基于反常热传导的分数阶热弹耦合理论  

New insights on fractional thermoelasticity from anomalous heat conduction

作　　者：尉亚军 吴华 邓子辰 Ya-Jun Yu;Hua Wu;Zi-Chen Deng(School of Mechanics,Civil Engineering and Architecture,Northwestern Polytechnical University,Xi'an 710072,China;MIIT Key Laboratory of Dynamics and Control of Complex Systems,Northwestern Polytechnical University,Xi'an 710072,China;State Key Laboratory for Strength and Vibration of Mechanical Structures,Xi'an Jiaotong University,Xi'an 710049,China)

机构地区：[1]不详

出　　处：《Acta Mechanica Sinica》2024年第5期141-149,共9页力学学报（英文版）

基　　金：This work was supported by the National Natural Science Foundation of China(Grant No.11802242);the Fundamental Research Funds for the Central Universities(Grant No.D5000230066).

摘　　要：原子模拟和实验研究表明低维纳米材料具有反常热传导特性,建立唯象反常热弹耦合模型,研究强热传导导致的热力耦合响应,对微纳器件的安全稳定运行至关重要.本文旨在从反常热传导的视角重新审视分数阶热弹耦合模型。反常热传导由第二声效应引起,考虑波动热传导理论,建立了波动热传导与粘弹性理论的类比,阐明了Cattaneo-Vernotte模型和Green-Naghdi模型的关联.通过引入分数阶导数,构建了基于Cattaneo-Vernotte和Green-Naghdi理论的分数阶热弹耦合模型。数值结果表明:对分数阶次区间[O,1],分数阶Cattaneo-Vernotte(FCV)I模型和分数阶Green-Naghdi(FGN)I-III模型都能预测反常热弹耦合响应:比经典热弹性理论更高的温度和应力;温度和应力随时间的变化显示:FGNII模型在所有时间范围内都能得到反常响应.对Green-Naghdi模型及其分数阶模型的进一步系统研究,有助于揭示反常热传导和热弹耦合机理,并促进纳米材料的广泛应用.Anomalous heat transport of low-dimensional nanomaterial,e.g.,divergent effective thermal conductivity,has been observed from both atomistic simulations and experimental studies.It is greatly urgent to establish the phenomenological anomalous thermoelastic model,and study the thermoelastic coupling due to strong heat transfer.The aim of this work is to revisit fractional wavetype thermoelastic models from the anomalous heat conductive viewpoint.Firstly,it has been suggested that anomalous heat conduction is due to the second sound,hence wave-type heat conduction is considered:the analogy between wave-type heat conduction and viscoelastic model is given,and thus the connection between Cattaneo-Vernotte and Green-Naghdi heat conductive models is clarified.Secondly,it has been recognized that the divergent thermal conductivity of one-dimensional systems satisfies the fractional order power law,therefore fractional derivative should be incorporated:Fractional order thermoelastic models based on Cattaneo-Vernotte and Green-Naghdi theories are summarized and compared,theoretically.Numerical investigations are conducted by using Laplace transform method,and the plot of thermoelastic responses vs.fractional order parameter shows:for all fractional order range[0,1],fractional Cattaneo-Vernotte(FCV)I model and fractional Green-Naghdi(FGN)I-III models can predict anomalous thermoelastic responses,i.e.,higher temperature and compressive stress than classical thermoelasticity.Furthermore,the history of temperature or stress indicates:FGN II model can predict anomalous responses for all time range.Further systematical studies are expected for Green-Naghdi model and its fractional versions to shed light on anomalous heat conduction and thermoelastic coupling,and to facilitate the applications of nanomaterials due to such anomalous behaviors.

关 键 词：粘弹性理论 分数阶导数 热弹耦合 热弹性理论 热传导 低维纳米材料 分数阶模型 微纳器件 

分 类 号：TB383[一般工业技术—材料科学与工程]