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Extended multiscale FE approach for steady-state heat conduction analysis of 3D braided composites
Composites Science and Technology 151 (2017) 317e324
Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: http://www.elsevier.com/locate/compscitech
Extended multiscale FE approach for steady-state heat conduction analysis of 3D braided composites Jun-jun Zhai a, Su Cheng a, Tao Zeng a, *, Zhi-hai Wang a, Dai-ning Fang b a b
Department of Engineering Mechanics, Harbin University of Science and Technology, Harbin 150080, PR China School of Aerospace Engineering, Beijing Institute of Technology, 100081, Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 July 2017 Received in revised form 27 August 2017 Accepted 28 August 2017 Available online 1 September 2017
A new multiscale finite element (FE) calculation procedure is presented to predict the thermal conductive performance of 3D braided composites by a combined approach of the multiscale asymptotic expansion homogenization (MAEH) method and multiphase finite element (MPFE) approach. The analysis was performed based on a homogeneous macrostructure model and a heterogeneous microstructure representative unit cell model. The heat flux distribution, temperature distribution of 3D braided composites under different boundary conditions are basically predicted. It is found that the heat flux transmission is mainly along fiber orientation direction and the heat flux of the braiding yarns in the center region is much higher than that in the outer region. Effective coefficients of thermal conductivity (CTC) of 3D braided composites is predicted and compared with experiment data available in the literature to demonstrate the accuracy and reliability of the present multiscale FE approach. The procedure, which can be implemented into commercial finite element codes, is an efficient tool for the design and analysis of a heterogeneous material with anisotropic properties or complex geometries. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Thermal properties Finite element analysis (FEA) Multiscale modeling Braiding
1. Introduction Three-dimensional (3D) braided composites have been widely studied for its excellent mechanical performance and potential industry application. In order to take full advantages of the 3D braided composites efficiently, safely and reasonably, it is necessary to develop modeling approaches and computational methods capable of predicting the mechanical behavior of 3D braided composites. To end this, a variety of micro-structural models were proposed in the recent years, the typical micro-structural models were the fiber interlock model, the fiber inclination model and the fabric geometric model established by Ma et al. [1], Yang et al. [2] and Byun et al. [3]. It is well known that thermal conductive behavior is one of the important thermo-mechanical properties affecting the failure and destruction of 3D braided composite under high temperature environment. Therefore, it is particularly significant to investigate the thermal conductive properties of 3D braided composite. Recently, many researches [4e14] have been conducted on this subject both numerically and experimentally. Liu et al. [4] set up a
* Corresponding author. E-mail address: [email protected] (T. Zeng). http://dx.doi.org/10.1016/j.compscitech.2017.08.030 0266-3538/© 2017 Elsevier Ltd. All rights reserved.
new microstructure model and investigated the thermal conductivity of 3D four-directional braided composites. Fang et al. [5] analyzed the effective thermal conductive properties of 3D fourdirectional braided composites by using the lattice Boltzmann method. Jiang et al. [6] investigated the thermal conductivity and temperature distribution of 3D four-directional braided composites by a modified finite element model. Gou et al. [7] studied the effective thermal conductivity and temperature distribution of 3D four-directional braided composites with appropriate boundary conditions. Gou et al. [8] predicted the temperature distributions and the effective thermal conductivities of 3D four-directional braided composites based on the full, quarter and eighth unit cells. Li et al. [9] calculated the transverse and longitudinal thermal conductivity coefficients of 3D 5-directional braided composites by the finite element method. Xia and Lu [10,11] investigated the thermophysical properties of 3D four-directional braided composites by considering and not considering the yarn/matrix interface. Cheng et al. [12] reported the thermal property of the 3D fourdirectional braided composites by experiment and numerical methods. Shi et al. [13] presented a procedure for predicting equivalent thermal property parameter through a combined approach of the generalized method of cells and multiscale heat transfer analysis. Dong et al. [14] presented a research on the
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thermal conductive behaviors of 3D braided composites both at meso-scale and full-scale structure level. As mentioned above, most of the researches on the thermal conductive behavior of 3D braided composites are based on the representative unit cell (RUC) model. MAEH approach, which is a distinct advantage over other homogenizing techniques, e.g. rule of mixtures, have capability to study the behavior of structural components built with composite materials by microscopic scale and macroscopic scale. It has been employed to study the mechanical behavior of 3D textile composites by many scholars [15e21]. Feng et al. [15] and Wang et al. [16] used MAEH methods to predict the effective modulus of 3D braided composites. Visrolia et al. [17] used a MAEH method to determine
qi ðxÞ ¼ ε1 kij ðx; yÞ m
m
T ε ðxÞ ¼ T ð0Þ ðx; yÞ þ εT ð1Þ ðx; yÞ þ ε2 T ð2Þ ðx; yÞ þ …
(1)
The heat flux vector qεi , given by the Fourier's law in terms of temperature gradient, can be written as
qεi ¼ kεij
vT ε vxj
(2)
where kεij is the thermal conductivity tensor and T ε is the change in temperature from the reference state. For the analysis of 3D braied composite, Eq. (3) can be obtained by introducing the asymptotic expansion Eq. (1) into Eq. (2)
vT ð0Þ ðx; yÞ vT ð1Þ ðx; yÞ vT ð1Þ ðx; yÞ vT ð2Þ ðx; yÞ vT ð0Þ ðx; yÞ m m εkij ðx; yÞ þ :::; i; j ¼ 1; 2; 3 and m kij ðx; yÞ þ þ vyj vxj vyj vxj vyj
¼ Y e ; M e ; Mixe
the local response of 3D weave composite by incorporating a continuum damage model. Yu and Cui [18] studied the stiffness and elasticity strength parameters of 3D braided composites by MAEH method, and the influences of the braiding angle and the fiber volume fraction on the strength of 3D braided composites were discussed. Dasgupta et al. [19] and Nasution et al. [20,21] calculated the homogenized thermo-mechanical properties of 3D textile composite materials based on MAEH method. However, few studies have been investigated the thermal conductive behavior of 3D braided composites from the perspective of macroscopic and microscopic scales. In our previous work, a multiscale FE method [22] has been proposed for investigating the thermo-mechanical properties of 3D braided composites. The main purpose of this work is to extend the multiscale FE method for predicting the heat flux distribution, temperature distribution and effective thermal conductivity of 3D braided composites based on macroscopic and microscopic models. The resulting thermal conductive behaviors of 3D braided composites under different temperature boundary conditions were also analyzed and discussed. Such an effort could also be extended to investigate the thermal conductive behaviors of the 3D multidirectional braided composites even the heterogeneous materials with periodic structures.
2. Extension 2.1. MAEH formulation for steady-state heat conduction In order to implement the homogenization technique into the investigation of heat conduction in 3D braided composites, the RUC for real composites with periodic structure are employed. Fig. 1 shows the composition of the idealized geometry of a RUC, which is subdivided by 20-node rectangular isoparametric elements. In MAEH approach, the microscopic scale is related to the length of the repeat pattern and the macroscopic scale is related to the length of the structure. 3D braided composite as a periodic structure, the microscopic RUC scale (y) to the macroscopic structure scale (x) is assumed to be a small value ε ¼ x/y (0<ε<<1). When ε approaches zero, the heterogeneous macrostructure of 3D braided composite can be regarded as a homogeneous macrostructure. For a steady-state heat conduction problem, asymptotic expansion is assumed for the temperature field
(3)
where Y e ,Me and Mixe represent the yarn element, matrix element and mixed element, respectively [6]. The heat flux in Eq. (3) under steady state conditions are governed by the thermal equilibrium equation, which can be written as
vqε i ¼0 vxi
(4)
Substituting Eq. (3) into Eq. (4), the following equation is obtained by collecting the terms with the same order of ε:
h h i i ε2 L1 T ð0Þ ðx; yÞ þ ε1 L1 T ð1Þ ðx; yÞ þ L2 T ð0Þ ðx; yÞ h i þ ε0 L1 T ð2Þ ðx; yÞ þ L2 T ð1Þ ðx; yÞ þ L3 T ð0Þ ðx; yÞ þ ε1 ½… þ ε2 ½… þ…¼0 (5) where
v m v L1 ¼ kij ðx; yÞ vyi vyi
(6)
v m v v m v k ðx; yÞ L2 ¼ kij ðx; yÞ vyi vxj vxi ij vxj
(7)
v m v L3 ¼ kij ðx; yÞ vxi vxj
(8)
2.2. MAEH solution for steady-state heat conduction For Eq. (5) to hold for arbitrarily small ε, each portion of the equation associated with a particular power of ε must be equalto zero. Thus, it can be readily shown that T ð0Þ ðx; yÞ ¼ T ð0Þ ðxÞ. Hence the first expansion term in Eq. (1) does not depend on the microscopic coordinate y. According to the conventional homogenization method [23], the following decomposition is assumed for the term of ε1 in Eq. (5)
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319
Fig. 1. The unit cell of 3D four-dimensional braided composites (a) Boundary condition I (b) Boundary condition II.
T ð1Þ ðx; yÞ ¼ wj ðyÞ
according to Eqs. (10) and (11), it is very important to calculate the
vT ð0Þ ðxÞ vxj
(9)
where wj ðyÞ is the arbitrary characteristic function which is independent of macroscopic scale coordinate system and have the periodicity with respect to microscopic scale coordinate system. Substituting Eq. (9) into Eq. (2), and integrating the thermal equilibrium Eq. (2) over the RUC yields the homogenized effective conductivity tensor Kij ðx; yÞ.
Kij ðx; yÞ ¼
1 j gj
Z ( g
Then the microscopic heat flux vector using the averaging technique ð0Þ
qi ðx; yÞ ¼ Kij ðx; yÞ
L1 T ð1Þ ðx; yÞ þ L2 T ð0Þ ðx; yÞ ¼ 0
(10)
are calculated by
(12)
Furthermore, the weak form equation for wj ðyÞ is obtained by multiplying Eq. (12) by virtual displacement dwj ðyÞ, and conducting integration within the region of RUC yields
Z
) vwj ðyÞ m m kij ðx; yÞ kij ðx; yÞ dg vyj ð0Þ qi ðx; yÞ
correctors wj ðyÞ effectually. According to section 2.2, it can be readily shown that
g
vdwj ðyÞ m vwj ðyÞ kij ðx; yÞ dg ¼ vyi vyj
Z g
vdwj ðyÞ m kij ðx; yÞdg vyi
(13)
Then, the finite element discretized form for homogenization of heat conduction can be written as
Hw ¼ Q
vT ð0Þ ðxÞ
(11)
vxj
(14)
where
Z H¼ g
½BT ½km ½Bdg
(15)
e
2.3. Finite element discretization for the microstructural fields To obtain the homogenization thermal conductive performance
Table 1 Influence of the elements number on the calculated CTCs of 3D braided composites (q ¼ 20 , Vf ¼0.35). Sum of elements 32
256
1323
2035
3240
4000
5203
lx ¼ ly ðW=m CÞ 0.2514 0.1951 0.2259 0.2286 0.2304 0.2321 0.2311 2.3695 1.8334 1.9897 1.9451 2.0320 2.0633 2.0442 lz ðW=m CÞ
Table 2 Comparison of the CTCs for 3D four-directional braided composites obtained from calculations and experiments. Fiber volume fraction
Braiding angle (º)
45% 53% 54%
48 42 41
lx ¼ ly ðW=m CÞ
lz ðW=m CÞ
Experiments [12]
Predictions
Predictions
0.770 0.750 0.694
0.712 0.657 0.563
1.239 1.334 1.356
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Z Q¼ g
½BT ½km dg
(16)
e
Here, the index e indicating a given element quantities from the meshed RUC domain. ½km is the thermal conductivity tensor. [B] is a matrix containing the spatial derivatives of the shape functions. It should be noted that Q is not a vector but a matrix. Each column of the matrix wj ðyÞ corresponds to a column of the matrix Q. In order to solve the matrix wj ðyÞ in Eq. (14), the MPFE method [6] was used in this work. Up to this stage, once the wj ðyÞ have been determined, then the homogenized effective conductivity tensor Kij ðx; yÞ will be known, and it is possible to recover the microscale response under macroscale boundary conditions.
3. Mesh convergence
Fig. 2. Boundary conditions used for macrostructure model.
Finite element solutions are more accurate with finer meshes, and this requirement also applies to this multiscale FE simulations. In the present work, a convergent test for 3D four-directional braided composites is carried out and the CTCs of the components used in the following simulation are excerpted from Ref. [6]. The calculated effective CTCs of 3D four-directional braided
Fig. 3. Temperature distributions in macrostructure under boundary condition I.
ð0Þ
Fig. 4. Heat flux qZ
distributions in macrostructure under boundary condition I.
J.-j. Zhai et al. / Composites Science and Technology 151 (2017) 317e324
ð0Þ
Fig. 5. Localization heat flux qi
321
ð0Þ
ð0Þ
ð0Þ
distributions in the three directions of the specific RUC under boundary condition I: (a) qX ,(b) qY ,(c) qZ .
composites (q ¼ 20 , Vf ¼ 0.35) with different number of elements are shown in Table 1. As can be seen in Table 1, the CTCs vary little as the mesh is refined beyond the level where it contains about 3240 elements. The obtained numerical results shown that the CTCs of 3D four-directional carbon/epoxy braided composites are independent of mesh refinement in this work. 4. Application and discussions 4.1. Coefficient of thermal conductivity For the 3D four-directional braided composites studied in this
work, the matrix is epoxy resin TDE-86 and the fiber is T300 carbon [6]. The effective CTCs evaluated by multiscale FE approach are compared with the experimental data [12] as shown in Table 2. The relatively good agreement gives a strong support to the reliability of the numerical method. 4.2. Thermal conductive behaviors under different temperature boundary conditions In order to fully exploit the advantages of the 3D braided composites, the thermal conductive behaviors of 3D braided composite are investigated by the extended multiscale FE approach. To
Fig. 6. Temperature distributions in macrostructure under boundary condition II: (a) external contours, (b) internal contours.
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J.-j. Zhai et al. / Composites Science and Technology 151 (2017) 317e324
ð0Þ
Fig. 7. Heat flux qi
ð0Þ
ð0Þ
ð0Þ
ð0Þ
distributions in the three directions of macrostructure under boundary condition II: (a) qX ,(b) qY ,(c) qZ ,(d) internal contours of. qZ .
complete the multiscale procedure, the definition of the homogenised heat conduction properties of the macroscale material is done using the homogenised thermal conductivity matrix, which results from the homogenization procedure. Considering two kinds of boundary conditions as shown in Fig. 2, a 3D braided composite macrostructure with the dimensions of 15 15 50 mm is studied in this work. The braiding angle and fiber volume fraction are q ¼ 20 and Vf ¼ 0.35, respectively. 4.2.1. Temperature boundary conditions I: macrostructure with adiabatic surfaces As shown in Fig. 2(a), a heating plate is assumed to be placed on the bottom surface (Z ¼ 50 mm) of the macrostructure, which provides a heat flux for going through along its axial direction. The temperature of bottom surface is assumed to be T and do not change with time. The symmetrical surface (Z ¼ 0 mm) is assumed to be mounted on a cool plate, which provides a heat sink. The plate is water cooled to a constant temperature Tsink. Others surfaces were assumed to be tightly wrapped and set as the adiabatic surfaces. When the constant temperature Tsink of the plate for the 3D braided composite structure is set as zero, Fig. 3 presents the temperature distribution of the 3D braided composite macrostructure when T ¼ 60 C. It can be found that temperature deð0Þ
creases gradually along the Z direction. Fig. 4 gives heat flux qZ distributions of 3D braided composite. From Fig. 4, it is found that the heat flux distributions are almost uniform in the interior of the 3D braided composite macrostructure. In order to give a detailed description of the microstructure heat flux distribution of 3D braided composites, the specific region A in the center of the braided composite is selected as shown in Fig. 2.
ð0Þ
Fig. 5 shows the localization heat flux qi distributions in the three directions (X, Y and Z) of the specific RUC, which are obtained by application of the current boundary conditions. It can be seen from ð0Þ
Figs. 4 and 5 that the distributions of heat flux qi in microscopic coordinate system are generally larger than that in macroscopic coordinate system, the ratio of the microscale heat flux component ð0Þ
qZ to the average macroscale heat flux in Z direction can be as high as 9.44 at certain locations for this particular heterogeneous microstructure in z direction. The higher value of heat flux exists in the yarns of Z direction due to the bigger effective thermal conductivity in the specific RUC. The microscale heat flux component ð0Þ
ð0Þ
qX ¼ qY need not be zero although the average macroscale heat flux component in X and Y directions is zero, this is due to the complex heat flux patterns within the heterogeneous microstructure of 3D braided composite. The figures indicate that heat flux patterns at the microscale are complex even when the macroscopic heat flux state is very simple. 4.2.2. Temperature boundary conditions II: macrostructure with nonadiabatic surfaces Different with the boundary condition I, the surrounding surfaces are assumed as nonadiabatic boundary, and the temperature remains at room temperature (23 C), which is called boundary condition II, as shown in Fig. 2(b). Based on the temperature boundary condition II, Fig. 6 shows the temperature distribution of the 3D braided composite structure. It can be seen that distribution of temperature is hourglass shape along the Z direction. Fig. 7 gives heat flux distributions in the three directions (X, Y and Z) of macrostructure. It can be found that the higher value of heat flux exists in the Z direction due to the
J.-j. Zhai et al. / Composites Science and Technology 151 (2017) 317e324
ð0Þ
Fig. 8. Localization heat flux qi
323
distributions in the Z direction of the specific RUC, which are related to the specific regions A, B and C.
larger temperature gradient, and the heat flux were not uniform distributions at the cross section of 3D braided composite macrostructure. Similarly, three specific regions A, B, C are selected in order to ð0Þ
describe the microscale heat flux qi distributions of 3D braided composites exactly, as shown in Fig. 2(b). By applying the temð0Þ
perature boundary conditions II, the microscale heat flux qi distributions related to the specific regions A, B, C in the Z direction were achieved as shown in Fig. 8. It can be seen from Fig. 8(a) that the heat flux is mainly concentrated in the yarns and the heat flux of the braiding yarns in the center region is much higher than that in the outer region. These phenomena can also be seen from Fig. 8(b) and (c). Comparing Fig. 8(a) and 8(c), it is observed that ð0Þ
the microscale heat flux qZ near the hot source (region C) is higher than the microscale heat flux away from the hot source (region B), although this phenomenon is not apparent at the macroscale. 5. Conclusions In this paper, an explicit and efficient multiscale FE calculation procedure for describing the heat flux distribution, temperature distribution and effective thermal conductivity of 3D braided composites, based on a homogeneous macrostructure model and a heterogeneous microstructure RUC model, has been developed. Multiscale FE numerical models are established and the effective thermal conductive behaviors of 3D braided composites under two kinds of boundary conditions are predicted. The main conclusions can be listed as follows:
(1) The thermal conductive behavior of the 3D four-directional braided composites is obviously anisotropic. (2) Temperature distributions of 3D four-directional braided composite structure resulted from the boundary conditions I are obviously different from that of boundary conditions II. (3) The heat flux distributions of 3D four-directional braided composite are dependent on the boundary conditions. Larger microscale heat flux belong to the related regions of macrostructure are obtained. It is found that the heat flux in the yarn regions is bigger than that in the matrix region, and the heat flux of the braiding yarns in the center region is much higher than that in the outer region. (4) There is a good agreement between the prediction and experimental results. For future works, more experiments will be completed to prove the validity of the conclusions obtained by the present multiscale FE method. Acknowledgments The authors would like to thank the National Natural Science Foundation of China (11432005). References [1] C.L. Ma, J.M. Yang, T.W. Chou, Elastic stiffness of three-dimensional braided textile structural composites, in: Composite Materials: Testing and Design, Seventh Conference, Philadelphia, 1986, pp. 404e421. [2] J.M. Yang, C.L. Ma, T.W. Chou, Fiber inclination model of three-dimensional textile structural composites, J. Compos. Mater 20 (5) (1986) 472e483. [3] J.H. Byun, G.W. Du, T.W. Chou, Analysis and modeling of three-dimensional textile structural composites, High-Tech Fibrous Mater. 457 (1991) 22e33. [4] Z.G. Liu, H.G. Zhang, Z.X. Lu, D.S. Li, Investigation on the thermal conductivity
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J.-j. Zhai et al. / Composites Science and Technology 151 (2017) 317e324 of 3-dimensional and 4-directional braided composites, Chin. J. Aeronaut. 20 (4) (2007) 327e331. W.Z. Fang, L. Chen, J.J. Gou, W.Q. Tao, Predictions of effective thermal conductivities for three-dimensional four-directional braided composites using the lattice Boltzmann method, Int. J. Heat. Mass Transf. 92 (2015) 120e130. L.L. Jiang, G.D. Xu, S. Cheng, X.M. Lu, T. Zeng, Predicting the thermal conductivity and temperature distribution in 3D braided composites, Compos. Struct. 108 (2014) 578e583. J.J. Gou, H. Zhang, Y.J. Dai, S. Li, W.Q. Tao, Numerical prediction of effective thermal conductivities of 3D four-directional braided composites, Compos. Struct. 125 (2015) 499e508. J.J. Gou, W.Z. Fang, Y.J. Dai, S. Li, W.Q. Tao, Multi-size unit cells to predict effective thermal conductivities of 3D four-directional braided composites, Compos. Struct. 163 (2017) 152e167. D.S. Li, Z.X. Lu, Z.G. Liu, Z.P. Li, Finite element analysis of thermal conductivity of three dimensional and five directional braided composites, J. Aerosp. Power 23 (8) (2008) 1455e1460. B. Xia, Z.X. Lu, Finite element analysis on thermo-physical properties of 3D braided composites, Acta Aeronaut. Astronaut. Sin. 32 (6) (2011) 1040e1049. Z.X. Lu, C.Y. Wang, B. Xia, Y. Zhou, Effect of interfacial properties on thethermophysical properties of 3D braided composites: 3D multiscale finite element study, Polym. Compos 35 (9) (2013) 1690e1700. W. Cheng, S. Zhao, Z. Liu, Z. Feng, C. Yao, R. Yu, Thermal property of 3-D braided fiber composites: experimental and numerical results, Acta Aeronaut. Astronaut. Sin. 23 (2) (2002) 102e105. Y.A. Shi, L.X. He, B. Qiu, L. Zeng, C.R. Geng, D. Wei, Multiscale heat transfer analysis of Z-directional carbon fiber reinforced braided composites, Acta Aeronaut. Astronaut.Sin 37 (4) (2016) 1207e1217.
[14] K. Dong, J. Zhang, L. Jin, B.H. Gu, B. Sun, Multi-scale finite element analyses on the thermal conductive behaviors of 3D braided composites, Compos. Struct. 143 (2016) 9e22. [15] M.L. Feng, C.C. Wu, A study of three-dimensional four-step braided piezoceramic composites by the homogenization method, Compos. Sci. Technol. 61 (13) (2001) 1889e1898. [16] X.F. Wang, X.W. Wang, G.M. Zhou, C.W. Zhou, Multi-scale analyses of 3D woven composite based on periodicity boundary conditions, J. Compos. Mater 41 (2007) 1773e1788. [17] A. Visrolia, M. Meo, Multi-scale damage modeling of 3D weave composite by asymptotic homogenization, Compos. Struct. 95 (2013) 105e113. [18] X.G. Yu, J.Z. Cui, The prediction on mechanical properties of 4-step braided composites via two-scale method, Compos. Sci. Technol. 67 (2007) 471e480. [19] A. Dasgupta, R.K. Agarwal, S.M. Bhandarkar, Three-dimensional modeling of woven-fabric composites for effective thermo-mechanical and thermal properties, Compos. Sci. Technol. 56 (1996) 209e223. [20] M.R.E. Nasution, N. Watanabe, A. Kondo, A. Yudhanto, Thermomechanical properties and stress analysis of 3-D textile composites by asymptotic expansion homogenization method, Compos. B Eng. 60 (2014) 378e391. [21] M.R.E. Nasution, N. Watanabe, A. Kondo, A. Yudhanto, A novel asymptotic expansion homogenization analysis for 3-D composite with relieved periodicity in the thickness direction, Compos. Sci. Technol. 97 (2014) 63e73. [22] J.J. Zhai, S. Cheng, T. Zeng, Z.H. Wang, L.L. Jiang, Thermo-mechanical behavior analysis of 3D braided composites by multiscale finite element method, Compos. Struct. 176 (2017) 664e672. [23] A.J. Goupee, S.S. Vel, Multiscale thermoelastic analysis of random heterogeneous materials: Part II: direct micromechanical failure analysis and multiscale simulations, Comput. Mat. Sci. 48 (1) (2010) 39e53.
Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: http://www.elsevier.com/locate/compscitech
Extended multiscale FE approach for steady-state heat conduction analysis of 3D braided composites Jun-jun Zhai a, Su Cheng a, Tao Zeng a, *, Zhi-hai Wang a, Dai-ning Fang b a b
Department of Engineering Mechanics, Harbin University of Science and Technology, Harbin 150080, PR China School of Aerospace Engineering, Beijing Institute of Technology, 100081, Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 July 2017 Received in revised form 27 August 2017 Accepted 28 August 2017 Available online 1 September 2017
A new multiscale finite element (FE) calculation procedure is presented to predict the thermal conductive performance of 3D braided composites by a combined approach of the multiscale asymptotic expansion homogenization (MAEH) method and multiphase finite element (MPFE) approach. The analysis was performed based on a homogeneous macrostructure model and a heterogeneous microstructure representative unit cell model. The heat flux distribution, temperature distribution of 3D braided composites under different boundary conditions are basically predicted. It is found that the heat flux transmission is mainly along fiber orientation direction and the heat flux of the braiding yarns in the center region is much higher than that in the outer region. Effective coefficients of thermal conductivity (CTC) of 3D braided composites is predicted and compared with experiment data available in the literature to demonstrate the accuracy and reliability of the present multiscale FE approach. The procedure, which can be implemented into commercial finite element codes, is an efficient tool for the design and analysis of a heterogeneous material with anisotropic properties or complex geometries. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Thermal properties Finite element analysis (FEA) Multiscale modeling Braiding
1. Introduction Three-dimensional (3D) braided composites have been widely studied for its excellent mechanical performance and potential industry application. In order to take full advantages of the 3D braided composites efficiently, safely and reasonably, it is necessary to develop modeling approaches and computational methods capable of predicting the mechanical behavior of 3D braided composites. To end this, a variety of micro-structural models were proposed in the recent years, the typical micro-structural models were the fiber interlock model, the fiber inclination model and the fabric geometric model established by Ma et al. [1], Yang et al. [2] and Byun et al. [3]. It is well known that thermal conductive behavior is one of the important thermo-mechanical properties affecting the failure and destruction of 3D braided composite under high temperature environment. Therefore, it is particularly significant to investigate the thermal conductive properties of 3D braided composite. Recently, many researches [4e14] have been conducted on this subject both numerically and experimentally. Liu et al. [4] set up a
* Corresponding author. E-mail address: [email protected] (T. Zeng). http://dx.doi.org/10.1016/j.compscitech.2017.08.030 0266-3538/© 2017 Elsevier Ltd. All rights reserved.
new microstructure model and investigated the thermal conductivity of 3D four-directional braided composites. Fang et al. [5] analyzed the effective thermal conductive properties of 3D fourdirectional braided composites by using the lattice Boltzmann method. Jiang et al. [6] investigated the thermal conductivity and temperature distribution of 3D four-directional braided composites by a modified finite element model. Gou et al. [7] studied the effective thermal conductivity and temperature distribution of 3D four-directional braided composites with appropriate boundary conditions. Gou et al. [8] predicted the temperature distributions and the effective thermal conductivities of 3D four-directional braided composites based on the full, quarter and eighth unit cells. Li et al. [9] calculated the transverse and longitudinal thermal conductivity coefficients of 3D 5-directional braided composites by the finite element method. Xia and Lu [10,11] investigated the thermophysical properties of 3D four-directional braided composites by considering and not considering the yarn/matrix interface. Cheng et al. [12] reported the thermal property of the 3D fourdirectional braided composites by experiment and numerical methods. Shi et al. [13] presented a procedure for predicting equivalent thermal property parameter through a combined approach of the generalized method of cells and multiscale heat transfer analysis. Dong et al. [14] presented a research on the
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thermal conductive behaviors of 3D braided composites both at meso-scale and full-scale structure level. As mentioned above, most of the researches on the thermal conductive behavior of 3D braided composites are based on the representative unit cell (RUC) model. MAEH approach, which is a distinct advantage over other homogenizing techniques, e.g. rule of mixtures, have capability to study the behavior of structural components built with composite materials by microscopic scale and macroscopic scale. It has been employed to study the mechanical behavior of 3D textile composites by many scholars [15e21]. Feng et al. [15] and Wang et al. [16] used MAEH methods to predict the effective modulus of 3D braided composites. Visrolia et al. [17] used a MAEH method to determine
qi ðxÞ ¼ ε1 kij ðx; yÞ m
m
T ε ðxÞ ¼ T ð0Þ ðx; yÞ þ εT ð1Þ ðx; yÞ þ ε2 T ð2Þ ðx; yÞ þ …
(1)
The heat flux vector qεi , given by the Fourier's law in terms of temperature gradient, can be written as
qεi ¼ kεij
vT ε vxj
(2)
where kεij is the thermal conductivity tensor and T ε is the change in temperature from the reference state. For the analysis of 3D braied composite, Eq. (3) can be obtained by introducing the asymptotic expansion Eq. (1) into Eq. (2)
vT ð0Þ ðx; yÞ vT ð1Þ ðx; yÞ vT ð1Þ ðx; yÞ vT ð2Þ ðx; yÞ vT ð0Þ ðx; yÞ m m εkij ðx; yÞ þ :::; i; j ¼ 1; 2; 3 and m kij ðx; yÞ þ þ vyj vxj vyj vxj vyj
¼ Y e ; M e ; Mixe
the local response of 3D weave composite by incorporating a continuum damage model. Yu and Cui [18] studied the stiffness and elasticity strength parameters of 3D braided composites by MAEH method, and the influences of the braiding angle and the fiber volume fraction on the strength of 3D braided composites were discussed. Dasgupta et al. [19] and Nasution et al. [20,21] calculated the homogenized thermo-mechanical properties of 3D textile composite materials based on MAEH method. However, few studies have been investigated the thermal conductive behavior of 3D braided composites from the perspective of macroscopic and microscopic scales. In our previous work, a multiscale FE method [22] has been proposed for investigating the thermo-mechanical properties of 3D braided composites. The main purpose of this work is to extend the multiscale FE method for predicting the heat flux distribution, temperature distribution and effective thermal conductivity of 3D braided composites based on macroscopic and microscopic models. The resulting thermal conductive behaviors of 3D braided composites under different temperature boundary conditions were also analyzed and discussed. Such an effort could also be extended to investigate the thermal conductive behaviors of the 3D multidirectional braided composites even the heterogeneous materials with periodic structures.
2. Extension 2.1. MAEH formulation for steady-state heat conduction In order to implement the homogenization technique into the investigation of heat conduction in 3D braided composites, the RUC for real composites with periodic structure are employed. Fig. 1 shows the composition of the idealized geometry of a RUC, which is subdivided by 20-node rectangular isoparametric elements. In MAEH approach, the microscopic scale is related to the length of the repeat pattern and the macroscopic scale is related to the length of the structure. 3D braided composite as a periodic structure, the microscopic RUC scale (y) to the macroscopic structure scale (x) is assumed to be a small value ε ¼ x/y (0<ε<<1). When ε approaches zero, the heterogeneous macrostructure of 3D braided composite can be regarded as a homogeneous macrostructure. For a steady-state heat conduction problem, asymptotic expansion is assumed for the temperature field
(3)
where Y e ,Me and Mixe represent the yarn element, matrix element and mixed element, respectively [6]. The heat flux in Eq. (3) under steady state conditions are governed by the thermal equilibrium equation, which can be written as
vqε i ¼0 vxi
(4)
Substituting Eq. (3) into Eq. (4), the following equation is obtained by collecting the terms with the same order of ε:
h h i i ε2 L1 T ð0Þ ðx; yÞ þ ε1 L1 T ð1Þ ðx; yÞ þ L2 T ð0Þ ðx; yÞ h i þ ε0 L1 T ð2Þ ðx; yÞ þ L2 T ð1Þ ðx; yÞ þ L3 T ð0Þ ðx; yÞ þ ε1 ½… þ ε2 ½… þ…¼0 (5) where
v m v L1 ¼ kij ðx; yÞ vyi vyi
(6)
v m v v m v k ðx; yÞ L2 ¼ kij ðx; yÞ vyi vxj vxi ij vxj
(7)
v m v L3 ¼ kij ðx; yÞ vxi vxj
(8)
2.2. MAEH solution for steady-state heat conduction For Eq. (5) to hold for arbitrarily small ε, each portion of the equation associated with a particular power of ε must be equalto zero. Thus, it can be readily shown that T ð0Þ ðx; yÞ ¼ T ð0Þ ðxÞ. Hence the first expansion term in Eq. (1) does not depend on the microscopic coordinate y. According to the conventional homogenization method [23], the following decomposition is assumed for the term of ε1 in Eq. (5)
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319
Fig. 1. The unit cell of 3D four-dimensional braided composites (a) Boundary condition I (b) Boundary condition II.
T ð1Þ ðx; yÞ ¼ wj ðyÞ
according to Eqs. (10) and (11), it is very important to calculate the
vT ð0Þ ðxÞ vxj
(9)
where wj ðyÞ is the arbitrary characteristic function which is independent of macroscopic scale coordinate system and have the periodicity with respect to microscopic scale coordinate system. Substituting Eq. (9) into Eq. (2), and integrating the thermal equilibrium Eq. (2) over the RUC yields the homogenized effective conductivity tensor Kij ðx; yÞ.
Kij ðx; yÞ ¼
1 j gj
Z ( g
Then the microscopic heat flux vector using the averaging technique ð0Þ
qi ðx; yÞ ¼ Kij ðx; yÞ
L1 T ð1Þ ðx; yÞ þ L2 T ð0Þ ðx; yÞ ¼ 0
(10)
are calculated by
(12)
Furthermore, the weak form equation for wj ðyÞ is obtained by multiplying Eq. (12) by virtual displacement dwj ðyÞ, and conducting integration within the region of RUC yields
Z
) vwj ðyÞ m m kij ðx; yÞ kij ðx; yÞ dg vyj ð0Þ qi ðx; yÞ
correctors wj ðyÞ effectually. According to section 2.2, it can be readily shown that
g
vdwj ðyÞ m vwj ðyÞ kij ðx; yÞ dg ¼ vyi vyj
Z g
vdwj ðyÞ m kij ðx; yÞdg vyi
(13)
Then, the finite element discretized form for homogenization of heat conduction can be written as
Hw ¼ Q
vT ð0Þ ðxÞ
(11)
vxj
(14)
where
Z H¼ g
½BT ½km ½Bdg
(15)
e
2.3. Finite element discretization for the microstructural fields To obtain the homogenization thermal conductive performance
Table 1 Influence of the elements number on the calculated CTCs of 3D braided composites (q ¼ 20 , Vf ¼0.35). Sum of elements 32
256
1323
2035
3240
4000
5203
lx ¼ ly ðW=m CÞ 0.2514 0.1951 0.2259 0.2286 0.2304 0.2321 0.2311 2.3695 1.8334 1.9897 1.9451 2.0320 2.0633 2.0442 lz ðW=m CÞ
Table 2 Comparison of the CTCs for 3D four-directional braided composites obtained from calculations and experiments. Fiber volume fraction
Braiding angle (º)
45% 53% 54%
48 42 41
lx ¼ ly ðW=m CÞ
lz ðW=m CÞ
Experiments [12]
Predictions
Predictions
0.770 0.750 0.694
0.712 0.657 0.563
1.239 1.334 1.356
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Z Q¼ g
½BT ½km dg
(16)
e
Here, the index e indicating a given element quantities from the meshed RUC domain. ½km is the thermal conductivity tensor. [B] is a matrix containing the spatial derivatives of the shape functions. It should be noted that Q is not a vector but a matrix. Each column of the matrix wj ðyÞ corresponds to a column of the matrix Q. In order to solve the matrix wj ðyÞ in Eq. (14), the MPFE method [6] was used in this work. Up to this stage, once the wj ðyÞ have been determined, then the homogenized effective conductivity tensor Kij ðx; yÞ will be known, and it is possible to recover the microscale response under macroscale boundary conditions.
3. Mesh convergence
Fig. 2. Boundary conditions used for macrostructure model.
Finite element solutions are more accurate with finer meshes, and this requirement also applies to this multiscale FE simulations. In the present work, a convergent test for 3D four-directional braided composites is carried out and the CTCs of the components used in the following simulation are excerpted from Ref. [6]. The calculated effective CTCs of 3D four-directional braided
Fig. 3. Temperature distributions in macrostructure under boundary condition I.
ð0Þ
Fig. 4. Heat flux qZ
distributions in macrostructure under boundary condition I.
J.-j. Zhai et al. / Composites Science and Technology 151 (2017) 317e324
ð0Þ
Fig. 5. Localization heat flux qi
321
ð0Þ
ð0Þ
ð0Þ
distributions in the three directions of the specific RUC under boundary condition I: (a) qX ,(b) qY ,(c) qZ .
composites (q ¼ 20 , Vf ¼ 0.35) with different number of elements are shown in Table 1. As can be seen in Table 1, the CTCs vary little as the mesh is refined beyond the level where it contains about 3240 elements. The obtained numerical results shown that the CTCs of 3D four-directional carbon/epoxy braided composites are independent of mesh refinement in this work. 4. Application and discussions 4.1. Coefficient of thermal conductivity For the 3D four-directional braided composites studied in this
work, the matrix is epoxy resin TDE-86 and the fiber is T300 carbon [6]. The effective CTCs evaluated by multiscale FE approach are compared with the experimental data [12] as shown in Table 2. The relatively good agreement gives a strong support to the reliability of the numerical method. 4.2. Thermal conductive behaviors under different temperature boundary conditions In order to fully exploit the advantages of the 3D braided composites, the thermal conductive behaviors of 3D braided composite are investigated by the extended multiscale FE approach. To
Fig. 6. Temperature distributions in macrostructure under boundary condition II: (a) external contours, (b) internal contours.
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ð0Þ
Fig. 7. Heat flux qi
ð0Þ
ð0Þ
ð0Þ
ð0Þ
distributions in the three directions of macrostructure under boundary condition II: (a) qX ,(b) qY ,(c) qZ ,(d) internal contours of. qZ .
complete the multiscale procedure, the definition of the homogenised heat conduction properties of the macroscale material is done using the homogenised thermal conductivity matrix, which results from the homogenization procedure. Considering two kinds of boundary conditions as shown in Fig. 2, a 3D braided composite macrostructure with the dimensions of 15 15 50 mm is studied in this work. The braiding angle and fiber volume fraction are q ¼ 20 and Vf ¼ 0.35, respectively. 4.2.1. Temperature boundary conditions I: macrostructure with adiabatic surfaces As shown in Fig. 2(a), a heating plate is assumed to be placed on the bottom surface (Z ¼ 50 mm) of the macrostructure, which provides a heat flux for going through along its axial direction. The temperature of bottom surface is assumed to be T and do not change with time. The symmetrical surface (Z ¼ 0 mm) is assumed to be mounted on a cool plate, which provides a heat sink. The plate is water cooled to a constant temperature Tsink. Others surfaces were assumed to be tightly wrapped and set as the adiabatic surfaces. When the constant temperature Tsink of the plate for the 3D braided composite structure is set as zero, Fig. 3 presents the temperature distribution of the 3D braided composite macrostructure when T ¼ 60 C. It can be found that temperature deð0Þ
creases gradually along the Z direction. Fig. 4 gives heat flux qZ distributions of 3D braided composite. From Fig. 4, it is found that the heat flux distributions are almost uniform in the interior of the 3D braided composite macrostructure. In order to give a detailed description of the microstructure heat flux distribution of 3D braided composites, the specific region A in the center of the braided composite is selected as shown in Fig. 2.
ð0Þ
Fig. 5 shows the localization heat flux qi distributions in the three directions (X, Y and Z) of the specific RUC, which are obtained by application of the current boundary conditions. It can be seen from ð0Þ
Figs. 4 and 5 that the distributions of heat flux qi in microscopic coordinate system are generally larger than that in macroscopic coordinate system, the ratio of the microscale heat flux component ð0Þ
qZ to the average macroscale heat flux in Z direction can be as high as 9.44 at certain locations for this particular heterogeneous microstructure in z direction. The higher value of heat flux exists in the yarns of Z direction due to the bigger effective thermal conductivity in the specific RUC. The microscale heat flux component ð0Þ
ð0Þ
qX ¼ qY need not be zero although the average macroscale heat flux component in X and Y directions is zero, this is due to the complex heat flux patterns within the heterogeneous microstructure of 3D braided composite. The figures indicate that heat flux patterns at the microscale are complex even when the macroscopic heat flux state is very simple. 4.2.2. Temperature boundary conditions II: macrostructure with nonadiabatic surfaces Different with the boundary condition I, the surrounding surfaces are assumed as nonadiabatic boundary, and the temperature remains at room temperature (23 C), which is called boundary condition II, as shown in Fig. 2(b). Based on the temperature boundary condition II, Fig. 6 shows the temperature distribution of the 3D braided composite structure. It can be seen that distribution of temperature is hourglass shape along the Z direction. Fig. 7 gives heat flux distributions in the three directions (X, Y and Z) of macrostructure. It can be found that the higher value of heat flux exists in the Z direction due to the
J.-j. Zhai et al. / Composites Science and Technology 151 (2017) 317e324
ð0Þ
Fig. 8. Localization heat flux qi
323
distributions in the Z direction of the specific RUC, which are related to the specific regions A, B and C.
larger temperature gradient, and the heat flux were not uniform distributions at the cross section of 3D braided composite macrostructure. Similarly, three specific regions A, B, C are selected in order to ð0Þ
describe the microscale heat flux qi distributions of 3D braided composites exactly, as shown in Fig. 2(b). By applying the temð0Þ
perature boundary conditions II, the microscale heat flux qi distributions related to the specific regions A, B, C in the Z direction were achieved as shown in Fig. 8. It can be seen from Fig. 8(a) that the heat flux is mainly concentrated in the yarns and the heat flux of the braiding yarns in the center region is much higher than that in the outer region. These phenomena can also be seen from Fig. 8(b) and (c). Comparing Fig. 8(a) and 8(c), it is observed that ð0Þ
the microscale heat flux qZ near the hot source (region C) is higher than the microscale heat flux away from the hot source (region B), although this phenomenon is not apparent at the macroscale. 5. Conclusions In this paper, an explicit and efficient multiscale FE calculation procedure for describing the heat flux distribution, temperature distribution and effective thermal conductivity of 3D braided composites, based on a homogeneous macrostructure model and a heterogeneous microstructure RUC model, has been developed. Multiscale FE numerical models are established and the effective thermal conductive behaviors of 3D braided composites under two kinds of boundary conditions are predicted. The main conclusions can be listed as follows:
(1) The thermal conductive behavior of the 3D four-directional braided composites is obviously anisotropic. (2) Temperature distributions of 3D four-directional braided composite structure resulted from the boundary conditions I are obviously different from that of boundary conditions II. (3) The heat flux distributions of 3D four-directional braided composite are dependent on the boundary conditions. Larger microscale heat flux belong to the related regions of macrostructure are obtained. It is found that the heat flux in the yarn regions is bigger than that in the matrix region, and the heat flux of the braiding yarns in the center region is much higher than that in the outer region. (4) There is a good agreement between the prediction and experimental results. For future works, more experiments will be completed to prove the validity of the conclusions obtained by the present multiscale FE method. Acknowledgments The authors would like to thank the National Natural Science Foundation of China (11432005). References [1] C.L. Ma, J.M. Yang, T.W. Chou, Elastic stiffness of three-dimensional braided textile structural composites, in: Composite Materials: Testing and Design, Seventh Conference, Philadelphia, 1986, pp. 404e421. [2] J.M. Yang, C.L. Ma, T.W. Chou, Fiber inclination model of three-dimensional textile structural composites, J. Compos. Mater 20 (5) (1986) 472e483. [3] J.H. Byun, G.W. Du, T.W. Chou, Analysis and modeling of three-dimensional textile structural composites, High-Tech Fibrous Mater. 457 (1991) 22e33. [4] Z.G. Liu, H.G. Zhang, Z.X. Lu, D.S. Li, Investigation on the thermal conductivity
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