Available online at www.sciencedirect.com
Procedia Engineering 31 (2012) 967 – 972
International Conference on Advances in Computational Modeling and Simulation
Comparisons of equivalent and detailed models of metallic honeycomb core structures with in-plane thermal conductivities Donghuan Liua*, Lei Jinb, Xinchun Shanga a
Department of Applied Mechanics, University of Science & Technology Beijing, Beijing 100083, China b China Aero-Polytechnology Establishment, Beijing 100028, China
Abstract Equivalent laminate model of the metallic honeycomb structure is given in the present paper firstly. The effective thermal model of the honeycomb core is anisotropic, and both in-plane and thickness direction effective thermal conductivity are deduced using the Swann-Pittman model. The effective mechanical properties of the honeycomb core are determined using the mechanics of materials method, and the effective continuum properties are then used with classical laminate theory to construct an equivalent laminate plate to simulate the response of three-dimensional honeycomb core structure. Then the thermal and free vibration behaviors of the equivalent laminate plate are compared with that of detailed model of the honeycomb core plate using finite element method. Numerical results show that the equivalent model is in good agreement with the detailed model in heat transfer and modal analysis.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology Open access under CC BY-NC-ND license. Keywords: honeycomb core structure; equivalent model; in-plane thermal conductivity
1. Introduction Metallic thermal protection structures (MTPS) are a key technology that may help achieve the goal of * Corresponding author. Tel.: +86-010-62332985. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.1128
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reducing the cost of space access for reusable launch vehicles (RLV) [1]. A typical MTPS consists of the radiative surface panel, the insulation package and the support structure. The radiative surface panel is always made of lightweight superalloy honeycomb. Finite element method is used to predict the thermal and thermomechanical responses of the honeycomb panel under aerodynamic heating and pressure. The accuracy of response quantities predicted by the three-dimensional computational model is high, but the computational effort associated with it increases very rapidly with the increase in the number of the cells in the panel core. So it is very necessary to development equivalent models of the honeycomb structures to decrease the size and complexity of the computational model, while maintaining an acceptable level of accuracy, which is very beneficial for support of design trade studies in the conceptual design phase for MTPS [2]. There are many published papers devoted to this topic [3-7], but few researchers pay attention to the effect of the equivalent in-plane thermal conductivities of the honeycomb core. The present paper firstly give the equivalent in-plane thermal conductivities of a honeycomb cell with the consideration of the heat conduction across the air inside the honeycomb core and neglecting of the radiation heat transfer, as well as the equivalent density and capacity, and the honeycomb core sandwich panel is modelled as a three-layer laminate plate with an equivalent orthotropic layer. Then a detailed three-dimensional model of the sandwich panel is given to verify the accuracy of the equivalent laminate plate model. The results of equivalent and detailed models such as the maximum temperature, the temperature history and free vibration frequencies and their corresponding modes are compared under many different cases, including steady and transient thermal environment, constant aerodynamic heating and gradient aerodynamic heating and so on. 2. Equivalent continuum model The following section outlines the methods for estimation of equivalent thermal and mechanical properties of the honeycomb core. A typical honeycomb plate consists of the radiative surface panel, the insulation package and the support structure. The radiative surface panel is always made of lightweight superalloy honeycomb. A typical honeycomb plate and a representative honeycomb cell is given in Fig.1, and the parameters used in the analyses are also graphically presented in Fig.1.
Fig. 1. A typical honeycomb plate and a representative honeycomb cell
In thermal analysis, equivalent thermal parameters include density, capacity and conductivity. The effective density Uc is given by: Wc 8 t Uc U c0 (1) Vall 3 3 l Where Vall 2hc (l l cos T )l sin T is the whole volume of the representative cell, for hexagonal cell T 60$ , Wc 4 U c0lthc is the mass of the honeycomb core, U c0 is the density of the honeycomb core material.
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For transient thermal analysis, the effect of the thermal capacity should also be taken into account which gives: 8 t Ucc U c0 c (2) 3 3l The effective thermal conductivity of the honeycomb core in thickness direction is calculated using the Swann-Pittman semi-analytical model. The effect of solid conduction through the honeycomb wall, gas conduction in the honeycomb cells and radiation is considered: § A · A (3) kcz,eff kcm cz kc,gas ¨1 cz ¸ kcz,rad Aall © Aall ¹ Where k cm is the thermal conductivity of the honeycomb material, kc,gas is the thermal conductivity of material inside the honeycomb cell, kc,rad represents the radiative effect through the honeycomb core which is neglect in the present research. Acz and Aall represent the area of honeycomb wall and the whole honeycomb cell: 3 3 2 (4) l 2 Most researches consider the honeycomb plate as a one-dimensional structure in thermal analysis, which implies that the in-plane thermal response can be neglected, and this assumption works only the in-plane thermal flux input is uniform, otherwise the effect of in-plane thermal conductivity should be considered as well. The in-plane effective area along x-direction and y-direction is given by: Acx 3 tc Acy 3 tc ˈ (5) 2 l Aall 3 l Aall Following the Swann-Pittman semi-analytical model, the in-plane effective thermal conductivity can then be obtained. In mechanical analysis, the honeycomb core is considered as an orthotropic material, and the equivalent structural parameters are given by [8]: Acz
4ltcˈAall
3
Ex,eff Gxy,eff
Q xy,eff
3
4 §t · ¨ ¸ Ec , Ey,eff 3©l ¹
4 §t · ¨ ¸ Ec , Ez,eff 3©l ¹
3
3t Gc , Gyz ,eff 2 l
3§t · ¨ ¸ Ec , Gxz ,eff 17 © l ¹ 1,Q zy,eff Q zx,eff Q c
8 t Ec 3 3l 3t Gc 3 l
(6)
Then the honeycomb core structure can be modelled as an equivalent three layer laminate plate using the effective thermal and structural parameters obtained before. 3. Numerical results
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Temperature at left side of the outer face sheet (K)
In this section, the thermal and free vibration behaviour of the equivalent laminate plate is compared with that of detailed model of the honeycomb core plate under different circumstances. The detailed model R . In the detailed model, the and equivalent model are both analyzed using the FE analysis code ANSYS ƻ whole honeycomb structure is modelled by solid elements in full detail. In the equivalent model, the equivalent laminate plate is modelled by shell elements. The length and width of the plate is 285 mm and 169 mm respectively, the thickness of both outer and interior face sheet is 1 mm. The honeycomb is a regular hexagon with side length 5 mm, thickness 0.15 mm and height 10 mm. The material of honeycomb structure is GH600 [9]. In thermal analysis, the outer face sheet of the honeycomb panel is subjected to transient heat flux with a linear gradient from left side (10% more) to the right side (10% less), and the mid-value of the heat flux history is given in Fig.2 (a). The initial temperature of the entire plate is assumed to be 300 K. Thermal boundary condition is characterized by the radiation of the outer surface to an ambient temperature of 300 K with an emissivity factor of 0.8. The radiative heat exchange between the face sheet and the honeycomb core is neglect. A convection boundary condition is assumed at the back surface of the interior face sheet, with a convection heat transfer coefficient 5 WK-1 m-2 and ambient temperature of 300 K. Comparisons between the temperature history obtained by using the equivalent model and detailed model is shown in Fig.2 (b). This figure shows the time history of temperature at the left side of the outer face sheet. Numerical results show that the equivalent model has a good agreement with the detailed model on the history of maximum temperature. 250
-2
Media aerodynamic heat flux (kWm )
(a) 200
150
100
50
(b) 1400
1200
Detailed model Equivalent model
1000
800
600
400
0 0
200
400
600
Time (s)
800
1000
0
200
400
600
800
1000
x (s)
Fig. 2. (a) Representative aerodynamic heat flux profile; (b) Comparison of the temperature at left side of the outer face sheet
Comparisons between free vibration frequencies obtained by using equivalent and detailed model are presented below. The first 10 free vibration eigenfrequencies and corresponding eigenmodes of both models are extracted. The mode and its corresponding frequency of the first order vibration of detailed and equivalent model are shown in Fig.3. Results of other orders of vibration provide the same evidence that the present equivalent model is feasible in capturing the vibration frequencies and their corresponding modes in structural analysis.
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1
NODAL SOLUTION
JUN 12 2011 12:37:26
STEP=1 SUB =1 FREQ=172.185 USUM (AVG) RSYS=0 DMX =2.009 SMX =2.009
NODAL SOLUTION
JUN 11 2011 23:09:45
STEP=1 SUB =1 FREQ=171.737 USUM (AVG) RSYS=0 DMX =2.005 SMX =2.005
MN
Y
MX
Z
Y
Z
X
MN
X
MX
0
.223233
.446465
.669698
.892931
1.116
1.339
1.563
1.786
0
2.009
.222793
.445585
.668378
.89117
1.114
1.337
1.56
1.782
2.005
Fig. 3. Comparison of the first order vibration: (a) detailed model; (b) equivalent model
It also should be pointed out that the computational time for the thermal and thermomechanical analyses is dramatically reduced by replacing the detailed three-dimensional model with its equivalent twodimensional laminate plate model, and the computational time reduction is more than 100 times of the given example. 4. Conclusions Numerical results show that the predictions of the present equivalent laminated plate model of the honeycomb panel has a good agreement with the detailed three-dimensional model both in thermal and free vibration analysis, with a great reduction of the CPU time. It is also found that the equivalent model becomes much more accurate when the transient effect becomes less dominant and the internal gradient has had time to develop. At the same time, it is very necessary to consider the in-plane thermal conductivities when the in-plane gradient of aerodynamic heating is ignorable. Future work consists of predicting the effective properties of the honeycomb core in terms of its geometric and material characteristics by means of mechanics of material approach, then the effective elastic properties are used in conduction with the classical laminate theory and the temperature field obtained here to determine the structural behaviour of the entire sandwich structure. Acknowledgements This research was supported by the Fundamental Research Funds for the Central Universities (FRF-BR10-007A, FRF-AS-09-001A) and the National Natural Science Foundation of China (10872104, 10772024). References [1] Blosser ML. Fundamental modeling and thermal performance issues for metallic thermal protection system concept. J Spacecraft Rockets, 2004; 41:195-206. [2] Fatemi J, Lemmen MHJ. Effective thermal/mechanical properties of honeycomb core panels for hot structure applications. J Spacecraft Rockets. 2009, 46:514-525. [3] Swann RT, Pittman CM. Analysis of effective thermal conductivities of honeycomb-core and corrugated-core sandwich panels. NASA TN D-714, 1961.
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[4] Nagahisa O, Masaki S, Qiang Y, et al. Estimation of orthotropic thermal conductivity of honeycpmb material. Heat Tran Asian Res. 2002, 31:617-625. [5] Daryabeigi K. Heat transfer in adhesively bonded honeycomb core panel. journal of thermophysics and heat transfer. J Thermophys Heat Tr, 2002, 16:217-221. [6] Bezazi A, Remillat C, Innocenti P, Scarpa F. In-plane mechanical and thermal conductivity properties of a rectangular±hexagonal honeycomb structure. Compos Struct. 2008, 84:248-255. [7] Martinez OA, Sankar BV, Haftka RT, Bapanapalli SK. Micromechanical analysis of composite corrugated -core sandwich panels for integral thermal protection systems. AIAA J. 2007, 45:2323-2336. [8] Gibson LJ, Ashby MF. Cellular Solids: Structure and Properties. New York: Cambridge University Press, 1997 [9] Liu Donghuan, Zheng Xiaoping, Wang Fei, et al. Mechanism of thermomechanical coupling of high temperature heat pipe cooled C/C composite material thermal protection structure. Acta Mater Compos Sin, 2010; 27:43-49. (in Chinese)
