Experimental and finite element simulation study of the mechanical behaviors of aluminum foam-filled single/double tubes

Aluminum foam-filled tubes are complicated structures created by filling one or more thin-walled metal tubes with varying shapes in cross-sections with aluminum foam. We optimized the structure of aluminum foam-filled tubes using software simulation, compression, and three-point bending experiments. Filling aluminum foam not only improves the axial compressive performance and bending strength of the thin-walled metal tube but also eliminates the disadvantage of the low strength of the aluminum foam. The aluminum foam-filled single tube exhibited significant improvements in load-bearing and energy absorption, with a one-time increase in load-bearing and five times increase in energy absorption compared with an empty tube. The aluminum foam-filled single tube with a smaller diameter-to-thickness ratio has a higher load-bearing capacity. In contrast, the length-to-diameter ratio has a lower impact on load-bearing capacity. Once the filling length reaches the effective filling length, the structure can still support higher loads and effectively reduce its overall weight. The compression and bending properties in the double-tube structure filled with foam aluminum improved significantly compared with the empty tube and single-tube structure filled with aluminum foam. The total compressive energy absorption capacity in the double-tube structure filled with aluminum foam is 2.01 times that of the empty tube and 1.81 times that in the single-tube-filled structure. When the wall thickness of the filled stainless steel tube is 1.0 mm, the total bending energy absorption and the specific absorption energy of the aluminum foam-filled double tube structure are 1.5 and 2.1 times that of the corresponding aluminum foam-filled single tube, respectively.


Introduction
The aluminum foam-filled tube is a structural and functional integrated material formed by filling one or more thin-walled metal tubes with aluminum foam of different cross-sectional shapes. Aluminum foam has positive energy absorption characteristics and excellent vibration reduction performance; the aluminum foam-filled tube has a better energy absorption effect and safety reliability than aluminum foam. The filling of aluminum foam not only improves the axial compression performance and bending resistance of thin-walled metal tubes but also avoids the disadvantage of the low strength of aluminum foam itself [1][2][3][4][5][6]. Therefore, the aluminum foam-filled tube has shown broad application prospects in the safety design of automobile and high-speed railways [7][8][9][10][11], spacecraft recovery [12][13][14], aerospace [15,16], and other fields [17,18].
The cushioning and energy absorption characteristics of aluminum foam-filled tubes are affected by the shape, the size of the filling tubes, the filling process, and the structure of the filled aluminum foam, etc A single square tube and a single round tube are the most common aluminum foam-filled tubes. Goel concluded that round tubes' energy absorption was higher than square tubes [19]. Li et al conducted axial impact tests on aluminum foam-filled multi-wall structures; the wall thickness was proportional to the maximum bearing force of the structure [20]. Liu et al investigated experimentally and numerically the dynamic response and energy absorption performance of foam-filled tubes under transverse external blast loads. They found that adding the aluminum foam core material had a significant effect on the energy absorption capacity of the structure. With an energy absorption contribution of 33%-45%, the foam core is an integral part of the structure [21]. Cui et al also found that the aluminum foam filling increased the flexural load capacity and limited the lateral displacement of thin-walled tubes [22]. Venkat Chillaet al conducted compression and bending experiments on empty highpressure tubes, non-in situ aluminum foam-filled tubes, and in situ aluminum foam-filled tubes; the specific energy absorption of in situ aluminum foam-filled tubes increased by 61% and 10% compared with the empty and non-in situ aluminum foam-filled tubes, respectively [23]. The superior mechanical absorption capacity of foam aluminum-filled tubes compared with metallic foam and empty tubes is due to the formation of a boundary layer between the tube wall and the foam aluminum [24]. Therefore, the performance of aluminum foam-filled tubes not only derived from the thin-walled tubes and the aluminum foam, but also from the additional forces, confirming the interactions between the aluminum foam and the thin-walled tubes.
To improve the average crushing force and mass energy absorption efficiency of aluminum foam-filled tube structure, researchers have developed double tube filled structure and multi-tube filled structure (figure 1). The double-tube structure improved the energy absorption efficiency of aluminum foam-filled tubes by quasi-static compression tests [25]. Also, the shape of the thin-walled tube affects the performance, and the round tube is better than the square tube. Lu et al studied the impact resistance of aluminum foam sandwich panels [26]. It is found that the energy absorption efficiency of the double-layer structure is lower than that of the single-layer structure, but the deformation of the structure under impact load is reduced and the impact resistance of the structure is improved. Cenk Kılıçaslan studied the double corrugated tube filled with aluminum foam; the increase of corrugation in the inner tube reduces the average force of the filled tube and the specific energy absorption (SEA) value [27]. The SEA value varies with the inner tube radius and wall. With the increase of thickness, the crushing force efficiency (CFE) showed a trend of increasing first and then decreasing.
A multi-tube structure filled with aluminum foam usually has side-by-side arrangements and concentric arrangements. The multi-tube structures have different properties compared to the single and double-tube structures. Guden investigated the axial quasi-static compression behaviors of pure aluminum empty tubes, double tubes, and multi-tubes filled aluminum foam with quadrilateral or hexagonal arrangements; the side-byside arrangement of aluminum foam-filled tubes is better than the simple thin-walled tube arrangement and aluminum foam-filled single tubes in terms of energy absorption, while it is lower than the thin-walled metal tube arrangement in terms of specific mass energy absorption [16]. In addition, the energy absorption of foamfilled single tubes was lower under dynamic loading. In contrast, double and triple tube structures showed improved energy absorption for both square and round types. Hashem Ghariblu found that the energy absorption of functionally graded foam (FGF)-filled multi tubes was higher than that of homogeneous foamfilled equivalent structures using simulation [28].
Aluminum foam-filled tubes have been of interest as a significant energy-consuming structure. Both the thin-walled tubes used for filling and the foamed aluminum filled in them play complex supporting and cushioning roles under different loads. However, most studies on the mechanical properties of aluminum foamfilled tubes have focused on the uniaxial compressive behavior. In this research, axial compression and threepoint bending experiments investigate the energy absorption performance of empty tubes, aluminum foamfilled single tubes, and aluminum foam-filled double tubes. Many parameters of the filled tubes, including diameter-to-thickness ratio, length-to-diameter ratio, filling length, and wall thickness, were studied to optimize the structure of aluminum foam-filled tubes. Meanwhile, we used MARC software to simulate the mechanical behaviors of aluminum foam-filled tubes.

Materials and methods
2.1. Finite element model of the structure MARC software was used to simulate the axial compression performance of an aluminum foam-filled 6063 round tube and the three-point bending performance of an aluminum foam-filled 301 square tube. Software UG was used for the modeling of aluminum foam. Aluminum foam is a kind of metal with many internal voids, and the solid part still meets the fundamental law of plastic deformation during the deformation. During the simulation, we supposed the aluminum foam pore is a ball, and the diameters are 3.0 and 1.3 mm, distributed evenly in the aluminum foam cell model ( figure 2(a)). The pore diameter approximates the actual pore-size distribution, and the porosity of the cell is 80%. Software Hyper Mesh was used for meshing, and the mesh type was tetrahedral, with an average mesh size of 0.4 mm. The Elastic-Plastic Isotropic material model was used and Solid 134 four-node element was selected. The foam aluminum mesh model is shown in figure 2 The quasi-static axial compression was simulated using the MSC.MARC ( figure 3). The tube structure has good symmetry during compression, so only half the model is selected for finite element simulation analysis. The outer tube's diameter, length, and wall thickness are 25, 52, and 2 mm, respectively. The Elastic-Plastic Isotropic model was used for the pipe wall materials, and Solid 17 six-node element was selected. The modulus of elasticity, the Poisson's ratio, the density and the yield strength are 72 GPa, 0.33, 2.8 ×10 3 kg m −3 , and 196 MPa, respectively. The boundary conditions are set for the nodes in the symmetry plane to limit the displacements in the average direction of indenter motion. The upper and lower indenters are simplified to rigid surfaces with dimensions of 40×40 mm, which can ensure that the filled tubes do not exceed the range of toughness properties during compression. The coefficient of friction of the contact is 0.3. The aluminum foam and the tube wall material are set up as a deformable body. Figure 4 shows the finite element simulation model of quasi-static three-point bending deformation. The outer tube is 200 mm long and 30 mm wide. The pipe wall is also the Elastic-Plastic Isotropic material model from Marc Mentat, and the element type is Solid17 six-node element. The pipe's elastic modulus, Poisson's  ratio, and yield strength are 206 GPa, 0.29, and 463 MPa, respectively. We set the upper and lower indenter heads to rigid bodies and the aluminum foam and filler tubes to deformable bodies. Also, we set contact between aluminum foam and pipe wall, aluminum foam and upper and lower indenter, and pipe wall and upper and lower indenter. The aluminum foam is self-contact, and the friction coefficient is 0.3. The size of the model is taken by half the actual size, and the filling structure has good symmetry, so the symmetry boundary condition is set to the rigid surface (consistent with the round tube part), and the limit node is displaced along the vertical center axis in the direction of symmetry plane movement. During the simulation, the loading speed of the indenter, the loading time, and the number of loading steps are 2 mm min −1 , 15 min, and 90, respectively.

Materials
The closed-cell aluminum foam was used; its porosity and pore sizes are 80% and 1-3 mm, respectively. There are two kinds of filled tubes: Cylindrical tube and square tube, which were made of a commercial 6063 aluminum alloy and 301 stainless steel, respectively. There are three specifications of 6063 aluminum alloy round tubes: Ø16 × 3 mm, Ø19 × 1 mm, and Ø25 × 2 mm (table 1); the radius of gyration (r return) is 3.9, 6.4, and 8.0, respectively; the ratio of diameter thickness (D/t) is 4.3, 18 and 11.5, respectively. Ø25 × 2 mm and Ø16 × 3 mm are also used to form the outer and inner tubes for double tube filling (figure 5). The filled square tube of bending test is 30 × 30 mm, and the thickness is 0.8 mm or 1.0 mm (table 2 and figure 6).

Compression and three-point bending tests
WDW-100 E electronic universal testing machine was used for quasi-static axial compression and three-point bending test. The axial compression test parameters were: the loading speed of the indenter was 2.0 mm min −1 , and the compression amount was 20 mm. The three-point bending test parameters were: the loading speed of the indenter was 120 mm min −1 , the span of the three-point bending was 120 mm, and the compression amount was 25 mm. The load-displacement curve was recorded, and the energy absorption (EA) and specific energy absorption (SEA) were calculated to measure the effectiveness of the filled tube. The SEA calculation equation is as follows.
Where E is the total energy absorption capacity; F is the load-carrying capacity of the structure; U is the stroke displacement, which is taken as 1/2 of the length of the structure, and M is the total mass of the structure.    wall near the head and tail of the aluminum foam-filled single tube ( figure 7(a)) which is evident from the stress map. The tube wall is under most pressure during the compressive deformation, while the filled aluminum foam is under less pressure. Since the material was predetermined to be elastoplastic, no material failure constitutive equation was introduced; consequently, the tube wall did not crack simulatively, while the tearing of the tube wall occurred in the experiment. The aluminum foam-filled double tube structure bored more stress than the foam-filled single tube (nearly twice figure 7(b)). The differences between experimental and simulative loads in figure 9 can be attributed to the inhomogeneity of the specimen material and geometrical defects. Figure 10 shows the simulation deformation of the filled structure after loading. The pressurized area of the square tube has a severe deformation, with apparent dents, and the square tube is turned outward; the tube wall is stretched at the lower part of the central area, and the middle part is concave upward. Except for the central region, the tube walls at two cylindrical supports also deformed slightly, and the other parts tilt toward the center region. Since the failure of aluminum foam and tube wall material is not introduced in this calculation, tearing the square tube does not appear in the calculation results.

Results and discussion
4.1. Effect of diameter-thickness ratio (D/t) on compression performance for aluminum foam-filled single tube 4.1.1. Effects of diameter-thickness ratio (D/t) on the morphology of empty and filled tubes after compression Figure 9 shows the macroscopic deformation of empty aluminum tubes with different value of D/t. Figure 9(a) shows axisymmetric deformation; figures 9(b) and (c) shows non-axisymmetric deformation, and (b) is more folded and irregularly deformed than (c). Under the same L/r, the folded deformation of the tube's wall increased with the increase of D/t, and the folds changed from regular to irregular distribution. When the round tube's height-to-diameter ratio (L/r) is known, the tube's wall is more susceptible to wrinkling as the diameterto-thickness ratio (D/t) rises. When the height-to-diameter ratio of the round tube is determined, with the diameter-to-thickness ratio increases, the wall is more prone to wrinkle deformation.   Figure 11 shows the aluminum foam-filled single tube first folds at both ends of the tube during compression due to the supporting effect of aluminum foam. The outer tube of the aluminum foam-filled single tube produced cracks. The aluminum foam-filled double tube has an extra inner tube; therefore, folds are made only at the ends of the filled tube during compression, without significant deformation in the middle part of the filled tube. The aluminum foam-filled double tube also cracked at the outer tube, and the crack expanded faster than the single tube under the same loading condition. With the D/t increasing to 18, the tube' 's wall tends to form folds but no cracking. Figure 12 depicts the load-displacement curve of quasi-static compression. Under an axial static load, an aluminum alloy round tube started to undergo elastic deformation. Then the tube wall experienced local yield instability. When the load first reached its peak, the load began to decline, and the load was at its lowest point before the fold of the upper and lower sections of the tube wall contact closely. When the loading displacement increases further, the load-displacement curve rises, meaning that the loaded wall is equivalent to another elastic buckling. When the load again reaches a certain peak, the wall once more exhibits local yield instability. As a result, the load-displacement curve gradually decreases, and the wall slowly forms the second fold. The load fluctuates intermittently as the displacement rises, and the load-displacement curve exhibits an unsteady fluctuation behavior. Each pair of peaks on the compression load-displacement curve of empty aluminum alloy round tubes is connected to a fold (curves C and E in figure 12). The curve exhibits unstable behavior and repeated morphology ( figure 9).  Comparing the empty tubes with different diameter-thickness ratios (curves A, C, and E), the maximum initial yield load of the circular tube with a diameter-thickness ratio of 11.5 is 44.6 KN, which is 6.37 times higher than that of the diameter-thickness ratio of 18.0. The larger the diameter-thickness ratio, the smaller the average compressive load of the aluminum alloy tube, and also the more likely the tube is to be unstable. When the diameter-thickness ratio is 4.3, the diameter-thickness ratio is small enough that the static compression buckling behavior of the circular tube is similar to the solid compression behavior. After the critical collapse stress, the stresses decreased for both diameter-thickness ratios of 11.5 and 18.0, the platform collapse stress was nearly equal to the increase in compression displacement, and there was no stress enhancement during the platform collapse phase. The load-displacement curve for the diameter-thickness ratio of 4.3 has almost no plateau, and the load keeps increasing with the increase of displacement. That is, the compressive load-displacement characteristics of the empty aluminum tube are closely related to its structural parameters. When the heightdiameter ratio of the circular tube is determined, the smaller the maximum initial yield load is as the diameterthickness ratio increases, the greater the possibility of crease deformation of the tube wall.

Effect of diameter-thickness ratio (D/t) on compression properties of the empty and filled tube after compression
Curve B and curve D correspond to the aluminum-filled tubes of C and E. The aluminum foam filling can significantly increase the structure's load-bearing capacity because the aluminum foam can bear part of the pressure during compression; the interaction between the aluminum foam and the aluminum alloy round tube also helps to increase the load-bearing capacity. From figure 11(a), it can be seen that the significant reduction of load in the middle section of curve B is due to the cracks in the outer wall of the outer tube after aluminum foam filling. However, in the subsequent compression, the aluminum foam-filled tube's load-carrying capacity is still more significant than the empty tube. Comparing curves D and E, it is evident that the load-bearing capacity is enhanced after aluminum foam filling. The number of buckling in the aluminum foam-filled tube is more significant than that of the empty tube, which indicates that the number of buckling on the wall of the tube increases and the length of the buckling becomes smaller after aluminum foam filling, which helps to improve the stability of the structure.

4.2.
Effect of filling tube's wall thickness and filling length on bending performance for aluminum foamfilled single tube 4.2.1. Effect of the wall thickness of filling tubes on bending properties Figure 13 shows the load-displacement curves of 301 empty tubes and filled tubes with different wall thicknesses. For empty tubes, increasing the wall thickness prolongs the elastic deformation time of tubes and improves the ultimate bearing capacity of tubes. The load of 1.0 mm thickness 301 stainless steel filling tube (Curve A in figure 13) is higher than that of 0.8 mm thickness (curve B in figure 13). Increasing the wall thickness leads to the foam aluminum-filled stainless steel tube obtaining better bearing capacity, and the filled tube shape changes more stably. Therefore, increasing the wall thickness reduces the elongation ratio of the sidewall and affects the load path until it reaches the initial peak load. The reduction in the sidewall's elongation ratio minimizes the buckling sensitivity [29]. The local buckling occurs under the elastoplastic state of the extruded material before the initial peak load. Furthermore, the stress of thin-walled tubes is much lower than that of thick tubes. The thicker the steel tube is, the higher the energy absorption rate is.

Effect of the filling length on bending properties
The load-displacement curve of the partially filled 304 stainless steel tube is shown in figure 14. Comparing the bending curve of the partially filled tube and empty tube, the partially filled aluminum foam tube in the more extended displacement section maintains a higher load; and the ultimate load point appears much later than the ultimate load point of the stainless steel empty tube. After adding foamed aluminum to the empty tube, the filled tube has a stress transfer process in the force process. The outer tube first bears the load; the load is transferred to the foamed aluminum filler, leading to the outer tube and foamed aluminum synergetic deformation. When the partially filled tube has the same deflection deformation as the empty tube, the amount of the filled tube's local pressure head on the pressurized surface is smaller than that of the empty tube. Therefore, the loss of the cross section's bending strength was reduced due to the filling foamed aluminum. The bending load capacity of a partially filled aluminum foam tube is improved.
Compared with the fully filled tube (Curve A in figure 13), the aluminum foam partially filled tube can still withstand higher loads and bending moments while reducing the overall weight, which is more suitable for manufacturing lightweight structural components. The filled aluminum foam only in the middle part of the  plastic deformation region can bear the more significant stress and strain to support and enhance the load; the other parts can bear minimal stress with almost no effect.
The effective filling length (L f ) was investigated numerically and experimentally by Santosa et al. The calculating formula is following [30]: Whereη=M uf /M u , which is the ratio between the ultimate bending moment of the filled section and the empty section, b is the width, and t is the thickness. In the optimization procedure, the foams' actual deformed length is considered the effective length. According to the parameters of the bending experiment, the calculated effective filling length is 32 mm, between 30 mm and 40 mm. There is one optimal L f (here 60) where the double optimum tube design.  Figure 15 shows the load-displacement curves for empty tubes made of 6063 aluminum alloy, single tubes filled with aluminum foam, and double tubes filled with aluminum foam under quasi-static loading conditions. The aluminum foam-filled double-tube construction has a significantly higher load-bearing capacity than the aluminum foam-filled single-tube structure. The performance of the aluminum foam in the core area is inferior to that of the aluminum foam in contact with the inner tube wall part when the aluminum foam-filled single tube structure is compressed in the axial direction. This is because the aluminum foam in the core area is only compressed in the axial direction, and the radial compression by the tube wall is negligible. Therefore, a double tube filling structure, which was produced by replacing the aluminum foam in the core with an aluminum alloy tube, significantly improved the load-bearing capacity. In contrast to the filled single tube structure, the aluminum foam-filled double tube structure exhibits more obvious tearing of the outer tube wall during compression (figures 11(a) (b)), but this does not affect the structure's ability to support loads. The double-tubefilled structure has a load-bearing capacity that is 1.7 times more than the single-tube-filled structure (figure 15), consistent with the simulation results. According to equation (1), table 3 displays the results of the energy absorption (EA) and specific energy absorption (SEA). The double-tube-filled structure has a substantially better specific mass energy absorption efficiency than the single-tube-filled structure (table 3).

Bending properties
The load-displacement curves for the stainless-steel tube with aluminum foam filling and the empty stainlesssteel tube are shown in figure 16. The three curves' elastic phases essentially coincide; the stainless-steel tube is the primary load-bearing part during the elastic deformation phase for the foam aluminum-filled stainless-steel tube. As the amount of compression increases, the corner of the stainless-steel empty square tube is tearing. When the thin wall of the side deformation is transmitted to the bottom surface, the load-bearing capacity is raised to a maximum, then essentially unaltered. Due to the mutual extrusion of the filled aluminum foam and stainless-steel square tube and the supporting of filled aluminum foam, the elastic deformation phase and plastic deformation phase are prolonged compared to an empty tube. As the amount of compression increases, the filled structure's corner tears, and the bearing capacity gradually decreases; however, due to the aluminum foam's support role, the aluminum foam-filled tube bending strength and bearing capacity continue to rise until the aluminum foam fails after compression; the load starts to drop. A double tube filled with aluminum foam remains more stable and durable than a single tube. When the pressure increased until the inner and outer tubes came into touch, the foam aluminum between the inner and outer tubes entirely failed (dashed line in figure 16). Table 4 shows energy absorption (EA) and specific energy absorption (SEA) of aluminum foam-filled stainlesssteel square tube, determined by equation (1). The filled aluminum foam improved the energy absorption effect of the aluminum foam-filled stainless-steel tube, and the energy absorption capacity per unit mass of the aluminum foam-filled structure is much larger than that of the empty stainless steel square tube. The energy absorption performance of the double tube-filled structure is better than that of the single tube-filled structure. When the wall thickness of the filled stainless steel tube is 1 mm, the total energy absorption is 1.5 times greater than that of the corresponding aluminum foam-filled single tube, and the specific absorption energy is 2.1 times.

Conclusions
(1) The bending load and absorption energy of the aluminum foam-filled tube are significantly higher than that of the empty tube. The load-carrying capacity is better for aluminum foam-filled single-tube structures with a smaller diameter to -to-thickness ratio. In contrast, the length-to-diameter ratio has less influence on the load-carrying capacity.