Crashworthiness Optimization Design of Aluminum Alloy Thin-Walled Triangle Column Based on Bioinspired Strategy

Aluminum alloy thin-walled structures have been well used in applications of energy absorption. In the present work, a bioinspired design strategy for aluminum alloy thin-walled structures is proposed to improve the performance of out-of-plane crashworthiness by altering the material distribution. According to the proposed strategy, a novel fractal thin-walled triangle column (FTTC) is designed, which is composed by iteratively applying the affine transformation of a base triangle up to 2nd-order. The finite element model is established to investigate the out-of-plane crashworthiness of FTTC and validated by experiment results. The numerical analysis of the crashworthiness of FTTC with different fractal orders (0th, 1st and 2nd) are performed, and the results show that 1st- and 2nd-order FTTC enhance the energy absorption of structures and crush force efficiency. In particular, 2nd-order FTTC has better energy absorption ability due to the optimal distribution of materials, which are efficiently organized by the proposed bioinspired design strategy. In addition, a parameter study is performed to investigate the effect of FTTC geometric details on the crushing procedure. The collapse mode shows that it tends to change from unstable to stable with the increase in thickness and side length and the decrease in height. Moreover, a positive relevant relationship is identified between the thickness and the crashworthiness for FTTC.


Introduction
Aluminum alloy thin-walled structures, due to their high energy absorption efficiency and lightweight structures, are an essential requirement in many industries. Aerospace, elevator, automotive, offshore structures, and liquid storage tanks inevitably use energy absorbers [1][2][3][4][5][6]. Different section shapes of aluminum alloy thin-walled structures such as square and hexagonal have been well documented [7,8], and the majority of research focuses on the even sides. However, odd sides' tubes such as triangular are also widely used in bridges and buildings et al. Therefore, more attention has been paid to study these structures in recent years. Alexander [9] presented the first theoretical model of the thin-walled structures' plastic collapse, which modeled the procedure as one stationary hinge and folding wave. Abramowicz [10] expanded the theory by proposing the effective crushing distance concept. Wierzbicki [11,12] proposed the Super Folding Element (SFE) theory, which combined the concepts from plasticity with moving hinge line collapse theories. Chen [13] proposed a simplified SFE based on energy mechanisms. Zhang [14] studied the collapse performance of X-shaped elements with various angles and modified the inextensional mode of corner element. Later, he established

Process Overview
The goal of this study is to develop a bioinspired design method to improve the crashworthiness performance of aluminum alloy thin-walled structures. Based on Ref [33], a top-down bionic process is developed, which starts from technical problems and finds solutions through natural systems. The main steps of the proposed method are shown in Figure 1. For a given profile of a thin-walled column, the design method is organized in 3 steps. Firstly, the insufficiencies of the existing method are found and refined into some improvement objectives. Secondly, the promotion goal is mapped into a real biological prototype. Finally, by studying how biology copes with this problem, a corresponding design strategy is extracted to guide the fabrication.

Process Overview
The goal of this study is to develop a bioinspired design method to improve the crashworthiness performance of aluminum alloy thin-walled structures. Based on Ref [33], a top-down bionic process is developed, which starts from technical problems and finds solutions through natural systems. The main steps of the proposed method are shown in Figure 1. For a given profile of a thin-walled column, the design method is organized in 3 steps. Firstly, the insufficiencies of the existing method are found and refined into some improvement objectives. Secondly, the promotion goal is mapped into a real biological prototype. Finally, by studying how biology copes with this problem, a corresponding design strategy is extracted to guide the fabrication.

Bioinspired Strategy
Fracture structures have emerging applications in strength requirement conditions, which are inspired by natural systems such as the wings of dragonflies, as shown in Figure 2. The definition of a fractal is a shape that repeats itself in a different range of scales. A fractal is a subset of an ndimensional metric space that is approximately or perfectly self-similar, whose Hausdorff dimension exceeds its topological dimension. Based on the contraction mapping theory [34], the fractal set as an attractor Δ can be obtained from a set of affine transformations F. The attractor Δ is a union set that is scaled copies of the initial set, which can be expressed as follows： where n is the number iterations, f is the subset of transformation, m is the number of transformations.

Bioinspired Strategy
Fracture structures have emerging applications in strength requirement conditions, which are inspired by natural systems such as the wings of dragonflies, as shown in Figure 2. The definition of a fractal is a shape that repeats itself in a different range of scales. A fractal is a subset of an n-dimensional metric space that is approximately or perfectly self-similar, whose Hausdorff dimension exceeds its topological dimension. Based on the contraction mapping theory [34], the fractal set as an attractor ∆ can be obtained from a set of affine transformations F. The attractor ∆ is a union set that is scaled copies of the initial set, which can be expressed as follows: where n is the number iterations, f is the subset of transformation, m is the number of transformations.
where n is the number iterations, f is the subset of transformation, m is the number of transformations.  Based on contraction mapping theory, the bio-inspired design strategy is illustrated as follows: Step 1. Define the affine transformation matrix F.
The two-dimensional affine transformation f i in Euclidean XY plane is defined as: where µ 1i and µ 2i are reflection control coefficient, respectively. θ i is the rotation angle, δ xi and δ yi are the displacement along the x-axis and y-axis, respectively.
Step 2. Perform the Iteration According to the Affine Transformation Matrix F. After the nth iteration (n initial to be 0) of Step 2, the new pattern based on the previous one is got. The pattern vertices are donated by vector p. According to the practical needs, the hierarchical transformation L is applied, which connects the vertices of the present and the previous one.
Step 3. Set Operation. Select whether to retain the intersection part (if any). If it is kept, Equation (5) is employed to fulfill the procedure. Otherwise, use Eqaution (6).
In case the graph after Step 3 does not satisfy the requirement, go Step 1 and come again. If the expected iteration steps reached, finish. Otherwise, n = n + 1, go Step 2 and perform the next iteration. Figure 3 shows an example of acquiring the designed shape through two iterations of the above procedure. Based on contraction mapping theory, the bio-inspired design strategy is illustrated as follows: Step 1. Define the affine transformation matrix F.
The two-dimensional affine transformation fi in Euclidean XY plane is defined as: where 1i  and 2i  are reflection control coefficient, respectively. i  is the rotation angle, xi  and yi  are the displacement along the x-axis and y-axis, respectively.
Step 2. Perform the Iteration According to the Affine Transformation Matrix F.
After the nth iteration (n initial to be 0) of Step 2, the new pattern based on the previous one is got. The pattern vertices are donated by vector p. According to the practical needs, the hierarchical transformation L is applied, which connects the vertices of the present and the previous one.
Select whether to retain the intersection part (if any). If it is kept, Equation (5) is employed to fulfill the procedure. Otherwise, use Eqaution (6).
In case the graph after Step 3 does not satisfy the requirement, go Step 1 and come again. If the expected iteration steps reached, finish. Otherwise, n=n+1, go Step 2 and perform the next iteration. Figure 3 shows an example of acquiring the designed shape through two iterations of the above procedure.

Geometry Model
According to the bioinspired fractal design strategy, the fractal pattern of dragonfly wings is employed. Based on the odd side's regular triangle, a series of fractal sections with different orders are created as shown in Table 1. They are named FTTC series, which is an abbreviation of fractal thinwalled triangle column. The initial triangle column is expressed as the 0th-order structure. D is the side length of the smallest triangle. Q is the total side number of all triangles. L is the total length of the sides. t is the thickness. More configuration details are shown in Table 1.

Geometry Model
According to the bioinspired fractal design strategy, the fractal pattern of dragonfly wings is employed. Based on the odd side's regular triangle, a series of fractal sections with different orders are created as shown in Table 1. They are named FTTC series, which is an abbreviation of fractal thin-walled triangle column. The initial triangle column is expressed as the 0th-order structure. D is the side length of the smallest triangle. Q is the total side number of all triangles. L is the total length of the sides. t is the thickness. More configuration details are shown in Table 1.

FE Models and Validation
To investigate the crashworthiness of FTTC structure, total energy absorption, specific energy absorption, and crush force efficiency are employed [35,36].
Total energy absorption can be obtained by the integral of the load-displacement curve. For a given displacement d, it can be calculated as: where F(x) is the crushing force at crushing distance x. Specific energy absorption is the per unit mass energy absorbed of a specimen. It is the main indicator to distinguish the energy absorption efficiency of different materials. It is formulated as: where m is the total mass of the structure. EA is the value of energy absorption.
The mean crush force that can be obtained as: Thus, the crush force efficiency definition is calculated as: where PCF is the peak force during collapse procedure. CFE is an indicator that reflects the load uniformity of a structure. The FE model of the novel fractal structures of FTTC in the out-of-plane crush is shown in Figure  4. The 0th fractal column is picked to do the FE model validation. The height He and the side length D0 of the 0th fractal column are 300 mm and 100 mm, respectively. The thickness t0 is set to 2 mm. The bottom of the column is supported by a rigid body. A rigid impactor is moving to the top end of the column with a constant speed of 5 mm/min. The total collapse distance is 150 mm.

FE Models and Validation
To investigate the crashworthiness of FTTC structure, total energy absorption, specific energy absorption, and crush force efficiency are employed [35,36].
Total energy absorption can be obtained by the integral of the load-displacement curve. For a given displacement d, it can be calculated as: where F(x) is the crushing force at crushing distance x. Specific energy absorption is the per unit mass energy absorbed of a specimen. It is the main indicator to distinguish the energy absorption efficiency of different materials. It is formulated as: where m is the total mass of the structure. EA is the value of energy absorption.
The mean crush force that can be obtained as: Thus, the crush force efficiency definition is calculated as: where PCF is the peak force during collapse procedure. CFE is an indicator that reflects the load uniformity of a structure. The FE model of the novel fractal structures of FTTC in the out-of-plane crush is shown in Figure  4. The 0th fractal column is picked to do the FE model validation. The height He and the side length D0 of the 0th fractal column are 300 mm and 100 mm, respectively. The thickness t0 is set to 2 mm. The bottom of the column is supported by a rigid body. A rigid impactor is moving to the top end of the column with a constant speed of 5 mm/min. The total collapse distance is 150 mm.

FE Models and Validation
To investigate the crashworthiness of FTTC structure, total energy absorption, specific energy absorption, and crush force efficiency are employed [35,36].
Total energy absorption can be obtained by the integral of the load-displacement curve. For a given displacement d, it can be calculated as: where F(x) is the crushing force at crushing distance x. Specific energy absorption is the per unit mass energy absorbed of a specimen. It is the main indicator to distinguish the energy absorption efficiency of different materials. It is formulated as: where m is the total mass of the structure. EA is the value of energy absorption.
The mean crush force that can be obtained as: Thus, the crush force efficiency definition is calculated as: where PCF is the peak force during collapse procedure. CFE is an indicator that reflects the load uniformity of a structure. The FE model of the novel fractal structures of FTTC in the out-of-plane crush is shown in Figure  4. The 0th fractal column is picked to do the FE model validation. The height He and the side length D0 of the 0th fractal column are 300 mm and 100 mm, respectively. The thickness t0 is set to 2 mm. The bottom of the column is supported by a rigid body. A rigid impactor is moving to the top end of the column with a constant speed of 5 mm/min. The total collapse distance is 150 mm.

FE Models and Validation
To investigate the crashworthiness of FTTC structure, total energy absorption, specific energy absorption, and crush force efficiency are employed [35,36].
Total energy absorption can be obtained by the integral of the load-displacement curve. For a given displacement d, it can be calculated as: where F(x) is the crushing force at crushing distance x. Specific energy absorption is the per unit mass energy absorbed of a specimen. It is the main indicator to distinguish the energy absorption efficiency of different materials. It is formulated as: where m is the total mass of the structure. EA is the value of energy absorption.
The mean crush force that can be obtained as: Thus, the crush force efficiency definition is calculated as: where PCF is the peak force during collapse procedure. CFE is an indicator that reflects the load uniformity of a structure. The FE model of the novel fractal structures of FTTC in the out-of-plane crush is shown in Figure  4. The 0th fractal column is picked to do the FE model validation. The height He and the side length D0 of the 0th fractal column are 300 mm and 100 mm, respectively. The thickness t0 is set to 2 mm. The bottom of the column is supported by a rigid body. A rigid impactor is moving to the top end of the column with a constant speed of 5 mm/min. The total collapse distance is 150 mm.

FE Models and Validation
To investigate the crashworthiness of FTTC structure, total energy absorption, specific energy absorption, and crush force efficiency are employed [35,36].
Total energy absorption can be obtained by the integral of the load-displacement curve. For a given displacement d, it can be calculated as: where F(x) is the crushing force at crushing distance x. Specific energy absorption is the per unit mass energy absorbed of a specimen. It is the main indicator to distinguish the energy absorption efficiency of different materials. It is formulated as: where m is the total mass of the structure. EA is the value of energy absorption.
The mean crush force that can be obtained as: Thus, the crush force efficiency definition is calculated as: where PCF is the peak force during collapse procedure. CFE is an indicator that reflects the load uniformity of a structure. The FE model of the novel fractal structures of FTTC in the out-of-plane crush is shown in Figure  4. The 0th fractal column is picked to do the FE model validation. The height He and the side length D0 of the 0th fractal column are 300 mm and 100 mm, respectively. The thickness t0 is set to 2 mm. The bottom of the column is supported by a rigid body. A rigid impactor is moving to the top end of the column with a constant speed of 5 mm/min. The total collapse distance is 150 mm.

FE Models and Validation
To investigate the crashworthiness of FTTC structure, total energy absorption, specific energy absorption, and crush force efficiency are employed [35,36].
Total energy absorption can be obtained by the integral of the load-displacement curve. For a given displacement d, it can be calculated as: where F(x) is the crushing force at crushing distance x. Specific energy absorption is the per unit mass energy absorbed of a specimen. It is the main indicator to distinguish the energy absorption efficiency of different materials. It is formulated as: where m is the total mass of the structure. EA is the value of energy absorption.
The mean crush force that can be obtained as: Thus, the crush force efficiency definition is calculated as: where PCF is the peak force during collapse procedure. CFE is an indicator that reflects the load uniformity of a structure. The FE model of the novel fractal structures of FTTC in the out-of-plane crush is shown in Figure  4. The 0th fractal column is picked to do the FE model validation. The height He and the side length D0 of the 0th fractal column are 300 mm and 100 mm, respectively. The thickness t0 is set to 2 mm. The bottom of the column is supported by a rigid body. A rigid impactor is moving to the top end of the column with a constant speed of 5 mm/min. The total collapse distance is 150 mm.

FE Models and Validation
To investigate the crashworthiness of FTTC structure, total energy absorption, specific energy absorption, and crush force efficiency are employed [35,36].
Total energy absorption can be obtained by the integral of the load-displacement curve. For a given displacement d, it can be calculated as: where F(x) is the crushing force at crushing distance x. Specific energy absorption is the per unit mass energy absorbed of a specimen. It is the main indicator to distinguish the energy absorption efficiency of different materials. It is formulated as: where m is the total mass of the structure. EA is the value of energy absorption. The mean crush force that can be obtained as: Thus, the crush force efficiency definition is calculated as: where PCF is the peak force during collapse procedure. CFE is an indicator that reflects the load uniformity of a structure. The FE model of the novel fractal structures of FTTC in the out-of-plane crush is shown in Figure 4. The 0th fractal column is picked to do the FE model validation. The height H e and the side length D 0 of the 0th fractal column are 300 mm and 100 mm, respectively. The thickness t 0 is set to 2 mm. The bottom of the column is supported by a rigid body. A rigid impactor is moving to the top end of the column with a constant speed of 5 mm/min. The total collapse distance is 150 mm. To simulate the collapse procedure of FTTC series, the nonlinear explicit FE code LS-DYNA is employed. All FE models are based on Belytschko-Tsay four-node shell elements. It has five integration points across the thickness and one integration point in the plane [35]. To acquire a fair balance between the numerical analysis computational cost and the accuracy, convergence analyses are carried out. For the buckling of the specimen under axial loading, an automatic single-surface contact is used. The automatic surface-to-surface contact is picked to simulate the contact between the rigid wall and the specimen. The friction coefficients of the dynamic and static are set as 0.2 and 0.3, respectively [27]. The material of specimens is aluminum alloy A6061-O with properties [37] listed in Table 2. The specimen is represented by a piecewise linear plastic strain hardening material (MAT_24). The impactor and rigid wall are modeled by rigid material (Mat_20). Additionally, due to the weak strain sensitivity of the material, the effect of strain rate is neglected [32]. The hourglass energy of the columns is checked on account of the element high distortion [38]. It is found that it is less than 2% of the internal energy, which means the correctness of the analysis is ensured. Figure 5 shows the SEA convergence behaviors of the 0th-order FTTC with various element size. It can be observed that the SEA gradually converges when the element dimension decreases to 5.0 mm × 5.0 mm. Therefore, this 5.0 mm × 5.0 mm element size is selected.  To simulate the collapse procedure of FTTC series, the nonlinear explicit FE code LS-DYNA is employed. All FE models are based on Belytschko-Tsay four-node shell elements. It has five integration points across the thickness and one integration point in the plane [35]. To acquire a fair balance between the numerical analysis computational cost and the accuracy, convergence analyses are carried out. For the buckling of the specimen under axial loading, an automatic single-surface contact is used. The automatic surface-to-surface contact is picked to simulate the contact between the rigid wall and the specimen. The friction coefficients of the dynamic and static are set as 0.2 and 0.3, respectively [27]. The material of specimens is aluminum alloy A6061-O with properties [37] listed in Table 2. The specimen is represented by a piecewise linear plastic strain hardening material (MAT_24). The impactor and rigid wall are modeled by rigid material (Mat_20). Additionally, due to the weak strain sensitivity of the material, the effect of strain rate is neglected [32]. The hourglass energy of the columns is checked on account of the element high distortion [38]. It is found that it is less than 2% of the internal energy, which means the correctness of the analysis is ensured. Figure 5 shows the SEA convergence behaviors of the 0th-order FTTC with various element size. It can be observed that the SEA gradually converges when the element dimension decreases to 5.0 mm × 5.0 mm. Therefore, this 5.0 mm × 5.0 mm element size is selected. To verify the accuracy of the FE model, the experiment is performed. The 0th-order column is fabricated by cutting and welding. The quasi-static compression test is performed using WDW-100 compression testing general machine as shown in Figure 6a. The specimen is placed between two flat plates, and 5 mm/min stroke is used in the test. The collapse displacement is 150 mm. The displacement and crushing force data are recorded by the sensor. The deformed shape and the force-displacement curves of simulation and experiment are illustrated in Figure 6b,c. All the forces experience a sharp rise in the beginning, followed by a fluctuation during the collapse period. Even though the experimental curve is not strictly consistent with the numerical curve, it still captures the general trend. Moreover, the comparison of energy absorption indicators is shown in Table 3. The experimental test indicators are consistent with numerical with the maximum relative error of 12.7%. Therefore, the FE model of the FTTC structure is reliable in predicting crushing behavior. To verify the accuracy of the FE model, the experiment is performed. The 0th-order column is fabricated by cutting and welding. The quasi-static compression test is performed using WDW-100 compression testing general machine as shown in Figure 6a. The specimen is placed between two flat plates, and 5 mm/min stroke is used in the test. The collapse displacement is 150 mm. The displacement and crushing force data are recorded by the sensor. The deformed shape and the forcedisplacement curves of simulation and experiment are illustrated in Figure 6b,c. All the forces experience a sharp rise in the beginning, followed by a fluctuation during the collapse period. Even though the experimental curve is not strictly consistent with the numerical curve, it still captures the general trend. Moreover, the comparison of energy absorption indicators is shown in Table 3. The experimental test indicators are consistent with numerical with the maximum relative error of 12.7%. Therefore, the FE model of the FTTC structure is reliable in predicting crushing behavior.   To verify the accuracy of the FE model, the experiment is performed. The 0th-order column is fabricated by cutting and welding. The quasi-static compression test is performed using WDW-100 compression testing general machine as shown in Figure 6a. The specimen is placed between two flat plates, and 5 mm/min stroke is used in the test. The collapse displacement is 150 mm. The displacement and crushing force data are recorded by the sensor. The deformed shape and the forcedisplacement curves of simulation and experiment are illustrated in Figure 6b,c. All the forces experience a sharp rise in the beginning, followed by a fluctuation during the collapse period. Even though the experimental curve is not strictly consistent with the numerical curve, it still captures the general trend. Moreover, the comparison of energy absorption indicators is shown in Table 3. The experimental test indicators are consistent with numerical with the maximum relative error of 12.7%. Therefore, the FE model of the FTTC structure is reliable in predicting crushing behavior.

Theoretical Model Establishment
Based on the Super Folding Element theory [12], the external work done during the deformation is equal to the energy dissipation. The dissipation of energy consists of bending energy and membrane energy. It can be expressed as: (10) where P m denotes the average crush force, H is the half fold length, η represents the effective crushing distance coefficient. E bending and E membrane are the bending dissipation energy and the membrane dissipation energy respectively. In the experiment, the mean crush force is calculated by: where P e is the experimental mean crush force, S d is the effective crushing length, P(x) is the crushing force at the distance of x. The bending energy is evaluated by the energy dissipation in the plastic hinge lines [13]. Due to theoretical deduction, the panel should be perfect flatten during the collapse. The bending energy for a multi-cell column is obtained as: where M 0 = σ 0 t 2 /4 represents the wholly plastic bending moment per unit width, b is the panel width, p is the number of the panel in the multi-cell column. σ 0 refers to the flow stress of the material, and t is the thickness of the column.
where σ y , σ u , are the yield stress of the material and the ultimate stress of the material respectively. n is power law exponent. For the 0th FTTC, it has three panels width D 0 and the bending energy is calculated as: For the 1st FTTC, it has nine panels width D 1 and the bending energy is calculated as: For the 2nd FTTC, it has 27 panels width D 2 and the bending energy is calculated as: According to Ref [12,39], the irregularity section is divided into a small element and the fractal columns' cross-section consists of two-panel and four-panel angle element, as shown in Figure 7. The membrane energy can be calculated as: where α and β are the angles in the element.
where α and β are the angles in the element. There are three two-panel angle elements in the 0th profile of the column. Therefore, for the 0thorder FTTC, the membrane energy is calculated as: For the 1st-order FTTC, there are three two-panel angle elements and three four-panel angle elements. As a result, the membrane energy is calculated as: The profile of the 2nd-order FTTC constitutes of three two-panel elements and twelve four-panel elements. The membrane energy is calculated as: The calculation of theoretical solution of the average crushing force is introduced in this section, taking the 0th fractal column as an example. Substituting Equation (15) and Equation (20) into Equation (11). The mean crush force can be obtained as: The half folding wavelength is determined by the stationary condition of the mean crushing force: There are three two-panel angle elements in the 0th profile of the column. Therefore, for the 0th-order FTTC, the membrane energy is calculated as: For the 1st-order FTTC, there are three two-panel angle elements and three four-panel angle elements. As a result, the membrane energy is calculated as: The profile of the 2nd-order FTTC constitutes of three two-panel elements and twelve four-panel elements. The membrane energy is calculated as: The calculation of theoretical solution of the average crushing force is introduced in this section, taking the 0th fractal column as an example. Substituting Equation (15) and Equation (20) into Equation (11). The mean crush force can be obtained as: The half folding wavelength is determined by the stationary condition of the mean crushing force: Therefore Substituting Equation (25) into Equation (23), the average crush force in quasi-static loading can be calculated as: Using the same procedure, the mean crush force of the other fractal columns can be expressed as:

Crashworthiness Comparison
For the sake of maintaining the same material usage, all the specimens have the same mass (same section area). According to Table 1, the dimensions of the FTTC series are listed in Table 4. The comparison of the FEA and theoretical prediction of the average crush force is illustrated in Table 5. It can be seen the maximum relative error of theoretical to the simulation is 14.54%. For the progressive folding 0th-order one, the error is relatively low. Therefore, the theoretical model is sufficient accuracy to predict PCF. The deformation process of the three columns is illustrated in Figure 8. As well as the corresponding force-displacement curves of the three columns are illustrated in Figure 9. It can be observed that columns have an acute peak at the beginning stage, then followed by a fluctuating load, which is the most energy dissipated place. The 0th-order column exhibits a typical progressive folding. The column shows a bending tendency with the increase of the fractal order. Thus, the force-displacement behave more irregular. The comparison of the indicators of the crashworthiness is shown in Table 6. Energy absorption is a key indicator that reflects energy absorption performance and SEA is employed to evaluate the energy absorption performance. As shown in Table 6, the 0th-order FTTC has the smallest SEA of 3.76 kJ/kg, and the 2nd-order FTTC has the biggest SEA of 7.13 kJ/kg, which is 89.6% higher than the 0th-order. There is an average increase of 30%-40% SEA via increasing the fractal order. CFE represents the load uniformity of a structure. The 2nd-order FTTC shows the most uniformity load due to its high rigidity. It can be observed that the CFE becomes larger with the increase of the fractal order.

Collapse Modes
According to Ref [27,40,41], there are two different collapse modes during bucking. Stable mode is the expected form of progressive top-down bucking. Unstable mode represents the local or global bending during the bucking. There are three main design variables that affect the collapse mode, initial side length, column height, and thickness. It is necessary to investigate the relationship between collapse mode and design variables. Initial side length D0 takes the value of 80 mm, 100 mm, and 120 mm. Column height He takes the value of 250 mm, 300 mm, and 350 mm. Thickness t0 takes the value of 1.5 mm, 2 mm, 2.5 mm. The relationship of between design variables and the collapse mode is illustrated in Figure 10. Figure 10a shows the sampling points of the three parameters. Figure  10b shows the collapse mode diagram of 0th-order column. Figure 10c shows collapse mode diagram of 1st-order column. Figure 10d shows collapse mode diagram of the 2nd-order column. It can be found that the fractal order has a significant influence on the collapse mode of the FTTC. With the increase of fractal order, FTTC has an unstable tendency. Moreover, increasing the thickness and side length contributes to a stable collapse.

Collapse Modes
According to Ref [27,40,41], there are two different collapse modes during bucking. Stable mode is the expected form of progressive top-down bucking. Unstable mode represents the local or global bending during the bucking. There are three main design variables that affect the collapse mode, initial side length, column height, and thickness. It is necessary to investigate the relationship between collapse mode and design variables. Initial side length D 0 takes the value of 80 mm, 100 mm, and 120 mm. Column height H e takes the value of 250 mm, 300 mm, and 350 mm. Thickness t 0 takes the value of 1.5 mm, 2 mm, 2.5 mm. The relationship of between design variables and the collapse mode is illustrated in Figure 10. Figure 10a shows the sampling points of the three parameters. Figure 10b shows the collapse mode diagram of 0th-order column. Figure 10c shows collapse mode diagram of 1st-order column. Figure 10d shows collapse mode diagram of the 2nd-order column. It can be found that the fractal order has a significant influence on the collapse mode of the FTTC. With the increase of fractal order, FTTC has an unstable tendency. Moreover, increasing the thickness and side length contributes to a stable collapse.

Collapse Modes
According to Ref [27,40,41], there are two different collapse modes during bucking. Stable mode is the expected form of progressive top-down bucking. Unstable mode represents the local or global bending during the bucking. There are three main design variables that affect the collapse mode, initial side length, column height, and thickness. It is necessary to investigate the relationship between collapse mode and design variables. Initial side length D0 takes the value of 80 mm, 100 mm, and 120 mm. Column height He takes the value of 250 mm, 300 mm, and 350 mm. Thickness t0 takes the value of 1.5 mm, 2 mm, 2.5 mm. The relationship of between design variables and the collapse mode is illustrated in Figure 10. Figure 10a shows the sampling points of the three parameters. Figure  10b shows the collapse mode diagram of 0th-order column. Figure 10c shows collapse mode diagram of 1st-order column. Figure 10d shows collapse mode diagram of the 2nd-order column. It can be found that the fractal order has a significant influence on the collapse mode of the FTTC. With the increase of fractal order, FTTC has an unstable tendency. Moreover, increasing the thickness and side length contributes to a stable collapse.

Effect of Thickness
In the 2nd-order fractal column, changing the panel thickness of the 1st-and 2nd-order provides an additional way in managing the material distribution. The coefficient that reflects the effect of wall thickness change between fractal orders is defined as:

Effect of Thickness
In the 2nd-order fractal column, changing the panel thickness of the 1st-and 2nd-order provides an additional way in managing the material distribution. The coefficient that reflects the effect of wall thickness change between fractal orders is defined as: where t 0 is the 0th-order wall thickness, t 1 is the 1st-order wall thickness, and t 2 is the 2nd-order wall thickness, as shown in Figure 11. t 0 is set 0.889 mm as the initial value, and the factor design of five levels (0.756 mm, 0.822 mm, 0.889 mm, 0.956 mm, 1.022 mm) is utilized to sample the design space of t 1 , and t 2 .
Materials 2020, 13, 666 14 of 17 where to is the 0th-order wall thickness, t1 is the 1st-order wall thickness, and t2 is the 2nd-order wall thickness, as shown in Figure 11. t0 is set 0.889 mm as the initial value, and the factor design of five levels (0.756 mm, 0.822 mm, 0.889 mm, 0.956 mm, 1.022 mm) is utilized to sample the design space of t1, and t2. Figure 11. Illustration of the thickness of different level walls.
The simulation results of SEA and PCF are fitted by a cubic surface as illustrated in Figure 12. Generally, the bigger φ1 and φ2, the bigger SEA and PCF. The slopes are steeper along the φ1 axis. Accordingly, it can be summarized that the wall thickness change between 0th-order and 1st-order has a bigger effect on the SEA and PCF. The simulation results of SEA and PCF are fitted by a cubic surface as illustrated in Figure 12. Generally, the bigger ϕ 1 and ϕ 2 , the bigger SEA and PCF. The slopes are steeper along the ϕ 1 axis. Accordingly, it can be summarized that the wall thickness change between 0th-order and 1st-order has a bigger effect on the SEA and PCF. where to is the 0th-order wall thickness, t1 is the 1st-order wall thickness, and t2 is the 2nd-order wall thickness, as shown in Figure 11. t0 is set 0.889 mm as the initial value, and the factor design of five levels (0.756 mm, 0.822 mm, 0.889 mm, 0.956 mm, 1.022 mm) is utilized to sample the design space of t1, and t2. Figure 11. Illustration of the thickness of different level walls.
The simulation results of SEA and PCF are fitted by a cubic surface as illustrated in Figure 12. Generally, the bigger φ1 and φ2, the bigger SEA and PCF. The slopes are steeper along the φ1 axis. Accordingly, it can be summarized that the wall thickness change between 0th-order and 1st-order has a bigger effect on the SEA and PCF.

Conclusions
A bioinspired design strategy has been proposed in the present work. Based on that, a novel fractal thin-walled triangle column (FTTC) has been designed. It is investigated up to the second-order and shows a great improvement in the crash indicators. A parameter study has been conducted to study the influence of geometric details. The conclusions are summed up as follows: (1) A bioinspired design strategy is proposed and shows great potential to advance the crashworthiness of the thin-walled column. The fractal order has a major influence on performance. In particular, SEA of the 2nd-order FTTC is 89.6% higher than that of the 0th-order.
(2) The theoretical models have been deduced to predict the average crush force of FTTC, and it shows a good agreement with the FEA model.
(3) The different collapse mode of the FTTC has been studied. It finds that the fractal order has a significant influence on the collapse mode. With the increase of t and D 0 , it shows a tendency to transfer from the unstable to stable, whereas the change of parameter H e shows the opposite trend.
(4) Changing the fractal thickness of different orders has an effect on crashworthiness by rearranging the material distribution. The wall thickness change between 0th-order and 1st-order has a bigger effect on the SEA and PCF.