考虑应变率的广义压电热弹理论及其应用

李吉伟; 何天虎

doi:10.6052/0459-1879-20-120

考虑应变率的广义压电热弹理论及其应用

doi: 10.6052/0459-1879-20-120

兰州理工大学理学院, 兰州 730050

基金项目: 1)国家自然科学基金资助项目(11972176)

详细信息

通讯作者:
何天虎

中图分类号: O343.6
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- 被引次数: 0
出版历程
- 收稿日期: 2020-04-15
- 刊出日期: 2020-10-10

A GENERALIZED PIEZOELECTRIC-THERMOELASTIC THEORY WITH STRAIN RATE AND ITS APPLICATION

Li Jiwei,
He Tianhu

School of Science, Lanzhou University of Technology, Lanzhou 730050, China

摘要

摘要: 工程中大量材料的形变介于弹性与黏性之间, 既具有弹性固体特性, 又具有黏性流体特点, 即为黏弹性. 黏弹性使得材料出现很多力学松弛现象, 如应变松弛、滞后损耗等行为. 在研究受热载荷作用的多场耦合问题的瞬态响应时, 考虑此类问题中的热松弛和应变松弛现象, 对准确描述其瞬态响应尤为重要. 针对广义压电热弹问题的瞬态响应, 尽管已有学者建立了考虑热松弛的广义压电热弹模型, 但迄今, 尚未计入应变松弛. 本文中, 考虑到材料变形时的应变松弛, 通过引入应变率, 在Chandrasekharaiah广义压电热弹理论的基础之上, 经拓展, 建立了考虑应变率的广义压电热弹理论. 借助热力学定律, 给出了理论的建立过程并得到了相应的状态方程及控制方程. 在本构方程中, 引入了应变松弛时间与应变率的乘积项, 同时, 分别在本构方程和能量方程中引入了热松弛时间因子. 其后, 该理论被用于研究受移动热源作用的压电热弹一维问题的动态响应问题. 采用拉普拉斯变换及其数值反变换, 对问题进行了求解, 得到了不同应变松弛时间和热源移动速度下的瞬态响应, 即无量纲温度、位移、应力和电势的分布规律, 并重点考察了应变率对各物理量的影响效应, 将结果以图形形式进行了表示. 结果表明: 应变率对温度、位移、应力和电势的分布规律有显著影响.
- 应变率 /
- 松弛时间 /
- 非傅里叶效应 /
- 瞬态响应 /
- 压电热弹性
Abstract: The deformation of a large number of materials in engineering is between elasticity and viscosity, which exhibits both features of elastic solid and viscous fluid, that is, viscoelasticity. Viscoelasticity causes many mechanical relaxation phenomena, such as strain relaxation and hysteresis loss. In the investigation of transient response of multi-field problems subjected to thermal loading, it is especially important to take the phenomena such as thermal relaxation and the strain relaxation into consideration to accurately describe their transient response. For the transient response of the generalized piezoelectric-thermoelastic problems, although the generalized piezoelectric-thermoelastic model was established by taking into account the thermal relaxation, no strain relaxation is included so far. In present paper, by considering the strain relaxation in the process of deformation, a generalized piezoelectric-thermoelastic theory is theoretically established by extending Chandrasekharaiah's theory through taking strain rate into consideration. By means of the thermodynamic laws, the theory is formulated and the corresponding state equations and governing equations are obtained. In constitutive equation, a term of the product of a strain relaxation time and the strain rate is introduced, meanwhile, thermal relaxation time factors are included in constitutive equation and energy equation respectively. Subsequently, this theory is applied to investigating the dynamic response of a one-dimensional piezoelectric-thermoelastic problem subjected to a moving heat source. The Laplace transform and its numerical inverse transform are used to solve the problem, and the transient response under different strain relaxation time and heat source moving speed is obtained, that is, the distribution law of dimensionless temperature, displacement, stress and electric potential. The effect of strain rate on each physical quantity was investigated, and the results were presented in graphical form. The results show that the strain rate has a significant effect on the distributions of temperature, displacement, stress and electric potential.
- strain rate /
- relaxation times /
- non-Fourier's effect /
- transient response /
- piezoelectric-thermoelasticity

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参考文献(1)

[1]	Afrin N, Zhang Y, Chen JK. Dual-phase lag behavior of a gas-saturated porous-medium heated by a short-pulsed laser. International Journal of Thermal Sciences, 2014,75:21-27
[2]	Chen JK, Tzou DY, Beraun JE. A semiclassical two-temperature model for ultrafast laser heating. International Journal of Heat & Mass Transfer, 2006,49(1/2):307-316
[3]	Hosoya N, Kajiwara I, Inoue T, et al. Non-contact acoustic tests based on nanosecond laser ablation: Generation of a pulse sound source with a small amplitude. Journal of Sound & Vibration, 2014,333(18):4254-4264
[4]	Abd-alla AN, Giorgio I, Galantucci L, et al. Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity. Continuum Mechanics Thermo-dynamics, 2016,28(1--2):67-84
[5]	Cattaneo C. Sulla conduzione del calore. Atti Sem Mat Fis Univ Modena, 1948,3:83-101
[6]	Partap G, Chugh N. Thermoelastic damping in microstretch thermoelastic rectangular plate. Microsystem Technologies, 2017,23:5875-5886
[7]	Hosseini SM. Shock-induced nonlocal coupled thermoelasticity analysis (with energy dissipation) in a MEMS/NEMS beam resonator based on Green-Naghdi theory: A meshless implementation considering small-scale effects. Journal of Thermal Stresses, 2017,40(7-9):1134-1151
[8]	Hosseini SM. Analytical solution for nonlocal coupled thermoelasticity analysis in a heat-affected MEMS/NEMS beam resonator based on Green-Naghdi theory. Applied Mathematical Modelling, 2018,57(5):21-36
[9]	Liu S, Sun Y, Ma J, et al. Theoretical analysis of thermoelastic damping in bilayered circular plate resonators with two-dimensional heat conduction. International Journal of Mechanical Sciences, 2017: S0020740317305957
[10]	Cattaneo C. On the conduction of heat. Atti Sem Mat Fis Univ Modena, 1948,3:3-21
[11]	Vernotte P. Some possible complications in the phenomena of thermal conduction. Compte Rendus, 1961,252:2190-2191
[12]	Lord HW, Shulman Y. A generalized dynamical theory of thermoelstiicity. Journal of Mechanics and Physics of Solids, 1967,15:299-309
[13]	Green AE, Lindsay KA. Thermoelasticity. Journal of Elasticity, 1972,2(1):1-7
[14]	Green AE, Naghdi PM. On undamped heat waves in an elastic solid. Journal of Thermal Stresses, 1992,15(2):253-264
[15]	Green AE, Naghdi PM. Thermoelasticity without energy dissipation. Journal of Elasticity, 1993,31:189-20
[16]	Quintanilla R. Instability and non-existence in the nonlinear theory of thermoelasticity without energy dissipation. Continuum Mechanics & Thermodynamics, 2001,13(2):121-129
[17]	Kuang ZB. Variational principles for generalized dynamical theory of thermopiezoelectricity. Acta Mechanica, 2008,203(1--2):1-11
[18]	Wang YZ, Zhang XB, Song XN. A generalized theory of thermoelasticity based on thermomass and its uniqueness theorem. Acta Mechanica, 2014,225(3):797-808
[19]	Povstenko YZ. Fractional heat conduction equation and associated thermal stress. Journal of Thermal Stresses, 2004,28(1):83-102
[20]	Youssef HM. Theory of fractional order generalized thermoelasticity. Journal of Heat Transfer, 2010,132(6):061301
[21]	Sherief HH, El-Sayed AMA, El-Latief AM. Fractional order theory of thermoelasticity. International Journal of Solids & Structures, 2010,47(2):269-275
[22]	Ezzat MA. Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer. Physica, 2011,406(1):30-35
[23]	Yu YJ, Hu W, Tian XG. A novel generalized thermoelasticity model based on memory-dependent derivative. International Journal of Engineering Science, 2014,81:123-134
[24]	Yu YJ, Tian XG. Fractional order generalized electro-magneto-thermo-elasticity with magnetic monopoles and geometrical nonlinearity. Applied Mechanics & Materials, 2013,273:162-166
[25]	许光映, 王晋宝, 薛大文. 短脉冲激光加热分数阶导热及其热应力研究. 力学学报, 2020,52(2):491-502
[25]	( Xu Guangyin, Wang Jinbao, Xue Dawen. Investigations on the thermal behavior and associated thermal stresses of the fractional heat conduction for short pulse laser heating. Chinese Journal of Theoretical and Applied Mechanics, 2020,52(2):491-502 (in Chinese))
[26]	Sherief HH, Anwar MN. A problem in generalized thermoelasticity for an infinitely long annular cylinder. Journal of Engineering Mathematics, 1998,34(4):387-402
[27]	He TH, Tian XG, Shen YP. A generalized electromagneto-thermoelastic problem for an infinitely long solid cylinder. European Journal of Mechanics, A, Solids, 2005,24(2):349-359
[28]	张培, 何天虎. 考虑非局部效应和记忆依赖微分的广义热弹问题. 力学学报, 2018,50(3):64-72
[28]	( Zhang Pei, He Tianhu. A generalized thermoelastic problem with nonlocal effect and memory- dependent derivative. Chinese Journal of Theoretical and Applied Mechanics, 2018,50(3):508-516 (in Chinese))
[29]	李妍, 何天虎, 田晓耕. 超短激光脉冲加热薄板的广义热弹扩散问题. 力学学报, 2020,52(4):1-12
[29]	( Li Yan, He Tianhu, Tian Xiaogeng. A generalized thermoelastic diffffusion problem of thin plate heated by the ultrashort laser pulses. Chinese Journal of Theoretical and Applied Mechanics, 2020,52(4):1-12 (in Chinese))
[30]	Atashipour SA, Sburlati R. Electro-elastic analysis of a coated spherical piezoceramic sensor. Composite Structures, 2015,156:399-409
[31]	Shu C, Reed D, Button TW. A phase diagram of Ba$_{1- x}$Ca$_x$TiO$_3$ ($x =0$--0.30) piezoceramics by Raman spectroscopy. Journal of the American Ceramic Society, 2018,101(6):2589-2593
[32]	Mindlin RD. On the Equations of Motion of Piezoelectric Crystals. Radok (ed.), Problems of Continuum Mechanics. Society of Industrial and Applied Mathematics, Philadelphia, 1961: 282-290
[33]	Chandrasekharaiah DS. A generalized linear thermoelastieity theory for piezoelectric media. Acta Mechanica, 1988,71:39-49
[34]	He TH, Tian XG, Shen YP. Two-dimensional generalized thermal shock problem of a thick piezoelectric plate of infinite extent. International Journal of Engineering Science, 2002,40(20):2249-2264
[35]	Majhi MC. Discontinuities in generalized thermoelastic wave propagation in a semi-infinite piezoelectric rod. Technical Physics, 1995,36:269-278
[36]	He TH, Cao L, Li SR. Dynamic response of a piezoelectric rod with thermal relaxation. Journal of Sound and Vibration, 2007,306:897-907
[37]	Babaei MH, Chen ZT. Dynamic response of a thermo- piezoelectric rod due to a moving heat source. Smart Materials and Structures, 2009,18:1-9
[38]	Yu YJ, Xue ZN, et al. A modified Green--Lindsay thermoelasticity with strain rate to eliminate the discontinuity. Meccanica, 2018,53:2543-2554
[39]	Honig G, Hirdes U. A method for the numerical inversion of Laplace transforms. Journal of Computational and Applied Mathematics, 1984,10(1):113-132