Simplified Super Folding Element Theory Research Articles

Design and study of crashworthiness for square vicsek fractal hierarchical multicellular tubes under axial compression

Through extensive examination of thin-walled structures, the efficacy of enhancing the crashworthiness of thin-walled tubes via layered structures has been confirmed. In this study, we introduce a novel design called the square Vicsek fractal hierarchical multicellular tube (SVFHMT), which integrates principles from fractal science, specifically Vicsek fractals. Unlike conventional layered structures that align along a single direction, our design incorporates layers evenly distributed at five positions throughout the structure, resulting in a more uniform material distribution compared to traditional layering methods. In this paper, the verified finite element model is used to analyze the crashworthiness of the structure, and the effects of levels, wall thickness, wall thickness ratio of different levels and height to diameter ratio on the crashworthiness of the structure are discussed. The mean crushing force (MCF) of SVFHMT is predicted by Simplified Super Folding Element theory. Our findings highlight the significant influence of layering, wall thickness, and ratios of wall thickness between layers on the crashworthiness of the structure. Specifically, for structures of equal mass, the energy absorption of second-order SVFHMTs and third-order SVFHMTs demonstrates substantial improvements compared to square nine-cell tubes, with increases of 51.05% and 91.70%, respectively. Moreover, when maintaining equal mass for second-order SVFHMTs with differing ratios of wall thickness between layers, specific energy absorption ranges from 15.69 kJ/kg to 21.29 kJ/kg. Notably, the maximum error between theoretical and simulated values of the mean crushing force for each order of SVFHMT is only 7.34%. This underscores the effectiveness of our novel layer design strategy, which diverges significantly from conventional approaches and offers valuable insights for advancing layered structures further.

Energy absorption characteristics of fractal multi-cell square tubular structures under axial crushing

In this study, a new fractal multi-cell square tube (FMST) based on the fractal design of the Sierpinski carpet is proposed for energy absorption. The dynamic crushing performance and energy absorption characteristics of the FMST are numerically and theoretically investigated. Extensive numerical simulations are carried out for the FMST structures with different fractal orders and masses of the structures. The findings reveal that the specific energy absorption of the 3rd order FMST is significantly higher, exhibiting a remarkable 100 % increase compared to the 0th order FMST. Furthermore, the undulation of the load-carrying capacity of the 3rd order FMST is reduced by up to 88.5 % compared to a conventional square tube, indicating the substantial potential of the FMST for designing highly efficient energy absorbers. Comparative analysis against other hierarchical multi-cell square tubes reported in the literature confirms that the specific energy absorption of the FMST surpasses existing designs. In addition, a theoretical study is presented for the mean crushing force of the proposed FMST, employing the simplified super folding element theory. The theoretical predictions agree well with the numerical results, further validating the effectiveness of the proposed design. This study provides an innovative design of a multi-cell energy absorber with exceptional energy absorption efficiency. The incorporation of fractal principle in the FMST design holds promise for advancing the field of energy absorption, with potential applications in various industries.

Numerical simulation and theoretical analysis of the crashworthiness of a tree-like hierarchical multicellular bi-tube

Drawing inspiration from the branching structure of trees, a novel Tree-like Hierarchical Multi-cellular Bi-tube (THMBT) has been devised. This design merges the hierarchical distribution traits of tree branches with the superior crashworthiness of double square tube structures. Utilizing a validated finite element model, this study investigates the crashworthiness of THMBT, examining the impacts of wall thickness, bifurcation angle, and bifurcation length on its performance. The Simplified Super Folding Element theory is employed for theoretical analysis, facilitating the calculation of the mean crushing force of THMBT. Findings reveal that wall thickness exerts the most pronounced influence on crashworthiness. As wall thickness increases, THMBT-3 demonstrates a substantial 88.07% enhancement in specific energy absorption, accompanied by a notable increase in peak crushing force. Particularly noteworthy is the exceptional crashworthiness of THMBT-3 with a bifurcation angle (θ) of 60°. For instance, the crushing force efficiency of THMBT-3 with a wall thickness of 0.5 mm reaches 92.08%. Furthermore, when considering different bifurcation lengths under the same wall thickness and bifurcation coefficients (c) set to 1/3, both THMBT-2 and THMBT-3 exhibit superior impact resistance within their respective structures. This study marks a significant advancement by integrating biomimetic design with double square tubes, resulting in a substantial improvement in structural crashworthiness. The findings offer valuable insights for the further development of biomimetic and double square tubes.

Crushing performance evaluation of gradient Sierpinski triangular fractal column

Thin-Walled Structures with excellent mechanical properties are the key to the protective system of the vehicle. This paper proposes a novel Sierpinski triangular fractal column with graded thickness (STFC-GT) to improve the crushing stability and energy absorption of Thin-Walled Structures. The crashworthiness of the STFC-GT is investigated through the experimental test and numerical simulation under axial crushing load. The results show that the Sierpinski triangular fractal column contributes to the energy absorption, and the graded thickness design can decrease the initial peak force. Specifically, the energy absorption of STFC-GT improves by 144.22 % from 0th to 2nd. And the initial peak force of 2nd STFC-GT is 55.32 % lower than that of 2nd STFC with uniform thickness. In addition, a theoretical relationship between the parameter configurations and the mean crushing force of STFC-GT are developed using the simplified super folding element theory (SSFE). The maximum error is less than 7.89 % between the theoretical and numerical results. Finally, the influence of gradient types on crushing performance is investigated to determine the optimal gradient configuration and maximize the mechanical performance of the STFC. The research result provides reliable guidance for the design of a new vehicle safety protection device.

Numerical and theoretical analysis of the out-of-plane crushing behavior of a sinusoidal-shaped honeycomb structure with tunable mechanical properties

Honeycomb structures are widely utilized for energy absorption in engineering, showing superior crushing strength and energy absorption in the vertical direction, yet their potential as energy absorbers is limited due to a high initial peak force (IPF) in that direction. In this study, a novel honeycomb structure with tunable out-of-plane mechanical properties is developed by incorporating sinusoidal-shaped features into traditional vertically straight-walled cells. Crushing behavior and energy absorption performance of sinusoidal-shaped honeycomb structure (SSHS) under quasi-static out-of-plane compression are investigated numerically and theoretically. Results demonstrate that the IPF and the fluctuation of crushing response of SSHS can be effectively reduced for the introduction of sinusoidal-curved configuration. The predictable collapse mode of SSHS could be achieved for the buckling-induced features of trough and peak of sinusoidal waves. Parametric analyses were carried out to explore the influence of the geometrical topology of sinusoidal waves, namely, wave amplitude (A = 0.6–1.6 mm), wave number (N = 1–6) and the relative density (ρ = 0.05–0.25) on the out-of-plane mechanical properties of SSHS, including IPF, mean crushing force (MCF), specific energy absorption (SEA) and crushing force efficiency (CFE). The results imply properly adjusting topology parameters can determine the collapse patterns and the stress distribution, thereby substantially improving the mechanical properties. Moreover, theoretical investigation was also performed based on the simplified super folding element theory. The calculation of MCF shows a good agreement between the theoretical predictions and the numerical results. The sinusoidally curved-walled topology design strategy provides insights into the development of lightweight structures with controllable and improved crushing performance.

Effect of cross-sectional shape evolvement on the energy absorption performances of multicellular square tube

This study systematically investigated the effect of position evolvement of four ribs in multicellular square tube on its crashworthiness performance. Results show that the multicellular square tubes with the shape factors of k = 0.25 and k = 0.75 have better energy absorption performances than other multicellular square tubes under the same wall thickness or the same mass. The mean crush force is derived through the simplified super folding element theory. Moreover, multi-objective optimization of multicellular square tube is carried out and the optimal result is validated by the finite element model.

Energy absorption characteristics of bio-inspired hierarchical multi-cell bi-tubular tubes

Previous studies have shown that the multi-cell bi-tubular tubes exhibit superior energy absorption compared to the multi-cell tubes. To further improve the energy absorption of the multi-cell bi-tubular tubes, in this study, we propose new bio-inspired hierarchical multi-cell bi-tubular (BHMB) tubes mimicking the tree-like structures in nature. Unlike the conventional multi-cell bi-tubular tubes, the webs in the BHMB are innovatively constructed based on the tree-like structures found in natural structures such as giant water lily leaves. The energy absorption performances of the BHMB tubes with different hierarchical orders, inner diameters and loading angles are numerically investigated. The results demonstrate that the specific energy absorption (SEA) increases with the hierarchical order and inner diameters, and the SEA of the 2nd order BHMB tube is about 55% and 81% higher than that of the conventional multi-cell circular tube and circular tube under axial loading, respectively; 46% and 72% higher under oblique loading with the loading angle of 10°. A theoretical analysis based on the simplified super folding element theory is also developed to determine the mean crushing force (MCF) of the proposed BHMB tubes. The theoretical predictions of the MCF agree well with the numerical results. The findings of this study provide an effective guide for the design of multi-cell bi-tubular tubes with high energy absorption efficiency.

Performance analysis and multi-objective optimization of bionic dendritic furcal energy-absorbing structures for trains

Rail vehicles offer the characteristics of large carrying capacity and high running speed. Train accidents usually result in substantial personal casualties and economic losses. Energy-absorbing structures can efficiently and quickly absorb and disperse the energy generated during collisions. In this paper, the performances with different cross-sections and partings of the bionic dendritic furcal structure (BDFS) inspired by the furcal branch and bifurcation structure of neurons are analyzed and compared by numerical simulation. The simplified super folding element theory and experiments verify that the finite element models are effective. A hybrid multi-stage optimization decision system, i.e., multi-criteria decision-making (MCDM)- multi-objective optimization (MOP)- MCDM, which integrates the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method, the multiple objective particle swarm optimization-crowding distance (MOPSO-CD) evolutionary algorithm and repetitive VIKOR method, is proposed to find the optimal structure and obtain the optimal parameter alternative of the structure. Comparisons and analyses of experiments and simulations are carried out to verify the effectiveness of the proposed method. The results show that BDFS with six furcal ribs and 3rd-order parting display the best comprehensive crashworthiness under the working condition of 10 m/s specified in EN15227 standard. When the single target requirement is considered, specific energy absorption (SEA) can be increased by 16.18%, peak crushing force (PCF) can be decreased by 67.07%, and undulation of the load-carrying capacity (ULC) can be decreased by 14.02%. This study provides an effective tool for the analysis and optimization of energy-absorbing structures.

Quasi-static and dynamic out-of-plane crashworthiness of 3D curved-walled mixed-phase honeycombs

Honeycombs are widely used in engineering protections due to desirable crashworthiness. However, the gap between peak and mean stresses remains to be narrowed, and effect of interaction between the cells should be enhanced. To break these limits, novel three-dimensional curved-walled mixed-phase honeycombs are proposed in this paper. The specimens manufactured by three-dimensional printing method using 316L stainless steel are used to conduct out-of-plane crushing experiments, and finite element method simulations are carried out by ABAQUS/Explicit. According to experimental and simulation results, theoretical model is established based on the Simplified Super Folding Element theory to estimate the out-of-plane effective mean stress. From the results, the proposed honeycombs display efficient progressive folding mode and desirable mechanical properties, and the gap between peak and mean stresses is effectively narrowed. Compared with traditional straight-walled honeycomb, total energy absorption, specific energy absorption, energy absorption efficiency and force efficiency can be enhanced respectively by 57.8%, 33.8%, 3.4% and 55.3% after implementing 3D curved-walled mixed-phase design, which can be further improved by parameter optimization and thickness gradient design. The work represents an effective approach for designing and optimizing high-performance honeycombs through curved-walled design and mixed-phase mechanism.

Research on out-of-plane impact characteristics of self-similar gradient hierarchical quasi-honeycomb structure

From the perspective of bionics, this paper proposes a new self-similar hierarchical quasi-honeycomb (SHQ) structure. It specifically analyzes the hierarchy and gradient distribution’s influence on the impact characteristics under different impact loads by using a combination of numerical simulation, theoretical analysis and experimental validation. Research found that the hierarchical design can improve the out-of-plane impact performance of the honeycomb, and a reasonable out-of-plane gradient distribution can greatly reduce the peak crushing force (PCF) of the out-of-plane impact. In addition, based on the simplified super folding element theory, an out-of-plane impact theoretical model of the SHQ was established, which can better predict the mean crushing force (MCF) and energy absorption (EA).

Crashworthiness analysis of gradient hierarchical multicellular columns evolved from the spatial folding

Based on the gradient hierarchical distribution of vascular bundles of bamboo, a gradient hierarchical multicellular column (GHMC) evolved from the spatial folding was proposed. Crashworthiness analysis of GHMC was carried via Abaqus/Explicit. Results show that the proposed GHMC has better energy absorption ability compared with the square tube under the same thickness or the same mass. The folding wavelength of GHMC is significantly decreased and it can produce more number of folds compared with the square tube. Under the condition of the same mass, the peak crushing force of GHMC decreases meanwhile its energy absorption increase by 10.1% to 72.44% compared with the square tube. Effect of hierarchical level and cell wall thickness on the energy absorption and their deformation mode under different height-length ratio (H/L) and cell wall thickness have been analyzed. Simplified super folding element theory was used to predict the mean crush force of GHMC, and the results were in good agreement with the simulations.

Crashworthiness analysis of bio-inspired fractal tree-like multi-cell circular tubes under axial crushing

This study proposes novel bio-inspired fractal multi-cell circular (BFMC) tubes for energy absorption. The inner structures of the proposed BFMC tubes were constructed based on the fractal tree-like forms found in many biological structures such as giant water lily and dragon blood tree. The crashworthiness performances of the proposed structures with different fractal orders and mass were numerically investigated. The numerical results indicated that the specific energy absorption (SEA) increased with the fractal order and the SEA of the 2nd-order BFMC tube was 35.43% higher than that of the conventional multi-cell circular tube. Furthermore, the complex proportional assessment (COPRAS) method was adopted to optimize the performance of the BFMC. The results demonstrated that the proposed structure with four number of tree-like branches and 2nd-order fractal provided the best performance. Finally, a theoretical derivation of the mean crushing force (MCF) was developed for the proposed tubes based on the simplified super folding element theory. The theoretical results of MCF agreed well with the numerical results. The findings of this study provide an effective guide for using the biomimetic approach with the fractal tree-like forms for the design of a multi-cell energy absorber with high energy absorption efficiency.

Out-of-plane mechanical design of bi-directional hierarchical honeycombs

A bi-directional hierarchical strategy is proposed to design a novel hierarchical honeycomb. Three kinds of bi-directional hierarchical honeycombs (BDHH), including hierarchical triangular honeycomb, hierarchical quadrangular honeycomb and hierarchical hexagonal honeycomb, are developed to investigate their out-of-plane crashworthiness. The compression test of a hierarchical hexagonal honeycomb fabricated by selective laser melting (SLM) method is performed to investigate the deformation mechanism and validate the finite element (FE) model. Compared with corresponding regular honeycombs, BDHHs are characterized by the superior material distribution across the network, which contributes to the crushing force efficiency and the energy absorption. In addition, numerical investigation is carried out to reveal the influence of hierarchical factors and thickness-to-length ratio (relative density) on the out-of-plane mechanical response of the BDHH. It is found that bi-directional hierarchical factors and the thickness-to-length ratio can promote the energy absorption of the BDHH. Furthermore, the analytical model of the mean crushing strength based on the simplified super folding element theory is derived to predict the mean crushing strength of bi-directional hierarchical hexagonal honeycomb. As a conclusion, there is a good consistency between theory solution and simulation result. All findings can be considered as effective guidance for the development of a lightweight protective structure.

Crushing responses and optimization of novel sandwich columns

The paper proposes a design method of the sandwich column with the corrugated sinusoidal core and develops the triangular corrugated sandwich column (TCSC), quadrangular corrugated sandwich column (QCSC) and hexagonal corrugated sandwich column (HCSC). The experimental tests of the TCSC and QCSC with A = 8 and N = 1 are conducted to estimate the crushing responses and the reliability of the numerical model of the sandwich column. The crushing responses are numerically investigated of the TCSC, QCSC and HCSC with different geometrical characteristics and the same mass under the axial impacting load. The results show that the energy absorption of the sandwich column is significantly affected by the geometric parameters A and N of the sinusoidal core tube, and the sandwich design has the large advantage of the energy absorption over corresponding members. Furthermore, the theoretical model of the mean crushing force is derived using the simplified super folding element theory, and the comparisons on experimental result and the finite element analysis demonstrate high prediction accuracy of the theoretical model. Lastly, a numerical optimization is proposed by combining orthogonal experimental design and discrete optimization to obtain the optimal crashworthiness of the sandwich column. Optimization results show that the discrete optimization method can be fast convergence and obtain the optimal design parameters. The energy absorption of the optimal sandwich column is about 10% higher than the initial design. Furthermore, the HCSC has the highest specific energy absorption under the same mass and design constraints.

Theoretical analysis and numerical simulation of square honeycombs

Aluminum honeycomb is widely used in energy absorption field for its special structure and excellent performance. The three-dimensional finite element models of aluminum honeycombs are developed in order to explore the mechanical behaviors under out-of-plane and in-plane compression. The simulation method is validated based on the experimental results of hexagonal honeycombs. Based on the simplified super folding element theory, we study the platform stress of square honeycomb under out-of-plane and in-plane compression. The theoretical formulas of platform stress under two conditions are provided in this study. Finally, compared with the simulation results of square honeycomb compression, the maximum error between the simulation and the theory for the platform stress under out-of-plane and in-lane compressions is less than 10%, and 15%, respectively. Both the results predicted by the theoretical formulas show high precision.

Theoretical prediction and analysis of hybrid material hat-shaped tubes with strengthened corner structures under quasi-static axial loading

This paper proposes an innovative hybrid material hat-shaped tube with strengthened corner structures (HMHT-SCS) to overcome the disadvantages of the limited energy absorption capacity and lightweight potential of conventional hat-shaped tubes. First, an improved theoretical prediction model of a corner element is established by modifying the simplified super folding element theory, and analytical formulas for energy absorption are derived under quasi-static axial loading. Second, a prediction model of the HMHT-SCS is developed based on the improved prediction model of corner element. Analytical formulas for the mean crushing force (MCF) of single-hat HMHT-SCS and double-hat HMHT-SCS are derived. Then, the accuracy of the analytical formulas is validated through axial crushing tests and simulation analysis. Finally, a discussion is performed. The total effect generation method is proposed to visually characterise the effects of input parameters on output responses. The parametric study showed that the thickness of the hat element of HMHT-SCS has the strongest effect on the MCF, and the thickness and material type of the hat element have similar effects on mass. The thickness of the corner element has a noticeable effect on MCF, while its section width has the weakest effect on the MCF and mass. Furthermore, the comparative analysis showed that the specific energy absorption (SEA) of the HMHT-SCS is larger than that of conventional hat-shaped tubes with one single material. Additionally, the SEA of the ‘aluminium & steel’ HMHT-SCS is larger than that of the ‘steel & aluminium’ HMHT-SCS under the same flow stress.

An investigation on the energy absorption characteristics of a multi-cell hexagonal tube under axial crushing loads.

A multi-cell tube enhances the energy absorption considerably compared to the absorption of a single tube under the same conditions. A novel tube configuration, namely, a multi-cell hexagonal tube, was proposed in this paper. The multi-cell tubes consist of three basic elements: a 2-panel element and two 3-panel elements (I and II). Simplified super folding element theory was utilized to estimate the energy dissipation of the basic elements. Based on this estimation, a theoretical expression for the mean crushing force was developed for the proposed tubes. The relative errors between a simulation, an experiment and theoretical results were no more than 5%. The effects of the hexagonal tube size and wall thickness on the crashworthiness of the multi-cell tubes were investigated. To a certain extent, the energy absorption and peak crushing force increased as the tube size and thickness increased. The response surface method (RSM) and the multi-objective non-dominated sorting genetic algorithm (NSGA-II) were used to improve the crashworthiness of the tube, and Pareto fronts were achieved. Finally, it was concluded that the optimal solution is C = 45 mm, t1 = 3.0 mm, and t2 = 2.35 mm, and the corresponding SEA and PCF were 16.52 kJ/kg and 411.36 kN, respectively.

Out-of-plane mechanical behaviors of a side hierarchical honeycomb

Hierarchical honeycombs have been extensively used as protective devices due to their superior mechanical performance. In this paper, a novel side hierarchical triangular honeycomb (SHTH) constructed by sequentially arranging a certain number of similar subtriangles on the geometrical side of an ordinary triangular honeycomb (OTH) is proposed to enhance structural mechanical performance. Experimental specimens and finite element models of SHTHs are developed to explore their mechanical behaviors under out-of-plane compression, and the numerical models are validated based on the results of crushing tests of SHTHs. Numerical comparisons of the SHTH, OTH and double-triangular honeycomb (DTH) are performed and illustrate that the SHTH has a higher collision force level and better energy absorption than the OTH and DTH due to more stable collapse deformation. Moreover, a parametric investigation of the SHTH is performed to explore the influences of the relative density, hierarchical factor and layout of the honeycomb on crashworthiness. The results show that the crashworthiness of the SHTH is improved with increasing relative density and number of layouts. Furthermore, the stable energy absorption and load bearing efficiency of the SHTH are observed when the hierarchical factor is close to 10. A theoretical model of the SHTH is also derived and validated based on simplified super folding element theory (SSFE) and numerical analysis. The findings of this study provide a new method for designing an energy absorber with excellent protective performance.

Experimental and theoretical study on crashworthiness of star-shaped tubes under axial compression

Thin-walled tubes play a very important role in energy absorbing for car crash. Current studies mainly focus on the design theory and performance improvement by proposing different structures. However, the practical performances might be influenced by manufacturing process, but few studies on this aspect could be found. In this paper, the crashworthiness of several star-shaped tubes, i.e., the hexagonal, octagonal and twelve corners, which were made with different material and fabrication methods, are compared through the experimental analysis. It is found that the energy absorption of octagonal star-shaped tubes is the best and that of the twelve corners star-shaped tubes is the lowest. Under quasi-static compression, the Pm of the octagonal star-shaped tubes increases by 7.94% with respect to the hexagonal star-shaped tubes, while the Pm of star-shaped tubes with twelve corners decreases by 15.75% with respect to the octagonal star-shaped tubes. The star-shaped tubes manufactured by different processing methods have some influence on the deformation mode, but the effect on overall crashworthiness is limited. Finally, the axial mean crushing force of the star-shaped tubes was analyzed theoretically based on the simplified Super Folding Element theory. A theoretical solution for the mean crushing force of the star-shaped tubes was derived, and the theoretical solutions show an excellent agreement with the experimental results.

A study on nested two-tube structures subjected to lateral crushing

The paper studies on experimental and theoretical investigations into the crushing response of the nested rectangular-square tube structures under lateral loading. Lateral crushing behavior of these nested tube structures is described in this work. The effect of peak crushing load on mean crushing load is pointed out for each nested tube structure. The theoretical formula is proposed based on the simple superposition principle and the modification of the simplified super folding element theory. The mean crushing load depends on the length of the plastic hinge line, wall thickness, and flow stress of material. The value of the mean crushing load obtained from calculation is compared with the results of the tests, resulting in good agreement with acceptable errors.