Multi-objective optimization of functionally graded thickness tubes under external inversion over circular dies
- Omid Mohammadiha^{1}Email author and
- Hashem Ghariblu^{1}
https://doi.org/10.1186/s40712-016-0061-3
© The Author(s). 2016
Received: 2 August 2016
Accepted: 15 September 2016
Published: 27 September 2016
Abstract
This paper optimizes the crushing response and energy absorption of functionally graded thickness (FGT) inversion tubes using the multi-objective optimization method. The numerical results, which are validated by the experimental tests, confirm that optimizing geometry of FGT inversion tube improves energy absorption capacity with respect to ordinary uniform tube UT with the same weight. Our study shows that the die radius r, coefficient of friction μ _{d} between the die and tube, and thickness distribution function have great influence on the responses of FGT inversion tubes. The specific energy absorption (SEA), peak crushing force \( {P}_{max} \), and dynamic amplification factor (DAF) are selected as the objectives of crashworthiness optimal design. Finally, the weighted average method, multi design optimization (MDO) technique, constrained single-objective optimization, and geometrical average method were employed to find the optimal configuration of the proposed inversion tube. The results give new design ideas to improve crashworthiness performance of inversion tubes.
Keywords
Inversion tube Functionally graded thickness Finite element Multi-objective optimizationBackground
Energy crisis and environmental concerns place higher requirement to decrease the weight of vehicles and achieve a highest possible safety and quality with minimum cost. In the last few decades, there has been a continuous focus on design optimization as a primary requirement in the design of automotive structures. Lightweight materials, such as aluminum and magnesium alloys, are gradually finding their place in vehicle engineering (Miller et al. 2000). Thin-walled structures are widely used to dissipate the vehicle’s kinetic energy in terms of plastic deformation in collisions. Many applications employ thin-walled tubes to enhance the crashworthiness of structure such as energy absorption devices at the front of cars and trains (Marsolek & Reimerdes 2002), aircraft sub floor structures (Bisagni), and rollover protective structures (ROPS) of heavy vehicles, such as bulldozers and tractors (Ahmad & Thambiratnam 2009a). Thin-walled tubes with different geometries and materials are commonly used to absorb kinetic energy through plastic material deformation. The widespread use of thin-walled tubes as energy absorbers is due to their good performance under dynamic loading, availability, low manufacturing cost, and efficiency. Numerous efforts have been made in the past decades to improve the crashworthiness performance of the crush absorber tubes, such as foam filled tubes (Hanssen et al. 2000; Ahmad & Thambiratnam 2009b; 2009c; Aktay et al. 2008; Mirfendereski et al. 2008), introducing different patterns (Zhang et al. 2007), grooves (Zhang & Huh 2009; Saleh ghaffari et al. 2010), multi-cells (Zhang & Zhang 2013; Zhang & Zhang 2014), and functionally graded structures (Mohammadiha & Ghariblu 2016; Sun et al. 2010). Relative merits of conical tubes with graded thickness subjected to oblique impact loads are investigated by Zhang (Zhang & Zahng 2015).
Inversion is a mode of plastic deformation which develops when a thin-walled tube is compressed between a flat plate at one end and filleted die at the other (Al-Hassani et al. 1972a; Reddy 1992a). Therefore, analysis of external inversion process is equally important for manufacturing technology as well as for impact and crashworthiness studies. Tube curling mode is a case of forming behavior between flaring and buckling failures; only meeting certain forming conditions can the tube curl to produce an axially symmetric second wall (Reddy 1992a). Recently, many applications of inversion tubes have been made, such as force actuating collapsible steering wheels, cushioning air drop cargo, helicopter seats, and soft landing of spacecraft (Guist & Marble 1966). Experimental studies and theoretical analysis on tube inversion have been conducted by many researchers in the recent decades. Al-Hassani et al. (1972b), Reid (Karagiozova et al. 2000), and Reddy (1992b) made experimental investigations and theoretical analyses for the deformation behavior and forming load of external inversion. Miscow and AL-Qureshi (1997) studied the theoretical and experimental studies of the static and dynamic inversion process in circular tubes. This theoretical analysis is valuable as the first formula that predicts the axial force versus the axial displacement during the inversion process. But theoretical curve that sketched based on Miscow’s theory shows an intensive increment at the onset of loading with a large difference, comparing the experimental curve. The effects of strain rate and inertia during dynamic free inversion process were further investigated by Colokoglu and Reddy (1996). However, the prediction process is very complicated and generally, theory and experiments have not been in acceptable agreements with each other. Accordingly, the predicted quasi-static inversion load is significantly lower than experimental value while the predicted dynamic mean load is overestimated. Chirwa (1993) investigated the plastic collapse of a tapered thin-walled metal Inver buckle energy absorbing tube subjected to axial impact. In this work, predicted specific energies and loads of collapse modes had been in good agreement with the experimental results. Recently, Masmoudi et al. (2016) carried out numerical and experimental analyses of external curling of thin-walled round tubes. Their model provides an accurate prediction on the forming kinematics and the deformed shape.
Reviewing the literature presents an effective way to increase the energy absorption of a structure, which is to harden the materials and simultaneously design the optimal structures. Up to now, the multi-objective optimization of functionally graded thickness (FGT) inversion tubes has never been investigated or presented elsewhere, even though optimal shapes of die and tube are very important in the crashworthiness of external inversion of tubes.
The aim of this study is to address the crashworthiness design issues of the various FGT inversion tubes under axial loading through the several quasi-static tests by following the multi-objective optimization procedure. A finite element (FE) model is developed to simulate the graded thickness in FGT tubes, and this FE model is validated through experimental tests. An integration of finite element modeling (FEM) with the response surface method (RSM) for design of experiments (DOE) was employed for generating the design guidelines for such inversion tube as energy absorbing devices. Finally, multi-objective optimization for different FGT are carried out, while specified energy absorption and peak force are selected as objective functions. Another focus was the relative merits of the optimized inversion tubes under dynamic loads, which has been raised by some pioneer researchers (Miscow & AL-Qureshi 1997) and not yet been disclosed. The results will demonstrate that the optimal FGT inversion tubes can be recommended as efficient energy absorbers. The primary outcome of this study is new design optimum information on the energy absorption performance of inversion tubes.
Methods
Objective functions of structural crashworthiness
where, \( {m}_{\mathrm{c}} \) is the total mass of the component and E is the energy absorbed. SEA is a key indicator to distinguish energy absorption capabilities for various structures and weights. The total strain energy absorbed by the structure during the deformation includes not only the elastic component but also the plastic strain energy.
On the other hand, the peak crushing force P _{max} is sometimes considered one of the critical design objectives to prevent the occupant’s body from severe biomechanical injury (Qiao et al. 2004). In the load-displacement curve, the peak crushing force P _{max} is defined as the maximum collapse load during the crushing strike. In this study, SEA and P _{max} are selected as the independent parameters in multi-objective optimization.
Finite element models of FGT inversion tubes
Here, all of the tubes have the same baseline dimensions with length L = 85 mm and outer diameter d = 50 mm. It is first assumed that the FGT tubes have a linearly graded thickness from t _{1} = 1 mm to t _{2} = 2 mm, and the UT tubes have an equivalent thickness of t = 2 mm. Also, the terms α and L _{α} in Fig. 1 are the angle and length of tube section with distributed thickness. Meanwhile, x is the distance from bottom to the height L _{α} and m is the order of thickness distribution, with m = 1 for linear variation of tube thickness.
The automatic single surface contact algorithm is used to account the contact force between the crush zone surfaces or metal folds. The node to surface algorithm models the contact between the rigid plate (die) and the crushed components. The friction coefficients at the different interfaces have been taken μ = 0.2. In all simulations, the die radius is r = 3 mm. To obtain the appropriate value of friction factor μ _{d} between die and tube interface, various values for μ _{d} were tested. It is seen that when μ _{d} = 0.02 was utilized, the load-displacement curves and deformed shapes of tubes are similar to the experimental tests. Therefore, the value of friction coefficient μ _{d} = 0.02 assumed for subsequent study.
Material properties
The tensile mechanical properties of aluminum alloy AA6061
Mechanical properties | Young’s modulus E | Yield strength σ_{0.2} | Ultimate tensile strength | Poison ratio ϑ | Failure strain % |
---|---|---|---|---|---|
Value | 68.4 GPa | 75.8 MPa | 130 MPa | 0.3 | 22 |
Validation of finite element models
Experimental and FE simulation results for various types of tubes
Tube case | P _{max} (kN) | P _{m} (kN) | E(kJ) | SEA(kJ/kg) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Exp. | FE | Diff. (%) | Exp. | FE | Diff. (%) | Exp. | FE | Diff. (%) | Exp. | FE | Diff. (%) | |
UT | 47.9 | 52.4 | 8.5 | 28.1 | 33.4 | 15.8 | 1.6 | 1.9 | 15.7 | 32.4 | 38.5 | 15.8 |
FGT (α=1°) | 47.4 | 53.9 | 12.6 | 39.3 | 42.9 | 7.2 | 2.1 | 2.5 | 16 | 43.9 | 48.6 | 9.6 |
FGT (α=2°) | 43.6 | 49.2 | 11.3 | 37.8 | 45.2 | 16.3 | 1.9 | 2.3 | 17.3 | 38.2 | 42.3 | 10.7 |
FGT (α=3°) | 28.4 | 33._{=}6 | 15.4 | 31.1 | 35.6 | 7.5 | 1.8 | 2.1 | 14.2 | 37.6 | 41.9 | 10.2 |
From experimental results represented in Fig. 9 and Table 2, it is seen that load-displacement curve in the complete inversion process for FGT tube with α = 1° is smoother than the UT tube with folding crush behavior, and meanwhile SEA = 43.9 and mean force P _{m} = 39.3 kN of FGT tube with α = 1° is meaningfully more than SEA = 32.4 P _{m} = 28.1 kN for UT tube with the same overall dimensions. Beside on, initial peak force in the FGT tubes is lower than in the UT tube. Therefore, using optimal FGT tubes guarantees full inversion process with better crush characteristics and behavior. The validated FE model will be further used for optimization designs in the following sections.
Multi-objective optimization design
Design methodology
Error evaluation of the surrogate model
Accuracy of different polynomial RSM models based on (a) CCD and (b) LHD
RS model | R ^{2} | R _{adj} ^{2} | RMSE | RE interval (%) |
---|---|---|---|---|
(a) | ||||
Quadratic polynomial | 0.9957 | 0.9942 | 0.0921 | [−2.18,2.24] |
Cubic polynomial | 0.9966 | 0.9956 | 0.0723 | [−1.03,2.31] |
Quartic polynomial | 0.9984 | 0.9977 | 0.0316 | [−0.78,1.17] |
(b) | ||||
Quadratic polynomial | 0.9990 | 0.9990 | 0.0893 | [−1.76,2.04] |
Cubic polynomial | 0.9992 | 0.9990 | 0.0714 | [−0.74,1.58] |
Quartic polynomial | 0.9999 | 0.9997 | 0.0292 | [−0.39,1.08] |
Optimization of die radius r and angle \( \alpha \) for linear thickness variation
Design optimization for coefficient of friction μ _{d} and exponent gradient \( m \)
Optimum friction \( {\mu}_d \) and exponent gradient m obtained through MDO for FGT inversion tube
Number of cycle | Design parameter interval | Optimal exponent gradient \( m \) | Optimal friction \( {\mu}_{\mathrm{d}} \) | SEA (kJ/kg) | P _{max} (kN) |
---|---|---|---|---|---|
1 | 3 < m < 4 0.01 < μ _{d} < 0.02 | 4 | 0.02 | 44.16 | 49.51 |
2 | 4 < m < 5 0.02 < μ _{d} < 0.03 | 5 | 0.029 | 48.2 | 43.29 |
3 | 5 < m < 6 0.03 < μ _{d} < 0.04 | 5.8 | 0.031 | 51.3 | 39.43 |
4 | 6 < m < 8 0.03 < μ _{d} < 0.035 | 7.3 | 0.03 | 54.6 | 35.18 |
5 | 7 < m < 8 0.03 < μ _{d} < 0.033 | 7.3 | 0.031 | 57.9 | 33.2 |
Design optimization for die radius r and thickness exponent gradient \( m \)
Optimization results of SEA for FGF filled conical tube with peak crushing force constraint
Type of optimization method | Optimal design variable | SEA (kJ/kg) | F _{max}(kN) |
---|---|---|---|
Peak load constraint | m = 7.33, r = 3.13 mm | 64.8 | 34.2 |
SEA constraint | m = 7.41, r = 3.18 mm | 58.2 | 31.7 |
Dynamic loading
Design optimization of DAF for coefficient of friction \( {\mu}_{\mathrm{d}} \) and exponent gradient m
Optimum die radius r and exponent gradient m obtained through MDO for FGT inversion tube
Number of cycle | Design parameter interval | Optimal exponent gradient \( m \) | Optimal die radius \( r\;\left(\mathrm{mm}\right) \) | DAF | P _{max} (kN) |
---|---|---|---|---|---|
1 | 3 < m < 4 0.01 < μ _{d} < 0.02 | 4 | 0.02 | 1.2 | 59.66 |
2 | 4 < m < 5 0.02 < μ _{d} < 0.03 | 4.8 | 0.028 | 1.3 | 57.34 |
3 | 5 < m < 6 0.03 < μ _{d} < 0.04 | 5.7 | 0.031 | 1.4 | 52.23 |
4 | 6 < m < 8 0.03 < μ _{d} < 0.035 | 6.3 | 0.031 | 1.5 | 50.42 |
5 | 6 < m < 7 0.03 < μ _{d} < 0.033 | 6.1 | 0.033 | 1.5 | 50.01 |
Design optimization of DAF for die radius r and exponent gradient m
The influence of die radius and exponent gradient m on dynamic response of FGT inversion tube is analyzed based on the approximate function derived by surrogate model. In order to find the best inversion tube configuration, the geometrical average method has been applied to maximize DAF(m, r) and minimize dynamic P _{max} (m, r) of the FGT tubes with coefficient of die radius r and exponent gradient m as design variable. Based on optimization study carried out in the “Design optimization of DAF for coefficient of friction \( \mu \) _{d} and exponent gradient m ” section, coefficient of friction μ _{ d } is selected 0.033.
Optimum die radius and gradient thickness obtained through the geometrical average method
Optimal values | F _{ g }(m, r) | d _{ DAF } | DAF | P _{ max } (kN) | |
---|---|---|---|---|---|
m = 6.4, μ _{d}=.034 | 0.7358 | 0.8451 | 0.8342 | 1.75 | 44.32 |
Conclusions
- (1)
Sensitivity analysis indicates that several design parameters (die radius r, coefficient of friction \( {\mu}_{\mathrm{d}} \), tube thickness variation pattern) have significant effect on SEA and peak force, which justify the selection of these parameters in multi-objective optimization.
- (2)
Friction between the die-tube interfaces plays a key role in the overall development of inversion process. In fact, a successful inversion mode of deformation easily switches into an unacceptable mode by simply changing this parameter. Optimization results show that the reduction of friction coefficient to less than 0.04 during inversion leads to the drastic improvement of crashworthiness performance.
- (3)
Experimental and numerical results present that the peak load in FGT inversion tubes obviously decreases, and simultaneously the load level increases steadily.
- (4)
The results achieved from multi-objective crashworthiness optimization show that using concave function for thickness distribution of tubes improves SEA without significant growth of initial peak force.
- (5)
The nonlinear thickness distribution exponent gradient m and die radius r are important parameters to control the DAF of FGT inversion response under dynamic loading. The optimum DAF of FGT inversion tube has different behavior and it increases at the dynamic deformation.
Briefly, this research introduces new designs of inversion tubes as energy absorber with superior characteristics with respect to UT tubes, under uniaxial loading. The multi-objective optimization method introduced here gives valuable information to develop and design of FGT inversion tubes in applications involving crush loading, such as automotive, aerospace, transportation, and defense industries.
Declarations
Authors’ contributions
Both authors read and approved the final manuscript.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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