Magneto-Thermoelastic Coupling Dynamic Responses of Narrow Long Thin Plates Under Memory Effects and Size Effects
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摘要:
引入记忆依赖微分的双相滞后热弹性理论能较完善地描述非Fourier导热现象，然而迄今尚未发现该理论综合考虑微尺度效应和磁、热、弹等多场耦合效应对材料力学行为的影响。通过考虑记忆依赖效应和非局部效应修正了双相滞后广义热弹性理论，基于改进后的理论研究了受周期性变化热源作用时窄长薄板的磁-热弹性耦合问题。首先建立问题的控制方程；然后结合边界条件与初值条件，利用Laplace变换和反变换技术对该问题进行求解；最后分别考察了磁场、相位滞后、时间延迟因子、核函数、非局部效应、时间对各无量纲量的影响，为微尺度材料的动态响应提供了有力参考依据。
Abstract:The dual-phase-lag thermoelasticity theory with the memory-dependent differential can perfectly describe the phenomenon of non-Fourier heat conduction, nevertheless, it has not been comprehensively considered: the mechanical responses of materials under the size-dependent effects and the multiphysics coupling effects such as magnetic, thermal and elastic fields. A modified dual-phase-lag thermoelasticity theory with memory-dependent effects and non-local effects was established. Based upon this theory, the magneto-thermoelastic coupling problem of narrow long thin plates subjected to cyclical heat sources was investigated. First, the governing equations for the problem were formulated. Then with the boundary conditions and initial conditions, the solution to the problem was obtained through the Laplace transform and the inverse transform techniques. Finally, the influences of the magnetic field, the phase lag, the time-delay, the kernel function, non-local effect and the time on the dimensionless quantities were investigated respectively. The work provides a powerful reference for the dynamic responses of micro-scale materials.
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图 2 磁场取不同值时温度θ的变化情况
Figure 2. The variation of temperature θ for different values of the magnetic field
图 4 磁场取不同值时应力σ的变化情况
Figure 4. The variation of stress σ for different values of the magnetic field
图 3 磁场取不同值时位移u的变化情况
Figure 3. The variation of displacement u for different values of the magnetic field
图 5 核函数取不同形式时温度θ的变化情况
Figure 5. The variation of temperature θ for different forms of the kernal function
图 6 核函数取不同形式时位移u的变化情况
Figure 6. The variation of displacement u for different forms of the kernal function
图 7 核函数取不同形式时应力σ的变化情况
Figure 7. The variation of stress σ for different forms of the kernal function
图 8
$ {\tau _q} $ ，$ {\tau _\theta } $ 和$ \omega $ 取不同值时温度θ的变化情况Figure 8. The variation of temperature θ for different values of
$ {\tau _q} $ ,$ {\tau _\theta } $ and$ \omega $ 
图 9
$ {\tau _q} $ ，$ {\tau _\theta } $ 和$ \omega $ 取不同值时位移u的变化情况Figure 9. The variation of displacement u for different values of
$ {\tau _q} $ ,$ {\tau _\theta } $ and$ \omega $ 
图 10
$ {\tau _q} $ ，$ {\tau _\theta } $ 和$ \omega $ 取不同值时应力σ的变化情况Figure 10. The variation of stress σ for different values of
$ {\tau _q} $ ,$ {\tau _\theta } $ and$ \omega $ 
图 11
$ {e_0}a $ 取不同值时温度θ的变化情况Figure 11. The variation of temperature θ for different values of
$ {e_0}a $ 
图 12
$ {e_0}a $ 取不同值时位移u的变化情况Figure 12. The variation of displacement u for different values of
$ {e_0}a $ 
图 13
$ {e_0}a $ 取不同值时应力σ的变化情况Figure 13. The variation of stress σ for different values of
$ {e_0}a $ 
图 14 t取不同值时温度θ的变化情况
Figure 14. The variation of temperature θ for different values of t
图 15 t取不同值时位移u的变化情况
Figure 15. The variation of displacement u for different values of t
表 1 相关参数
Table 1. Related parameters
parameter value thermal conductivity K/(N·K−1·s−1) 386 specific heat at constant strain $ {C_E} $/($ {m^2} $/K) 383.1 Lamé constant $ \mu $/(N/m2) 3.86 × 1010 Lamé constant $ \lambda $/(N/m2) 7.76 × 1010 density $ \rho $/(kg/m3) 8 954 magnetic permeability in vacuum $ {\mu _0} $/( N·s2/C2) 1.256 × 10-6 electric permittivity in vacuum $ {\varepsilon _0} $/( C2·N−1·m−2) 10−9/(36π) reference temperature $ {T_0} $/K 293 
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