This study consists of four stages; FG thickness optimization, rolling, deep drawing, and crush analysis. In order to perform these stages, a series of FE analyses conducted using commercial FE software, LS-DYNA. In the first phase of this study, the thickness of TWST was optimized using LS-OPT software to obtain the comparison results to be used in determining the coupled effect of optimization and plastic forming. Rolling process was then conducted to achieve the intended thickness gradient on the plate rather than directly importing optimized thickness values to the FE model. Later, deep drawing was applied to the plate having the desired thickness gradient after rolling to shape the thin-walled square tube. Finally, crush analysis under high velocity impact load was carried out to determine the coupled effect of optimization and plastic forming on the crashworthiness performance of the TWST.
2.1 Finite element modeling
In this study, mild steel (E = 207 GPa, ν = 0.3) was used for energy absorber material [2]. The stress-strain values given as tabular data in reference 2 were arranged and given graphically in Fig. 1. Because of its strain rate sensitivity and suitability of crash modeling [3–5, 25], piecewise linear plasticity was selected for material model. Effect of the strain rate was implemented by activating VP (Formulation for rate effects) option which uses Cowper-Symonds constitutive model. Cowper-Symonds equation is expressed by scaling two factors; strain and strain rate factors [26];
Figure 2 illustrates the base FE model for TWST. The base model of the TWST has a constant thickness of 1.5 mm and, it corresponds to a mass of 1 kg. The tube geometry was partitioned into 12 pieces to perform thickness optimization. The side length of the tube is 90 mm, while 240 mm is the total length. The TWST was meshed using a 5x5 mm Belytschko-Tsay shell element with five integration points through the thickness. The TWST is fixed from bottom and a solid-rigid wall with 100 kg mass impacts the tube with a speed of 54 km/h. For the contacts between lobes of the TWST and between the TWST and the rigid wall, automatic single surface contact algorithm was used. A constant friction coefficient of 0.2 was applied to all contacts surfaces. Also, hourglass was controlled via control hourglass card.
2.2 Validation of the finite element model
Before proceeding with the actual analysis, it should be shown that the FE model created for the problem simulation is correct. Two finite element analyses (FEA), namely mesh convergence and model validation studies, were conducted within this purpose. Three different element sizes of 1.25, 2.5, and 5 mm were evaluated in order to assess the effect of element size. Figure 3 shows the results of the mesh convergence studies. For each element size, crush force – crush displacement curves showed a good agreement except for small fluctuations through the end and more importantly, the peak crushing forces (PCF) are nearly identical. Similarly, the absorbed energy versus crush displacement plots initially follow almost the same path, although there is a deviation towards the end of the process, the difference is negligible. It has been shown that the FE model is not affected by the element sizes tested in this study. Also, for validation purposes, the FE model in the Nagel and Thambiratnam’s study [2] reproduced and the results were compared. While the results were in perfect harmony for the peak force at the beginning of the process, as seen in Fig. 4, this consistency decreased at the end showing little fluctuations. After the convergence and verification studies, it was concluded that the FE model created could be used in the studies that constitute future stages of this research.
Prior to thickness optimization, the FEA of TWST was performed with a constant thickness of 1.5 mm for subsequent comparison studies (will be named as single-piece tube). As can be seen from the Fig. 5 that the preliminary results for the TWST with constant thickness were obtained to be 240 kN and 9.9 kJ for the PCF and the absorbed energy, respectively
2.3 Effect of thickness optimization
After carrying out the FEA of TWST with constant thickness, thickness optimization of the TWST was performed using LS-OPT software. For each section of the tube, a thickness parameter; a total of 12 parameters were defined and the mass of the tube, absorbed energy, and peak crushing force was calculated. LS-OPT offers six different metamodeling techniques namely; Polynomial, Sensitivity, Feed Forward Neural Network (FNN), Radial Basis Function Network (RBFN) and Kriging. However, when using polynomial-based response surfaces, the user has to choose the order of polynomial and there is a considerable possibility of bias error [30]. Meanwhile, Kriging has fitting problems [31] and considerably sensitive to noise [32]. Distinctly, RBFN and FNN have not serious deficiencies and they are similar in terms of accuracy. Moreover it is known that FNN is better than RBFN for some problems [33]. In this study, thickness values were therefore determined by the FNN method to maximize absorbed energy and 1/PCF values. For each thickness, 20 simulation points with a space-filling scheme were selected. Also, with regards to the result obtained from the TWST with constant thickness, the mass of the tube was constrained by 1 kg while the peak crush force between 0–5 ms was constrained by 240 kN. Figure 6 shows the flowchart of the optimization process. Figure 6 shows the flowchart of the optimization process while the scatter plot of the simulation results is illustrated in Fig. 7.
Optimum thickness values of the TWST obtained from multi-objective optimization are given in Table 1 while Fig. 8 shows the crush responses obtained from the FEA of the optimized model. The optimized model showed a significant improvement in terms of both peak crush force (PCF) and absorbed energy. While the PCF was obtained as 150 kN with a 37.5% decrease, the amount of absorbed energy increased by 18% to 11.68 kJ. However, after 100 mm of deformation, the crush force of the optimized TWST starts to increase and exceeds the crush force of the TWST with constant thickness after 130 mm. This is inevitable behavior due to the nature of the TWST with graded thickness. If the thickness of the tube is constant, the crash force shows only small peaks afterward of deformation initiation. However, for the case of optimized tube, deformation initiates from the thinner section of the tube (t2-t7) and the crush force does not make another considerable peak until the thinner section completely deforms. With the increased thickness sections of the TWST (t8-t12), there is more force needed to initiate new deformation.
Table 1
Thickness values of the FG-TWST obtained from multi-objective optimization
Optimum Thickness Values (mm)
|
t1
|
t2-t7
|
t8
|
t9
|
t10
|
t11
|
t12
|
2.26
|
0.96
|
1.13
|
1.73
|
1.78
|
2.41
|
2.29
|
2.4 Effect of plastic forming
Apparently, there is quite an improvement in the crashworthiness performance of the TWST in optimization process needed to achieve such a graded thickness. However, the results given above and in most of the literature have been obtained without considering the effects of the manufacturing processes used in the optimization method. Therefore, the validity of the results obtained in this way for the crashworthiness performance of the TWST can be questionable.
The most convenient and economical manufacturing process to achieve graded thickness is rolling. However, it is not possible to produce the TWST as a whole with rolling process only. For this reason, after achieving variable thickness via rolling, the tube geometry was produced in two parts by deep drawing process. As a result, rolling and deep drawing processes were sequentially performed in order to investigate the coupled effect of optimization and plastic forming on the TWST's crashworthiness performance in this study. Subsequently these two parts was welded to form the square tube for crush analysis. Each forming analysis connected to each other via dynain file until crush analysis of the tube.
2.4.1 Rolling
The FE model of the rolling was constructed in two dimensional. Both rolls and workpiece were meshed using plane strain elements. In order to reduce complexity it was assumed that workpiece kept constant while movement was applied to rolls modelled as rigid. For the contacts between rolls and the workpiece 2D automatic surface to surface contact algorithm was used. The friction coefficient between the rolls and the workpiece was considered to be zero in order to simulate rotating effect.
The thickness of the plate was initially 2.5 mm. Considering thickness values obtained from multi-objective optimization the workpiece was partitioned into 12 sections. The length of each section was determined with regard to the conservation of volume. Maximum thickness reduction was limited to 40% for a single pass and multiple passes applied to the sections to obtain required thickness. To control thickness reduction, contact between the rolls and the workpiece was only activated for the related section of the workpiece. Table 2 shows thickness distribution of the sheet after rolling process.
Table 2
Thickness distribution of the workpiece after rolling process
Thickness Values (\(\pm 0.02\)mm)
|
t1
|
t2-t7
|
t8
|
t9
|
t10
|
t11
|
t12
|
2.28
|
0.98
|
1.14
|
1.74
|
1.78
|
2.40
|
2.32
|
2.4.2 Deep drawing
Deep drawing process was used to produce a half of the TWST from the plate with graded thickness. The finite element model of the deep drawing consists of four parts: Workpiece, punch, die and binders. Schematic illustration of deep drawing process is given in Fig. 9.
In deep drawing process, a trapezoidal punch velocity profile with maximum speed of 2 mm/ms was used as given in Fig. 10. Tool motion defined with the boundary prescribed motion rigid keyword.
All parts were meshed with Belytschko-Tsay shell element and 5 integration points through the thickness of workpiece was utilized. Using control shell keyword, shell element thickness change and warping stiffness were activated. A stiffness type of hourglass control implemented to the FE model with control hourglass keyword. The punch, the die and binders were modelled as rigid while for workpiece piecewise linear plasticity material model was used. Contacts between the workpiece and other parts created using forming one way surface to surface contact algorithm with friction coefficient of 0.1. Shell thickness offsets were activated with control contact keyword excluding rigid bodies. The workpiece meshed with 5x5 mm elements and adaptive mesh refinement activated by adpopt flag in the part keyword. Finally, springback analysis was established using interface springback seamless keyword. Figure 11 shows the effective plastic strain values of the workpiece obtained from the springback analysis.
After springback analysis surplus sections of the workpiece was trimmed using element trim keyword and the final form of the workpiece is given in Fig. 12.
2.5 Crush Analysis with coupled effect
Prior to this section thickness gradient of the TWST was achieved by performing rolling process while deep drawing process was then used to form the tube geometry. In this section, crash analysis was then carried out by considering the coupled effects of optimization and plastic forming carried out for thickness optimization. After rolling and deep drawing processes, configuration of tube has minor changes due to nature of the forming processes, e.g., the corners of the tube are smooth due to deep drawing process. After deep drawing process, there are now two symmetric half-tubular parts with graded thickness to build up TWST. These two symmetric parts were then connected to each other using 22 spot welds. Spot welds were modelled using constrained spot weld keyword and considered as rigid. Failure of the spot welds has not been taken into account in this study. The FE model was the same as the base FE model given above. The thickness distribution and the residual plastic strains of the tube prior to crush analysis were given in Fig. 13.
Although the geometry of the built up TWST, named as two-piece tube, did not change significantly compared to single-piece TWST, FE analyzes were rerun to determine individual effects of plastic forming and optimization. Figure 14 shows only the plastic forming effect on crashworthiness performance of the TWST by comparing the single-piece and two-piece tubes with constant thickness. The values of PCF and total absorbed energy dropped from 240 kN to 215 kN and from 9.9 kJ to 9.5 kJ by showing 9% and 4% decrease from a single-piece tube to two-piece tube, respectively. In addition to this, the results showing only the optimization effect in one-piece and two-piece tubes with graded thickness were given in Fig. 15. The PCF decreased from 150 kN to 139 kN while the total amount of absorbed energy also dropped from 11.68 kJ to 10.4 kJ from the single-piece tube to the two-piece tube, respectively. The decrease in the PCF and the absorbed energy corresponds to 7% and 11%, respectively. As mentioned above, all these results are just individual effects and independent of the coupled effect of optimization and plastic forming.
The final stage of this study has therefore been carried out by using two-piece tube to assess the coupled effect of optimization and plastic forming history on the TWST with graded thickness. More precisely, Fig. 16 shows the coupled effect on the crashworthiness performance of two-piece tube rolled to obtain optimized thickness. It is very clear from the figure that considering only the optimization effect, the obtained PCF and the absorbed energy were 139 kN and 10.4 kJ, respectively. After the coupled effect, which took the plastic forming effect into account, the PCF increased by 18% to 164 kN while the absorbed energy increased by 27% to 13.2 kJ. On other hand, these results correspond to 24% decrease in the PCF from 215 kN to 164 kN and 39% increase from 9.5 kJ to 13.2 kJ in total absorbed energy compared to the two-piece tube with constant thickness as shown in Fig. 17. Moreover, after the optimization process of the two-piece tube, the PCF decreased by 35% (215 kN to 139 kN) and the amount of absorbed energy increased by 9% (9.5 kJ to 10.4 kJ). As clearly seen from Fig. 17 that the results of this study showed that plastic forming history, when evaluated together with thickness optimization, has significant effects on the crashworthiness performance of the TWST.