低维功能复合材料均匀化方法研究进展

Research progress in homogenization methods for low-dimensional functional composites

摘要

低维功能复合材料由于其优异的功能性能，已在航空航天、能源工业等领域中得到广泛应用。本研究重点讨论了低维功能复合材料有效功能性能的理论和数值均匀化模型。首先对使用较为广泛的静态、动态和数值均匀化方法进行了介绍，并对涉及到的填料、频率相关电界面效应和热界面效应进行了详细的论述。随后，介绍了低维功能复合材料的功能性能包括电性能、电磁屏蔽性能、储能性能、热性能和多场耦合性能的理论及数值均匀化模型。最后，对低维功能复合材料未来的发展方向进行了展望。

Abstract

Low-dimensional functional composites due to their exceptional functional properties have found critical applications in various fields such as aerospace and energy industries. This review focuses on theoretical and numerical homogenization models for the effective functional properties of low-dimensional functional composites. First，an introduction to widely used static，dynamic，and numerical homogenization methods is provided，along with a detailed discussion of the involved fillers，frequency-dependent electric interfacial effects，and thermal interfacial effects. Then，theoretical and numerical homogenization models for the functional properties of low-dimensional functional composites are presented，including electrical properties，electromagnetic shielding performance，energy storage properties，thermal properties and multi-field coupling properties. Last，future development directions for low-dimensional functional composites are prospected.

关键词

低维功能复合材料 / 功能性能 / 均匀化方法 / 理论及数值模型

Key words

low-dimensional functional composite / functional property / homogenization method / theoretical and numerical models

参考文献

[1] 倪虹， 白云， 郭霞， 等. 碳纳米复合材料力学性能测试技术的研究及展望[J]. 分析仪器， 2023（6）： 1-9.

[2] NI Hong， BAI Yun， GUO Xia， et al. Research and prospects of mechanical property testing techniques for carbon nanocomposites[J]. Analytical instrumentation， 2023（6）： 1-9（in Chinese）.

[3] UYOR U O， POPOOLA A P， POPOOLA O， et al. Energy storage and loss capacity of graphene-reinforced poly（vinylidene fluoride）nanocomposites from electrical and dielectric properties perspective：a review[J]. Advances in polymer technology， 2018， 37（8）： 2838-2858.

[4] 钱昀达， 马欣雨， 张国强， 等. 二维富勒烯分子网络结构稳定性及电子特性[J]. 应用力学学报， 2025， 42（1）： 99-108.

[5] QIAN Yunda， MA Xinyu， ZHANG Guoqiang， et al. Structural stability and electronic properties of two-dimensional fullerene molecular network structures[J]. Chinese journal of applied mechanics， 2025， 42（1）： 99-108（in Chinese）.

[6] DRESSELHAUS M S， DRESSELHAUS G， SAITO R. Physics of carbon nanotubes[J]. Carbon， 1995， 33（7）： 883-891.

[7] RAMANATHAN T， ABDALA A A， STANKOVICH S， et al. Functionalized graphene sheets for polymer nanocomposites[J]. Nature nanotechnology， 2008， 3（6）： 327-331.

[8] BERA S， SINGH M， THANTIRIGE R， et al. 2D-nanofiller-based polymer nanocomposites for capacitive energy storage applications[J]. Small science， 2023， 3（7）： 2300016.

[9] DANG Z M， ZHENG M S， ZHA J W. 1D/2D carbon nanomaterial-polymer dielectric composites with high permittivity for power energy storage applications[J]. Small， 2016， 12（13）： 1688-1701.

[10] CHEN S， SKORDOS A， THAKUR V K. Functional nanocomposites for energy storage：chemistry and new horizons[J]. Materials today chemistry， 2020， 17： 100304.

[11] 王静， 李闯， 耿闻， 等. 碳纤维复合材料高温界面性能研究进展[J]. 高分子材料科学与工程， 2024， 40（3）： 163-171.

[12] WANG Jing， LI Chuang， GENG Wen， et al. Progress of high temperature interfacial properties of carbon fiber composites[J]. Polymer materials science & engineering， 2024， 40（3）： 163-171（in Chinese）.

[13] 王敏洁， 沈将华， 陈彪， 等. 铝基碳纳米管增强复合材料的动态力学行为及应变率效应[J]. 现代应用物理， 2021， 12（4）： 149-155.

[14] WANG Minjie， SHEN Jianghua， CHEN Biao， et al. Dynamic mechanical behavior and strain rate effect of al-based carbon nanotubes reinforced composites[J]. Modern applied physics， 2021， 12（4）： 149-155（in Chinese）.

[15] XIA X D， LIU Y， LI J， et al. Review and perspective on the calculations of mechanical and functional properties of low-dimensional nanocomposites[J]. Journal of micromechanics and molecular physics， 2021， 6（4）： 67-87.

[16] 容敏智， 章明秋， 郑永祥， 等. 纳米SiO₂增韧增强聚丙烯的界面效应与逾渗行为 [J]. 复合材料学报， 2002， 19（ 1）： 1-4.

[17] RONG Minzhi， ZHANG Mingqiu， ZHENG Yongxiang， et al. Interfacial effects and percolation behavior in silica nanoparticles filled polypropylene composites[J]. Acta materiae compositae Sinica， 2002， 19（ 1）： 1-4（in Chinese）.

[18] JIA T M， HAO Y L， QI X S， et al. Interface engineering and impedance matching strategy to develop core@shell urchin-like NiO/Ni@carbon nanotubes nanocomposites for microwave absorption[J]. Journal of materials science & technology， 2024， 176： 1-12.

[19] YU Z H， LI F， XIANG Q J. Carbon dots-based nanocomposites for heterogeneous photocatalysis[J]. Journal of materials science & technology， 2024， 175： 244-257.

[20] ZHENG M， SONG E， LIU X. Transmission properties of anisotropic pentamode material and filtering application on elastic wave[C]// The 21st International Conference on Composite Materials. Shaanxi：Chinese Society for Composite Materials， 2017： 362-363.

[21] MORI T， TANAKA K. Average stress in matrix and average elastic energy of materials with misfitting inclusions[J]. Acta metallurgica， 1973， 21（5）： 571-574.

[22] BRUGGEMAN V D. The calculation of various physical constants of heterogeneous substances. I. The dielectric constants and conductivities of mixtures composed of isotropic substances[J]. Annals of physics， 1935， 416： 636-791.

[23] CASTAÑEDA P P， WILLIS J R. The effect of spatial distribution on the effective behavior of composite materials and cracked media[J]. Journal of the mechanics and physics of solids， 1995， 43（12）： 1919-1951.

[24] WENG G J. A dynamical theory for the Mori-Tanaka and Ponte Castañeda-Willis estimates[J]. Mechanics of materials， 2010， 42（9）： 886-893.

[25] HASHIN Z， SHTRIKMAN S. A variational approach to the theory of the elastic behaviour of polycrystals[J]. Journal of the mechanics and physics of solids， 1962， 10（4）： 343-352.

[26] STROUD D. The effective medium approximations：some recent developments[J]. Superlattices and microstructures， 1998， 23（3/4）： 567-573.

[27] JAMES CLERK-MAXWELL M A F R S. A treatise on electricity and magnetism[J]. Nature， 1873， 7（182）： 478-480.

[28] NAYAK J K， ROY CHAUDHURI P， RATHA S， et al. A comprehensive review on effective medium theories to find effective dielectric constant of composites[J]. Journal of electromagnetic waves and applications， 2023， 37（2）： 282-322.

[29] GARNETT J C M. Colours in metal glasses and in metallic films[J]. Proceedings of the Royal Society of London， 1904， 73（488/496）： 443-445.

[30] WEIGLHOFER W S， LAKHTAKIA A， MICHEL B. Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium[J]. Microwave and optical technology letters， 1997， 15（4）： 263-266.

[31] BRUGGEMAN D A G. Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitäts-konstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[J]. Annalen der physik， 1935， 416（7）： 636-664.

[32] LAKHTAKIA A， SLEPYAN G Y， MAKSIMENKO S A， et al. Effective medium theory of the microwave and the infrared properties of composites with carbon nanotube inclusions[J]. Carbon， 1998， 36（12）： 1833-1839.

[33] GARCÍA-VIDAL F J， PITARKE J M， PENDRY J B. Effective medium theory of the optical properties of aligned carbon nanotubes[J]. Physical review letters， 1997， 78（22）： 4289-4292.

[34] WEIGLHOFER W S， LAKHTAKIA A， MONZON J C. Maxwell-garnett model for composites of electrically small uniaxial objects[J]. Microwave and optical technology letters， 1993， 6（12）： 681-684.

[35] DUNN M L， TAYA M. Electromechanical properties of porous piezoelectric ceramics[J]. Journal of the American Ceramic Society， 1993， 76（7）： 1697-1706.

[36] RAJU B， HIREMATH S R， ROY MAHAPATRA D. A review of micromechanics based models for effective elastic properties of reinforced polymer matrix composites[J]. Composite structures， 2018， 204： 607-619.

[37] MARTÍNEZ-AYUSO G， FRISWELL M I， ADHIKARI S， et al. Homogenization of porous piezoelectric materials[J]. International journal of solids and structures， 2017， 113-114： 218-229.

[38] KURUKURI S， ECKARDT S. A review of homogenization techniques for heterogeneous materials[Z]. Weimar，Germany， 2004.

[39] ZAOUI A. Continuum micromechanics：survey[J]. Journal of engineering mechanics， 2002， 128（8）： 808-816.

[40] WILLIS J R. Variational principles for dynamic problems for inhomogeneous elastic media[J]. Wave motion， 1981， 3（ 1）： 1-11.

[41] WILLIS J R. The nonlocal influence of density variations in a composite[J]. International journal of solids and structures， 1985， 21（7）： 805-817.

[42] 宋娥， 刘晓宁， 胡更开. 周期介质Willis动态均匀化数值方法[C]// 中国力学大会-2017暨庆祝中国力学学会成立60周年大会论文集（A）会议论文集. 中国北京： 中国力学学会， 2017： 1063-1072.

[43] SABINA F J， GANDARRILLA-PÉREZ C A， OTERO J A， et al. Dynamic homogenization for composites with embedded multioriented ellipsoidal inclusions[J]. International journal of solids and structures， 2015， 69-70： 121-130.

[44] 潘燕环， 嵇醒， 薛松涛. 复合材料中的渐近均匀化方法[J]. 上海力学， 1997， 18（4）： 290-297.

[45] PAN Yanhuan， JI Xing， XUE Songtao. The method of asymptotic homogenization in the research of composite materials[J]. Chinese quarterly mechanics， 1997， 18（4）： 290-297（in Chinese）.

[46] 张锐， 文立华， 杨淋雅， 等. 复合材料热传导系数均匀化计算的实现方法[J]. 复合材料学报， 2014， 31（6）： 1581-1587.

[47] ZHANG Rui， WEN Lihua， YANG Linya， et al. Realization methods of computational homogenization for thermal conductivity coefficient of composites[J]. Acta materiae compositae Sinica， 2014， 31（6）： 1581-1587（in Chinese）.

[48] CHEN Q， WANG G N， PINDERA M J. Homogenization and localization of nanoporous composites-a critical review and new developments[J]. Composites part b：engineering， 2018， 155： 329-368.

[49] ALLEN M P， TILDESLEY D J. Computer simulation of liquids[M]. 2nd ed. Oxford： Oxford University Press， 2017.

[50] FRENKEL D， SMIT B. Understanding molecular simulation：from algorithms to applications[M]. 3rd ed. New York：Academic Press， 2023.

[51] METROPOLIS N， ROSENBLUTH A W， ROSENBLUTH M N， et al. Equation of state calculations by fast computing machines[J]. The journal of chemical physics， 1953， 21（6）： 1087-1092.

[52] ZENG Q H， YU A B， LU G Q. Multiscale modeling and simulation of polymer nanocomposites[J]. Progress in polymer science， 2008， 33（2）： 191-269.

[53] FIROOZ S， STEINMANN P， JAVILI A. Homogenization of composites with extended general interfaces：comprehensive review and unified modeling[J]. Applied mechanics reviews， 2021， 73（4）： 040802.

[54] RUDD R E， BROUGHTON J Q. Concurrent coupling of length scales in solid state systems[J]. Physica status solidi（b）， 2000， 217（1）： 251-291.

[55] STARR F W， GLOTZER S C. Simulations of filled polymers on multiple length scales[J]. MRS Proceedings， 2000， 661： KK4.1.

[56] CHARALAMBAKIS N. Homogenization techniques and micromechanics. A survey and perspectives[J]. Applied mechanics reviews， 2010， 63（3）： 030803.

[57] BARTHÉLÉMY J F， BIGNONNET F. The Eshelby problem of the confocal N-layer spheroid with imperfect interfaces and the notion of equivalent particle in thermal conduction[J]. International journal of engineering science， 2020， 150： 103274.

[58] WANG Y， WENG G J， MEGUID S A， et al. A continuum model with a percolation threshold and tunneling-assisted interfacial conductivity for carbon nanotube-based nanocomposites[J]. Journal of applied physics， 2014， 115（19）： 193706.

[59] DYRE J C. A simple model of ac hopping conductivity in disordered solids[J]. Physics letters a， 1985， 108（9）： 457-461.

[60] WINKLER J R， GRAY H B. Long-range electron tunneling[J]. Journal of the American Chemical Society， 2014， 136（8）： 2930-2939.

[61] WENTZEL G. Eine verallgemeinerung der quantenbedingungen für die zwecke der wellenmechanik[J]. Zeitschrift für physik， 1926， 38（6）： 518-529.

[62] KRAMERS H A. Wellenmechanik und halbzahlige quantisierung[J]. Zeitschrift für physik， 1926， 39（10）： 828-840.

[63] BRILLOUIN L. La mécanique ondulatoire de Schrödinger；une méthode générale de resolution par approximations successives[J]. Comptes rendus hebdomadaires des seances de 1′academie des sciences， 1926， 183（11）： 24-26.

[64] HUANG Z H， FEUCHTWANG T E， CUTLER P H， et al. Wentzel-Kramers-Brillouin method in multidimensional tunneling[J]. Physical review a， 1990， 41（1）： 32-41.

[65] SIMMONS J G. Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film[J]. Journal of applied physics， 1963， 34（6）： 1793-1803.

[66] WANG Y， SHAN J W， WENG G J. Percolation threshold and electrical conductivity of graphene-based nanocomposites with filler agglomeration and interfacial tunneling[J]. Journal of applied physics， 2015， 118（6）： 065101.

[67] WAGNER K W. Erklärung der dielektrischen nachwirkungsvorgänge auf grund maxwellscher vorstellungen[J]. Archiv für elektrotechnik， 1914， 2（9）： 371-387.

[68] SILLARS R W. The properties of a dielectric containing semiconducting particles of various shapes[J]. Journal of the Institution of Electrical Engineers， 1937， 80（484）： 378-394.

[69] XIA X D， ZHONG Z， WENG G J. Maxwell-Wagner-Sillars mechanism in the frequency dependence of electrical conductivity and dielectric permittivity of graphene-polymer nanocomposites[J]. Mechanics of materials， 2017， 109： 42-50.

[70] BAO W S， MEGUID S A， ZHU Z H， et al. Tunneling resistance and its effect on the electrical conductivity of carbon nanotube nanocomposites[J]. Journal of applied physics， 2012， 111（9）： 093726.

[71] KIL T， BAE J H， PARK J， et al. A multiscale modeling framework for predicting strain-dependent electrical conductivity of carbon nanotube-incorporated nanocomposites considering the electron tunneling effect[J]. Polymer composites， 2024， 45（8）： 7551-7563.

[72] LU X X， YVONNET J， DETREZ F， et al. Multiscale modeling of nonlinear electric conductivity in graphene-reinforced nanocomposites taking into account tunnelling effect[J]. Journal of computational physics， 2017， 337： 116-131.

[73] ALAMUSI， LIU Y L， HU N. Numerical simulations on piezoresistivity of CNT/polymer based nanocomposites[J]. Computers，materials & continua， 2010， 20（2）： 101-118.

[74] MANTA A， GRESIL M， SOUTIS C. Simulated electrical response of randomly distributed and aligned graphene/polymer nanocomposites[J]. Composite structures， 2018， 192： 452-459.

[75] HE D L， WANG Y， SONG S L， et al. Polymer-based nanocomposites employing Bi₂S₃@SiO₂ nanorods for high dielectric performance：understanding the role of interfacial polarization in semiconductor-insulator core-shell nanostructure [J]. Composites science and technology， 2017， 151： 25-33.

[76] BLAUCH D N， SAVEANT J M. Dynamics of electron hopping in assemblies of redox centers. Percolation and diffusion[J]. Journal of the American Chemical Society， 1992， 114（9）： 3323-3332.

[77] GODOVSKI D Y. Electron behavior and magnetic properties of polymer nanocomposites[M]// Thermal and Electrical Conductivity of Polymer Materials. Berlin，Heidelberg：Springer Berlin Heidelberg， 1995： 79-122.

[78] MOTT N F. Conduction in non-crystalline materials：III. Localized states in a pseudogap and near extremities of conduction and valence bands[J]. The philosophical magazine：a journal of theoretical experimental and applied physics， 1969， 19（160）： 835-852.

[79] ÉFROS A L， SHKLOVSKII B I. Coulomb gap and low temperature conductivity of disordered systems[J]. Journal of physics c：solid state physics， 1975， 8（4）： L49-L51.

[80] JONSCHER A K. A new understanding of the dielectric relaxation of solids[J]. Journal of materials science， 1981， 16（8）： 2037-2060.

[81] CHANNEL C V， JOG J. Dielectric relaxation in PVDF/BaTiO₃ nanocomposites [J]. Express polymer letters， 2008， 4（2）： 294-301.

[82] DEBYE P. Results of a kinetic theory of dielectrics[J]. Physikalische zeitschrift， 1912， 13： 976-1000.

[83] COLE K S， COLE R H. Dispersion and absorption in dielectrics I. Alternating current characteristics[J]. The journal of chemical physics， 1941， 9（4）： 341-351.

[84] FRIEDRICH C， BRAUN H. Generalized cole-cole behavior and its rheological relevance[J]. Rheologica acta， 1992， 31（4）： 309-322.

[85] HAVRILIAK S， NEGAMI S. A complex plane representation of dielectric and mechanical relaxation processes in some polymers[J]. Polymer， 1967， 8： 161-210.

[86] CHAURASIA A K， REN X， SEIDEL G D. Computational micromechanics analysis of electron hopping and interfacial damage induced piezoresistive response in carbon nanotube-polymer nanocomposites[J]. Smart materials and structures， 2014， 23（7）： 075023.

[87] CHAURASIA A K， SEIDEL G D. Computational micromechanics analysis of electron-hopping-induced conductive paths and associated macroscale piezoresistive response in carbon nanotube-polymer nanocomposites[J]. Journal of intelligent material systems and structures， 2014， 25（17）： 2141-2164.

[88] RAMOS A， PEZZIN S H， FARIAS H D， et al. Conductivity analysis of epoxy/carbon nanotubes composites by dipole relaxation and hopping models[J]. Physica b：condensed matter， 2016， 499： 57-63.

[89] ZHAO H， LI Y， BRINSON L C， et al. Dielectric spectroscopy analysis using viscoelasticity-inspired relaxation theory with finite element modeling[J]. IEEE transactions on dielectrics and electrical insulation， 2017， 24（6）： 3776-3785.

[90] HERNÁNDEZ M， CARRETERO-GONZÁLEZ J， VERDEJO R， et al. Molecular dynamics of natural rubber/layered silicate nanocomposites as studied by dielectric relaxation spectroscopy[J]. Macromolecules， 2010， 43（2）： 643-651.

[91] GHORBEL N， KALLEL A， BOUFI S. Molecular dynamics of poly（vinyl alcohol）/cellulose nanofibrils nanocomposites highlighted by dielectric relaxation spectroscopy[J]. Composites part a：applied science and manufacturing， 2019， 124： 105465.

[92] RUAN K P， SHI X T， GUO Y Q， et al. Interfacial thermal resistance in thermally conductive polymer composites：a review[J]. Composites communications， 2020， 22： 100518.

[93] LÜ F， QIN M M， ZHANG F， et al. High cross-plane thermally conductive hierarchical composite using graphene-coated vertically aligned carbon nanotubes/graphite[J]. Carbon， 2019， 149： 281-289.

[94] KAPITZA P. The study of the specific resistance of bismuth crystals and its change in strong magnetic fields and some allied problems[J]. Proceedings of the royal society of london. series a，containing papers of a mathematical and physical character， 1928， 119（782）： 358-443.

[95] YANG X T， ZHU J H， YANG D， et al. High-efficiency improvement of thermal conductivities for epoxy composites from synthesized liquid crystal epoxy followed by doping BN fillers[J]. Composites part b：engineering， 2020， 185： 107784.

[96] SWARTZ E T. Solid-solid thermal boundary resistance[D]. Ithaca： Cornell University， 1987.

[97] PRASHER R S， PHELAN P E. A scattering-mediated acoustic mismatch model for the prediction of thermal boundary resistance[J]. Journal of heat transfer， 2001， 123（1）： 105-112.

[98] FOYGEL M， MORRIS R D， ANEZ D， et al. Theoretical and computational studies of carbon nanotube composites and suspensions：electrical and thermal conductivity[J]. Physical review b， 2005， 71（10）： 104201.

[99] LI Y， WEI W， WANG Y， et al. Construction of highly aligned graphene-based aerogels and their epoxy composites towards high thermal conductivity[J]. Journal of materials chemistry c， 2019， 7（38）： 11783-11789.

[100] MA J K， SHANG T Y， REN L L， et al. Through-plane assembly of carbon fibers into 3D skeleton achieving enhanced thermal conductivity of a thermal interface material[J]. Chemical engineering journal， 2020， 380： 122550.

[101] ZENG X L， SUN J J， YAO Y M， et al. A combination of boron nitride nanotubes and cellulose nanofibers for the preparation of a nanocomposite with high thermal conductivity[J]. ACS nano， 2017， 11（5）： 5167-5178.

[102] HAMILTON R L， CROSSER O K. Thermal conductivity of heterogeneous two-component systems[J]. Industrial & engineering chemistry fundamentals， 1962， 1（3）： 187-191.

[103] HASSELMAN D P H， JOHNSON L F. Effective thermal conductivity of composites with interfacial thermal barrier resistance[J]. Journal of composite materials， 1987， 21（6）： 508-515.

[104] WANG J J， YI X S. Effects of interfacial thermal barrier resistance and particle shape and size on the thermal conductivity of AlN/PI composites[J]. Composites science and technology， 2004， 64（10/11）： 1623-1628.

[105] SHIN H， YANG S， CHANG S， et al. Multiscale homogenization modeling for thermal transport properties of polymer nanocomposites with Kapitza thermal resistance[J]. Polymer， 2013， 54（5）： 1543-1554.

[106] HE B， MORTAZAVI B， ZHUANG X Y， et al. Modeling Kapitza resistance of two-phase composite material[J]. Composite structures， 2016， 152： 939-946.

[107] KOTHARI R， SUN C T， DINWIDDIE R， et al. Experimental and numerical study of the effective thermal conductivity of nano composites with thermal boundary resistance[J]. International journal of heat and mass transfer， 2013， 66： 823-829.

[108] ALOSIOUS S. Study of Kapitza resistance in nano-confined fluids[D]. Chennai：Indian Institute of Technology， 2021.

[109] PAPAVASSILIOU D V， BUI K， NGUYEN H. Thermal boundary resistance effects in carbon nanotube composites[M]//SILVESTRE N. Advanced Computational Nanomechanics. Hoboken，NJ，USA：John Wiley & Sons，Inc， 2016： 281-294.

[110] BUI K， GRADY B P， SAHA M C， et al. Effect of carbon nanotube persistence length on heat transfer in nanocomposites：a simulation approach[J]. Applied physics letters， 2013， 102（20）： 203116.

[111] 马连华， 周靖博， 曹亚阔. 界面效应对纳米多孔材料宏观弹塑性力学性能的影响[J]. 河北大学学报（自然科学版）， 2021， 41（2）： 161-165.

[112] MA Lianhua， ZHOU Jingbo， CAO Yakuo. Effects of interface effects on the macroscopic elastoplastic properties of nanoporous materials[J]. Journal of Hebei University（natural science edition）， 2021， 41（2）： 161-165（in Chinese）.

[113] XIA X D， HAO J， WANG Y， et al. Theory of electrical conductivity and dielectric permittivity of highly aligned graphene-based nanocomposites[J]. Journal of physics：condensed matter， 2017， 29（20）： 205702.

[114] ESHELBY J D. The determination of the elastic field of an ellipsoidal inclusion，and related problems[J]. Proceedings of the Royal Society of London. series a. mathematical and physical sciences， 1957， 241（1226）： 376-396.

[115] MAZAHERI M， PAYANDEHPEYMAN J， KHAMEHCHI M. A developed theoretical model for effective electrical conductivity and percolation behavior of polymer-graphene nanocomposites with various exfoliated filleted nanoplatelets[J]. Carbon， 2020， 169： 264-275.

[116] HASHEMI R， WENG G J. A theoretical treatment of graphene nanocomposites with percolation threshold，tunneling-assisted conductivity and microcapacitor effect in AC and DC electrical settings[J]. Carbon， 2016， 96： 474-490.

[117] XIA X D， WANG Y， ZHONG Z， et al. A frequency-dependent theory of electrical conductivity and dielectric permittivity for graphene-polymer nanocomposites[J]. Carbon， 2017， 111： 221-230.

[118] XIA X D， DU Z J， ZHANG J J， et al. Modeling the impact of glass transition on the frequency-dependent complex conductivity of CNT-polymer nanocomposites[J]. Mechanics of materials， 2022， 165： 104195.

[119] XIA X D， WENG G J， HOU D， et al. Tailoring the frequency-dependent electrical conductivity and dielectric permittivity of CNT-polymer nanocomposites with nanosized particles[J]. International journal of engineering science， 2019， 142： 1-19.

[120] XIA X D， ZHAO S J， WANG J， et al. Tuning the AC electric responses of decorated PDA@MWCNT/PVDF nanocomposites[J]. Composites science and technology， 2022， 222： 109398.

[121] ZHANG J J， WENG G J， XIA X D， et al. A theory of frequency dependence and sustained high dielectric constant in functionalized graphene-polymer nanocomposites[J]. Mechanics of materials， 2020， 144： 103352.

[122] GARCÍA-MACÍAS E， D'ALESSANDRO A， CASTRO-TRIGUERO R， et al. Micromechanics modeling of the electrical conductivity of carbon nanotube cement-matrix composites[J]. Composites part b：engineering， 2017， 108： 451-469.

[123] TANG Z H， LI Y Q， HUANG P， et al. A new analytical model for predicting the electrical conductivity of carbon nanotube nanocomposites[J]. Composites communications， 2021， 23： 100577.

[124] ZARE Y， RHEE K Y. Development of a conventional model to predict the electrical conductivity of polymer/carbon nanotubes nanocomposites by interphase，waviness and contact effects[J]. Composites part a：applied science and manufacturing， 2017， 100： 305-312.

[125] ZARE Y， RHEE K Y. Electrical conductivity of interphase zone in polymer nanocomposites by carbon nanotubes properties and interphase depth[J]. Journal of applied polymer science， 2021， 138（17）： 50313.

[126] ZARE Y， MUNIR M T， RHEE K Y. A novel approach to predict the electrical conductivity of nanocomposites by a weak interphase around graphene network[J]. Scientific reports， 2024， 14（1）： 21514.

[127] VEISKARAMI A， SARDARI D， MALEKIE S， et al. Computational prediction of electrical percolation threshold in polymer/graphene-based nanocomposites with finite element method[J]. Journal of polymer engineering， 2022， 42（10）： 936-945.

[128] LU X X， ZHANG A N， DUBRUNFAUT O， et al. Numerical modeling and experimental characterization of the AC conductivity and dielectric properties of CNT/polymer nanocomposites[J]. Composites science and technology， 2020， 194： 108150.

[129] MANTA A， GRESIL M， SOUTIS C. Predictive model of graphene based polymer nanocomposites：electrical performance[J]. Applied composite materials， 2017， 24（2）： 281-300.

[130] TALLMAN T N， HASSAN H. A computational exploration of the effect of alignment and aspect ratio on alternating current conductivity in carbon nanofiber-modified epoxy[J]. Journal of intelligent material systems and structures， 2020， 31（5）： 756-770.

[131] WANG S， HUANG Y F， CHANG E， et al. Evaluation and modeling of electrical conductivity in conductive polymer nanocomposite foams with multiwalled carbon nanotube networks[J]. Chemical engineering journal， 2021， 411： 128382.

[132] ZABIHI Z， ARAGHI H. Monte Carlo simulations of effective electrical conductivity of graphene/poly（methyl methacrylate）nanocomposite：Landauer-Buttiker approach[J]. Synthetic metals， 2016， 217： 87-93.

[133] FANG C， ZHANG J J， CHEN X Q， et al. Calculating the electrical conductivity of graphene nanoplatelet polymer composites by a Monte Carlo method[J]. Nanomaterials， 2020， 10（6）： 1129.

[134] MATOS M A S， TAGARIELLI V L， PINHO S T. On the electrical conductivity of composites with a polymeric matrix and a non-uniform concentration of carbon nanotubes[J]. Composites science and technology， 2020， 188： 108003.

[135] DUAN X H， ZHANG H， LIU J， et al. Optimizing the electrical conductivity of polymer nanocomposites under the shear field by hybrid fillers：insights from molecular dynamics simulation[J]. Polymer， 2019， 168： 138-145.

[136] XIA X D， WANG Y， ZHONG Z， et al. A theory of electrical conductivity，dielectric constant，and electromagnetic interference shielding for lightweight graphene composite foams[J]. Journal of applied physics， 2016， 120（8）： 085102.

[137] XIA X D， MAZZEO A D， ZHONG Z， et al. An X-band theory of electromagnetic interference shielding for graphene-polymer nanocomposites[J]. Journal of applied physics， 2017， 122（2）： 025104.

[138] XIA X D， LI Y， LONG L， et al. Modeling for the electromagnetic properties and EMI shielding of C_f/mullite composites in the gigahertz range [J]. Journal of the European Ceramic Society， 2020， 40（9）： 3423-3430.

[139] XIA X D， DU Z J， ZHANG J J， et al. The influence of ambient temperature and X-band frequency on EMI shielding performance of graphene/silica nanocomposites[J]. Mechanics of materials， 2022， 173： 104419.

[140] XIA X D， WENG G J. Dual percolations of electrical conductivity and electromagnetic interference shielding in progressively agglomerated CNT/polymer nanocomposites[J]. Mathematics and mechanics of solids， 2021， 26（8）： 1120-1137.

[141] XIA X D， LIU Y， ZHANG J J， et al. Tailoring electromagnetic interference shielding effectiveness of SiO₂-decorated MWCNT/polymer nanocomposites [J]. Mechanics of materials， 2024， 191： 104949.

[142] HUYNEN I， QUIÉVY N， BAILLY C， et al. Multifunctional hybrids for electromagnetic absorption[J]. Acta materialia， 2011， 59（8）： 3255-3266.

[143] DUBEY P， GUPTA M， BANSODP V， et al. Carbon nanotube-based nanocomposite for effective electromagnetic interference shielding in aerospace applications：a finite element study[M]//PALANIAPPAN S K，LAKSHMINARASIMHAN R，RANGAPPA S M， et al. Finite Element Analysis of Polymers and Composites. Cambridge，UK：Woodhead Publishing， 2024： 569-597.

[144] DUBEY P， GUPTA M， KUNDALWAL S I. Conductive polymer nanocomposite incorporated with carbon nanotubes for effective electromagnetic interference shielding：a numerical study[J]. Polymer composites， 2024， 45（4）： 3576-3590.

[145] PRABHU S S， BHAVANI R， THONNUTHODI G K， et al. A computational approach to determine shielding effectiveness of carbon nanotube-based nanocomposites for EMC application[J]. Computational materials science， 2017， 126： 400-406.

[146] ARJMAND M， SADEGHI S， OTERO NAVAS I， et al. Carbon nanotube versus graphene nanoribbon：impact of nanofiller geometry on electromagnetic interference shielding of polyvinylidene fluoride nanocomposites[J]. Polymers， 2019， 11（6）： 1064.

[147] SANTHOSI B V S R N， RAMJI K， RAO N B R M. Design and development of polymeric nanocomposite reinforced with graphene for effective EMI shielding in X-band[J]. Physica b：condensed matter， 2020， 586： 412144.

[148] KARIMI P， OSTOJA-STARZEWSKI M， JASIUK I. Experimental and computational study of shielding effectiveness of polycarbonate carbon nanocomposites[J]. Journal of applied physics， 2016， 120（14）： 145103.

[149] XUE J M， YIN X W， YE F， et al. Theoretical prediction and experimental verification on EMI shielding effectiveness of dielectric composites using complex permittivity[J]. Ceramics international， 2017， 43（18）： 16736-16743.

[150] LI J Y， ZHANG L， DUCHARME S. Electric energy density of dielectric nanocomposites[J]. Applied physics letters， 2007， 90（13）： 132901.

[151] XIA X D， XU B X， XIAO X Z， et al. Modeling the dielectric breakdown strength and energy storage density of graphite-polymer composites with dielectric damage process[J]. Materials & design， 2020， 189： 108531.

[152] WANG L W， HUANG X Y， ZHU Y K， et al. Enhancing electrical energy storage capability of dielectric polymer nanocomposites via the room temperature coulomb blockade effect of ultra-small platinum nanoparticles[J]. Physical chemistry chemical physics， 2018， 20（7）： 5001-5011.

[153] PERUANI F， SOLOVEY G， IRURZUN I M， et al. Dielectric breakdown model for composite materials[J]. Physical review e， 2003， 67（6）： 066121.

[154] CHAITANYA PITIKE K， HONG W. Phase-field model for dielectric breakdown in solids[J]. Journal of applied physics， 2014， 115（4）： 044101.

[155] HONG W， PITIKE K C. Modeling breakdown-resistant composite dielectrics[J]. Procedia IUTAM， 2015， 12： 73-82.

[156] SHEN Z H， WANG J J， LIN Y H， et al. High-throughput phase-field design of high-energy-density polymer nanocomposites[J]. Advanced materials， 2018， 30（2）： 1704380.

[157] SHEN Z H， WANG J J， JIANG J Y， et al. Phase-field modeling and machine learning of electric-thermal-mechanical breakdown of polymer-based dielectrics[J]. Nature communications， 2019， 10（1）： 1843.

[158] WANG Y F， WANG L X， YUAN Q B， et al. Ultrahigh energy density and greatly enhanced discharged efficiency of sandwich-structured polymer nanocomposites with optimized spatial organization[J]. Nano energy， 2018， 44： 364-370.

[159] SU Y， LI J J， WENG G J. Theory of thermal conductivity of graphene-polymer nanocomposites with interfacial Kapitza resistance and graphene-graphene contact resistance[J]. Carbon， 2018， 137： 222-233.

[160] WANG J， LI J J， WENG G J， et al. The effects of temperature and alignment state of nanofillers on the thermal conductivity of both metal and nonmetal based graphene nanocomposites[J]. Acta materialia， 2020， 185： 461-473.

[161] WANG J， LI C， LI J， et al. A multiscale study of the filler-size and temperature dependence of the thermal conductivity of graphene-polymer nanocomposites[J]. Carbon， 2021， 175： 259-270.

[162] LI C， WANG J， SU Y. A dual-role theory of the aspect ratio of the nanofillers for the thermal conductivity of graphene-polymer nanocomposites[J]. International journal of engineering science， 2021， 160： 103453.

[163] HASSANZADEH-AGHDAM M K， MAHMOODI M J， JAMALI J. Effect of CNT coating on the overall thermal conductivity of unidirectional polymer hybrid nanocomposites[J]. International journal of heat and mass transfer， 2018， 124： 190-200.

[164] HASSANZADEH-AGHDAM M K， MAHMOODI M J， JAMALI J， et al. A new micromechanical method for the analysis of thermal conductivities of unidirectional fiber/CNT-reinforced polymer hybrid nanocomposites[J]. Composites part b：engineering， 2019， 175： 107137.

[165] HE J， WANG H， QU Q Q， et al. Construction of interconnected SiC particles attached rGO structure in epoxy composites to achieve significant thermal conductivity enhancement[J]. Materials today communications， 2020， 25： 101584.

[166] CONG C N， CHEN Y Q， HUANG Z P， et al. A micromechanics model to predict effective thermal conductivity of rGO/MMT/polymer composites[J]. Journal of applied physics， 2021， 129（15）： 155108.

[167] NGO I L， BYON C， LEE B J. Analytical study on thermal conductivity enhancement of hybrid-filler polymer composites under high thermal contact resistance[J]. International journal of heat and mass transfer， 2018， 126，Part B： 474-484.

[168] NGO I L， PRABHAKAR VATTIKUTI S V， BYON C. A modified Hashin-Shtrikman model for predicting the thermal conductivity of polymer composites reinforced with randomly distributed hybrid fillers[J]. International journal of heat and mass transfer， 2017， 114： 727-734.

[169] FÉREC J， BERTEVAS E， KHOO B C， et al. Thermal or electrical bulk properties of rod-filled composites[J]. International journal of engineering science， 2018， 133： 219-230.

[170] LIU B K， VU-BAC N， ZHUANG X Y， et al. Stochastic multiscale modeling of heat conductivity of polymeric clay nanocomposites[J]. Mechanics of materials， 2020， 142： 103280.

[171] NGO I L， BYON C. A novel correlation for predicting the thermal conductivity of heterogeneous nanofiller polymer composites under effects of thermal contact resistance[J]. International journal of heat and mass transfer， 2017， 106： 539-545.

[172] QIAN L J， PANG X M， ZHOU J Q， et al. Theoretical model and finite element simulation on the effective thermal conductivity of particulate composite materials[J]. Composites part b：engineering， 2017， 116： 291-297.

[173] RAZZAGHI L， KHALKHALI M， RAJABPOUR A， et al. Effect of graphene and carbon-nitride nanofillers on the thermal transport properties of polymer nanocomposites：a combined molecular dynamics and finite element study[J]. Physical review E， 2021， 103（1）： 013310.

[174] ZHANG Y F， ZHAO Y H， BAI S L， et al. Numerical simulation of thermal conductivity of graphene filled polymer composites[J]. Composites part b：engineering， 2016， 106： 324-331.

[175] DI PIERRO A， MORTAZAVI B， NOORI H， et al. A multiscale investigation on the thermal transport in polydimethylsiloxane nanocomposites：graphene vs. borophene[J]. Nanomaterials， 2021， 11（5）： 1252.

[176] LI M Y， ZHOU H L Z， ZHANG Y， et al. Effect of defects on thermal conductivity of graphene/epoxy nanocomposites[J]. Carbon， 2018， 130： 295-303.

[177] MATSUBARA H， OHARA T. Effect of the in-plane aspect ratio of a graphene filler on anisotropic heat conduction in paraffin/graphene composites[J]. Physical chemistry chemical physics， 2021， 23（21）： 12082-12092.

[178] CHEN S H， LIU Q， GORBATIKH L， et al. Does thermal percolation exist in graphene-reinforced polymer composites？A molecular dynamics answer[J]. The journal of physical chemistry C， 2021， 125（1）： 1018-1028.

[179] CHINKANJANAROT S， TOMASI J M， KING J A， et al. Thermal conductivity of graphene nanoplatelet/cycloaliphatic epoxy composites：multiscale modeling[J]. Carbon， 2018， 140： 653-663.

[180] WANG Y F， CHANG Z C， GAO K， et al. Designing high thermal conductivity of polydimethylsiloxane filled with hybrid h-BN/MoS₂ via molecular dynamics simulation [J]. Polymer， 2021， 224： 123697.

[181] XIA X D， ZHAO S J， FANG C， et al. Modeling the strain-dependent electrical resistance and strain sensitivity factor of CNT-polymer nanocomposites[J]. Mathematical methods in the applied sciences， 2020： 1-15.

[182] XIA X D， ZHAO S J， ZHANG J J， et al. A unified investigation into the tensile and compressive sensing performance in highly sensitive MWCNT/epoxy nanocomposite strain sensor through loading-dependent tunneling distance[J]. Composites science and technology， 2022， 230，Part 1： 109723.

[183] XIA X D， ZHAO S J， ZHANG J J， et al. Revealing the time-dependent electromechanically coupled performances of viscoelastic MWCNT/polyethylene nanocomposite stress sensors[J]. Composites science and technology， 2024， 247： 110417.

[184] XIA X D， NIEKAMP R， BRANDS D， et al. A hybrid computational scheme on electromechanically coupled behaviors of aligned MWCNT/polymer nanocomposite sensors with strain-dependent tunneling effect[J]. Archive of applied mechanics， 2023， 93（12）： 4305-4325.

[185] ZHANG J Z， FENG C， FAN Y C， et al. Improved thermoelectric properties of graphene reinforced multiphase cement composites：experiments and modeling[J]. Journal of sustainable cement-based materials， 2024， 13（10）： 1452-1469.

[186] ALIAN A R， MEGUID S A. Multiscale modeling of the coupled electromechanical behavior of multifunctional nanocomposites[J]. Composite structures， 2019， 208： 826-835.

[187] CHEN X， ALIAN A R， MEGUID S A. Coupled electromechanical modeling of piezoresistive behavior of CNT-reinforced nanocomposites with varied morphology and concentration[J]. European journal of mechanics-a/solids， 2020， 84： 104053.

[188] MATOS M A S， TAGARIELLI V L， BAIZ-VILLAFRANCA P M， et al. Predictions of the electro-mechanical response of conductive CNT-polymer composites[J]. Journal of the mechanics and physics of solids， 2018， 114： 84-96.

[189] ALIDOUST A， HAGHGOO M， ANSARI R Z， et al. A finite element percolation tunneling approach on the electrical properties of carbon nanotube elastomer nanocomposite pressure sensors[J]. Composites part a：applied science and manufacturing， 2024， 180： 108111.

[190] KOO G M， TALLMAN T N. Frequency-dependent alternating current piezoresistive switching behavior in self-sensing carbon nanofiber composites[J]. Carbon， 2021， 173： 384-394.

[191] HAGHGOO M， HASSANZADEH-AGHDAM M K， ANSARI R. A comprehensive evaluation of piezoresistive response and percolation behavior of multiscale polymer-based nanocomposites[J]. Composites part a：applied science and manufacturing， 2020， 130： 105735.

[192] TALLMAN T， WANG K W. An arbitrary strains carbon nanotube composite piezoresistivity model for finite element integration[J]. Applied physics letters， 2013， 102（1）： 011909.

[193] ZHU S J， WEN P F， LI Y， et al. Micromechanical model for effective thermoelectric properties of thermoelectric composite containing ellipsoidal inclusions with interfacial resistances[J]. International journal of heat and mass transfer， 2023， 217： 124713.

[194] ARORA G， PATHAK H. Fracture and elastoplastic behavior of polymer-carbon nanotube composites under thermomechanical environment：an integrated dual-scale modeling and experimental study[J]. Journal of materials engineering and performance， 2022， 31（9）： 7120-7137.

[195] CHANG S， YANG S， SHIN H， et al. Multiscale homogenization model for thermoelastic behavior of epoxy-based composites with polydisperse SiC nanoparticles[J]. Composite structures， 2015， 128： 342-353.

基金资助

国家自然科学基金资助项目(12372158)

湖南省自然科学基金资助项目(2024JJ4049)