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Investigation on the temperature dependent out-of-plane quasi-static compressive behavior of metallic honeycombs
Thin–Walled Structures 149 (2020) 106625
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Investigation on the temperature dependent out-of-plane quasi-static compressive behavior of metallic honeycombs Shubin Wang a, b, Weiguo Li a, b, *, Yong Tao a, **, Xin Zhang a, Xuyao Zhang a, Ying Li a, Yong Deng a, Zheng Shen a, Xi Zhang a, Jiasen Xu a, Liming Chen a, Daining Fang c a b c
College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing, 400044, China Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing, 100081, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Metallic honeycombs Temperature dependent Out-of-plane Plateau stress Energy absorption
This study investigates the effect of temperature on the out-of-plane quasi-static compressive behavior of metallic honeycombs by numerical and theoretical methods. Adopting a temperature dependent elastic-perfectly plastic constitutive relation, the temperature dependent out-of-plane quasi-static compressive mechanical performance of Al5052 honeycomb is studied with the finite element method. The results show that the plateau stress, peak stress and specific energy absorption (SEA) of Al5052 honeycomb decrease gradually with increasing temper ature, while the crush load efficiency (CLE) and deformation pattern are temperature insensitive. Then, based on the numerical results in this study and super folding element theory, the temperature dependent theoretical models without fitting parameters for the plateau stress and SEA of metallic honeycombs are developed, respectively. These theoretical models are successfully verified against numerical simulations and experiments. Based on these theoretical models, the effects of parent material properties on the plateau stress and SEA at different temperatures are discussed. The results in this study provide useful guidelines in optimizing the choice of parent materials to fabricate metallic honeycombs with better energy absorption efficiency at different temperatures.
1. Introduction Cellular materials have many excellent properties such as high spe cific stiffness and strength and high energy absorption efficiency [1–3]. As a kind of typical cellular materials, metallic honeycombs are increasingly used in fields of automotive, packaging, and aerospace, etc. To better understand their mechanical behavior, much effort has been made on out-of-plane compressive behavior of metallic honeycombs [4–6]. Taking the deformation mechanics of hexagonal honeycombs into account properly, Wierzbicki [5] developed a reasonable model for the plateau stress. Zhang and Ashby [6] established a model for the plateau stress of regular hexagonal honeycomb with single cell wall thickness. A systematic study on the deformation mechanisms and me chanical properties of honeycomb materials was conducted by Gibson and Ashby [4]. Afterwards, a lot of theoretical analyses [7–10], nu merical simulations [11–16] and experiments [17–23] have been per formed on various honeycombs under different loading conditions.
The above researches mainly focus on the performances of honey combs at room temperature. For engineering applications such as in aerospace and nuclear industries, it is very necessary to have a full un derstanding of the mechanical response of honeycombs at different temperatures, since their responses at different temperatures are significantly different. However, up to now, a few investigations have been carried out on the mechanical behavior of honeycombs at different temperatures. Lee et al. [24] performed compressive and shear tests to study the deformation behavior and failure mechanism of honeycomb composite at the temperature range of 25–300 � C. Erickson et al. [25] conducted impact tests on both foam-filled and unfilled honeycomb core sandwich panels at three temperatures. Their results showed that the effect of temperature on the energy absorption and maximum force of honeycomb was significant. Salehi-Khojin et al. [26] also studied the effect of temperature on honeycomb sandwich composites with Kev lar/hybrid and carbon facesheets under impact loading. Xie et al. [27] reported an investigation of honeycomb sandwich panels subjected to
* Corresponding author. College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China. ** Corresponding author. E-mail addresses: [email protected] (W. Li), [email protected] (Y. Tao). https://doi.org/10.1016/j.tws.2020.106625 Received 4 July 2019; Received in revised form 9 January 2020; Accepted 18 January 2020 Available online 8 February 2020 0263-8231/© 2020 Elsevier Ltd. All rights reserved.
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high-velocity impact at elevated temperatures. Wang et al. [28] per formed experiments to analyze the cushioning property of paper hon eycomb sandwich panels under different temperatures and relative humilities. By using experimental and numerical methods, Bai at al [29]. analyzed the temperature effect on the modal characteristics of com posite honeycomb structure. The above investigations [24–29] on the effect of temperature on mechanical properties of honeycombs were conducted by experimental and numerical methods. Although those studies provided many instructive insights into the design of honeycombs at different temper atures, investigations on the temperature dependent behavior of hon eycombs are still very limited. For example, it is still unclear how the temperature affects the energy absorption performance of metallic honeycombs. Besides, it is well known that the plateau stress and SEA are the key performance indicators for the evaluation and design of honeycombs in practical applications, but theoretical models for the temperature dependent plateau stress and SEA are not available at present. Therefore, more investigations are needed to be conducted to get a clearer understanding on the temperature dependent behavior of honeycombs. In this study, the effect of temperature on the out-of-plane quasistatic compressive behavior of metallic honeycombs is systematically studied through numerical modeling and theoretical research. The rest of this study is organized as follows. In Section 2, the temperature dependent numerical model for metallic honeycombs is established, and the effect of temperature on the mechanical behavior of aluminum honeycomb is analyzed. Theoretical models without fitting parameters for predicting the temperature dependent plateau stress and energy absorption are developed and verified in Section 3, and the effects of parent material properties on the plateau stress and energy absorption of metallic honeycombs at different temperatures are discussed. The main conclusions are summarized in Section 4.
Fig. 1. (a) A schematic of out-of-plane compression test on a hexagonal hon eycomb and (b) Y-shaped finite element model and its boundary conditions for honeycombs under out-of-plane compression.
8 > > < EðTÞε ;
σ¼
2. Numerical modeling 2.1. Finite element model
ε�
> > : σ y ðTÞ ;
σy ðTÞ EðTÞ
ε>
(1)
σy ðTÞ EðTÞ
where EðTÞ and σy ðTÞ are the Young’s modulus and yield strength of parent material at temperature T, respectively. Based on force-heat equivalence energy density principle proposed by Li et al. [33], the temperature dependent yield strength of metallic materials can be calculated by Ref. [34]:
To investigate the compressive behavior of metallic honeycombs at different temperatures, the ABAQUS/Explicit, an explicit finite element code, is used to simulate the out-of-plane quasi-static compression process. As shown in Fig. 1a, the x3 direction is usually defined as the out-of-plane direction and the other two directions (x1 and x2) are referred to in-plane directions for honeycombs [6,8,13,30]. In this study, a Y-shaped honeycomb finite element model (Fig. 1b) is employed for simulations. The parent material and geometric parameters of the honeycomb are adopted from the Hexcel data sheet [31], and the Al5052 honeycomb with cell wall thickness of 0.01778 mm and side length of 2.75 mm (specification: 3/16-5052-0.0007) is used. The cor responding relative density is 0.01. The initial height of honeycomb is 8.5 mm. The honeycomb is placed between two rigid plates, and four-node doubly curved shell elements (type S4R in ABAQUS) with reduced integration are adopted to model the cell walls of honeycomb. The honeycomb is attached to the bottom rigid plate via tie constraint, and all degrees of freedom of the bottom rigid plate are fixed. A constant velocity is applied to the top rigid plate to crush the honeycomb. Since a Y-shaped model is employed, symmetric boundary conditions in local coordinate system are applied on the non-intersecting edge of each honeycomb cell wall [11,32]. To avoid interpenetration during compression, surface-to-surface contact is used to simulate the contact between the honeycomb and two rigid plates, and general contact is adopted for all cell walls of honeycomb. The frictional coefficients defined in the surface-to-surface contact and general contact are 0.15 and 0, respectively. In this study, to include the effect of temperature, a temperature dependent elastic-perfectly plastic constitutive model is employed:
σ y ðTÞ ¼
" 1 þ μT0 EðTÞ 1 1 þ μT EðT0 Þ
RT T
CP ðTÞdT
T0
CP ðTÞdT
R T0m
!#0:5
σ y ðT0 Þ
(2)
where μT0 , EðT0 Þ and σy ðT0 Þ are the Poisson’s ratio, Young’s modulus and yield strength at reference temperature T0, respectively. μT and CP ðTÞ are the Poisson’s ratio and specific heat capacity at temperature T, respectively. And Tm is the melting point of parent material. In this study, T0 is set as 293 K (room temperature) because it is convenient to obtain the experimental data at this temperature. The parent material of honeycombs is aluminum alloy Al5052, whose material parameters are listed in Table 1 [35,36]. The Poisson’s ratio at room temperature is employed in simulations due to its weak temperature dependence [34]. In general, the loading velocity in actual quasi-static experiments is to slow for the quasi-static simulation of honeycombs, namely it can be very time-consuming when an actual quasi-static loading velocity is adopted. To accelerate the simulation, an increased loading velocity is usually used [12,15,18,37]. However, when an increased loading ve locity is used, the inertial effect is enlarged at the same time, which may break the quasi-static condition and lead to an inaccurate result. Ac cording to previous studies [37,38], when an artificially increased loading velocity is used, quasi-static simulation still can be achieved by satisfying two conditions. Firstly, the kinetic energy should be very small compared with the internal energy of the model. Secondly, the stress-strain response must be independent from the loading velocity 2
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Table 1 Material properties of aluminum alloy Al5052 [35,36]. Material parameter 3
Value or expression
Density: ρs (kg/m ) Young’s modulus at T0: EðT0 Þ (GPa)
2680 69.3
Yield strength at T0: σy ðT0 Þ (MPa)
265
Melting point: Tm (K) Young’s modulus: EðTÞ (GPa)
775 61 þ 0:07844T
Specific heat capacity: Cp ðTÞ (J/ (kg ⋅ K))
1001:78 10 4 T2
Poisson’s ratio at T0: uT0
0.33 0:0001713T2
0:03045T
7882620T
2
þ 3:23 �
when the velocity is increased to accelerate the simulation. To obtain a reasonable balance between the computational cost and accuracy of numerical simulation, the impact velocity of 4 m/s and an element size of 0.06 mm � 0.06 mm are found to be sufficient after a convergence study.
Fig. 3. Comparison between numerical and experimental results for Al5052 honeycombs (t ¼ 0.01778 mm, l ¼ 2.75 mm and ρ ¼ 0:01) at two different temperatures.
2.2. Validation of finite element model To validate the finite element model, the numerical results are compared with the experimental results of aluminum honeycombs re ported by Xu et al. [16,22]. The parent material of honeycomb in experiment is Al5052–H39, and the material properties adopted in the simulation are the same as those in references [16,22]. In the validation, the geometric parameters of finite element model are consistent with the actual honeycomb specimen (specification: 4.2-3/8-5052-0.003 N) [16, 22], whose cell wall thickness and side length are 0.08 mm and 5.4993 mm, respectively. The impact velocity is 5 m/s. As shown in Fig. 2, the numerical and experimental stress-strain curves have a reasonable agreement, and the corresponding plateau stresses are close to each other with a difference less than 10%. To further validate the finite element model, the numerical results are also compared with the experimental data [31] of Al5052 honeycombs at two different tem peratures. The material properties employed in the simulations are showed in Table 1. Numerical results of aluminum honeycombs at 243 K and 443 K are presented in Fig. 3, in which the experimental data [31] is also included for comparison. It can be seen that a good agreement between the plateau stresses obtained from simulations and experiments is achieved. Therefore, the numerical modeling method used in this
study is reliable and accurate, and it will be adopted to study the out-of-plane compressive behavior of metallic honeycombs at different temperatures. 2.3. Key performance indicators A typical compressive stress-strain curve for honeycombs under outof-plane compression can be divided into three stages, namely linearelastic stage, plateau stage with almost constant stress, and densifica tion stage with rapidly rising stress [4]. The relative density is an important parameter to determine the performance of honeycombs, and it is defined as the ratio of the density of honeycomb material to that of parent material:
ρ¼
ρ* ρs
(3)
where ρ* and ρs are the density of honeycomb material and parent material, respectively. Under out-of-plane compression, the stress and strain of honeycombs can be expressed as:
σ¼
F ; A0
ε¼
δ δ0
(4)
where F is the compressive force, δ is the displacement, and A0 and δ0 are the initial cross section area and initial height of honeycomb, respec tively. The constant stress of honeycombs during compression is usually called plateau stress, which can be calculated by Refs. [30,32]: Z εD 1 σm ¼ σ ðεÞ dε (5)
εD
0
where εD is the onset strain of densification, and σðεÞ is the compressive stress at compressive strain ε. CLE indicates the uniformity of loaddisplacement (or stress-strain) curve, meaning that the higher the CLE, the more efficient the honeycombs. And CLE of honeycomb material can be defined as [39,40]: CLE ¼
Fm σm � 100% ¼ � 100% Fp σp
(6)
where σp is the initial peak stress, and Fm and Fp are the average load and initial peak load, respectively. For energy absorbers, the higher the SEA, the better the energy absorption capability [41]. Here, SEA is defined as the amount of energy per unit mass, which can be defined as:
Fig. 2. Comparison of stress-strain curves between numerical and experimental [16,22] results for aluminium honeycomb (t ¼ 0.08 mm, l ¼ 5.4993 mm and ρ ¼ 0:0213) at room temperature. 3
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UM ðδÞ ¼
UðδÞ M
Thin-Walled Structures 149 (2020) 106625
(7)
where M is the mass of the whole honeycomb structure. UðδÞ is the en ergy absorbed of honeycomb material, and it can be expressed as the area under the load-displacement curve [42,43]: Z δ UðδÞ ¼ FðδÞ dδ (8) 0
where FðδÞ is the compressive load at compressive displacement δ. 2.4. Effect of temperature To study the effect of temperature on out-of-plane quasi-static compressive behavior of metallic honeycombs, simulations of Al5052 honeycombs at different temperatures are conducted. The temperature dependent Young’s modulus and temperature dependent yield strength of Al5052 are used in the finite element model, while the other material properties and geometric parameters are kept the same as those in Sections 2.1 and 2.2. Fig. 4a–c shows the numerical deformation patterns of Al5052 honeycombs at temperatures of 293 K, 473 K and 743 K, respectively. It can be seen that the deformation of Al5052 honeycombs at different temperatures are characterized by localized and progressive buckling. The results show that the deformation patterns of Al5052 honeycombs at different temperatures within the studied temperature range are similar with that at room temperature. This indicates that temperature has little effect on the deformation of Al5052 honeycomb. For comparison, the experimental deformation patterns of Al5052 honeycomb [30] and Al3003 honeycomb [8] at room temperature are also presented in Fig. 4d and e, respectively. It can be seen that the numerical deformation pattern at room temperature (293 K) corresponds well with the exper imental results. Fig. 5 shows the number of folds for Al5052 honeycombs at different temperatures. The fold number changes slightly with the increase of temperature. It also indicates that the deformation pattern of Al5052 honeycomb is temperature insensitive within the studied
Fig. 5. Number of folds for Al5052 honeycombs at different temperatures.
temperature range. Fig. 6a–d shows the variation of peak stress, plateau stress, CLE and SEA of Al5052 honeycomb with temperature, respectively. As can be seen, the peak stress, plateau stress and SEA of Al5052 honeycombs decrease with temperature increasing. CLE changes a little with increasing temperature, owing to the fact that the peak stress and plateau stress have similar change tendency with temperature (as shown in Fig. 6a and b). 3. Temperature dependent models of honeycomb 3.1. Theoretical model for plateau stress The energy method proposed by Wierzbicki et al. [5,44,45] is widely used to theoretically characterize the plateau stress of metallic honey combs under out-of-plane compression. According to the method,
Fig. 4. Typical final deformation patterns of aluminum honeycombs under out-of-plane quasi-static compression: (a)–(c) numerical deformation patterns of Al5052 honeycombs (t ¼ 0:01778 mm, l ¼ 2:75 mm) at three different temperatures, experimental deformation patterns of (d) Al5052 honeycomb (t ¼ 0:0762 mm, l ¼ 5:5 mm) [30] and (e) Al3003 honeycomb (t ¼ 0:06 mm, l ¼ 2 mm) [8]. 4
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Fig. 6. (a) Peak stress, (b) plateau stress, (c) CLE and (d) curves of SEA versus strain for Al5052 honeycombs at different temperatures.
honeycombs can be considered as a structure consisting of angle ele ments, which is usually divided into four parts during the folding pro cess, including trapezoidal rigid plane, cylindrical surface, conical surface and toroidal surface. The absorbed energy of an angle element in the folding process consists of the dissipated energy of the toroidal surface E1, a horizontal hinge line on the cylindrical surface E2, and an incline hinge line on the conical surface E3. The three dissipated energy can be calculated by Refs. [5,44,45]: E1 ¼ 16M0
H0 b0 I1 ðψ Þ t0
Fm ⋅ 2H0 ¼ N1 E1 þ N3 E3 þ N2 E2
H 20 I3 ðψ Þ b0
(9)
1 tan ψ
Z 0
π
2
(11)
cos ϕ dϕ sin γ
(15)
The above equations focus on the case at room temperature. Ac cording to the numerical results in Section 2.4 and the experimental data in those literatures [26,27,46], there is a great difference for the plateau stress and SEA of metallic honeycombs at different temperatures. Thus it is important to study the temperature dependence of the mechanical properties of metallic honeycombs. The numerical results also show that, in terms of the deformation pattern, there is no significant differ ence for honeycombs in the studied temperature range. Therefore, by incorporating the influence of temperature, Eq. (15) can be modified as:
where H0 is the half-wavelength of the folding mode, b0 is the radius of curvature of toroidal surface, L0 is the length of horizontal hinge line, and M0 ¼ σy t20 =4 is the fully plastic bending moment per unit length, I1 ðψ Þ and I3 ðψ Þ are the functions of geometric parameters of folding mode, which be calculated by Refs. [5,44,45]: � � �� Z π2 π π 2ψ I1 ðψ Þ ¼ cos ϕ cos ψ cos ψ þ β dϕ (12) π ðπ 2ψ Þtan ψ 0 I3 ðψ Þ ¼
(14)
Substitute Eqs. (9)‒(11) into Eq. (14), then: Fm b0 L0 H0 ¼ 8N1 I1 þ N2 π þ N3 I3 M0 t0 H0 b0
(10)
E2 ¼ 2πM0 L0 E3 ¼ 2M0
toroidal surface, horizontal hinge line and incline hinge line, respec tively. According to the energy balance, the total energy dissipated in forming a fold in a honeycomb unit cell should be equal to the work of external force:
Fm ðTÞ bðTÞ LðTÞ HðTÞ ¼ 8N 1 I1 þ N2 π þ N3 I3 MðTÞ tðTÞ HðTÞ bðTÞ
(16)
where MðTÞ ¼ σy ðTÞ½tðTÞ�2 =4 is the temperature dependent fully plastic bending moment per unit length. The temperature dependent geometric parameters are calculated by fðTÞ ¼ f⋅λðTÞ, in which λðTÞ ¼ 1 þ αðT T0 Þ, f ¼ ½b0 ; t0 ; L0 ; H0 �, and α denotes the thermal expansion co efficient. In order to minimize the average compression load Fm, the following condition must be satisfied:
(13)
where β ¼ arctanðtan ϕ =tan ψ Þ, γ ¼ arctanðtan ψ =sin ϕÞ, ψ , ϕ, β and γ are the angle parameters. Generally speaking, the number of plastic defor mation zones contained in a unit cell of honeycombs with different configurations is different. Here, let N1, N2 and N3 be the number of
∂Fm ðTÞ ¼ 0; ∂HðTÞ 5
∂Fm ðTÞ ¼0 ∂bðTÞ
(17)
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Thus we can obtain: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi λðTÞ 3 N2 2 π 2 3 t0 L20 ; HðTÞ ¼ 2 N1 N3 I1 I3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi λðTÞ 3 π N2 N3 I3 3 2 t0 L0 bðTÞ ¼ 2 2 4 N1 I1
by comparing with the experimental and numerical results. The nu merical results are obtained from simulations of Al5052 honeycombs with different relative densities at different temperatures. The numerical models are the same as those in Section 2, expect that the cell wall thickness is changed to obtain finite element models with different relative densities. In the theoretical models, the temperature depen dence of Poisson’s ratio is neglected, due to the fact that the Poisson’s ratio of Al5052 is insensitive to the change of temperature [31]. Fig. 7 shows the theoretical predictions of plateau stress of Al5052 honeycombs at different temperatures, compared with the experimental data provided by Hexcel Corporation [31]. It can be seen that the pre dicted plateau stresses are in good agreement with the experimental data. In addition to the experimental validation, the comparison be tween the theoretical predictions and numerical results for Al5052 honeycombs with different relative densities is also included in Fig. 7. It can be observed that temperature has a significant effect on the plateau stress, and the temperature dependence of plateau stress can be well characterized by the established theoretical model. To verify the tem perature dependent theoretical SEA model (Eq. (25)), Fig. 8 shows a comparison between the SEAs provided by the theoretical model and numerical simulations. It shows that there is a good agreement between the theoretical and numerical results for Al5052 honeycombs with different relative densities at different temperatures.
(18)
Substituting Eq. (18) into Eq. (16), we can get: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5=3 3 Fm ðTÞ ¼ ½λðTÞ�2 σy ðTÞ 3 πN1 N2 N3 I1 I3 t0 L1=3 2
(19)
For hexagonal honeycombs with double thickness walls [5,32]: ψ ¼ 2:39. By substituting these parameters into Eq. (19), the temperature dependent average crushing force can be expressed as:
π=6 , N1 ¼ 2, N2 ¼ 6 , N3 ¼ 4, I1 ¼ 1:05 and I3 ¼
(20)
5=3 1=3
Fm ðTÞ ¼ 8:61½λðTÞ�2 σ y ðTÞt0 L0
Dividing the average crushing force Fm ðTÞ by the cross-sectional area over which the force is applied, one can obtain the following expression:
σ m ðTÞ ¼
� �53 Fm ðTÞ t0 ¼ 6:63σy ðTÞ l0 SðTÞ
(21)
pffiffiffi where SðTÞ ¼ 3 3½lðTÞ�2 =4, and t0 and l0 are the cell wall thickness and side length of honeycombs at temperature T0, respectively. Substituting the expression of temperature dependent yield strength (Eq. (2)) into Eq. (21), the temperature dependent plateau stress can be obtained: (
σ m ðTÞ ¼ 6:63
"
1 þ μT0 EðTÞ 1 1 þ μT EðT0 Þ
RT T
Cp ðTÞdT
T0
Cp ðTÞdT
R T0m
#)12
� �53 t σy ðT0 Þ 0 l0
4. Results and discussion As can be seen from Figs. 7 and 8, with the increase of temperature, both SEA and plateau stress decrease gradually for Al5052 honeycombs with different relative densities. Moreover, SEA and plateau stress have the similar variation tendency with temperature. This is reasonable since the energy absorption is mainly determined by the plateau stress and onset strain of densification [32]. Fig. 9a and b respectively show the normalized plateau stress
(22)
In this formula, the specific heat capacity can easily be found in material handbook, the temperature dependent Young’s modulus and Poisson’s ratio can be easily obtained using impulse excitation method or vibrating reed method. Therefore, the plateau stress of honeycomb at different temperatures can be conveniently predicted by the theoretical model. According to the above formula, it can be found that the tem perature affects the plateau stress by changing the Poisson’s ratio, Young’s modulus and specific heat capacity of parent material.
σ m ¼ σm ðTÞ =σρm¼ρ0 ðTÞ and normalized SEA UM ¼ UM ðTÞ =UρM¼ρ0 ðTÞ of Al5052 honeycombs at different temperatures, where σ m
ρ¼ρ0
ðTÞ and
Um ðTÞ are the plateau stress and SEA of honeycombs with relative density ρ ¼ ρ0 , respectively. Here, ρ0 is a constant, and its value is 0.03. From Fig. 9, it is interesting to find that, for honeycombs with the same relative density, both the normalized plateau stress and normalized SEA fluctuate slightly with the increase of temperature according to the numerical results. This can be easily proved by the temperature ρ¼ρ0
3.2. Theoretical model for energy absorption According to the definition of SEA, the following expression can be obtained: R εD � σðεÞdε ⋅V U UM ¼ ¼ 0 (23) M M where V is the volume of the whole honeycomb structure. By employing Eq. (5), the SEA can be further expressed as: UM ¼
ðσ m ⋅εD Þ⋅V σ m ⋅εD ¼ * M ρ
(24)
pffiffiffi where ρ* ¼ 8t0 ρs =ð3 3 l0 Þ is the density of honeycomb material. Substituting Eq. (22) and the expression of ρ* into Eq. (24), the tem perature dependent SEA can be obtained as follows: UM ðTÞ ¼ 4:306
" � �23 ( 1 þ μT0 EðTÞ 1 1 þ μT EðT0 Þ
εD t0 ρs l0
RT T R T0m T0
Cp ðTÞdT Cp ðTÞdT
#)12
σy ðT0 Þ
(25)
From the above formula, it can be found that the temperature affects the SEA by changing the Poisson’s ratio, Young’s modulus and specific heat capacity of parent material. 3.3. Validation of theoretical models In this section, the temperature dependent plateau stress model (Eq. (22)) and the temperature dependent SEA model (Eq. (25)) are verified
Fig. 7. Comparison of theoretical plateau stress with experimental and nu merical results for Al5052 honeycombs at different temperatures. 6
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parent material properties on out-of-plane compressive performance of honeycombs at different temperatures, the plateau stress and SEA of Al5052 honeycomb and stainless steel 304L (SS304L) honeycomb are compared in Figs. 10 and 11. Both numerical and theoretical results are presented in the two figures, and the related material parameters of SS304L [47,48] are listed in Table 2. For comparison, two different cases are discussed. In the first case, the relative densities of two kinds of honeycombs are kept the same, which is achieved by keeping geometric dimensions with the same. The plateau stress and SEA in this case are shown in Fig. 10. It can be seen from Fig. 10a that the plateau stress of SS304L honeycombs is always larger than that of Al5052 honeycombs at different temperatures. SEA of SS304L honeycomb is smaller compared with that of Al5052 honeycomb when temperature is lower than the critical temperature, and the exact opposite result is obtained above the critical temperature (Fig. 10b). In the second case, the masses of two kinds of honeycombs are kept the same, which is realized by changing cell wall thicknesses and keeping side length with the same. The plateau stress and SEA in this case are shown in Fig. 11. As can be seen from this figure, when the temperature is lower than the critical temperature, the plateau stress and SEA of SS304L honeycomb are smaller compared with those of Al5052 honeycomb. On the contrary, when the temperature is higher than the critical temperature, the plateau stress and SEA of SS304L honeycomb are larger than those of Al5052 honeycomb. From the comparisons in these two cases, it can be found that, in terms of SEA, the Al5052 honeycomb is superior to SS304L honeycomb below the critical temperature. This is coincide with the quasi-static experimental results at room temperature reported by Baker et al. [49]. Interestingly, the
Fig. 8. Comparison of theoretical SEA with numerical results for Al5052 honeycombs at different temperatures.
dependent theoretical models (Eqs. (22) and (25)). The theoretical re sults, which are coincident with the numerical results, are also included in Fig. 9. It is well-known that the parent material properties can significantly affect the performance of honeycombs [4]. To explore the effect of
Fig. 9. (a) Normalized plateau stress and (b) normalized SEA for Al5052 honeycombs at different temperatures.
Fig. 10. Comparison of (a) plateau stress and (b) SEA between Al5052 honeycombs (t ¼ 0.01778 mm, l ¼ 2.75 mm and ρ ¼ 0:01) and SS304L honeycombs (t ¼ 0.01778 mm, l ¼ 2.75 mm and ρ ¼ 0:01) at different temperatures. 7
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Fig. 11. Comparison of (a) plateau stress and (b) SEA between Al5052 honeycombs (t ¼ 0.01778 mm, l ¼ 2.75 mm and ρ ¼ 0:01) and SS304L honeycombs (t ¼ 0.006 mm, l ¼ 2.75 mm and ρ ¼ 0:0034) at different temperatures (the masses of two kinds of honeycombs are kept the same).
This study will help to understand the temperature dependent behavior of metallic honeycombs under out-of-plane quasi-static compression. And the discussion regarding the effect of parent material properties on the performance of metallic honeycombs also provides useful guidelines for designing metallic honeycombs with better energy absorption efficiency at different temperatures.
Table 2 Material properties of stainless steel 304L (SS304L) [47,48]. Material parameter 3
Density: ρs (kg/m ) Young’s modulus at T0: EðT0 Þ (GPa)
Value or expression 7900 198.5
Yield strength at T0: σy ðT0 Þ (MPa)
485
Melting point: Tm (K) Young’s modulus: EðTÞ (GPa)
1753 232:131
Specific heat capacity: Cp ðTÞ (J/(kg ⋅ K))
438:3 þ 0:17T 10 6 T2
Poisson’s ratio at T0: uT0
Declaration of competing interest
0.29 0:1242T þ 0:00004527T2 1777226:43T
2
We declare that we have no conflicts of interest.
8:62 �
CRediT authorship contribution statement Shubin Wang: Methodology, Software, Validation, Data curation, Writing - original draft. Weiguo Li: Conceptualization, Supervision. Yong Tao: Conceptualization, Software, Writing - review & editing. Xin Zhang: Investigation, Writing - review & editing. Xuyao Zhang: Investigation, Data curation. Ying Li: Software. Yong Deng: Validation. Zheng Shen: Validation. Xi Zhang: Investigation. Jiasen Xu: Data curation. Liming Chen: Writing - review & editing. Daining Fang: Supervision.
SS304L honeycomb performs better than Al5052 honeycomb in terms of SEA above the critical temperature. These results can provide useful insights into choosing an appropriate parent material to fabricate metallic honeycombs with better energy absorption efficiency at different temperatures. 5. Conclusions In this study, the temperature dependent behavior of metallic hon eycombs under out-of-plane quasi-static compression is investigated numerically and theoretically. Adopting the temperature dependent elastic-perfectly plastic constitutive relation, a temperature dependence finite element model is developed to study the out-of-plane quasi-static compressive behavior of metallic honeycombs at different temperatures. The effect of temperature on the deformation pattern, mechanical properties and energy absorption of Al5052 honeycomb is studied by the finite element model. Then the temperature dependent theoretical models without fitting parameters to predict the plateau stress and SEA of metallic honeycombs are developed and verified by simulations and previous experiments. Moreover, the effect of parent material properties on the plateau stress and SEA of honeycombs at different temperatures are discussed. Main conclusions are given as follows:
Acknowledgements This work was supported by the National Natural Science Foundation of China (11727802, 11672050, 11602043), the Fundamental Research Funds for the Central Universities (2019CDQYHK016), the Postdoctoral Innovation Talents Support Program of Chongqing and the China Post doctoral Science Foundation (2018M633313). References [1] C. Ge, Q. Gao, L. Wang, Theoretical and numerical analysis of crashworthiness of elliptical thin-walled tube, Int. J. Mech. Sci. 148 (2018) 467–474. [2] Q. Gao, L. Wang, Y. Wang, F. Guo, Z. Zhang, Optimization of foam-filled double ellipse tubes under multiple loading cases, Adv. Eng. Software 99 (2016) 27–35. [3] N. Qiu, Y. Gao, J. Fang, Z. Feng, G. Sun, Q. Li, Theoretical prediction and optimization of multi-cell hexagonal tubes under axial crashing, Thin Wall. Struct. 102 (2016) 111–121. [4] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge university press, 1999. [5] T. Wierzbicki, Crushing analysis of metal honeycombs, Int. J. Impact Eng. 1 (1983) 157–174. [6] J. Zhang, M.F. Ashby, The out-of-plane properties of honeycombs, Int. J. Mech. Sci. 34 (1992) 475–489. [7] Z. Bai, H. Guo, B. Jiang, F. Zhu, L. Cao, A study on the mean crushing strength of hexagonal multi-cell thin-walled structures, Thin-Walled Struct. 80 (2014) 38–45. [8] Y. Tao, M. Chen, H. Chen, Y. Pei, D. Fang, Strain rate effect on the out-of-plane dynamic compressive behavior of metallic honeycombs: experiment and theory, Compos. Struct. 132 (2015) 644–651. [9] M. Fu, F. Liu, L. Hu, A novel category of 3D chiral material with negative Poisson’s ratio, Compos. Sci. Technol. 160 (2018) 111–118.
(1) The plateau stress, peak stress and SEA of Al5052 honeycomb decrease gradually with the increase of temperature; while the CLE and deformation mode are temperature insensitive. (2) The theoretical predictions, as well as numerical results in this study reveal that both the normalized plateau stress and normalized SEA are independent with temperature for Al5052 honeycombs with the same relative density. (3) At temperatures lower than the critical temperature, the Al5052 honeycomb is superior to SS304L honeycomb in terms of SEA, while SS304L honeycomb performs better at temperatures above critical temperature. 8
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Full length article
Investigation on the temperature dependent out-of-plane quasi-static compressive behavior of metallic honeycombs Shubin Wang a, b, Weiguo Li a, b, *, Yong Tao a, **, Xin Zhang a, Xuyao Zhang a, Ying Li a, Yong Deng a, Zheng Shen a, Xi Zhang a, Jiasen Xu a, Liming Chen a, Daining Fang c a b c
College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing, 400044, China Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing, 100081, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Metallic honeycombs Temperature dependent Out-of-plane Plateau stress Energy absorption
This study investigates the effect of temperature on the out-of-plane quasi-static compressive behavior of metallic honeycombs by numerical and theoretical methods. Adopting a temperature dependent elastic-perfectly plastic constitutive relation, the temperature dependent out-of-plane quasi-static compressive mechanical performance of Al5052 honeycomb is studied with the finite element method. The results show that the plateau stress, peak stress and specific energy absorption (SEA) of Al5052 honeycomb decrease gradually with increasing temper ature, while the crush load efficiency (CLE) and deformation pattern are temperature insensitive. Then, based on the numerical results in this study and super folding element theory, the temperature dependent theoretical models without fitting parameters for the plateau stress and SEA of metallic honeycombs are developed, respectively. These theoretical models are successfully verified against numerical simulations and experiments. Based on these theoretical models, the effects of parent material properties on the plateau stress and SEA at different temperatures are discussed. The results in this study provide useful guidelines in optimizing the choice of parent materials to fabricate metallic honeycombs with better energy absorption efficiency at different temperatures.
1. Introduction Cellular materials have many excellent properties such as high spe cific stiffness and strength and high energy absorption efficiency [1–3]. As a kind of typical cellular materials, metallic honeycombs are increasingly used in fields of automotive, packaging, and aerospace, etc. To better understand their mechanical behavior, much effort has been made on out-of-plane compressive behavior of metallic honeycombs [4–6]. Taking the deformation mechanics of hexagonal honeycombs into account properly, Wierzbicki [5] developed a reasonable model for the plateau stress. Zhang and Ashby [6] established a model for the plateau stress of regular hexagonal honeycomb with single cell wall thickness. A systematic study on the deformation mechanisms and me chanical properties of honeycomb materials was conducted by Gibson and Ashby [4]. Afterwards, a lot of theoretical analyses [7–10], nu merical simulations [11–16] and experiments [17–23] have been per formed on various honeycombs under different loading conditions.
The above researches mainly focus on the performances of honey combs at room temperature. For engineering applications such as in aerospace and nuclear industries, it is very necessary to have a full un derstanding of the mechanical response of honeycombs at different temperatures, since their responses at different temperatures are significantly different. However, up to now, a few investigations have been carried out on the mechanical behavior of honeycombs at different temperatures. Lee et al. [24] performed compressive and shear tests to study the deformation behavior and failure mechanism of honeycomb composite at the temperature range of 25–300 � C. Erickson et al. [25] conducted impact tests on both foam-filled and unfilled honeycomb core sandwich panels at three temperatures. Their results showed that the effect of temperature on the energy absorption and maximum force of honeycomb was significant. Salehi-Khojin et al. [26] also studied the effect of temperature on honeycomb sandwich composites with Kev lar/hybrid and carbon facesheets under impact loading. Xie et al. [27] reported an investigation of honeycomb sandwich panels subjected to
* Corresponding author. College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China. ** Corresponding author. E-mail addresses: [email protected] (W. Li), [email protected] (Y. Tao). https://doi.org/10.1016/j.tws.2020.106625 Received 4 July 2019; Received in revised form 9 January 2020; Accepted 18 January 2020 Available online 8 February 2020 0263-8231/© 2020 Elsevier Ltd. All rights reserved.
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high-velocity impact at elevated temperatures. Wang et al. [28] per formed experiments to analyze the cushioning property of paper hon eycomb sandwich panels under different temperatures and relative humilities. By using experimental and numerical methods, Bai at al [29]. analyzed the temperature effect on the modal characteristics of com posite honeycomb structure. The above investigations [24–29] on the effect of temperature on mechanical properties of honeycombs were conducted by experimental and numerical methods. Although those studies provided many instructive insights into the design of honeycombs at different temper atures, investigations on the temperature dependent behavior of hon eycombs are still very limited. For example, it is still unclear how the temperature affects the energy absorption performance of metallic honeycombs. Besides, it is well known that the plateau stress and SEA are the key performance indicators for the evaluation and design of honeycombs in practical applications, but theoretical models for the temperature dependent plateau stress and SEA are not available at present. Therefore, more investigations are needed to be conducted to get a clearer understanding on the temperature dependent behavior of honeycombs. In this study, the effect of temperature on the out-of-plane quasistatic compressive behavior of metallic honeycombs is systematically studied through numerical modeling and theoretical research. The rest of this study is organized as follows. In Section 2, the temperature dependent numerical model for metallic honeycombs is established, and the effect of temperature on the mechanical behavior of aluminum honeycomb is analyzed. Theoretical models without fitting parameters for predicting the temperature dependent plateau stress and energy absorption are developed and verified in Section 3, and the effects of parent material properties on the plateau stress and energy absorption of metallic honeycombs at different temperatures are discussed. The main conclusions are summarized in Section 4.
Fig. 1. (a) A schematic of out-of-plane compression test on a hexagonal hon eycomb and (b) Y-shaped finite element model and its boundary conditions for honeycombs under out-of-plane compression.
8 > > < EðTÞε ;
σ¼
2. Numerical modeling 2.1. Finite element model
ε�
> > : σ y ðTÞ ;
σy ðTÞ EðTÞ
ε>
(1)
σy ðTÞ EðTÞ
where EðTÞ and σy ðTÞ are the Young’s modulus and yield strength of parent material at temperature T, respectively. Based on force-heat equivalence energy density principle proposed by Li et al. [33], the temperature dependent yield strength of metallic materials can be calculated by Ref. [34]:
To investigate the compressive behavior of metallic honeycombs at different temperatures, the ABAQUS/Explicit, an explicit finite element code, is used to simulate the out-of-plane quasi-static compression process. As shown in Fig. 1a, the x3 direction is usually defined as the out-of-plane direction and the other two directions (x1 and x2) are referred to in-plane directions for honeycombs [6,8,13,30]. In this study, a Y-shaped honeycomb finite element model (Fig. 1b) is employed for simulations. The parent material and geometric parameters of the honeycomb are adopted from the Hexcel data sheet [31], and the Al5052 honeycomb with cell wall thickness of 0.01778 mm and side length of 2.75 mm (specification: 3/16-5052-0.0007) is used. The cor responding relative density is 0.01. The initial height of honeycomb is 8.5 mm. The honeycomb is placed between two rigid plates, and four-node doubly curved shell elements (type S4R in ABAQUS) with reduced integration are adopted to model the cell walls of honeycomb. The honeycomb is attached to the bottom rigid plate via tie constraint, and all degrees of freedom of the bottom rigid plate are fixed. A constant velocity is applied to the top rigid plate to crush the honeycomb. Since a Y-shaped model is employed, symmetric boundary conditions in local coordinate system are applied on the non-intersecting edge of each honeycomb cell wall [11,32]. To avoid interpenetration during compression, surface-to-surface contact is used to simulate the contact between the honeycomb and two rigid plates, and general contact is adopted for all cell walls of honeycomb. The frictional coefficients defined in the surface-to-surface contact and general contact are 0.15 and 0, respectively. In this study, to include the effect of temperature, a temperature dependent elastic-perfectly plastic constitutive model is employed:
σ y ðTÞ ¼
" 1 þ μT0 EðTÞ 1 1 þ μT EðT0 Þ
RT T
CP ðTÞdT
T0
CP ðTÞdT
R T0m
!#0:5
σ y ðT0 Þ
(2)
where μT0 , EðT0 Þ and σy ðT0 Þ are the Poisson’s ratio, Young’s modulus and yield strength at reference temperature T0, respectively. μT and CP ðTÞ are the Poisson’s ratio and specific heat capacity at temperature T, respectively. And Tm is the melting point of parent material. In this study, T0 is set as 293 K (room temperature) because it is convenient to obtain the experimental data at this temperature. The parent material of honeycombs is aluminum alloy Al5052, whose material parameters are listed in Table 1 [35,36]. The Poisson’s ratio at room temperature is employed in simulations due to its weak temperature dependence [34]. In general, the loading velocity in actual quasi-static experiments is to slow for the quasi-static simulation of honeycombs, namely it can be very time-consuming when an actual quasi-static loading velocity is adopted. To accelerate the simulation, an increased loading velocity is usually used [12,15,18,37]. However, when an increased loading ve locity is used, the inertial effect is enlarged at the same time, which may break the quasi-static condition and lead to an inaccurate result. Ac cording to previous studies [37,38], when an artificially increased loading velocity is used, quasi-static simulation still can be achieved by satisfying two conditions. Firstly, the kinetic energy should be very small compared with the internal energy of the model. Secondly, the stress-strain response must be independent from the loading velocity 2
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Table 1 Material properties of aluminum alloy Al5052 [35,36]. Material parameter 3
Value or expression
Density: ρs (kg/m ) Young’s modulus at T0: EðT0 Þ (GPa)
2680 69.3
Yield strength at T0: σy ðT0 Þ (MPa)
265
Melting point: Tm (K) Young’s modulus: EðTÞ (GPa)
775 61 þ 0:07844T
Specific heat capacity: Cp ðTÞ (J/ (kg ⋅ K))
1001:78 10 4 T2
Poisson’s ratio at T0: uT0
0.33 0:0001713T2
0:03045T
7882620T
2
þ 3:23 �
when the velocity is increased to accelerate the simulation. To obtain a reasonable balance between the computational cost and accuracy of numerical simulation, the impact velocity of 4 m/s and an element size of 0.06 mm � 0.06 mm are found to be sufficient after a convergence study.
Fig. 3. Comparison between numerical and experimental results for Al5052 honeycombs (t ¼ 0.01778 mm, l ¼ 2.75 mm and ρ ¼ 0:01) at two different temperatures.
2.2. Validation of finite element model To validate the finite element model, the numerical results are compared with the experimental results of aluminum honeycombs re ported by Xu et al. [16,22]. The parent material of honeycomb in experiment is Al5052–H39, and the material properties adopted in the simulation are the same as those in references [16,22]. In the validation, the geometric parameters of finite element model are consistent with the actual honeycomb specimen (specification: 4.2-3/8-5052-0.003 N) [16, 22], whose cell wall thickness and side length are 0.08 mm and 5.4993 mm, respectively. The impact velocity is 5 m/s. As shown in Fig. 2, the numerical and experimental stress-strain curves have a reasonable agreement, and the corresponding plateau stresses are close to each other with a difference less than 10%. To further validate the finite element model, the numerical results are also compared with the experimental data [31] of Al5052 honeycombs at two different tem peratures. The material properties employed in the simulations are showed in Table 1. Numerical results of aluminum honeycombs at 243 K and 443 K are presented in Fig. 3, in which the experimental data [31] is also included for comparison. It can be seen that a good agreement between the plateau stresses obtained from simulations and experiments is achieved. Therefore, the numerical modeling method used in this
study is reliable and accurate, and it will be adopted to study the out-of-plane compressive behavior of metallic honeycombs at different temperatures. 2.3. Key performance indicators A typical compressive stress-strain curve for honeycombs under outof-plane compression can be divided into three stages, namely linearelastic stage, plateau stage with almost constant stress, and densifica tion stage with rapidly rising stress [4]. The relative density is an important parameter to determine the performance of honeycombs, and it is defined as the ratio of the density of honeycomb material to that of parent material:
ρ¼
ρ* ρs
(3)
where ρ* and ρs are the density of honeycomb material and parent material, respectively. Under out-of-plane compression, the stress and strain of honeycombs can be expressed as:
σ¼
F ; A0
ε¼
δ δ0
(4)
where F is the compressive force, δ is the displacement, and A0 and δ0 are the initial cross section area and initial height of honeycomb, respec tively. The constant stress of honeycombs during compression is usually called plateau stress, which can be calculated by Refs. [30,32]: Z εD 1 σm ¼ σ ðεÞ dε (5)
εD
0
where εD is the onset strain of densification, and σðεÞ is the compressive stress at compressive strain ε. CLE indicates the uniformity of loaddisplacement (or stress-strain) curve, meaning that the higher the CLE, the more efficient the honeycombs. And CLE of honeycomb material can be defined as [39,40]: CLE ¼
Fm σm � 100% ¼ � 100% Fp σp
(6)
where σp is the initial peak stress, and Fm and Fp are the average load and initial peak load, respectively. For energy absorbers, the higher the SEA, the better the energy absorption capability [41]. Here, SEA is defined as the amount of energy per unit mass, which can be defined as:
Fig. 2. Comparison of stress-strain curves between numerical and experimental [16,22] results for aluminium honeycomb (t ¼ 0.08 mm, l ¼ 5.4993 mm and ρ ¼ 0:0213) at room temperature. 3
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UM ðδÞ ¼
UðδÞ M
Thin-Walled Structures 149 (2020) 106625
(7)
where M is the mass of the whole honeycomb structure. UðδÞ is the en ergy absorbed of honeycomb material, and it can be expressed as the area under the load-displacement curve [42,43]: Z δ UðδÞ ¼ FðδÞ dδ (8) 0
where FðδÞ is the compressive load at compressive displacement δ. 2.4. Effect of temperature To study the effect of temperature on out-of-plane quasi-static compressive behavior of metallic honeycombs, simulations of Al5052 honeycombs at different temperatures are conducted. The temperature dependent Young’s modulus and temperature dependent yield strength of Al5052 are used in the finite element model, while the other material properties and geometric parameters are kept the same as those in Sections 2.1 and 2.2. Fig. 4a–c shows the numerical deformation patterns of Al5052 honeycombs at temperatures of 293 K, 473 K and 743 K, respectively. It can be seen that the deformation of Al5052 honeycombs at different temperatures are characterized by localized and progressive buckling. The results show that the deformation patterns of Al5052 honeycombs at different temperatures within the studied temperature range are similar with that at room temperature. This indicates that temperature has little effect on the deformation of Al5052 honeycomb. For comparison, the experimental deformation patterns of Al5052 honeycomb [30] and Al3003 honeycomb [8] at room temperature are also presented in Fig. 4d and e, respectively. It can be seen that the numerical deformation pattern at room temperature (293 K) corresponds well with the exper imental results. Fig. 5 shows the number of folds for Al5052 honeycombs at different temperatures. The fold number changes slightly with the increase of temperature. It also indicates that the deformation pattern of Al5052 honeycomb is temperature insensitive within the studied
Fig. 5. Number of folds for Al5052 honeycombs at different temperatures.
temperature range. Fig. 6a–d shows the variation of peak stress, plateau stress, CLE and SEA of Al5052 honeycomb with temperature, respectively. As can be seen, the peak stress, plateau stress and SEA of Al5052 honeycombs decrease with temperature increasing. CLE changes a little with increasing temperature, owing to the fact that the peak stress and plateau stress have similar change tendency with temperature (as shown in Fig. 6a and b). 3. Temperature dependent models of honeycomb 3.1. Theoretical model for plateau stress The energy method proposed by Wierzbicki et al. [5,44,45] is widely used to theoretically characterize the plateau stress of metallic honey combs under out-of-plane compression. According to the method,
Fig. 4. Typical final deformation patterns of aluminum honeycombs under out-of-plane quasi-static compression: (a)–(c) numerical deformation patterns of Al5052 honeycombs (t ¼ 0:01778 mm, l ¼ 2:75 mm) at three different temperatures, experimental deformation patterns of (d) Al5052 honeycomb (t ¼ 0:0762 mm, l ¼ 5:5 mm) [30] and (e) Al3003 honeycomb (t ¼ 0:06 mm, l ¼ 2 mm) [8]. 4
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Thin-Walled Structures 149 (2020) 106625
Fig. 6. (a) Peak stress, (b) plateau stress, (c) CLE and (d) curves of SEA versus strain for Al5052 honeycombs at different temperatures.
honeycombs can be considered as a structure consisting of angle ele ments, which is usually divided into four parts during the folding pro cess, including trapezoidal rigid plane, cylindrical surface, conical surface and toroidal surface. The absorbed energy of an angle element in the folding process consists of the dissipated energy of the toroidal surface E1, a horizontal hinge line on the cylindrical surface E2, and an incline hinge line on the conical surface E3. The three dissipated energy can be calculated by Refs. [5,44,45]: E1 ¼ 16M0
H0 b0 I1 ðψ Þ t0
Fm ⋅ 2H0 ¼ N1 E1 þ N3 E3 þ N2 E2
H 20 I3 ðψ Þ b0
(9)
1 tan ψ
Z 0
π
2
(11)
cos ϕ dϕ sin γ
(15)
The above equations focus on the case at room temperature. Ac cording to the numerical results in Section 2.4 and the experimental data in those literatures [26,27,46], there is a great difference for the plateau stress and SEA of metallic honeycombs at different temperatures. Thus it is important to study the temperature dependence of the mechanical properties of metallic honeycombs. The numerical results also show that, in terms of the deformation pattern, there is no significant differ ence for honeycombs in the studied temperature range. Therefore, by incorporating the influence of temperature, Eq. (15) can be modified as:
where H0 is the half-wavelength of the folding mode, b0 is the radius of curvature of toroidal surface, L0 is the length of horizontal hinge line, and M0 ¼ σy t20 =4 is the fully plastic bending moment per unit length, I1 ðψ Þ and I3 ðψ Þ are the functions of geometric parameters of folding mode, which be calculated by Refs. [5,44,45]: � � �� Z π2 π π 2ψ I1 ðψ Þ ¼ cos ϕ cos ψ cos ψ þ β dϕ (12) π ðπ 2ψ Þtan ψ 0 I3 ðψ Þ ¼
(14)
Substitute Eqs. (9)‒(11) into Eq. (14), then: Fm b0 L0 H0 ¼ 8N1 I1 þ N2 π þ N3 I3 M0 t0 H0 b0
(10)
E2 ¼ 2πM0 L0 E3 ¼ 2M0
toroidal surface, horizontal hinge line and incline hinge line, respec tively. According to the energy balance, the total energy dissipated in forming a fold in a honeycomb unit cell should be equal to the work of external force:
Fm ðTÞ bðTÞ LðTÞ HðTÞ ¼ 8N 1 I1 þ N2 π þ N3 I3 MðTÞ tðTÞ HðTÞ bðTÞ
(16)
where MðTÞ ¼ σy ðTÞ½tðTÞ�2 =4 is the temperature dependent fully plastic bending moment per unit length. The temperature dependent geometric parameters are calculated by fðTÞ ¼ f⋅λðTÞ, in which λðTÞ ¼ 1 þ αðT T0 Þ, f ¼ ½b0 ; t0 ; L0 ; H0 �, and α denotes the thermal expansion co efficient. In order to minimize the average compression load Fm, the following condition must be satisfied:
(13)
where β ¼ arctanðtan ϕ =tan ψ Þ, γ ¼ arctanðtan ψ =sin ϕÞ, ψ , ϕ, β and γ are the angle parameters. Generally speaking, the number of plastic defor mation zones contained in a unit cell of honeycombs with different configurations is different. Here, let N1, N2 and N3 be the number of
∂Fm ðTÞ ¼ 0; ∂HðTÞ 5
∂Fm ðTÞ ¼0 ∂bðTÞ
(17)
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Thin-Walled Structures 149 (2020) 106625
Thus we can obtain: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi λðTÞ 3 N2 2 π 2 3 t0 L20 ; HðTÞ ¼ 2 N1 N3 I1 I3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi λðTÞ 3 π N2 N3 I3 3 2 t0 L0 bðTÞ ¼ 2 2 4 N1 I1
by comparing with the experimental and numerical results. The nu merical results are obtained from simulations of Al5052 honeycombs with different relative densities at different temperatures. The numerical models are the same as those in Section 2, expect that the cell wall thickness is changed to obtain finite element models with different relative densities. In the theoretical models, the temperature depen dence of Poisson’s ratio is neglected, due to the fact that the Poisson’s ratio of Al5052 is insensitive to the change of temperature [31]. Fig. 7 shows the theoretical predictions of plateau stress of Al5052 honeycombs at different temperatures, compared with the experimental data provided by Hexcel Corporation [31]. It can be seen that the pre dicted plateau stresses are in good agreement with the experimental data. In addition to the experimental validation, the comparison be tween the theoretical predictions and numerical results for Al5052 honeycombs with different relative densities is also included in Fig. 7. It can be observed that temperature has a significant effect on the plateau stress, and the temperature dependence of plateau stress can be well characterized by the established theoretical model. To verify the tem perature dependent theoretical SEA model (Eq. (25)), Fig. 8 shows a comparison between the SEAs provided by the theoretical model and numerical simulations. It shows that there is a good agreement between the theoretical and numerical results for Al5052 honeycombs with different relative densities at different temperatures.
(18)
Substituting Eq. (18) into Eq. (16), we can get: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5=3 3 Fm ðTÞ ¼ ½λðTÞ�2 σy ðTÞ 3 πN1 N2 N3 I1 I3 t0 L1=3 2
(19)
For hexagonal honeycombs with double thickness walls [5,32]: ψ ¼ 2:39. By substituting these parameters into Eq. (19), the temperature dependent average crushing force can be expressed as:
π=6 , N1 ¼ 2, N2 ¼ 6 , N3 ¼ 4, I1 ¼ 1:05 and I3 ¼
(20)
5=3 1=3
Fm ðTÞ ¼ 8:61½λðTÞ�2 σ y ðTÞt0 L0
Dividing the average crushing force Fm ðTÞ by the cross-sectional area over which the force is applied, one can obtain the following expression:
σ m ðTÞ ¼
� �53 Fm ðTÞ t0 ¼ 6:63σy ðTÞ l0 SðTÞ
(21)
pffiffiffi where SðTÞ ¼ 3 3½lðTÞ�2 =4, and t0 and l0 are the cell wall thickness and side length of honeycombs at temperature T0, respectively. Substituting the expression of temperature dependent yield strength (Eq. (2)) into Eq. (21), the temperature dependent plateau stress can be obtained: (
σ m ðTÞ ¼ 6:63
"
1 þ μT0 EðTÞ 1 1 þ μT EðT0 Þ
RT T
Cp ðTÞdT
T0
Cp ðTÞdT
R T0m
#)12
� �53 t σy ðT0 Þ 0 l0
4. Results and discussion As can be seen from Figs. 7 and 8, with the increase of temperature, both SEA and plateau stress decrease gradually for Al5052 honeycombs with different relative densities. Moreover, SEA and plateau stress have the similar variation tendency with temperature. This is reasonable since the energy absorption is mainly determined by the plateau stress and onset strain of densification [32]. Fig. 9a and b respectively show the normalized plateau stress
(22)
In this formula, the specific heat capacity can easily be found in material handbook, the temperature dependent Young’s modulus and Poisson’s ratio can be easily obtained using impulse excitation method or vibrating reed method. Therefore, the plateau stress of honeycomb at different temperatures can be conveniently predicted by the theoretical model. According to the above formula, it can be found that the tem perature affects the plateau stress by changing the Poisson’s ratio, Young’s modulus and specific heat capacity of parent material.
σ m ¼ σm ðTÞ =σρm¼ρ0 ðTÞ and normalized SEA UM ¼ UM ðTÞ =UρM¼ρ0 ðTÞ of Al5052 honeycombs at different temperatures, where σ m
ρ¼ρ0
ðTÞ and
Um ðTÞ are the plateau stress and SEA of honeycombs with relative density ρ ¼ ρ0 , respectively. Here, ρ0 is a constant, and its value is 0.03. From Fig. 9, it is interesting to find that, for honeycombs with the same relative density, both the normalized plateau stress and normalized SEA fluctuate slightly with the increase of temperature according to the numerical results. This can be easily proved by the temperature ρ¼ρ0
3.2. Theoretical model for energy absorption According to the definition of SEA, the following expression can be obtained: R εD � σðεÞdε ⋅V U UM ¼ ¼ 0 (23) M M where V is the volume of the whole honeycomb structure. By employing Eq. (5), the SEA can be further expressed as: UM ¼
ðσ m ⋅εD Þ⋅V σ m ⋅εD ¼ * M ρ
(24)
pffiffiffi where ρ* ¼ 8t0 ρs =ð3 3 l0 Þ is the density of honeycomb material. Substituting Eq. (22) and the expression of ρ* into Eq. (24), the tem perature dependent SEA can be obtained as follows: UM ðTÞ ¼ 4:306
" � �23 ( 1 þ μT0 EðTÞ 1 1 þ μT EðT0 Þ
εD t0 ρs l0
RT T R T0m T0
Cp ðTÞdT Cp ðTÞdT
#)12
σy ðT0 Þ
(25)
From the above formula, it can be found that the temperature affects the SEA by changing the Poisson’s ratio, Young’s modulus and specific heat capacity of parent material. 3.3. Validation of theoretical models In this section, the temperature dependent plateau stress model (Eq. (22)) and the temperature dependent SEA model (Eq. (25)) are verified
Fig. 7. Comparison of theoretical plateau stress with experimental and nu merical results for Al5052 honeycombs at different temperatures. 6
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Thin-Walled Structures 149 (2020) 106625
parent material properties on out-of-plane compressive performance of honeycombs at different temperatures, the plateau stress and SEA of Al5052 honeycomb and stainless steel 304L (SS304L) honeycomb are compared in Figs. 10 and 11. Both numerical and theoretical results are presented in the two figures, and the related material parameters of SS304L [47,48] are listed in Table 2. For comparison, two different cases are discussed. In the first case, the relative densities of two kinds of honeycombs are kept the same, which is achieved by keeping geometric dimensions with the same. The plateau stress and SEA in this case are shown in Fig. 10. It can be seen from Fig. 10a that the plateau stress of SS304L honeycombs is always larger than that of Al5052 honeycombs at different temperatures. SEA of SS304L honeycomb is smaller compared with that of Al5052 honeycomb when temperature is lower than the critical temperature, and the exact opposite result is obtained above the critical temperature (Fig. 10b). In the second case, the masses of two kinds of honeycombs are kept the same, which is realized by changing cell wall thicknesses and keeping side length with the same. The plateau stress and SEA in this case are shown in Fig. 11. As can be seen from this figure, when the temperature is lower than the critical temperature, the plateau stress and SEA of SS304L honeycomb are smaller compared with those of Al5052 honeycomb. On the contrary, when the temperature is higher than the critical temperature, the plateau stress and SEA of SS304L honeycomb are larger than those of Al5052 honeycomb. From the comparisons in these two cases, it can be found that, in terms of SEA, the Al5052 honeycomb is superior to SS304L honeycomb below the critical temperature. This is coincide with the quasi-static experimental results at room temperature reported by Baker et al. [49]. Interestingly, the
Fig. 8. Comparison of theoretical SEA with numerical results for Al5052 honeycombs at different temperatures.
dependent theoretical models (Eqs. (22) and (25)). The theoretical re sults, which are coincident with the numerical results, are also included in Fig. 9. It is well-known that the parent material properties can significantly affect the performance of honeycombs [4]. To explore the effect of
Fig. 9. (a) Normalized plateau stress and (b) normalized SEA for Al5052 honeycombs at different temperatures.
Fig. 10. Comparison of (a) plateau stress and (b) SEA between Al5052 honeycombs (t ¼ 0.01778 mm, l ¼ 2.75 mm and ρ ¼ 0:01) and SS304L honeycombs (t ¼ 0.01778 mm, l ¼ 2.75 mm and ρ ¼ 0:01) at different temperatures. 7
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Thin-Walled Structures 149 (2020) 106625
Fig. 11. Comparison of (a) plateau stress and (b) SEA between Al5052 honeycombs (t ¼ 0.01778 mm, l ¼ 2.75 mm and ρ ¼ 0:01) and SS304L honeycombs (t ¼ 0.006 mm, l ¼ 2.75 mm and ρ ¼ 0:0034) at different temperatures (the masses of two kinds of honeycombs are kept the same).
This study will help to understand the temperature dependent behavior of metallic honeycombs under out-of-plane quasi-static compression. And the discussion regarding the effect of parent material properties on the performance of metallic honeycombs also provides useful guidelines for designing metallic honeycombs with better energy absorption efficiency at different temperatures.
Table 2 Material properties of stainless steel 304L (SS304L) [47,48]. Material parameter 3
Density: ρs (kg/m ) Young’s modulus at T0: EðT0 Þ (GPa)
Value or expression 7900 198.5
Yield strength at T0: σy ðT0 Þ (MPa)
485
Melting point: Tm (K) Young’s modulus: EðTÞ (GPa)
1753 232:131
Specific heat capacity: Cp ðTÞ (J/(kg ⋅ K))
438:3 þ 0:17T 10 6 T2
Poisson’s ratio at T0: uT0
Declaration of competing interest
0.29 0:1242T þ 0:00004527T2 1777226:43T
2
We declare that we have no conflicts of interest.
8:62 �
CRediT authorship contribution statement Shubin Wang: Methodology, Software, Validation, Data curation, Writing - original draft. Weiguo Li: Conceptualization, Supervision. Yong Tao: Conceptualization, Software, Writing - review & editing. Xin Zhang: Investigation, Writing - review & editing. Xuyao Zhang: Investigation, Data curation. Ying Li: Software. Yong Deng: Validation. Zheng Shen: Validation. Xi Zhang: Investigation. Jiasen Xu: Data curation. Liming Chen: Writing - review & editing. Daining Fang: Supervision.
SS304L honeycomb performs better than Al5052 honeycomb in terms of SEA above the critical temperature. These results can provide useful insights into choosing an appropriate parent material to fabricate metallic honeycombs with better energy absorption efficiency at different temperatures. 5. Conclusions In this study, the temperature dependent behavior of metallic hon eycombs under out-of-plane quasi-static compression is investigated numerically and theoretically. Adopting the temperature dependent elastic-perfectly plastic constitutive relation, a temperature dependence finite element model is developed to study the out-of-plane quasi-static compressive behavior of metallic honeycombs at different temperatures. The effect of temperature on the deformation pattern, mechanical properties and energy absorption of Al5052 honeycomb is studied by the finite element model. Then the temperature dependent theoretical models without fitting parameters to predict the plateau stress and SEA of metallic honeycombs are developed and verified by simulations and previous experiments. Moreover, the effect of parent material properties on the plateau stress and SEA of honeycombs at different temperatures are discussed. Main conclusions are given as follows:
Acknowledgements This work was supported by the National Natural Science Foundation of China (11727802, 11672050, 11602043), the Fundamental Research Funds for the Central Universities (2019CDQYHK016), the Postdoctoral Innovation Talents Support Program of Chongqing and the China Post doctoral Science Foundation (2018M633313). References [1] C. Ge, Q. Gao, L. Wang, Theoretical and numerical analysis of crashworthiness of elliptical thin-walled tube, Int. J. Mech. Sci. 148 (2018) 467–474. [2] Q. Gao, L. Wang, Y. Wang, F. Guo, Z. Zhang, Optimization of foam-filled double ellipse tubes under multiple loading cases, Adv. Eng. Software 99 (2016) 27–35. [3] N. Qiu, Y. Gao, J. Fang, Z. Feng, G. Sun, Q. Li, Theoretical prediction and optimization of multi-cell hexagonal tubes under axial crashing, Thin Wall. Struct. 102 (2016) 111–121. [4] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Cambridge university press, 1999. [5] T. Wierzbicki, Crushing analysis of metal honeycombs, Int. J. Impact Eng. 1 (1983) 157–174. [6] J. Zhang, M.F. Ashby, The out-of-plane properties of honeycombs, Int. J. Mech. Sci. 34 (1992) 475–489. [7] Z. Bai, H. Guo, B. Jiang, F. Zhu, L. Cao, A study on the mean crushing strength of hexagonal multi-cell thin-walled structures, Thin-Walled Struct. 80 (2014) 38–45. [8] Y. Tao, M. Chen, H. Chen, Y. Pei, D. Fang, Strain rate effect on the out-of-plane dynamic compressive behavior of metallic honeycombs: experiment and theory, Compos. Struct. 132 (2015) 644–651. [9] M. Fu, F. Liu, L. Hu, A novel category of 3D chiral material with negative Poisson’s ratio, Compos. Sci. Technol. 160 (2018) 111–118.
(1) The plateau stress, peak stress and SEA of Al5052 honeycomb decrease gradually with the increase of temperature; while the CLE and deformation mode are temperature insensitive. (2) The theoretical predictions, as well as numerical results in this study reveal that both the normalized plateau stress and normalized SEA are independent with temperature for Al5052 honeycombs with the same relative density. (3) At temperatures lower than the critical temperature, the Al5052 honeycomb is superior to SS304L honeycomb in terms of SEA, while SS304L honeycomb performs better at temperatures above critical temperature. 8
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