per (relative density)3/2 for different aluminum foams.
It can be seen from Fig. 3 that the absorption energy increases with the relative density. This is due to the fact that the yield stress increases with the relative density, so the area under the stress–strain curve also increases. To assess the effect of strain rate on the absorption energy of different cell-sizes foams, the comparison must be made on the basis of the same density. Figs. 6 and 7 show the absorption energy capacities per unit volume and mass of different cellsizes aluminum alloy foams after compression strains of 5%, 10%, 15%, 20%, 25%, 30%, and 35% at different strain rates. For strain rates ranging from 103 to 1400 s1, the absorption energy capacity of the larger cell size foam at a strain of 35% varies from 2.85 to 3.44 MJ/m3, or equivalently from 3.91 to 4.72 kJ/kg, which is about a 20% increase. The absorption energy capacity of the smaller cell size foam at a strain of 35% varies from 2.73 to 3.25 MJ/ m3 or from 3.74 to 4.46 kJ/kg for strain rates from 103 to 1400 s1, which is about a 19% increase. This is much
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range of the strain rates employed, the present aluminum alloy foams exhibits a slight strain rate dependence in both yield strength and absorption energy. The relation between the relative stress and relative density is in reasonable agreement with Gibson and AshbyÕs equation. It was concluded that density is by far the most important variable and the intrinsic properties of the cell wall material are overwhelmed by the extrinsic properties of the cell structure during dynamic compression. Mechanical properties of the present foams appear to be independent of the cell size. Therefore, intensive efforts to modify the cell morphology are probably not an effective way to improve the impact absorption energy capacity. Acknowledgements Fig. 6. Energy absorption of aluminum alloy foams with the 0.9 mm cell size at different strain rates.
This work described in this paper was supported by the National Natural Science Foundation of China through Grant No. 90205018, the Natural Science Foundation, and the Homecomings Foundation and the Youth Academic Leader of Shanxi Program through Grant No. 2004–006 and 2003–23. References
Fig. 7. Energy absorption of aluminum alloy foams with the 1.6 mm cell size at different strain rates.
more than the energy absorption capacity of high strength polymer foams made, for example, from polymethacrylimide, particularly for the energy absorption per volume ratio. The increased values of WV and WM are noted to be practically the same, despite the different cell sizes. It also can be observed from Figs. 6 and 7 that the present aluminum alloy foams exhibit slight strain rate dependence in both yield strength and absorption energy. Thus, it is essential to obtain the accurate strain rate sensitivity of a material before selecting it for a specific application for impact energy absorption. 5. Conclusions The quasi-static and dynamic compressive stress–strain curves of aluminum alloy foams for various relative densities and different cell sizes have been studied. Over the
[1] Gibson LJ, Ashby MF. Cellular solids: structures and properties. Cambridge: Cambridge University Press; 1997. [2] Kenny LD. Mechanical properties of particle stabilized aluminum foam. Mater Sci Forum 1996;217–222:1883. [3] Deshpande VS, Fleck NA. High strain rate compressive behavior of aluminum alloy foams. Int J Impact Eng 2000;24:277–98. [4] Lankford JJ, Dannemann KA. Strain rate effects in porous materials. In: Proc Materials Research Society Symp, Warrendale (PA), vol. 521; 1998:s p. 103–8. [5] Kanahashi H, Mukai T, Yamada Y, et al. Dynamic compression of an ultra-low density aluminium foam. Mater Sci Eng A 2000;280:349–53. [6] Mukai T, kanahashi H, Miyoshi T, et al. Experimental study of energy absorption in a close-celled aluminum foam under dynamic loading. Scripta Mater 1999;40:921–7. [7] Yamada Y, Shimojima K, Sakaguchi Y, et al. Compressive properties of open-cellular SG91A Al and AZ91 Mg. Mater Sci Eng A 1999;272:455–8. [8] Dannemann KA, Lankford JJ. High strain rate compression of closed-cell aluminium foams. Mater Sci Eng A 2000;293:157–64. [9] Lehmhus D, Banhart J. Properties of heat-treated aluminium foams. Mater Sci Eng A 2003;349:98–110. [10] Yamada Y, Shimojima K, Sakaguchi Y, et al. Effects of heat treatment on compressive properties of AZ91 Mg and SG91A Al foams with open-cell structure. Mater Sci Eng A 2000;280:225–8. [11] Kanahashi H, Mukai T, Yamada Y, et al. Experimental study for the improvement of crashworthiness in AZ91 magnesium foam controlling its microstructure. Mater Sci Eng A 2001;308:283–7. [12] Nieh TG, Higashi K, Wadsworth J, et al. Effect of cell morphology on the compressive properties of open-cell aluminum foams. Mater Sci Eng A 2000;283:105–10. [13] Andrews E, Sanders W, Gibson LJ. Compressive and tensile behaviour of aluminum foams. Mater Sci Eng A 1999;270:113–24. [14] Sathiamoorthy S. Compressive failure of composites. PhD Thesis, 1997. Department of Engineering, University of Cambridge. [15] Lindholm US, Bessey RL, Smith GV. Effect of strain rate on yield strength, tensile strength and elongation of three aluminum alloys. J Mater JMLSA 1971;6:19.
Studies on the dynamic compressive properties of open-cell aluminum alloy foams Wang Zhihua a
a,*
, Ma Hongwei
a,b
, Zhao Longmao a, Yang Guitong
a
Institute of Applied Mechanics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China b College of Science and Engineering, Jinan University, Guangzhou 510632, China Received 25 May 2005; received in revised form 20 August 2005; accepted 1 September 2005 Available online 6 October 2005
Abstract The compressive deformation behavior of open-cell aluminum foams with different densities and morphologies is assessed under quasi-static and dynamic loading conditions. The experimental results show that density, in contrast with cell size, is the primary variable characterizing the elastic modulus and yield strength of foams. Yield strength and energy absorption of foams are almost insensitive to strain rate, over a wide range of strain rates. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Aluminum alloy foams; Strain rate sensitivity; Compressive behavior; Energy absorption
1. Introduction Metal foams, as a new class of engineering materials, have potential for absorbing energy for they have an extended stress plateau in the compressive stress–strain curve. Recent improvements, as well as the development of production methods for metallic foams, have offered a variety of applications in fields such as the automobile, railway and aerospace industries. In these applications, the foam is subject to high-velocity deformations. Designing for these applications therefore demands a full characterization of their mechanical properties under a wide range of strain rates. The quasi-static mechanical properties of aluminum alloy foams, such as compressive strength and elastic modulus, have been extensively studied and reviewed [1]. However, related studies under dynamic conditions have been relatively limited due to the difficulty of characterizing the high strain rate behavior of aluminum alloy foams. Kenny [2] reported that the specific energy absorption of Alcan foam (open-cell) was independent of applied
*
Corresponding author. Tel.: +86 351 601 0560. E-mail address: [email protected] (W. Zhihua).
strain rate in the range 103–103 s1. No measurements were made however, of the stress–strain curve under dynamic and quasi-static loadings. In line with the findings of Kenny, Deshpande and Fleck [3] found that the strain rate dependence of mechanical strength was negligible for the Doucel (open-cell) and Alulight (closed-cell) foams. Lankford and Dannemann [4] also showed that open-cell aluminum foam did not exhibit the strain rate dependence of plateau stress. On the other hand, Mukai et al. reported that open-cell SG91A AL [5], closed-cell Alporas [6], and open-cell AZ91Mg [7], all had a high strain rate sensitivity of the plateau stress. Dannemann and Lankford [8] also demonstrated the strain rate dependence of plateau stress for Alporas (closed-cell) for high strain rates ranging from 4 · 102 to 2.5 · 103 s1. Thus, it is noted that, despite the fact that metallic foams are attractive materials for energy absorption, only limited data are available for dynamic strain rates. The deformation behavior of metallic foams can vary significantly by changing the intrinsic cell structures, such as wavy distortions of cell walls, microstructure of cell edge materials and cell morphology. For example, Lehmhus and Banhart [9] reported the effect of nine different heat treatments on the mechanical property of aluminum alloy
1359-6462/$ - see front matter 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2005.09.008
84
W. Zhihua et al. / Scripta Materialia 54 (2006) 83–87
foams at a static strain rate. They found that foams that fully heat-treated performed best with an increase of strength by 60–75% over the untreated samples. Kanahashi et al. [10,11] demonstrated that deformation behavior of open-cell alloy foams at both quasi-static and dynamic strain rates was strongly affected by heat treatment. These results suggest that heat treatment can improve the microstructure of the solid materials and, further, the dynamic response of metallic foams. Some studies have also been carried out to evaluate the effect of cell morphology on the mechanical properties of metallic foams. Nieh et al. [12] observed the effect of cell size and shape on the compressive behavior of open-cell AA6101-T6 aluminum foams with different relative density and showed that the former, in contrast to the latter, appeared to have a negligible effect on the strength of foam. It is noted that these studies were all carried out at static strain rates. The effect of cell morphology on the deformation behavior of metallic foams at dynamic strain rates has not been investigated. In the present paper, open-cell aluminum alloy foams with different cell sizes and relative densities were examined under quasi-static and dynamic conditions in order to estimate the effect of cell size on the compressive mechanical property and the energy absorption capability. 2. Experimental The open-cell aluminum alloy foams produced by the infiltration process were used in the present study. The composition of the cell wall material is Al–3 wt.%Mg– 8 wt.%Si–1.2 wt.%Fe. The relative density (defined as the density of the foam divided by the density of the cell wall material) of the foam ranged from 0.25 to 0.30 and the average cell sizes were 0.9 mm and 1.6 mm, respectively. Relevant properties of this material are listed in Table 1. Dynamic compression tests at strain rates in the range 102–104 s1 were performed at room temperature using the split Hopkinson pressure bar (SHPB) technique. We have assumed that the cellular material can be treated as a continuum: that is, that the sample dimensions should be as large as possible within the limitations of the Hopkinson pressure bar in order to obtain the actual properties of foam materials, However, to obtain the stress–strain relations with a large strain, including linear elasticity, collapse plateau and densification, the sample height needs to be minimized. AndrewsÕ study [13] showed that the height must be at least seven times the cell size, this being the minimum specimen size required to avoid edge effects, and the
experiments of Mukai [6] confirmed that the specific yield strength of a specimen with a height over 6 mm is almost constant. Therefore, the selected specimens are circular cylinders of diameter 35 mm and length 10 mm. With this choice of specimen dimensions, the specimens have at least 6–10 cells in all directions. A brief description of the experiment set-up is given below, with complete details given by Sathiamoorthy [14]. The striker, incident pressure and transmitter bars consisted of 37 mm diameter aluminum bars and their lengths were 800, 2000 and 2000 mm, respectively. The end surfaces were lubricated to reduce the frictional restraint. The compressive pulse is generated by axial impact on the incident pressure bar by the striker bar. When the compressive pulse reaches the specimen, a portion of the pulse is reflected from the interface, while the remainder is transmitted through to the transmitter bar. The incident pulse and reflected waves in the incident bar are recorded by the resistance strain gauge attached at the incident bar. The transmitted wave is also recorded by the semiconductor strain gauge attached at the transmitter bar. Typical incident, reflected and transmitted waves are shown in Fig. 1. Dynamic compressive stress–strain curves could be obtained from the measured results. Compressive tests were also performed at a quasi-static strain rate of 103 s1 using a servo-hydraulic test machine and specimens of 35 mm in diameter and 30 mm in height. Detailed experimental results are given in the following. 3. Results and discussion 3.1. Experimental results Fig. 2 shows the stress–strain curves for different cell sizes with the same relative densities under quasi-static and dynamic compression, respectively. The quasi-static deformation of open-cell aluminum alloy foams is uniform, which is different from the closed-cell foams with band
Table 1 Properties of open-cell aluminum alloy foam Property
Value
Composition of foam material Relative density (q*/qs) Average cell size Average wall thickness
Al–3Mg–8Si–1.2Fe (wt.%) 0.25–0.30 0.9 mm, 1.6 mm 0.6 mm
Fig. 1. Typical waves recorded from the incident and transmitter bars.
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tions of Nieh et al. [12], who reported a cell size effect at quasi-static strain rates in Doucel foams. Fig. 3 shows quasi-static and dynamic stress–strain curves for various densities of foams. The higher the foamsÕ relative densities are, the shorter the plateau region is, but higher densities also imply higher yield stresses. The relationship between the relative stress and the relative density for open-cell materials is given by Gibson and Ashby [1] as rpl 3=2 ¼ Cðq =qs Þ ; rys
ð1Þ
where rpl ; q are the plastic-collapse stress and the density of the cellular material, respectively, and rys, qs are the yield stress and the density of the cell wall material, respectively. Using this equation, the relationship between the relative stress and relative density for the presently studied aluminum alloy foams is plotted in Fig. 4, also included in the figure are the data from other aluminum foams [6]. The experimental data of the open-cell aluminum alloy foams,
Fig. 2. Stress–strain curves of aluminum alloy foams at different strain rates.
formation. At overall macroscopic strains greater than about 30% the whole specimen has crushed and uniform additional strain occurring as observed by Deshpande [3]. The compressive stress–strain curve of aluminum alloy foams, under either quasi-static or dynamic compression, exhibits three universal deformation characteristics: an initial linear-elastic region; an extended plateau region where the stress increases slowly as the cells deform plastically; and a final densification as collapsed cells are compacted together. These deformation characteristics of the aluminum alloy foams are similar to those of other metal foams [3,4]. The only significant difference between the dynamic and static stress versus strain curves is that, while the static curves are smooth, oscillations can be seen in the dynamic curves; the reason may be uninterrupted destabilization of aluminum alloy foams with the plastic collapse in cell walls. It can be seen from Fig. 2 that linear elasticity only appears at very low strain (smaller than about 0.05) and seems to be independent of the strain rate. It also can be readily observed in this figure that the cell size appears to have an insignificant effect on plastic collapse, even at the dynamic strain rates. This is very similar to the observa-
Fig. 3. Quasi-static and dynamic compression stress–strain curves of aluminum alloy foams of different densities and at different strain rates.
Fig. 4. Relation between the relative stress and relative density for different aluminum foams.
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W. Zhihua et al. / Scripta Materialia 54 (2006) 83–87
in the present investigation are shown to be in reasonable agreement with the values predicted for the case of C = 0.3. This further indicates that relative density is the most important variable determining the mechanical properties of metallic foams. 3.2. Strain rate sensitivity It is of interest to note in Fig. 2 that, despite of six orders of magnitude difference in strain rate, the curves at both the dynamic and quasi-static strain rates are practically the same. It appears that there is no strain rate effect on stress for the open-cell foam in the present study. Note that we adopt the criterion of a 20% elevation in strength to define strain rate sensitivity. This is consistent with the fact that metallic foams are highly heterogeneous imperfect materials with dispersion strengths of the order of 20%. The specific yield stress (yield stress per (relative density)3/2) of the open-cell foams as a function of strain rate is plotted in Fig. 5. As can be seen in this figure, the strengths for the presented foams at the dynamic strain rates are consistent, and essentially the same as that observed at the quasi-static strain rate. This result indicates that the yield stress exhibits a slight strain rate dependence. In contrast, the Alporas foam shows a remarkable dependence. Its yield strength nearly doubles with a six orders of magnitude increase in strain rate. Two reasons may contribute to the strain rate sensitivity of a cellular material: cell morphology and strain rate sensitivity of the cell wall material. To characterize the stress dependence on the strain rate of the cell materials, Lindholm [15] have performed dynamic tests on 6000-series aluminum alloys, with compositions close to those of the present foams. They found that the strength increases by less than 15% when the strain rate is increased from 104 to 103. This suggests the absence of a strain rate effect in the studied aluminum alloy foams is associated with the cellular structure not the microstructure of the cell edge material. A simplified theo-
retical analysis presented by Deshpande [3] also demonstrates that the effect of strain rate on the dynamic strength of the cell wall material in aluminium alloy foams is less than the scatter band of the strength of the foams. Apparently, the intrinsic properties of the cell wall material are overwhelmed by the extrinsic properties of the cell structure during dynamic compression. Only limited studies on the strain rate effect on densification strain are available for aluminium alloy foams. If the densification strain is significantly reduced at a high strain rate, the foams may become mildly beneficial in absorption of impact energy, so it is important to investigate the variety of densification strain at high strain rates. From Fig. 3, one can see that the densification strain is sensitive to the relative density and the densification strain decreases with an increase of relative density. This is due to the fact that the lower density foam affords more opportunity for cell walls to collapse and deform. It can be seen from Fig. 2 that the densification strain exhibits a tendency to slightly decrease with increasing strain rate. 4. Energy absorption The increasing demands for active and passive safety of vehicles, especially in the automotive industry, lead to a specific interest in the energy absorption capacity of metallic foam materials. One significant advantage for aluminum foams is the isotropy of energy absorption, which is important in many crash situations. The absorption energy per unit volume WV at the specific strain can be evaluated by integrating the area under the stress–strain curve, namely: Z e WV ¼ r de ð2Þ 0
Correspondingly, the absorption energy per unit mass, WM, can be obtained as follows: Z e W M ¼ W V =q ¼ r de=q ð3Þ 0
Fig. 5. Strain rate dependence of yield stress per (relative density)3/2 for different aluminum foams.
It can be seen from Fig. 3 that the absorption energy increases with the relative density. This is due to the fact that the yield stress increases with the relative density, so the area under the stress–strain curve also increases. To assess the effect of strain rate on the absorption energy of different cell-sizes foams, the comparison must be made on the basis of the same density. Figs. 6 and 7 show the absorption energy capacities per unit volume and mass of different cellsizes aluminum alloy foams after compression strains of 5%, 10%, 15%, 20%, 25%, 30%, and 35% at different strain rates. For strain rates ranging from 103 to 1400 s1, the absorption energy capacity of the larger cell size foam at a strain of 35% varies from 2.85 to 3.44 MJ/m3, or equivalently from 3.91 to 4.72 kJ/kg, which is about a 20% increase. The absorption energy capacity of the smaller cell size foam at a strain of 35% varies from 2.73 to 3.25 MJ/ m3 or from 3.74 to 4.46 kJ/kg for strain rates from 103 to 1400 s1, which is about a 19% increase. This is much
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range of the strain rates employed, the present aluminum alloy foams exhibits a slight strain rate dependence in both yield strength and absorption energy. The relation between the relative stress and relative density is in reasonable agreement with Gibson and AshbyÕs equation. It was concluded that density is by far the most important variable and the intrinsic properties of the cell wall material are overwhelmed by the extrinsic properties of the cell structure during dynamic compression. Mechanical properties of the present foams appear to be independent of the cell size. Therefore, intensive efforts to modify the cell morphology are probably not an effective way to improve the impact absorption energy capacity. Acknowledgements Fig. 6. Energy absorption of aluminum alloy foams with the 0.9 mm cell size at different strain rates.
This work described in this paper was supported by the National Natural Science Foundation of China through Grant No. 90205018, the Natural Science Foundation, and the Homecomings Foundation and the Youth Academic Leader of Shanxi Program through Grant No. 2004–006 and 2003–23. References
Fig. 7. Energy absorption of aluminum alloy foams with the 1.6 mm cell size at different strain rates.
more than the energy absorption capacity of high strength polymer foams made, for example, from polymethacrylimide, particularly for the energy absorption per volume ratio. The increased values of WV and WM are noted to be practically the same, despite the different cell sizes. It also can be observed from Figs. 6 and 7 that the present aluminum alloy foams exhibit slight strain rate dependence in both yield strength and absorption energy. Thus, it is essential to obtain the accurate strain rate sensitivity of a material before selecting it for a specific application for impact energy absorption. 5. Conclusions The quasi-static and dynamic compressive stress–strain curves of aluminum alloy foams for various relative densities and different cell sizes have been studied. Over the
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