Energy performance of metallic tubular systems under reverse complex loading paths

1 Introduction

Axial buckling of thin-walled cylindrical (TWCs) tubes is a popular subject due to its simplicity and energy performance. The state-of-the-art reveals several theoretical and experimental approaches, with ongoing research in various fields of transportation [1–9]. In recent decades, the regulatory concept of passive vehicle safety has emphasized the creation of lightweight, cost-effective, environmentally friendly, and fuel-efficient structures. The increasing number of accidents and casualties has further intensified the interest of researchers and manufacturers in more absorbent thin-walled tubular structures.

Increasing the absorption capacity of a vehicle’s mechanical components improves the impact resistance of its structure. The main vector for dissipating energy is the plastic deformation of tube walls, which is governed by the formation of plastic hinges. This process highlights macroscopic mechanisms, such as deformation modes, cracking, and friction phenomena, and microscopic physical phenomena [10–15]. It is important to note that TWC tubes generate well-known deformation modes, including axisymmetric AM, diamond DM, and/or mixed XM. Additionally, the geometric influence is determined by two parameters, namely, η and λ, which designate the radial and longitudinal ratios of each tube. These parameters have been extensively discussed in numerous works [8,16,17]. Many other works have also explored various types of thin-walled cylinders to enhance energy absorption capacity [15,16,18,19].

Furthermore, various techniques have been continuously deployed to enhance the energy capacity of these energy-absorbing systems, as evidenced by recent studies [20–23]. One of these techniques involves controlling plastic flow by favoring the more dissipative MA, which has been the subject of several investigations [19,20]. For instance, tube subdivision has significantly improved the energy absorption of TWC tubes by 22% [20]. Also, several corrective approaches have been adopted to attenuate the effect of the plasticity entry peak on the wall of TWC tubes, such as the introduction of grooves and corrugations of various dimensions [17,21–28].

Cross-sectional profiles can be used to improve shock absorbers. Plastic buckling was performed using stepwise thickness of square extruded members or multi-cell profiles with a square element added to the corner part of a given cross-section [29,30]. The amount of energy absorbed depends on the number of angle elements in the cross-section. Severe deformation is concentrated in these zones near the corner, providing higher energy absorption. Therefore, multicellular profiles are structures that are effective in terms of weight and can absorb up to 220% more energy, as demonstrated by Zhang and Zhang [31].

The use of functionally graded thickness in thin-walled tubes has been proposed as a non-conventional method to control plastic straining and its distribution. This approach can be applied to various structural shapes, including square and circular thin-walled tubes and frusta [32–35]. Additionally, the use of twin-tube TWCs has resulted in a significant improvement in specific energy absorption when compared to single-tube systems. For advanced industrial applications, composite metal foams have been used [36–44]. Non-conventional systems, such as TWC tubes made of pseudo-composite steel obtained by specific thermochemical treatment, have been shown to improve energy performance by 46% compared to conventional systems [45]. In another study [46], the use of the origami technique allowed the tube to follow a predetermined failure mode, resulting in improved energy absorption efficiency. The TWC spiral steel tube was developed as a new structural component for energy absorption under dynamic crushing in a recent study [47].

The investigation examines the effect of the loading path as a potential solution for improvement based on the causal relationship between the material’s behavior and the multiaxial nature of the imposed load, particularly its degree of complexity [48]. Studies have shown that TWC tubes made of copper and aluminum are sensitive to energy dissipation. As a result, over-cracking and energy dissipation have improved by 46 and 150%, respectively [49,50]. Several interesting published experimental works have also focused on other systems with square cross-sections, using the same approach and parameters but with greater complexity, leading to a significant improvement in the energy dissipation capacity of such systems [51].

The present investigation employs the same methodological approach as the original rig, studying the improvement in the energy dissipative capacity of TWC tubes from the same angle. This is achieved through a novel loading path induced by a new biaxial test rig to produce an intensified complexity load for plastic buckling. Several metal tubular structures made of copper and aluminum (CuTWC and AlTWC) were tested under similar quasistatic and dynamic crushing conditions. A detailed analysis was then carried out, focusing on the effect of complex multiaxial buckling on the limits of the energy dissipative capacity of such non-conventional systems.

2 Notation and structural indices of buckling resistance for EAS-TWCs

According to the state of the art, we adopted the main dynamic and energetic parameters listed by the four equations below, which are largely used to estimate the energy dissipating capacity of the TWC tube subjected to axial quasistatic and dynamic axial crushing for each of the two quasi-static and dynamic regimes [27,28,52].

• The symbol “EA” then denotes the total energy absorbed, estimated as the area under the load–displacement curve presented

  (1) EA = ∫ 0 δ F d l .

• The crushing stroke efficiency (SE) is defined as follows:

  (2) SE = L L 0 ,

  where L ₀ and L are the initial and final lengths, respectively.

• The crushing load efficiency is defined as the ratio between the average crushing load (F _av) and the peak load (F _max)

  (3) ℈ = F av F max .

• The specific energy absorption (₳) is expressed as

  (4) ₳ = EA M ,

  where M denotes the mass of the specimen.

  Also, throughout this study, the following additional designations are also adopted as symbols:

• L: effective length of the tubular structure.

• t: Tube wall thickness.

• R: Radius of the cylindrical tube.

• η = R/t: Radial geometric parameter of the tube.

• λ = R/L: Longitudinal geometric parameter of the tube.

• Dy: Dynamic buckling loading regime.

• Qs: Quasi-static buckling loading regime.

• AM: Axisymmetric mode of deformation.

• DM: Diamond mode.

• XM: Mixed mode.

• DXM: Predominantly diamond mixed mode.

• AXM: Predominantly axisymmetric mixed mode.

• TXM: Twisted mixed mode.

• Also, through this study, the following designation is also adopted to define each configuration discussed, e.g., “AlTWC-QsBi45S” referring to a AlTWC tube under quasistatic crushing (5 mm/min) in alternating torsion under 45° biaxial mode.

3 About the new ACTP-S device

The objective of this work is to create a new, non-traditional EAS-TWC. The complexity of the loading path is due to the two interchangeable cylindrical bodies, each designed with a regular helix inclination angle of either 45° or 60° (Figure 1). Both bodies have a rolling path with an “S” profile, resulting in two multiaxial configurations: Bi45S and Bi60S. These configurations can be classified into two categories of multiaxiality: moderate and severe. This difference in the rates of the torsion component is due to the two helix inclination angles. The choice of these configurations is based on the relevance of the results obtained in our previous studies using the initial ACTP device [49–51]. It is expected that this new variant extension will improve the absorbed energy compared to the original conception, by generating a maximum change in the mechanical behavior of the base material of the tubes.

Figure 1

(a) Development of three basic cylinders defining the three new ACTP-S alternating multi-axis configurations, (b) the same set of tube mounting and fixing accessories is used for both ACTP and ACTP-S devices, and (c) Bi45S test in preparation.

3.2 Experimental protocol

The tubular structures used in this study were loaded onto an Instron Universal Testing Machine (type 5582) between two parallel platens. The loading was done under a constant compressive crosshead speed of 5 mm/min at room temperature. The fixed extremities were considered as a uniaxial case, as shown in Figure 2. For the multiaxial case, the ACTP-S was fixed in the machine, with particular emphasis on centering the device to ensure load coaxiality, as shown in Figure 3. The machine is connected to an acquisition chain to record the force and corresponding displacement during the crushing process.

Figure 2

Illustration of running tests under the quasistatic regime of aluminum and copper EAS-TWCs: from tests of the reference uniaxial case (a) QsAlBi0° and (b) QsCuBi0°.

Figure 3

Illustration of running tests under quasistatic conditions of aluminum and copper SAE-TWCs from tests with alternating multiaxial configurations: (a) CuB45S and (b) AlB60S.

The influence of non-coaxiality, which is almost inevitable during specimen assembly, remains relatively low in practice. To ensure accuracy, each test is repeated at least three times under the same experimental conditions, including applied speed and room temperature. If the difference between the two responses exceeds 3%, another test should be performed to ensure widely acceptable reliability of the results.

The dynamic tests were conducted on a drop-weight crash test station (Figure 4), with a maximum effective velocity of impact of 9 m/s and a theoretical energy capacity of 2.5 kJ. The instrumentation mainly comprises a 20 ton dynamic force cell, a 5,000 g accelerometer, and a laser beam displacement transducer with a measuring range of 100 mm. The instruments are linked to a high-speed acquisition chain (2.5 MHz) that records the primary physical quantities that govern plastic deformation mechanisms, namely, force, acceleration, and displacement. The acquisition of these three quantities is synchronized by an optical sensor that includes two photocells for triggering acquisition.

Figure 4

Illustration of the executive environment for dynamic testing: the reference DyBi0° in phases (a) initial and (b) final; (c) the DyBi45S biaxial and (d) the data acquisition phase. (1) impact load; (2) Impactor carriage; (3) impactor and integrated accelerometer; (4) upper specimen embedding; (5) copper specimen; (6) impact zone and integrated force cell; (7) electromagnetic triggering suction cup; (8) laser beam displacement sensor; and (9) ACTPS device with embedded specimen.

Following several preliminary tests to determine the optimal crushing stroke for both materials, dynamic tests were conducted under identical experimental conditions: (i) drop height of 3.5 m and (ii) impact load of 47 kg. To ensure the reliability of the experimental results, the protocol requires redundancy of 4–6 tests. For each multi-axial configuration, almost twice as many tests were conducted compared to those under the quasistatic regime. A maximum deviation of approximately 5% is allowed for each series of tests, including all multiaxial configurations under dynamic conditions.

4 Analysis of experimental results

4.1 Influence of loading path complexity on plastic flow

It is well known that the energy absorbed remains governed essentially by plastic buckling mechanisms, leading to the formation of more or less continuous plasticity zones (known as plastic hinges or crowns). They materialize as distinct deformation modes, such as AM or DM, respectively, and other mixed modes AXM and DXM, or the new device with TXM (twisted mixed mode). In addition, changes in the known physical quantities according to each deformation mode are the target of the ultimate phase of this investigative approach. They are the maximum and average crushing loads, the crushed length, and the energy absorbed. This allows the final post-processing phase to analyze in depth the evolution of the main energy performance indicators of these non-conventual systems, which are given later.

Repetitive deformation modes recorded during the tests corresponding to the three simple multi-axial configurations are summarized and discussed. As a reminder, all the specimens are geometrically identical, made from the same two batches of materials – copper and aluminum – and subjected to the same quasistatic and dynamic loading conditions. The reference configuration used during this test corresponds to the uniaxial crushing of TWC tubes (denoted as Bi0°) with fixed ends.

Examination of the photos in Figures 5 and 6 reveals the classic plastic flow patterns –AM, XM, AXM, DXM, and twisted AM (TAM) and TXM, or twisted DM (TDM) – generated by all the configurations and for the two cross-end speeds of 5 mm/min and 9 m/s corresponding to the targeted quasistatic and dynamic regimes, respectively. However, the similarity between the two SAE-TWCs should be qualified for the quasistatic regime in particular (Figure 6), with a clear predominance of the AM deformation mode for copper systems and XM or DXM for aluminum. Indeed, careful observation of the samples from the multiaxial configurations shows not only in Figures 5 and 6 but also in almost all the results from the different series of samples. Relatively, different influences on the plastic deformation flows induced by the complexities are defined as uniaxial, moderate, and severe for both materials. A careful analysis of the two figures shows that, in the case of copper tubes, the plastic flow is virtually identical. The geometric parameters λ and ƞ define the useful crushing length of the used specimens. It is the preferred deformation mode, i.e., AM, which is systematically generated after plastic buckling on all specimens and for all configurations. As the most interesting mode in terms of energy dissipation, it is fully in line with the objectives of the investigation, which is to improve the absorbed energy of such systems. However, an important observation can be made in the case of severe dynamic crushing (Figure 7c), where only the most severe loading path (Bi60S) has a significant influence on the deformation mode, leading to induce a mixed XM mode. Thus, this could be due to the coupling of path loading complexity and crushing rate accentuating the sensitivity of CuTWC.

Figure 5

Summary of deformation modes obtained under quasistatic conditions for reference and alternating biaxial configurations: (a) uniaxial QsBi0°, (b) QsBi45S, and (c) QsBi60S for both materials.

Figure 6

Summary of deformation modes obtained under dynamic conditions for reference and alternating biaxial configurations: (a) uniaxial DyBi0, (b) DyBi45S, and (c) DyBi60S for both materials.

Figure 7

Load–displacement curves for alternating biaxial and reference uniaxial crushing of copper SAE-TWCs under quasistatic conditions (v = 5 mm/min).

In the case of aluminum tubes, the influence of the type of the degree of loading complexity, the state rate, and the coupling of these parameters is apparent between the six situations presented in Figures 5 and 6. It can also be noticed that the choice made regarding the initial geometry to consider the influence of plastic flow by the two basic geometric parameters ratios η, λ, and their interaction proves to be judicious. For the reference configuration Bi0°, AM systematically is generated whatever the loading regime applied, whether quasistatic or dynamic. In other words, for the AlTWC system, the strain rate does not affect the deformation mode.

On the other hand, the nature of the loading complexity plays a key role in the plastic flow. Thus, the two multiaxial configurations moderate Bi45S and severe Bi60S produce either DXM or DM for all aluminum tubes and for both regimes, unlike copper, which is due to its sensitivity to load coaxiality and the importance of its ductility. Regarding the influence of the torsional component of compound and multiaxial loading, it is clear that the two materials are affected in different proportions. Thus, plastic flow takes place in different modes. In addition to AM, AXM, TXM, and DM are the most common, regardless of crushing speed.

It should be noted, however, that the Cu-TWC system remains less affected than AlTWC, due to the greater rigidity of copper. The change in deformation mode is proportional to the degree of loading complexity of the rate. Thus, we pass, respectively, from AM for uniaxial Bi0°, to AXM or TAM in alternating multiaxial QsBi45S or even to DXM in QsBi60S, depending on the material type, as shown in Figures 5 and 6.

Furthermore, a fairly significant phenomenon of tube wall damage is visible in the form of several cracks, or microcracks, which are identifiable on aluminum tubes in particular. Indeed, these are highly visible to the naked eye and systematically revealed in the DM zone of aluminum tubes tested in QsBi60S in particular. This is not at all the case and not at all for CuTWC systems, implying the advent of an additional source of energy dissipation for AlTWC systems and the advent of a better specific absorption capacity.

At the end of this analysis of all the results of the three distinct configurations, the plastic flow of the two systems depends on the influence of the nature of the material, the loading complexity, and the coupling of all these factors. A comparison of the DyBi45S and DyBi60S cases shows that both materials are much more affected in AM situations. It is easy to see that the AM generated is more twisted (TAM) for copper and more transformed into XM or DM for aluminum. This may be explained by the greater sensitivity of the latter material to the loading path and is in line with the literature.

4.2 Mechanical properties systems

4.2.1 Multiaxial plastic buckling under quasistatic conditions

Analysis of the load–displacement curves in Figures 7 and 8 reveals classic patterns, as presented in the literature for EAS-TWC, under original quasistatic multiaxial loading applied via ACTP. Indeed, there is a clear influence on all tube responses, with significant differences in the first plasticity peaks and quite disparate values of the average load. This depends on the variant of the applied multiaxiality degree.

Figure 8

Load–displacement curves during quasistatic crushing (v = 5 mm/min) in alternating biaxial and reference uniaxial cases for aluminum EAS-TWCs.

Thus, under quasistatic crushing conditions, the results obtained in Figures 7 and 8 show a similarity in crushing load trends and a clear proportionality to the loading complexity. Therefore, the behavior of the two unconventional CuTWC and AlTWC systems is remarkably influenced by the loading complexity. For the same quantity of deformation governed by a buckling length limited to 50 mm, it is recorded progressive in the maximum and the mean loads that are of primary interest.

An important increase in mean loads was recorded under the same crushing speed of 5 mm/min of the order of 45 and 77% via the most severe configuration QsBi60S for the two unconventional CuTWC and AlTWC systems, respectively, compared to the uniaxial reference QsBi0°. Moreover, the smaller increases of 27 and 68% are captured under moderate multiaxial configuration QsBi45S.

This reveals a clear proportionality between the degree of loading complexity and the rate of change of the two systems, in other words, an obvious difference in the sensitivity of the two materials to the loading path. Thus, the undeniable and significant influence of the complexity of biaxial loading under alternating torsion on the mechanical behavior of copper and aluminum tubes could generate better energy dissipation, i.e., improve the capacity of the tubes.

In the light of these initial results and by way of a preliminary conclusion, this would prove the effectiveness of this new variant of the biaxial ACTP-S rig, giving rise to a new level of non-conventional system.

Of course, the same trend can be observed in the energy–displacement curves (Figures 9 and 10). They show interesting improvements in energy dissipation in favor of the most complex configurations, evolving linearly with the degree of loading complexity. These curves, for δ = 50 mm, lead to absorbed energy of 0.51, 0.68, and 0.79 kJ for the CuTWC and 0.38, 0.65, and 0.78 kJ for the AlTWC under Bi0°, QsBi45S, and QsBi60S, respectively.

Figure 9

Energy–displacement curves for alternating biaxial and reference uniaxial crushing of copper EAS-TWCs under quasistatic conditions (v = 5 mm/min).

Figure 10

Energy–displacement curves during quasistatic crushing (v = 5 mm/min) in alternating biaxial and reference uniaxial cases for aluminum EAS-TWCs.

This justifies the influence of the degree of complexity of the loading on the behavior of the crushed tubes, and consequently on the energy absorption capacity. As a result, the ability of tubes made from these two metallic materials to dissipate energy under severe multiaxial alternating loading has been considerably improved. The aluminum tubes show more sensitivity to the type of loading compared to copper.

This shows the new ACTP-S device. The difference in the recorded values of energy absorbed reflects the difference in sensitivity of the two materials used under this loading.

4.2.2 Multiaxial plastic buckling under dynamic conditions

It is widely known that materials become more resistant to plastic deformation if tested under a high strain rate. In addition to the stress complexity highlighted above, the influence of the strain rate on the responses of each of the two systems, CuTWC and AlTWC, combined with multiaxiality was examined. The tubes were tested under the same dynamic condition.

Their responses are analyzed and compared to each other, as well as to those collected under quasistatic conditions. The sensitivity of the two copper and aluminum EAS-TWCs to the dynamic loading was demonstrated. The interaction of the loading complexity with strain rate was estimated showing the performance of these unconventional systems under such conditions. Figures 11 and 12 show the evolution of the crushing load as a function of the crushing stroke required for each crash test. The recorded trends are classic and fairly similar. Two peculiarities were presented: (i) an amplification of the crushing load in the purely dynamic zone for both systems and (ii) the appearance of a densification phase on certain load-displacement curves, for the DyBi0° and DyBi45S cases for the AlTWC system (Figure 14).

Figure 11

Dynamic crushing load vs stroke curves for different configurations of copper EAS-TWC case.

Figure 12

Dynamic crushing load vs stroke curves for different configurations of the EAS-TWC in aluminum case.

The latter observation does not exist at all, neither in the DyBi60S curve nor in all three configurations of the CuTWC system. This is due to the greater strength developed in the latter cases, and therefore, necessarily the event of over-winding under the effect of this severe multiaxiality and the strength of copper’s mechanical properties, which are better than that of aluminum, hence the shortening for crushing length.

This quantity is of particular interest for the efficiency of the crushing stroke for both materials due to the crash-test nature of the plastic buckling load applied. As expected, the transition to dynamic plastic buckling of CuTWC and AlTWC systems is manifested by an undeniable intensification of the plasticity input peak (Figures 11 and 12). This becomes more intense for the severe biaxial (DyBi60S) of greater than 100 kN for both systems. This evolution leads directly to an increase in the energy absorbed, as it can be seen from the curves describing energy dissipation as shown in Figures 13 and 14.

Figure 13

Curves showing dissipated energy as a function of stroke under dynamic conditions for different configurations of copper EAS-TWCs.

Figure 14

Curves showing dissipated energy as a function of stroke under dynamic conditions for different configurations of aluminum SAE-TWCs.

These show positive evolutions for materials after the first plasticity peak. However, one can already conclude that, in addition to the amplification of the maximum load that triggers the plastic flow, there are impressive improvements in the mean loads that result, depending on the degree of loading complexity, for both systems. The two figures also show parallel increases in the mean load for the Bi0°, DyBi45S, and DyBi60S cases giving dynamic mean loads of 16.3, 19, and 24.3 kN and 14.5, 17, and 27.3 kN for copper and aluminum, respectively. This gives a gain of 47 and 91% in DyBi60S compared to DyBi0°, for CuTWC and AlTWC, respectively. While, under quasistatic load, the gain is 45 and 77% only. So, the materials exhibit further sensitivity to crushing strain in addition to loading complexity due to the ACTP-S loading path.

With the results for the quasistatic regime, Figures 10 and 13, derived by mathematical integration of the load–displacement curves, provide energy information absorbed as a function of axial displacement of the various specimens. They offer an interesting comparison of the plastic work developed as evolution for all three configurations, DyBi0°, DyBi45S, and DyBi60S. The results show a sharp increase in energy absorbed, always to the advantage, as desired, of multiaxial configurations giving the following energies for the three situations for δ = 45 mm and CuTWC and AlTWC: 0.74, 0.89, and 1.24 kJ, and 0.7, 0.89, and 1.34 kJ, respectively. This leads to a gain of 20 and 68% for copper tubes and 29 and 91% for aluminum tubes.

In addition, and to more closely approximate the change in behavior under essentially dynamic buckling of each tube for industrial use eventually, a treatment centered around the results obtained as a function of the length crushed around the first plasticity zone is thus addressed. The treatment targeting exclusively the energy fluxes generated according to the three configurations for the same plasticized length, bounded at δ ≤ 20 mm, is presented through the histograms in Figures 15 and 16. They show the energy absorbed for four selected deformed lengths: 5, 10, 15, and 20 mm, while considering dynamic crushing exclusively. Impact tests are then compared to those obtained in quasistatic for both CuTWC and AlTWC.

Figure 15

Energies absorbed by copper tubes under crushing regimes Qs and Dy for four crushing strokes for all regimes and all configurations.

Figure 16

Energy absorbed by aluminum tubes under crushing regimes Qs and Dy for four crushing strokes for all regimes and all configurations.

The incorporation of energy absorptions under quasistatic conditions is intended to highlight the influence of the coupling of the dynamic strain rate factor and the loading complexity. Indeed, an examination of reveals gradual and moderate increases in the energy absorbed, based solely on the results of the two regimes under loading Bi0°. The discrepancies are much greater as the loading complexity increases, reaching the highest value for Bi60S for both systems, particularly under dynamic conditions. This confirms the significant contribution of the coupling of the two parameters, strain rate and loading path complexity, to the improvement of the respective energy dissipation capacities of the two unconventional systems.

In conclusion, we can summarize:

1. Similar curves between the dynamic and quasistatic regimes for CuTWC and AlTWC but with an interesting effect of strain rate-loading complexity.

2. A strong influence on the dynamic maximum load and mean load, which are significantly greater with respect to quasistatic results.

3. Similar influence of loading configuration on behavior from the crushing loads, and energy absorbed, albeit with widely differing proportions for the two materials.

4. A clear advantage for aluminum tubes in terms of energy efficiency, as explained in Section 5.

5 Trends in the main structural energy performance indicators

It is worth noting that even if the dynamic crushing conditions were identical (same drop height h = 4 m and same impact mass m = 47 kg) for all configurations, the crushing strokes recorded under dynamic conditions, as shown in Figures 14 and 15, are more significant, particularly for aluminum SAE-TWCs. They differ according to the nature of the applied load, which is certainly and amply explained by the high ductility property of this material compared to copper. Insofar as this distinction also for the latter and the severe multiaxial Bi60S configuration leads to a shortening of the crushing length.

An analytical synthesis of the results of all the figures discussed above is summarized in Tables 3 and 4. This analytical approach highlights both the relevant and gradual effect of load path complexity. In this way, it gives a quantitative assessment of the crushing force efficiency ℈ , specific stroke SE, and specific energy absorption ₳ for both materials and regimes.

As a first observation, it should be noted that both tables show that the generation of the integral AM mode leads relatively to lower dynamic crushing strokes than the mixed mode or DM generated in multiaxial for both materials. This leads to distinct evolutions in the specific stroke parameter “SE” between the three configurations considered: uniaxial Bi0°, moderate Bi45S, and severe Bi60S. Moreover, the load–displacement curves systematically reveal, for aluminum EAS only, the advent of a third stage of densification for the uniaxial DyBi0° and the less complex DyBi45S configurations alone, unlike the DyBi60S. This is not the case for CuTWC. This is logically explained by the influence of loading complexity and the difference in ductility of the two materials. It is therefore important to look at the influence of the indicators SE, ℈, and ₳ for each of the two EAS. The two extreme cases of DyBi0° and DyBi60S, provide variations ranging from 69.1 to 60.6% for copper and from 81 to 75.6% for aluminum in the case of dynamic loading.

Results show values inversely proportional to the degree of complexity of the loading path, dropping from approximately 54 to 23 kN in the dynamic regime, e.g., when comparing the extreme cases of DyBi0° and DyBi60S. For the crushing stroke of 50 mm, it was revealed that ₳ in order of 71 and 76% for aluminum and 32 and 37% for copper in favor of the QsBi45S and QsBi60S quasistatic multiaxial configurations compared to the reference case QsBi0°. Significant increases in ₳ are recorded under dynamic conditions of 37 and 84% for aluminum and of 20 and 43% for copper tubes for DyBi45S and DyBi60S, respectively.

6 General conclusion

The experimental investigation revealed significant improvements in the performance of SAE-TWCs for metallic materials. The reason for this improvement could be explained by a change in the behavior of the metals under complex loading conditions. The study also found that aluminum mechanical components offer better advantages than copper-based ones from the absorbed energy standpoint. Overall, these experimental results demonstrate that the ACTP-S technique effectively improves energy dissipation capacity.

Future work will focus on microstructural analysis to identify and quantify the physical phenomena active at the plastic hinge level. The approach previously used by Drusina et al. [8], specifically transmission microscopy at the level of dislocations, will be adopted to gain a better understanding of the local behavior and interpret the macroscopic response of the tubular structures.

A numerical approach is also necessary to set up a finite element model that aims to reproduce the behavior of these unconventional SAE-TWCs. Modeling of plastically deformed zones, or plastic hinges, would reveal the influence of the degree of loading complexity on the strain-hardening rate.