1 Introduction and background of research

Thin-walled energy absorbers are one of the structures used in car bodies as a safety system [1]. This structure prevents collisions with the passengers during traffic accidents by converting the kinetic energy of the cars into plastic deformations that occur in the absorber body [2]. So far, different types of energy absorbers have been introduced. In the past, these absorbers were made using metal materials such as aluminum and steel, but today, due to advances in technology and materials, the number of manufacturing methods and materials used in energy absorbers is increasing every day. Fiberglass-reinforced composite materials [3, 4] as well as filaments used in 3D printers such as PLA [5, 6] and ABS [7, 8] are among the new materials that have been recently introduced. Researchers around the world are trying to identify the absorbers that work best for them. Among these, the performance of energy absorbers can be evaluated by examining several parameters. These parameters are mainly obtained from force–displacement curves. Thus, each energy absorber has a curve during its collapse process, according to which the performance indicators of the absorber can be extracted. One of the most important indicators of any energy absorber is the area below the force–displacement diagram, which shows the energy absorbed in the absorber (Energy or EA) [9]. Another indicator that is important in any force–displacement curve is the initial peak force. This force mainly represents the maximum force in the curve (Fm) [9]. By dividing the energy absorbed in the absorber by the mass of the absorber and the length of the absorber collapse, the specific energy absorption (SEA) [9] and the average force (Fa) [10] are obtained, respectively. It should be noted that by dividing the average force by the maximum force, the crush force efficiency (CFE) [10] is obtained. Research in the field of energy absorbers is often done in both quasi-static and dynamic forms. The following are some of the studies that have examined energy absorbers.

Wang et al. [11] investigated the energy absorption of sandwich panels containing intersecting circular sections with triangular cores under lateral loading. In their research, they used LS-DYNA software for simulation, and good agreement between experimental and numerical results was observed. In a study conducted in 2019 by Usta and Türkmen [12] on nested absorbers, the effect of adding ABSplus honeycomb cells on the collapse behavior of nested absorbers was investigated, their specimens were investigated under impact loading. It was revealed that using honeycomb as filler has a high influence on the crushing properties of single and nested tubular structures. In a 2019 study, Yang et al. [13] investigated elliptical cross-sectional absorbers analytically, experimentally, and numerically. In their analytical studies, they used a rigid-perfectly plastic material model. Fourfold-tube absorbers were introduced by Wang and Xu [14]. Their absorbers were tested under lateral quasi-static loading numerically and experimentally. LS-DYNA was used for simulation; the results showed that the fourfold-tube system (FT) can show the most efficient impact force. Nested tube structures with rectangular cross sections were investigated by Tran [15]. He did his article in both experimental and analytical methods. In his analytical investigation simple superposition principle was used. Baroutaji et al. [16,17,18] introduced different arrangements of nested tubes under lateral loading. Dynamic and quasi static tests were done in their investigations. Their works could be a good reference for lateral loading of nested tubes. Alavi Nia et al. [10, 19, 20] investigated nested multi-tubular absorbers under axial quasi static loading. In their works, they proposed different heights for their tubes which leads to lower maximum force. They optimized their structures based on RSM method. Elahi et al. [21] studied foam-filled absorbers under lateral loading. Their proposed theoretical model was based on Castigliano’s theorem. Chahardoli et al. [22, 23] investigated different types of nested energy absorbers under axial quasi-static loading. They used LS-DYNA in their simulations and showed that combining different circular and rectangular shells leads to better performance under axial and lateral loading. Tang et al. [24] studied the crushing properties of multicellular columns under axial impact loading. They investigated the effects of geometrical parameters on collapse properties, and it was revealed that thickness of wall, and also the number of cells along the radial and circumferential directions have a significant effect on absorbed energy. In another study which was done by Xiong Zhang and Hui Zhang [25], axial collapse of circular multi-cell columns was investigated experimentally and numerically under impact loading.

Given the history of energy absorbers, these structures are constantly being reviewed and updated. Since the issue of motor vehicle accidents has always been considered by researchers to reduce the damage caused by traffic accidents, the topic of energy absorption was chosen for further study. In this study, a new type of multi-cellular absorber has been proposed. These absorbers consist of 4 walls and a maximum of 3 concentric cylinders, which are more efficient than single cylinders that are subjected to quasi-static loading. Investigated absorbers of this manuscript have not been studied in previous studies. Also their force–displacement diagrams are compared in more detail and based on initial loads.

2 Raw materials

The raw materials required for this research are 6063 aluminum cylinders. Figure 1 shows the cylinders intended for experimental tests. Three different types of cylinders with diameters of 83, 58 and 32 mm were prepared, the thickness of the large, medium and small cylinders were 2, 2.1 and 1.1 mm, respectively, and their length was 83 mm. In order to ensure the validation of the tests, a replication of each sample was prepared. Figure 2 shows the true stress–strain curve for all three samples. To extract the stress–strain curve, dumbbell-shaped samples were extracted based on the dimensions presented in the reference [26] and were subjected to tensile testing in the machine. It must be noted that the approximate yield stress for Al6063 is around 150 MPa, in this investigation, in order to have a similar stress–strain curve for tubes they were annealed (they were subjected to a temperature of 420 °C for 2 h and then their temperature were decreased in room temperature).

Fig. 1
figure 1

Samples prepared for experimental studies

Full size image
Fig. 2
figure 2

True stress–strain curve for 6063 aluminum

Full size image

3 Numerical simulation

In order to simulate the proposed absorbers in this research, LS-DYNA finite element software was selected. Although LS-DYNA is appropriate for Impact problems but in may many studies it was used for quasi static loading [27, 28]. According to the model shown in Fig. 3A which is a finite element model of simulated absorber, to model the elements related to walls and pipes, a shell element with 2 * 2 mm dimensions was used. Selecting this dimension was based on a mesh convergence study which is presented in Fig. 3B; this figure shows the amounts of absorbed energy for a specimen with a code of C111 which would be introduced in the following paragraphs. As it can be seen after dimension of 2 × 2 mm2 the amount of absorbed energy doesn’t change significantly. Mat-piecewise-linear-plasticity material model was used to introduce the mechanical properties of aluminum to LS-DYNA. The required properties for this model were loaded into the software according to the stress–strain diagram of Fig. 2, which is for 6063 aluminum. The Young's modulus of aluminum is 70 GPa and the yield stress for the stress–strain curve shown in Fig. 2 is 43 MPa.

Fig. 3
figure 3

A: Finite element model simulated in LS-DYNA, B: Mesh convergence study for C111

Full size image

The simulated absorbers were placed between the upper and lower jaws as shown in Fig. 3. The lower jaw is fixed and the upper jaw moves downward in a quasi-static manner until the absorber completely collapses. The coefficients used to define static and dynamic friction were selected to be 0.3 and 0.2, respectively.

In this section, 27 different examples are simulated. In terms of the number of inner cylinders, these specimens can be divided into three different categories, the lateral cross-sectional geometry of which is shown in Fig. 4. Among these three categories, the diameters of large, medium and small cylinders were 83, 58 and 32 mm, respectively. It should be noted that the length of all three cylinders, like the experimental samples in Fig. 2, was equal to 83 mm.

Fig. 4
figure 4

General geometry of the simulated samples

Full size image

In each of the geometries presented in Fig. 4, three thicknesses of 1, 2 and 3 mm were considered for all walls and three thicknesses of 1, 2 and 3 for all cylinders. In naming the simulated samples, the initial letter and the number after it represent one of the geometries presented in Fig. 4, and the numbers that come after these geometries, show the wall thickness and the thickness of the cylinders in millimeters, respectively. For example, C231 represents an absorber that has the geometry of C2 and has a thickness of all its walls of 3 mm and a thickness of all its cylinders of 1 mm. Table 1 shows the collapse characteristics of all simulated samples.

Table 1 Decomposition characteristics of simulated samples under quasi-static lateral loading
Full size table

3.1 Validation of numerical simulations

In order to validate the simulations, the samples prepared in Sect. 2 were subjected to quasi-static compression tests. Figure 5 shows how these samples were placed between the jaws of the device. Figure 6 shows the force–displacement curves of these specimens under quasi-static lateral collapse, which is both experimental and numerical. The letters O1, O2, and O3 represent specimens with small, medium, and large diameters. Figure 7 compares the collapsed shell of the O1 sample in experimental and numerical modes. The collapse stages of the O1 sample are also shown in Fig. 8.

Fig. 5
figure 5

How to place the samples between the jaws of the device

Full size image
Fig. 6
figure 6

Force–displacement curves for validation of numerical results

Full size image
Fig. 7
figure 7

Comparison of the collapsed state of the O1 sample in experimental tests and numerical simulations

Full size image
Fig. 8
figure 8

Comparison of collapse stages of O2 sample in experimental and numerical cases

Full size image

Considering Fig. 6 shows that the initial slope of the force curve in the simulation mode is greater than in the experimental case. This is due to the completeness and uniformity of the thickness of the samples in numerical case. Also, in numerical mode, there is no distortion in the samples, and therefore, it is expected that all samples in numerical case have more initial force. A comparison of the area below the graph for the experimental and numerical cases shows that the same level is obtained for both cases, and this indicates that in both cases the samples absorb the same energy. Based on the comparison between the experimental and numerical results, it can be seen that LS-DYNA software can make accurate predictions about force–displacement curves, collapse states, and collapse parameters. Therefore, the results obtained in Table 1 can be analyzed with good confidence.

4 Results and discussion

In this part of the research, the collapse characteristics for the results obtained in Table 1 are investigated.

4.1 Investigation of the effect of wall and circle thickness on the collapse characteristics

As mentioned in the previous sections. In each instance, the number 2 at the end of each code indicates the thickness of the circular cylinders, which can be 1, 2, or 3. As can be seen, Fig. 9 indicates changes in the collapse characteristics in terms of changes in the thickness of the shells and walls (based on data presented in Table 1). According to these figures, it is clear that increasing the tube thickness leads to an increase in the collapse properties, also comparing absorbers in different groups shows that absorbers with higher wall thickness have higher amounts of collapse properties. Among the studied characteristics, increase in specific energy and crushing force efficiency are positive changes and indicate improvement in the collapse properties, but increase in maximum force is a negative change and should be avoided as much as possible.

Fig. 9
figure 9

Comparison of collapse properties of simulated samples

Full size image

4.2 Investigation of force–displacement curves and collapse states

According to Fig. 9, regarding the crushing force efficiency for the samples in groups C32 and C33, it can be seen that in these samples, the increase in the thickness of the cylinders from 2 to 3 did not create significant changes in the crushing force efficiency. To investigate this issue, the force–displacement curves for C322 and C323, as well as C332 and C333, are shown in Fig. 10. According to these two curves, it can be seen that although the average collapse force (Fa) for both C323 and C333 samples increased compared to C322 and C332 samples, but the maximum force also increased and the same increase in maximum force prevents the increase of CFE according to the definition of crushing force efficiency.

Fig. 10
figure 10

Comparison of force–displacement curves for C322 and C323 samples as well as C332 and C333 samples

Full size image

Reviewing Fig. 10 shows that in the process of collapse of the samples shown, there are sudden changes in the force–displacement curves. To further investigate this change, the force–displacement curves of all samples are given in the following sections.

Examining the diagrams in Fig. 11A, it can be seen that in these diagrams the initial maximum force occurred at the beginning of the collapse process. This maximum force can occur due to the collapse of the wall or the collapse of one of the cylinders that make up the structure. Figure 12 shows the first collapse stage for all C1 group samples. As can be seen in this figure, the collapse of all C1 group specimens occurred by buckling the upper part of the vertical wall. However, in C131 samples, wall collapse occurred at both the top and bottom. The higher wall thickness compared to cylinder thickness increases the probability of simultaneous collapse of both the upper and lower parts. A closer look at the force–displacement curves of Fig. 11A shows that in Group C1 specimens, the maximum force stability increases with increasing wall thickness from 1 to 3. To better understand this issue, the beginning of the force–displacement diagrams is shown in Fig. 11B. Figure 6, which is related to the lateral collapse of cylindrical shells, shows that these shells do not have any sudden increase in their force curve, and since increasing the wall thickness makes it stronger, so collapse at the beginning of the process for those specimens with a wall thickness of 3 mm is in the form of a combined collapse which involves the simultaneous collapse of both parts of the vertical wall (top and bottom) and the cylindrical shell.

Fig. 11
figure 11

A: Force–displacement curve for Group C1 specimens, B: Initial part of the force–displacement diagrams for Group C1 specimens

Full size image
Fig. 12
figure 12

A: Early stage of collapse of samples C111, C112, C113 and C21, C122, C123 and samples C132 and C133 B: Early stage of collapse of samples C131

Full size image

It should also be noted that often a sudden increase in the force–displacement diagram is due to the sudden collapse of the wall. Looking at the highlighted area shown in Fig. 11A, shows that the force curves of the C132 and C133 specimens have a sudden peak at the end of their collapse process. This peak is due to the sudden collapse of the lower part of the vertical wall. Figure 13 shows the buckling of the lower part of the vertical wall for both specimens.

Fig. 13
figure 13

Bending of the lower part of the vertical wall for samples C132 and C133

Full size image

Figure 14 shows the force–displacement curves for the C2 group samples. A distinctive feature of the force diagrams of these specimens is the presence of a peak force at a displacement of 30 mm. As can be seen in this figure, the curve can be divided into three different parts. In the initial part of the diagram, the force curve is almost uniform, in the middle part there is a sudden jump in the curve, and in the final part, while the force gradually increases, a secondary peak force occurs in the curve.

Fig. 14
figure 14

Force–displacement curve for Group C2 samples

Full size image

To justify the behavior of the curve in the C2 samples, it is useful to refer to the collapse pattern of these samples. Figure 15 shows the collapse steps for samples C212 and C213. As shown in Fig. 15, at displacements between 0 and 30 mm, the collapse of the absorbers is mainly due to the collapse of the cylinders, and in this collapse there is no deformation in the walls, but as the collapse progresses. At displacements of nearly 30 mm, the lower vertical wall collapses and the force curves experience their first peak. After a displacement of 30 mm, the collapse of the structure continues with the deformation of the cylinders and walls (except the upper wall). Then the upper wall buckles at the displacement of 50 mm and the curve experiences the second peak force. Another point that can be deduced from the diagrams in Fig. 14 is that at a constant thickness for the walls, the greater the thickness of the cylinders, the smaller the difference between the peak force of zone B and the uniform force of zone A.

Fig. 15
figure 15

Decomposition steps for samples C212 and C213

Full size image

Figure 16 shows the force curves of the C3 group samples. As can be seen in these diagrams, the force curves have a significant peak force at a displacement of 30 mm. The reason for this sudden increase was explained in the previous paragraphs. It should also be noted that the diameter of the innermost cylinder is 32 mm and in all samples with more than two cylinders (groups C2 and C3) after the collapse of the innermost cylinder, the first peak of force with buckling of one of the vertical walls occur.

Fig. 16
figure 16

Force–displacement curve for Group C3 specimens

Full size image

Among the diagrams in Fig. 16, the curve for the C313 sample is slightly different from the other curves. In this curve, the force diagram experiences an initial peak at the beginning of the collapse process. This is due to the fact that the cylindrical shells become more resistant with increasing thickness, so that in samples where the thickness of the cylinders is high, the displacement of the upper jaw cannot cause acceptable deformation in the structure of the cylinder and therefore the buckling of a 1 mm thick wall begins in the very early stages of collapse. Figure 17 shows the collapsed state of all specimens at a displacement of 60 mm. As can be seen in this figure, in all specimens except the C313 specimen, the inner cylinder has completely collapsed at a displacement of 60 mm.

Fig. 17
figure 17

Collapse of Group C3 specimens at a displacement of 60 mm

Full size image

5 Conclusion

In this study, energy absorbers made of nested cylinders were investigated. After validation of the results obtained in LS-DYNA software and observation of high agreement between the experimental and numerical results, the collapse properties of the structures under quasi-static loading were investigated. The results obtained in the study showed that increasing the thickness of the wall or cylinder leads to an increase in collapse properties. So the amounts of Fm, Energy, CFE and Fa for absorbers which had more thickness were more than thin ones. Studies of force–displacement curves have shown that most of the peak forces in the force curve are due to the collapse or buckling of vertical walls.

In the studies performed on the absorbers of group C1, it was found that by increasing the wall thickness from 1 to 3, the stability of the maximum force increases. It was also found that higher wall thickness increases the probability of simultaneous collapse of both the upper and lower parts of the wall as compared to cylindrical thickness.

In the study of the absorbers of group C2, it was found that the force–displacement curves of these samples could be divided into three zones A, B and C and as the ratio of the cylinder thickness to the wall thickness increases, the difference between the peak force of zone B and the uniform force of zone A decreases.

The gravitational force diagrams of group C3 showed almost the same behavior. In most of these absorbers, with the exception of absorber C313, the absorber collapse in the initial 30 mm is related to the collapse of the innermost cylinder, after which the collapse continues with the buckling of one of the walls.