Physical interpretation of mathematical homogenization method for thermomechanical problem
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摘要:
针对复合材料周期结构热力耦合问题，通过构造各阶摄动项的全解耦格式，推导了高阶数学均匀化方法（MHM）的数学表达式，并使用加权残量方法将其转换为易于编程实现的矩阵列式。将弹性影响函数和热影响函数分别比拟为弹性虚拟位移和热虚拟位移，通过弹性虚拟载荷和热虚拟载荷的自平衡特性、量纲分析及几何直观等角度揭示了各阶影响函数和摄动位移的物理意义，并指出二阶摄动位移对于细观结构分析的必要性。数值计算结果验证了高阶MHM矩阵列式及物理意义分析的正确性。
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关键词:
- 周期复合材料结构 /
- 数学均匀化方法(MHM) /
- 热力耦合 /
- 摄动位移 /
- 物理意义
Abstract:The mathematical expression of high-order mathematical homogenization method (MHM) is formulated by constructing decoupling form of each order perturbation for the thermomechanical problem of periodical composite structure, and it is converted into a matrix form by weighted residual method, which is convenient for use as standard finite element method. The elastic influence function and the heat influence function are respectively compared to the elastic virtual displacement and the thermal virtual displacement, and the physical interpretation of each order influence function and perturbation displacement are revealed by the self-balancing characteristics and dimensional analysis and geometric visualization. The second-order perturbation displacement is emphasized for the analysis of micro structure. The numerical results verify the correctness of high-order MHM matrix form and the analysis of physical interpretation.
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图 1 包含3块夹杂的二维周期复合材料单胞结构
Figure 1. 2D periodical composites unit cell structures with 3 inclusions
图 5 包含一块夹杂的周期复合材料杆单胞结构
Figure 5. Periodical composites rod unit cell structure with one inclusion
图 7 沿纵线A、B、C、D上节点位移曲线
Figure 7. Nodal displacement curves along longitudinal lines A, B, C and D
图 9 沿纵线A′、B′、C′、D′上节点位移曲线
Figure 9. Nodal displacement curves along longitudinal lines A′, B′, C′ and D′
表 1 摄动位移及影响函数控制方程
Table 1. Perturbation displacements and governing equation of influence functions
摄动位移 影响函数控制方程 
一阶弹性影响函数 
一阶热影响函数 
二阶弹性影响函数 
二阶热影响函数 
三阶弹性影响函数 
三阶热影响函数 
s阶弹性影响函数 
s阶热影响函数 
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表 2 二维周期复合材料单胞结构的势能泛函
Table 2. Potential energy function for 2D periodical composites unit cell
摄动位移 Π/(10-6J) 
MHM FEM MHM1 -2.475 15.8 MHM2 -2.916 -2.939 0.78 MHM3 -2.937 0.068
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表 3 二维周期复合材料多胞结构的势能泛函
Table 3. Potential energy function for 2D periodical composite multi-cell structure
摄动位移 Π/(10-8J) 
MHM FEM MHM1 -3.547 7 20 MHM2 -4.335 6 -4.436 1 2.27 MHM3 -4.396 8 0.89
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[1] BERTHELOT J M. Composite materials:Mechanical behavior and structural analysis[M]. New York:Springer, 1999. [2] KALIDINDI S R, ABUSAFIEH A.Longitudinal and transverse moduli and strengths of low angle 3-D braided composites[J]. Journal of Composite Materials, 1996, 30(8):885-905. doi: 10.1177/002199839603000802 [3] BABUSKA I.Solution of interface problems by homogenization, Parts I[J]. SIAM Journal on Mathematical Analysis, 1976, 7(5): 603-634. doi: 10.1137/0507048 [4] BENSSOUSAN A, LIONS J L.Asymptotic analysis for periodic structures[M]. Amsterdam:North-Holland, 1978. [5] STROUBOULIS T, BABUSKA I, COPPS K.The generalized finite element method:An example of its implementation and illustration of its performance[J]. International Journal for Numerical Methods in Engineering, 2000, 47(8):1401-1417. doi: 10.1002/(SICI)1097-0207(20000320)47:8<1401::AID-NME835>3.0.CO;2-8 [6] BABUSKA I, OSBOM J.Generalized finite element methods:Their performance and their relation to mixed methods[J]. SIAM Journal on Numerical Analysis, 1983, 20(3):510-536. doi: 10.1137/0720034 [7] HOU T Y, WU X H.A multiscale finite element method for elliptic problems in composite materials and porous media[J]. Journal of Computational Physics, 1997, 134(1):169-189. http://d.old.wanfangdata.com.cn/OAPaper/oai_arXiv.org_1211.3614 [8] HOU T Y, WU X H, CAI Z Q.Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients[J]. Mathematics of Computation, 1999, 68(227):913-943. doi: 10.1090/S0025-5718-99-01077-7 [9] E W, ENGQUIST B.The heterogeneous multiscale methods[J]. Communications in Mathematical Sciences, 2003, 1:87-132. doi: 10.4310/CMS.2003.v1.n1.a8 [10] E W, ENGQUIST B, LI X T, et al.Heterogeneous multiscale methods:A review[J]. Communications in Computational Physics, 2007, 2(3):367-450. http://d.old.wanfangdata.com.cn/Periodical/gtlxxb-e201101002 [11] XING Y F, YANG Y.An eigenelement method of periodical composite structures[J]. Composite Structures, 2011, 93(2):502-512. doi: 10.1016/j.compstruct.2010.08.029 [12] XING Y F, YANG Y, WANG X M.A multiscale eigenelement method and its application to periodical composite structures[J]. Composite Structures, 2010, 92:2265-2275. doi: 10.1016/j.compstruct.2009.08.006 [13] TERADA K, KURUMATANI M, USHIDAI N, et al.A method of two-scale thermo-mechanical analysis for porous solids with micro-scale heat transfer[J]. Computational Mechanics, 2010, 46(2):269-285. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=4f89028c5dee36d135f023f62a36c8eb [14] MUHAMMAD R, ERDATA N, NAOYUKI W, et al.A novel asymptotic expansion homogenization analysis for 3-D composite with relieved periodicity in the thickness direction[J]. Composites Science and Technology, 2014, 97:63-73. doi: 10.1016/j.compscitech.2014.04.006 [15] MUHAMMAD R, ERDATA N, NAOYUKI W, et al.Thermomechanical properties and stress analysis of 3-D textile composites by asymptotic expansion homogenization method[J]. Composites Part B:Engineering, 2014, 60:378-391. doi: 10.1016/j.compositesb.2013.12.038 [16] BARROQUEIRO B, DIAS-DE-OLIVEIRA J, PINHO-DA-CRUZ J, et al.Practical implementation of asymptotic expansion homogenizationin thermoelasticity using a commercial simulation software[J]. Composite Structures, 2016, 141:117-131. doi: 10.1016/j.compstruct.2016.01.036 [17] ZHAI J J, CHENG S, ZENG T, et al.Thermo-mechanical behavior analysis of 3D braided composites by multiscale finite element method[J]. Composite Structures, 2017, 176:664-672. doi: 10.1016/j.compstruct.2017.05.064 [18] 李志青, 冯永平.一类小周期结构热力耦合问题的双尺度渐近分析[J].广州大学学报, 2016, 15(2):27-32. http://d.old.wanfangdata.com.cn/Periodical/gzdxxb-zkb201602006LI Z Q, FENG Y P.Two-scale asymptotic analysis on one class of thermoelastic coupling problem in small periodic structure[J]. Chinese Journal of Guangzhou University, 2016, 15(2):27-32(in Chinese). http://d.old.wanfangdata.com.cn/Periodical/gzdxxb-zkb201602006 [19] YANG Z Q, CUI J Z, ZHOU S.Thermo-mechanical analysis of periodic porous materials with microscale heat transfer by multiscale asymptotic expansion method[J]. International Journal of Heat and Mass Transfer, 2016, 92:904-919. doi: 10.1016/j.ijheatmasstransfer.2015.09.055 [20] GUAN X F, LIU X, JIA X, et al.A stochastic multiscale model for predicting mechanical properties of fiber reinforced concrete[J]. International Journal of Solids and Structures, 2015, 56-57:280-289. doi: 10.1016/j.ijsolstr.2014.10.008 [21] YANG Z Q, CUI J Z, MA Q.The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiationin periodic porous materials[J]. Discrete and Continous Dynamical System-Series B, 2014, 19(3):827-848. doi: 10.3934/dcdsb.2014.19.827 [22] YANG Z Q, SUN Y, CUI J Z.A multiscale algorithm for heat conduction-radiation problems in porous materials with quasi-periodic structures[J]. Communications in Computational Physics, 2018, 24(1):204-233. [23] ALLAIRE G, HABIBI Z.Second order corrector in the homogenization of a conductive-radiative heat transfer problem[J]. Discrete and Continuous Dynamical System-Series B, 2013, 18(1): 1-36. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=2f321d65f48b55a82ee3504444f70394 [24] WAN X, CAO L Q, WONG Y S.Multiscale computation and convergence for coupled thermoelastic system in composite materials[J]. Multiscale Model & Simulation, 2015, 13(2):661-690. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=fe1fbc708d43da7895d1e22626940bca [25] YANG Z Q, CUI J Z, SUN Y, et al.Multiscale analysis method for thermo-mechanical performance of periodic porous materials with interior surface radiation[J]. International Journal for Numerical Methods in Engineering, 2016, 105(5):323-350. doi: 10.1002/nme.4964 [26] HAN F, CUI J Z, YU Y.The statistical second-order two-scale method for thermomechanical properties of statistically inhomogeneous materials[J]. Computational Materials Science, 2009, 46(3):654-659. doi: 10.1016/j.commatsci.2009.03.026 [27] HAN F, CUI J Z, YU Y.The statistical second-order two-scale method for mechanical properties of statistically inhomogeneous materials[J]. International Journal for Numerical Methods in Engineering, 2010, 84(8):972-988. doi: 10.1002/nme.2928 [28] XING Y F, CHEN L.Physical interpretation of multi-scale asymptotic expansion method[J]. Composite Structures, 2014, 116:694-702. doi: 10.1016/j.compstruct.2014.06.004 [29] 郑健龙, 李友云, 钱国平.多尺度计算方法-均匀化和平均化[M].北京:科学出版社, 2010.ZHENG J L, LI Y Y, QIAN G P.Multi-scale calculation methods-homogenization and averaging[M]. Beijing:Science Press, 2010(in Chinese). [30] XING Y F, GAO Y H, CHEN L, et al.Solution methods for two key problems in multiscale asymptotic expansion method[J]. Composite Structures, 2017, 160:854-866. doi: 10.1016/j.compstruct.2016.10.104 -
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