Optimal design of functionally graded foam material under impact loading

International Journal of Mechanical Sciences 68 (2013) 199–211

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Optimal design of functionally graded foam material under impact loading Xiong Zhang a,c, Hui Zhang b,n a b c

Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430073, Hubei, PR China Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Luoyu Road 1037, Wuhan 430074, China

a r t i c l e i n f o

abstract

Article history: Received 12 August 2012 Received in revised form 9 December 2012 Accepted 17 January 2013 Available online 29 January 2013

With appropriate design, functionally graded metallic foam can show definitely better properties than homogeneous counterpart due to its better designability. In the present work, functionally graded aluminum foam blocks subjected to ball impact are investigated numerically by using nonlinear finite element code. Blocks with different density gradient distributions, various geometric parameters and under different impact velocities are analyzed. The block with linear decreasing density gradient is found to possess excellent performance in energy absorption and outperform blocks with other density distributions under middle to high speed impact. To obtain the optimal design of the functionally graded foam block, a structural optimization problem with the objective of maximizing the crush force efficiency is solved by response surface method (RSM). The thickness and density of each layer are selected as design variables and it is interesting to find that the optimum design shows gradually decreasing density distribution. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Functionally graded material Aluminum foam Energy absorption Impact loads Response surface method

1. Introduction Cellular materials including honeycombs and foams were widely applied in various engineering fields and among these applications, the employment of them as structural materials of dampers or energy dissipation devices [1,2] for the traffic vehicles was one of the most focused. Although the energy absorption properties of foam materials have been addressed in numerous literatures [3–10], the energy absorption properties of functionally graded foam materials (FGFMs) attract the attention of researchers most recently [11–15]. From the viewpoint of a designer, the introduction of gradient distribution in a material offers us a bigger design domain and therefore provides a better designability. Consequently, it is certain that the graded material will outperform (or equally perform) its uniform counterpart. For FGFMs, this superior designability was not fully realized and utilized to increase the passive safety of structures or occupants under impact loads. This is partly due to the lag in fabrication technology of FGFMs. The concept of functionally graded materials (FGMs) was originated by a group of scientists in Japan [16] in 1984. However, in fact, the cellular materials encountered in nature are often functionally graded such as the bone of a human and the stalk of a plant. Functionally graded foam material is a type of bulk FGMs

n

Corresponding author. Tel.: þ86 27 87543538; fax: þ 86 27 87543501. E-mail address: [email protected] (H. Zhang).

0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.01.016

and recently significant advances have been made in syntactic foam technology to enable the fabrication of spatial grading of properties of the foam under laboratory conditions [17,18] by changing the properties of micro-balloons. Compared with polymeric foams, aluminum foams have higher strength and lower temperature sensitivity and therefore offer better performance in terms of energy absorption. However, functionally graded metal (like aluminum) foams are much more difficult to fabricate than syntactic foams and no reports are available on the processing method of them in the open literature. Just like general FGMs, there are two basic approaches to model FGFMs, one is the quasi-homogeneous multi-layer method with a piecewise variation of the density of foam and the other one is a continuous variation of the density of foam. The latter may be considered as a limiting case of the former and cannot be easily employed in a numerical model. Quasi-homogeneous multi-layer method will be employed in this paper to analyze the dynamic response of FGFM blocks under ball impact. In fact, the FGFMs with multi-layers are very attractive in terms of energy absorption and passive safety protection. As we know, dissipating the impact kinetic energy in a controlled manner is desirable in the design of energy-absorbing structures. In order to achieve this, researchers adopted various design and optimization methods including topology optimization [19] to obtain a desired energy absorption history which means a desired reaction force path in the force–displacement curve. This can, however, be easily realized by using FGFMs. Since the plateau stress of foam is proportional to a power function of the relative density, desired

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models applicable to aluminum foams were validated by Reyes et al. [26,27] and Hanssen et al. [28]. One of the most widely applied material models for foam was developed by Deshpande and Fleck [29]. It is a phenomenological isotropic constitutive model and has been implemented in LS-DYNA with material model 154 [30]. The aluminum foam produced by Hydro Aluminum AS was experimentally tested by Hanssen et al. [28] and the material parameters of the constitutive model were calibrated based on the test results. According to this model, the yield surface of the foam material can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi se 2 þ a2 sm 2 Y ¼ 0 1 þ ða=3Þ2

Fig. 1. Quasi-static stress–strain curves of FGFMs with different layers.

where Y is the yield strength, se and sm are the effective von Mises stress and the mean stress, respectively. The parameter a controls the shape of the yield surface and it is a function of the plastic Poisson0 s ratio np given as

a2 ¼ reaction force response can be obtained by adjusting the relative density and the thickness of each layer. Representative stress– strain curves of FGFMs with three and six layers are shown in Fig. 1. It is expected that in the limiting case, the stress–strain curve of FGFM will be approximately a power function curve. Under impact loads, highly nonlinear behaviors such as material nonlinearities, large deformation and contacts are always included during the deformation of structures and make the sensitivity-based optimization technology hard to apply in the design optimization of relevant structures. Surrogate optimization method is currently the best choice for structural optimization problems in crashworthiness design. Surrogate models can be established by different methods including RSM, Kriging, artificial neural networks etc. Several researches have been carried out to compare the efficiency of these methods on crashworthiness optimization problems [20–22]. Classical polynomial-based response surface method was found to be one of the most robust methods and was widely adopted by researchers [23–25]. In the present work, the design optimization of FGFM blocks under ball impact will be conducted based on the polynomial response surface method. The thickness and density of each layer will be defined as design variables and the crush force efficiency of the impact process will be the objective to be maximized. In this paper, the dynamic behavior of functionally graded aluminum foam blocks and sandwich structures with FGFM cores under ball impact is firstly studied by using the nonlinear finite element code LS-DYNA. The influence of gradient distribution of FGFMs, geometric parameters and impact velocity on the dynamic response is investigated. Secondly, the formulation of relevant optimization problem of the FGFM block is presented and the surrogate model by RSM is established and solved. The paper is organized as follows. In Section 2, the constitutive model and relate mechanical parameters of aluminum foam material are presented. Section 3 gives a full introduction of the finite element model and a validation of the finite element model is conducted in Section 4. The relevant results of the parametric studies are then presented in Section 5 and the structural optimization problem of FGFM block is discussed in Section 6. Finally, Section 7 summarizes the present work.

ð1Þ

9 ð12np Þ 2 ð1 þ np Þ

ð2Þ

The material model adopts the following strain hardening rule: Y ¼ sp þ g

e^ 1 þ a2 ln eD 1ðe^ =eD Þb

! ð3Þ

where e^ is the equivalent plastic strain, sp , a2 , g, eD , and b are material parameters and can be expressed as functions of the foam density: 

1



sp , a2 , g, ,E ¼ C 0 þ C 1 b



rf rs

n ð4Þ

Young0 s modulus is also found to correlate with the foam density by power-law relation [28] as Eq. (4) and the densification Table 1 Material parameters of constitutive model [18]. Parameters

sp (MPa)

a2 (MPa)

1=b

g (MPa)

E (MPa)

C0 C1 n

0 590 2.21

0 140 0.45

0.22 320 4.66

0 40 1.4

0 0.33e6 2.45

2. Constitutive model of aluminum foam Many constitutive models have been established to simulate the behavior of cellular metal foams and several constitutive

Fig. 2. Compressive stress–strain curves of aluminum foams with different densities.

X. Zhang, H. Zhang / International Journal of Mechanical Sciences 68 (2013) 199–211

strain eD is related to the density of foam by the equation   rf 9 þ a2 eD ¼  ln rs 3a2

ρ2

3. Finite element model To study the relative merits of FGFM under impact loads, the dynamic behavior of functionally graded aluminum foam blocks and sandwich structures with FGFM cores under ball impact is investigated. As shown in Fig. 3, FGFM blocks with the section of 200  200 mm2 and thickness t ranging from 6 mm to 48 mm are impacted by a rigid ball striker with the radius of 20 mm. The FGFM block is supported in the bottom by a fixed rigid block. The cross sectional area of the FGFM blocks is very much larger than that of the impact ball to simulate approximately the impact of a ball on an infinite block. The direction of the initial velocity of the impact ball coincides with the thickness direction of the blocks and the density distribution is also varied along this direction. The linear density distributions are primarily considered in this paper for the FGFM blocks, while other gradient distributions including triangular and inverse triangular distributions are also analyzed. An illustration of various gradient distributions of density versus normalized distance from top (incident) to bottom (distal) surface is shown in Fig. 4. The average density of the foam material is set to be 0.3 g/cm3 and r1 and r2 are set to be 0.12 g/cm3 and 0.48 g/cm3, respectively. The impact ball and the fixed rigid block are assumed as steel with Young0 s modulus, Poisson0 s ratio and mass density to be 210 GPa, 0.3 and 7.8 g/cm3, respectively. The impact ball is modeled by elastic material model and the fixed rigid block is modeled by rigid material model. As mentioned in the previous section, the aluminum foam materials will be modeled with material model 154 in LS-DYNA and all the mechanical parameters will be completely determined by the density of the foam. Automatic surface to surface contact is applied to simulate the

80 mm Velocity

Rigid block

Z

24 (mm)

Y

X

200 mm 200 mm

Fig. 3. Finite element model for impact test with FGFM.

Linear decreasing Linear increasing Uniform Triangular Inverse triangular

ð5Þ

It can be found that all the material parameters are functions of the foam density and the calibrated values were given by Reyes et al. [26] and Hanssen et al. [28]. In the present work, the material parameters employed are listed in Table 1 and the plastic Poisson0 s ratio is set to 0 (a ¼2.12) for all foam densities unless otherwise noted. Fracture is also considered and the approach is given in detail in Section 4. Now, the material properties of aluminum foam can be completely determined by one parameter, that is, the foam density. Therefore, the FGFM with multi-layers can be simulated by specifying the density of each layer. The representative compressive stress–strain curves of aluminum foams with different densities are plotted in Fig. 2.

FGFM

201

(ρ1+ρ2)/2

ρ1

0

0.5

1.0

Normalized distance Fig. 4. Distribution of density versus normalized distance from top to bottom surface.

interaction between the rigid ball and FGFM block and the contact between FGFM block and the fixed supporting block. A coulomb friction coefficient of 0.1 between all surfaces in contact is employed. All components are modeled by eight-node solid element and the characteristic size of element mesh is about 2 mm for rigid ball and rigid block. As for the FGFM block, a uniform mesh size of 1 mm is employed in the impact area (80  80 mm2 in the middle) and a gradually-transitional mesh generation with the mesh size varied from 1 mm in the center to about 5 mm in the ends. A total of six layers of aluminum foam with different density are found to be enough to simulate the FGFM block with linear density gradient and 12 layers are adopted for bilinear density gradient.

4. Validation of FE model The fracture of the foam is considered in the present model and the influence of foam fracture on the dynamic response of FGFM blocks is analyzed here. The influence of friction on the dynamic response of the blocks was also studied for various friction coefficients. In addition, mesh sensitivity study is carried out to check the accuracy of the simulations in this section. 4.1. Influence of foam fracture The mechanical behavior of aluminum foams is quite different in tension than in compression. They are quite brittle in tension and easily to fracture, which increases the difficulty for researchers to exactly simulate the behavior of them. In FEM analyses, fracture is generally modeled by removing (eroding) elements when the fracture criterion is satisfied and sometimes this also causes problems as will be mentioned further on. The fracture of aluminum foam has been comprehensively studied by Reyes et al. [26,27] and two fracture criterions based on plastic volumetric strain and principal stress, respectively, were proposed and validated by them. The former will be adopted here and the elements will fail when plastic volumetric strain reaches a critical value. As employed previously by Reyes et al. [26,27], a critical value of 2% is used in the present work. The FGFM blocks with three representative density gradients including linear decreasing density (LDD), linear increasing density (LID) and uniform density (UD) distributions are analyzed first. The thickness t of the blocks is 24 mm and the initial velocity of the impact ball is V¼35 m/s. It is found that the fracture behavior of foam in the blocks is closely associated with the density distribution of the block. As denoted by the blue dashed circles in Fig. 5, foam

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fracture concentrates on the middle and bottom region of the LDD block, while foam fracture only occurs in the top region of LID and UD blocks. In the bottom region of LDD block, some elements are highly distorted due to the elimination of the failed elements which are subjected to tension stress due to bending of the layer above a softer layer. An adjustment is suggested and adopted in the present model to avoid the distortion of the elements and even to get a better simulation. As shown in Fig. 6a, a truncated cone region (trapezoidal area in section view) just below the impact ball is set as reserve zone where the critical volume strain for foam fracture is set to 10%. The reason for the establishment of this zone is twofold. Firstly, the remove of fractured elements in this zone is not quite reasonable since the foam material in this region can still resist the compression and absorb certain amount of energy even if it is fractured. Secondly, the tensile volumetric strain in this region is small and therefore the influence of this assumption is negligible. As given in Fig. 6b and c, when 10% critical strain is set, no element fails and no distortion occurs in this zone. The top radius r of the reserve zone is set to be a small value of 8 mm and the angle a is set to about 721 with a tangent value of 3. The impact force response curves of foam blocks with and without reserve zone are shown in Fig. 7a. It is found that there is only a small difference between the curves. The dimension of the reserve zone should not be chosen randomly. It should be smaller than the influence zone of the deformation which is the zone below the fractured elements just as the truncated cone region in Fig. 6c. The influence of critical volumetric strain for foam fracture on the response of FGFM blocks is also analyzed for the three different density gradients. The foams are set to fracture at critical strain of 1%, 2% and no fracture. The force response curves are given in Fig. 7b–d for LDD, LID and UD block, respectively. The critical strain is found to have negligible influence for LID and UD

L DD

block. This is reasonable since the number of elements that fracture in these two blocks is much less than that of LDD block as shown in Fig. 5. For LDD block, the critical fracture strain has bigger influence on the response. Comparing to the case with no fracture, the curves of blocks with 1% and 2% critical strain have two peak values. The forces firstly increase to a certain value in the initial stage, and then drop slightly due to the fracture of elements, and after that they increase to a value higher than that of the block without fracture, and finally drop to 0. 4.2. Influence of friction on the interfaces The influence of the friction on the reaction force responses of the FGFM blocks is investigated here with three different friction conditions by changing the coefficient of friction m to be 0.1, 0.3 and 0.5. The reaction force–time curves of FGFM with t¼24 mm and V¼35 m/s are shown in Fig. 8. It can be found that the friction coefficient has only small influence on the reaction force response. It seems that the influence of friction on the response of LDD block is slightly higher than that of LID and UD block. This could be associated with the foam failure during loading since LDD block has more foam failure regions. The friction on the interface will affect the stress and deformation of the blocks and therefore the fracture of the foam materials. In a word, it can be concluded that the interface friction has negligible influence on the force response of the structure and the friction coefficient of 0.1 between all surfaces in contact is employed in the present work. 4.3. Mesh sensitivity analysis Mesh sensitivity analysis is carried out to check the accuracy of the present model. Different mesh sizes in the impact area both

L ID

UD

Fig. 5. Fracture of foam in FGFM block with different density gradients.

2r

α Reserve zone Fig. 6. (a) Illustration of reserve zone, (b) fracture of LDD block with reserve zone, and (c) general view of the fracture.

X. Zhang, H. Zhang / International Journal of Mechanical Sciences 68 (2013) 199–211

203

Fig. 7. Influence of reserve zone and critical fracture strain on the response of FGFM block: (a) LDD block with and without reserve zone, (b) LDD block, (c) LID block and (d) UD block.

Fig. 8. The influence of interface friction on force response of FGFM blocks (t ¼24 mm,V¼ 35 m/s).

in-plane (XY) and in the through-thickness (Z) direction are adopted to calculate the case with thickness t ¼24 mm and initial velocity of the impact ball V¼ 35 m/s. It is known that much finer mesh is required to get a converged solution when the fracture of foam (erosion of elements) is present in the model. However, the occurrence of fracture in the present model is associated with the density distribution of FGFM blocks and therefore the mesh sensitivity should be checked for different density gradients. The FGFM blocks with three linear density gradients in previous subsection are analyzed here. The mesh cases are marked as a mm  b mm, where a is the through-thickness mesh size and b is the in-plane mesh size in the impact area. As shown in Fig. 9a, for UD and LID blocks, an in-plane mesh size of 2 mm is enough to get a converged solution while this is not the case for LDD block. The reason for this is associated with the deformation behavior of

the LDD block and will be discussed in the next section. Three mesh sizes are analyzed and 1 mm in-plane mesh size is found to achieve simulation results with enough accuracy. Meanwhile, Fig. 9b shows that a through-thickness mesh size of 1 mm is appropriate to obtain good results. Consequently, the mesh size of 1 mm  1 mm is selected in the present model and 345,600 elements are generated for blocks with thickness of 24 mm.

5. Parametric study During vehicle accidents or other collision events, one of the most important indices concerned with passenger safety or object integrity is the peak force or acceleration (deceleration) which is directly associated with the force and then damage caused by the

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Fig. 9. Mesh sensitivity study: (a) different transverse meshes and (b) different thickness meshes.

Fig. 10. Force–time curves of FGFM blocks under ball impact with different velocity (t¼ 24 mm).

impact. The objects being protected may be the strikers or the objects being struck, for instance, the head of a helmeted rider falling during a racing accident or a bridge pier with protective devices being hit by a ship or rockfall. Therefore, both the peak force between the striker and the FGFM block and that between FGFM block and the supporting rigid block should be considered to provide more comprehensive information. Besides peak force, another two indices used to evaluate the performance of energy absorbers are considered here, one is the maximum stroke distance ds which is the maximum distance the impact ball can move along the thickness direction of the blocks after the contact with the incident face of them and the other is the crush force efficiency IU which is defined as the ratio of the mean force Fm to the peak (maximum) force Fp over the whole stroke distance [31]. The index IU should be as close to 100% as possible in practice and it is 100% for an ideal energy absorber. The FGFM blocks with linear density gradients are analyzed first. Under ball impact with different velocities, the striking forces at the incident face of FGFM blocks with the thickness of 24 mm are plotted versus time in Fig. 10 and are given versus displacement (stroke distance) in Fig. 11. In relatively low speed

V¼10 m/s, the LDD block shows the highest peak force, followed by UD block and finally the LID block. With the striking speed increasing, the peak force of all blocks increases. However, the peak force of LID blocks increases with the highest speed, followed by that of UD blocks and finally that of LDD blocks. When V ¼15 m/s, the peak force of LID block exceeds that of UD block while the peak force of LDD block is still the highest. At V¼20 m/s, the peak forces of LID and UD block exceed that of LDD block and at higher speed, the relative magnitude of the peak forces of the three FGFM blocks is unchanged while the differences of them become pretty large. Judging from the peak force, the LDD blocks are superior to UD and LID blocks under middle to high speed impact. The time duration of the impact also shows some interesting characteristics for the three FGFM blocks within the speed range investigated. With the increasing of impact velocity, the time duration increases for LDD blocks, while it decreases for LID blocks and UD blocks. It could be expected that the time duration will also decrease for LDD blocks with the further increasing of striking velocity since it is natural to have shorter response time for higher loading rate. The peak forces, the average forces and the

X. Zhang, H. Zhang / International Journal of Mechanical Sciences 68 (2013) 199–211

205

Fig. 11. Force–displacement curves of FGFM blocks under ball impact with different velocity (t ¼ 24 mm).

Table 2 Crush force efficiency and maximum stroke distance of FGFM blocks. V (m/s)

Blocks

Fm (kN)

Fp (kN)

IU

10

LDD LID UD

3.03 1.34 2.10

5.47 3.65 4.20

0.55 0.37 0.50

4.17 9.68 6.12

15

LDD LID UD

4.34 2.28 3.21

7.15 6.53 6.35

0.61 0.35 0.50

6.51 12.78 9.04

20

LDD LID UD

5.29 3.35 4.32

7.45 9.92 8.44

0.71 0.34 0.51

9.81 15.43 11.96

25

LDD LID UD

5.97 4.55 5.39

8.42 13.85 10.32

0.71 0.33 0.52

13.55 17.75 14.82

30

LDD LID UD

6.92 5.87 6.52

11.70 19.20 12.70

0.59 0.31 0.51

16.81 19.84 17.87

35

LDD LID UD

8.03 7.37 7.77

16.77 31.64 19.66

0.48 0.23 0.40

19.77 21.51 20.44

ds (mm)

crush force efficiency factors are listed in Table 2. In all cases, the crush force efficiency of LDD blocks is the highest and that of LID blocks is the lowest. As for the stroke distance ds, the striking rigid ball shows the shortest stroke distance for LDD blocks and the longest one for LID blocks as plotted in Fig. 11 and listed in Table 2. Consequently, judging by the above two indices of energy absorbers, the LDD block is the best one in the three blocks and the LID block is the worst one. This also indicates that the introduction of FGFM will not necessarily improve the crashworthiness performance of cushioning structures. The influence of block thickness on the response of the FGFM blocks is also investigated while the radius of the impact rigid ball is kept constant. The blocks with thickness of 6, 12, 36 and 48 mm are simulated for different foam density gradients and different impact

velocities. The results show that similar dynamic responses as the blocks with thickness of 24 mm are observed except for very small thickness t¼6 mm. The force–time curves of FGFM blocks with different thickness under relatively high speed impact are shown in Fig. 12. For larger thicknesses (t¼12, 36 and 48 mm), the dynamic responses are similar as the blocks with t¼ 24 mm. The major difference is that the definition of low, middle or high initial velocity (or kinetic energy) of the impact ball is varied due to the variation of the thickness. However, for t¼6 mm, the peak force of UD block is lower than that of LDD block, which is different from the case with t¼24 mm. Analyses with a lower impact velocity V¼5 m/s show that the relative magnitude of the peak forces is the same as the case with t¼24 mm. The different behavior of blocks with t¼6 mm may be caused by the size effect of the structure and will be discussed somewhere else. Although the thickness may have certain influence on the response of the blocks, it can still be concluded that the LDD blocks gain great advantages over other FGFM blocks in energy absorption under middle to high speed impact. The force response of the FGFM blocks under impact is closely associated with the deformation behavior of them. The deformation processes of FGFM blocks with linear density gradient and under relatively high speed impact are shown in Fig. 13. The thickness t of the blocks and the initial velocity V of the rigid ball are selected to be 24 mm and 35 m/s, respectively. For LDD block, the deformation is firstly localized in the region below the rigid ball and then concentrates in the area around the bottom surface until the densification of the foams. Two oblique crack zones are observed in the block below the impact ball. While for LID and UD block, the deformation is always localized in the region around the rigid ball and the foam materials coming into this region are simply densified with the moving of ball from top to the bottom. The only difference of the two blocks is that the extent of densification is somewhat different due to the density difference of aluminum foam. From the deformation process, it can be found that there are two reasons for the better performance of LDD block in energy absorption. Firstly, the influence zone of deformation is much larger and more foam material participates in the deformation process and secondly, more high density foam

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Fig. 12. Force–time curves of FGFM blocks with different thickness under relatively high speed ball impact.

material is deformed in LDD block since it is located in the top region of the block. With the increase of block thickness, the deformation process is similar except that more foam material in the top region of the block is fractured. The deformation processes of FGFM blocks with thickness of 48 mm and under the impact velocity of 70 m/s are shown in Fig. 14. It can be noticed that the influence zone of LDD block is much larger than that of LDD block with t¼24 mm. The influence zone is still approximately a truncated cone (with a trapezoidal section) and the foam material highly compressed in the bottom is significantly increased comparing to the case t ¼24 mm. For LID and UD block, the fracture is still present only on the top region and no fracture occurs in the bottom no matter whether the reserve zone is set or not. The above mentioned peak forces or force–time curves in Fig. 10 are only concerned with the striking force at the incident face of FGFM blocks. What about the interaction force between FGFM block and the supporting rigid block? A comparison of the force–time curves on the top and bottom surface of FGFM blocks is conducted in Fig. 15. It can be found that the force responses on the top and bottom surface of LID and UD block are very similar and only small difference exists. However, the LDD block shows a significant difference in the force responses on the two surfaces. The difference on the two surfaces is apparently associated with the propagation of the stress wave in the blocks and it is expected to related to the fracture of the foam materials. No effort is exerted here to deal with it. For LID and UD block, the force on the top surface is initially larger than that on the bottom surface while this situation is reversed in approximately the latter half time of the impact.

The bilinear density gradient distributions including triangular density (TD) and inverse triangular density (ITD) are also analyzed here. Both the mass and the material constitutions of TD and ITD block are completely same as LDD and LID block. Therefore the comparison between them can show more definitely the influence of material distributions without the disturbance caused by the nonlinear relationship between mechanical properties and density of constitutive materials. The impact force–time curves of FGFM blocks with linear and bilinear density gradient are shown in Fig. 16 with the initial velocity of the rigid ball to be 35 m/s. It is quite interesting that the response curves of blocks with bilinear gradient seem to be a blend of the curves of LDD and LID block. For TD block, the response is similar to that of LID block in the initial stage, and then the feature of the curve is shifted towards that of LDD block. A similar process is observed for ITD block whose response is similar to that of LDD block in the beginning and then approaches that of LID block. Judging from the curves in Fig. 16, it seems that the response of the block with bilinear density gradient is primarily determined by the density distribution in the bottom half thickness of the block. In addition, it is found that the peak force of TD and ITD block is slightly higher than that of LDD and LID block, respectively. Foam materials are always used as core of sandwich structures or filler of thin-walled structures. In the present work, the dynamic response of sandwich structure with FGFM core under ball impact is also analyzed with the same model as Fig. 3 except for an introduction of face sheet with the thickness of 1.5 mm. The face sheet is assumed to be perfectly attached to the foam core and the material of the face sheet is aluminum alloy AA6060

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207

Fig. 13. Deformation process of FGFM blocks with linear density gradient (t¼ 24 mm, V ¼ 35 m/s): (a) LDD block, (b) LID block and (c) UD block.

T4 whose mechanical properties were given in literature [32]. The force–time curves of sandwich structures with t¼ 24 mm and linear density gradient are given in Fig. 17 for V¼30, 40 and 50 m/s. Numerical analyses show that the characteristics of the striking force response of the sandwich structures are generally consistent with those of FGFM blocks. That is, foam core with LDD gradient is still found to be superior to the core with other density gradient distributions under middle to high speed impact. It can also be observed that the advantage of sandwich with LDD core is increased with the increase of impact velocity. It can be concluded that the face sheet does not have significant influence on the energy absorption of the foam core except that the initial velocity of the impact ball must be increased to achieve the same extent of deformation of the foam core due to the resistance of the face sheet.

6. Optimization of FGFM block 6.1. Formulation of the optimization problem The purpose of this section is to obtain the optimal structural design and optimum material distribution of the FGFM block to resist the ball impact with a relatively high velocity. According to the analysis in the previous section, the block with LDD gradient outperforms other blocks at middle to high speed impact. Is it the optimal design of the structure? In the above section, the foam

density of each layer is variable while the thickness of each layer is invariant. What is the optimal design if the thickness of each layer is also variable? In the design optimization of this section, the thickness and foam density of each layer are selected as design variables. Since the number of sampling points (and hence the time of finite element analyses) is increased significantly with the increase of the number of design variables for RSM optimization, a total of three layers are adopted here for the FGFM block and an illustration of the block is given in Fig. 18. As denoted in the figure, the thickness of each layer t1, t2, t3 and the foam density of each layer d1, d2, d3 are design variables. In the design process, the total mass and the whole thickness of the block are kept constant and the optimal design of the structure will reflect the best distribution of the foam material in the block. The total mass and the whole thickness of the block are not selected randomly. The total mass is set to be the mass of block with uniform foam at an average density of 0.3 g/cm3 and the whole thickness is set to 24 mm to guarantee that the whole kinetic energy KE of the ball is able to be absorbed by the FGFM block when subject to an impact velocity of 30 m/s. The objective of the optimization problem is to maximize the crush force efficiency during the ball impact. Since the whole kinetic energy will basically be absorbed and be converted to internal energy E (includes elastic strain energy and work done in permanent deformation) of foam material during impact, the mean force Fm is inversely proportional to the stroke distance ds. Consequently, the objective can be calculated as E/(dsFp).

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0.2 ms

0.4 ms

0.6ms

0.8 ms

Fig. 14. Deformation process of FGFM blocks with linear density gradient (t ¼48 mm, V ¼70 m/s): (a) LDD block, (b) LID block and (c) UD block.

Fig. 15. Force–time curves on the top and bottom surface of FGFM blocks with linear density gradient (V ¼35 m/s): (a) LDD block and (b) LID and UD block.

The optimization problem can be formulated as follows: Maximize IU ¼ E=ðds F p Þ

t 1 d1 þ t 2 d2 þt 3 d3 ¼ 0:3 g=cm3  24 mm

ð6Þ

6.2. Experimental design and RSM solution

subjected to : 4 mmr t 1 ,t 2 ,t 3 r 10 mm 0:2 g=cm3 r d1 ,d2 ,d3 r0:4 g=cm3 t 1 þ t 2 þ t 3 ¼ 24 mm

The RSM is a kind of statistical optimization method originated from the model fitting of experimental data. The first step for RSM is to assume the form of the approximation or basis function.

X. Zhang, H. Zhang / International Journal of Mechanical Sciences 68 (2013) 199–211

Then, enough number of design points in the design space must be selected for numerical analyses by experimental design techniques. The approximation function is then fitted to the analysis results at the selected design points using the least-squares method and finally the constructed approximate design surrogate problem is solved and the optimal design is obtained. In this paper, a full set of quadratic polynomial functions is selected as basis function. Since the whole thickness and total mass of the block are constrained to be constant, only t1, t2, d1 and d2 are determined as variables while t3 and d3 are calculated according to the constraint equations of thickness and mass. The basic functions are therefore to be 1, t1, t2, d1, d2, t21, t1t2, t1d1, t1d2, t22, t2d1, t2d2, d21, d1d2, d22. The optimization problem now has the following form:

209

is a cylindrical zone with radius r ¼10 mm. The relevant computational results Fp, ds, E and the objective value IU are presented in Table 3. By fitting these data, the response surface function of IU

V=30m/s

t3 t2 t1

MaximizeIU ¼ f ðt 1 ,t 2 ,d1 ,d2 Þ subjectedto : 4 mm rt 1 ,t 2 r10 mm

d3

0:2 g=cm3 rd1 ,d2 r 0:4 g=cm3 4 mm r24t 1 þ t 2 r 10 mm

d2

3

3

0:2 g=cm r ð7:2t 1 d1 þ t 2 d2 Þ=ð24t 1 þ t 2 Þ r 0:4 g=cm

d1

ð7Þ Many different types of experimental design are available for RSM sampling and central composite design is employed here. The number of experimental points is 25 for central composite design with four design variables. The design points are listed in Table 3 and numerical simulations are carried out on these points. The finite element model presented in Section 3 is employed here and the reserve zone to avoid the highly distortion of element in the bottom is also adopted. For simplicity, the top radius r of the reserve zone is set to be 10 mm and the angle a is set to 901, that

Fig. 16. Force–time curves of linear and bilinear gradient blocks (V ¼ 35 m/s).

Fig. 18. Design variables of FGFM block with three layers.

Table 3 Design points and relevant numerical results on these points. Point

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0

t1 (mm)

t2 (mm)

d1 (g/cm3)

d2 (g/cm3)

Fp (kN)

ds

7.0 8.5 5.5 8.5 5.5 8.5 5.5 8.5 5.5 8.5 5.5 8.5 5.5 8.5 5.5 8.5 5.5 10.0 7.0 7.0 7.0 4.0 7.0 7.0 7.0

7.0 8.5 8.5 5.5 5.5 8.5 8.5 5.5 5.5 8.5 8.5 5.5 5.5 8.5 8.5 5.5 5.5 7.0 10.0 7.0 7.0 7.0 4.0 7.0 7.0

0.30 0.35 0.35 0.35 0.35 0.25 0.25 0.25 0.25 0.35 0.35 0.35 0.35 0.25 0.25 0.25 0.25 0.30 0.30 0.40 0.30 0.30 0.30 0.20 0.30

0.30 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.30 0.30 0.30 0.40 0.30 0.30 0.30 0.20

12.71 18.72 15.76 15.77 15.12 12.98 13.61 12.51 13.12 13.83 12.87 14.16 13.30 11.28 10.49 10.49 11.64 12.71 12.71 16.91 15.13 12.71 12.71 10.30 10.76

17.89 18.89 18.92 18.92 18.82 17.09 17.69 17.11 17.60 18.56 17.94 18.52 18.10 18.42 17.28 17.28 16.64 17.89 17.89 19.38 18.57 17.89 17.89 16.58 18.89

Fig. 17. Force–time curves for sandwich structure with FGFM core (t¼ 24 mm).

E (J)

IU

111.79 112.09 111.92 111.91 111.68 112.26 111.96 112.25 111.86 112.03 111.93 111.76 111.81 108.83 112.81 112.81 112.46 111.79 111.79 111.57 112.26 111.79 111.79 112.86 110.80

0.492 0.317 0.375 0.375 0.392 0.506 0.465 0.524 0.484 0.436 0.485 0.426 0.465 0.524 0.622 0.622 0.581 0.492 0.492 0.340 0.399 0.492 0.492 0.661 0.545

(mm)

210

X. Zhang, H. Zhang / International Journal of Mechanical Sciences 68 (2013) 199–211

Fig. 19. Comparison of uniform design and optimal design: (a) force–time curves and (b) displacement–time curves.

can be determined as

6 mm Optimal design

IU ¼ f ðt 1 ,t 2 ,d1 ,d2 Þ ¼ 0:50741 þ 0:057023t 1 þ 0:056901t 2 0:72352d1

4 mm

þ0:40001d2 0:0010404t 1 2 0:0053t 1 t 2 0:15533t 1 d1 þ0:12467t 1 d2 0:0010363t 2 2 þ 0:041514t 2 d1 2

0:072069t 2 d2 0:039925d1 þ0:43158d1 d2 2:90325d2

2

ð8Þ Substitute Eq. (8) into Eq.(7), the optimization problem can now be easily solved and the optimum point is found to be (10, 4, 0.2, 0.3). The other two variables t3 and d3 are 10 mm and 0.4 g/cm3, respectively, according to the constraint equations of mass and thickness. Consequently, the optimal design of the FGFM block is the one with t1 ¼10 mm, t2 ¼4 mm, t3 ¼10 mm and d1 ¼0.2 g/cm3, d2 ¼0.3 g/cm3, d3 ¼0.4 g/cm3. Based on the approximate function Eq. (8), a predicted value of objective can also be given and it is IU ¼0.721. A verification analysis of the optimal design is conducted and at the same time, the block with uniform foam distribution (t1 ¼t2 ¼t3 ¼8 mm, d1 ¼d2 ¼d3 ¼0.3 g/cm3) is also analyzed to compare with the optimal design. The force–time and displacement– time curves of impact ball for the optimal design and uniform foam design are shown in Fig. 19. The crush force efficiency of the optimal design is IU ¼0.685 while the value is 0.492 for uniform foam block. Apparently, the crush force efficiency is increased by about 40% through the structural optimization. The computed value of IU is slightly lower than the predicted value by about 5% at the optimal design point. This can be overcome by adopting sequential RSM which decreases the error by decreasing the region of interest. Since qualitative analyses are conducted here, no effort is exerted to get higher accuracy. The optimal design shows a gradually decreasing density from top to the bottom. The foam density is 0.4 g/cm3, 0.3 g/cm3 and 0.2 g/cm3 for the top, middle and bottom layer, respectively. This is consistent with the parametric study in the previous section that has shown the relative merits of LDD block under relatively high speed impact. In addition, for the optimal design, the thickness of the top and bottom layer is equal to the upper bound 10 mm while the thickness of the middle layer is found to be the lower bound 4 mm. As shown in Fig. 20, the design can also be considered as a structure with two layers in the two ends with the density of 0.2 and 0.4 g/cm3, respectively, and a LDD gradient layer in the middle with the thickness of 12 mm. This shows that the application of FGFMs is quite promising to improve the crashworthiness performance of sandwich structures and structures with foam filler.

LDD block

0.4 g/cm3

4 mm 4 mm

0.4 g/cm3 0.3 g/cm3

6 mm

0.2 g/cm3 0.2 g/cm3 Fig. 20. Optimal design of FGFM block.

7. Conclusion Dynamic responses of FGFM blocks subjected to ball impact are parametrically studied with different density gradient distributions, various geometric parameters and under different impact velocities. The LDD blocks are found to be superior to FGFM blocks with other density gradient distributions under middle to high speed impact. When applied as core of sandwich structure, the foam core with LDD gradient also outperforms other alternatives. The introduction of FGFM will not necessarily improve the crashworthiness performance of cushioning structures. Appropriate structural design is indispensable. By RSM optimization, an optimal design of FGFM block under ball impact is found with the objective of maximizing the crush force efficiency. The optimum design shows a gradient density distribution and gains significant advantage over uniform foam design. There are still many problems to be further studied on the buffering characteristics and energy absorption of functionally graded foam materials. The application and design of FGFM as filler of thin-walled structures may be an important area of future research.

Acknowledgments The present work was supported by National Natural Science Foundation of China (no. 11002060) and the Fundamental Research Funds for the Central Universities, HUST (no. 2012QN025). References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Cambridge: Cambridge University Press; 1997.

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