四边简支载流纳米板的磁弹性稳定问题分析

Analysis of magneto-elastic stability of current-carrying nanoplates with four edges simply supported

研究四边简支载流纳米板在电磁场及机械载荷作用下的磁弹性稳定问题。结合非局部理论与板壳磁弹性理论,导出考虑尺度效应的纳米板的磁弹性动力学方程,得出载流纳米板在磁场及机械载荷作用下的磁弹性动力稳定方程;利用伽辽金原理将稳定性方程整理为特殊函数马丢方程的标准形式,根据其系数的本征值关系,判别磁弹性稳定问题的最低失稳临界状态;通过数值模拟得到纳米板失稳临界电流密度与相关参数之间的关系图及变化规律。结果表明:磁感应强度、板长与板厚、外加机械载荷及小尺度参数均会影响纳米板的稳定性;当小尺度参数取1nm时,板长在200nm^450nm、板厚在1nm^10nm的区间内,纳米板的稳定性随着磁感应强度、板长及板厚的变化而急剧变化。在这个灵敏区间内改变磁感应强度、板长及板厚的大小,可以有效地提高纳米板的稳定性。

Under electromagnetic and mechanical loads, the magneto-elastic stability of a current-carrying nano-plate with four sides simply supported is studied. Firstly, combining the nonlocal theory and the magneto-elastic theory of plates and shells, the magneto-elastic dynamic equations of nanoscale plate which considered the scale effects are derived, and the magneto-elastic dynamic stability equation of the current-carrying nano-plate under electromagnetic and mechanical loads are obtained. Then applying Galerkin’s principle, the stability equation is expressed as the standard form of special function Mathieu equation. According to the eigenvalues relation of the coefficients in Mathieu equation, the minimum critical instable state of magneto-elastic stability problem is determined. Finally, the relationship between critical current density and related parameters of nano-plate is obtained through numerical simulation. The results show that the stability of nanoplates is affected by magnetic induction intensity, plate length and thickness, applied mechanical loads and small scale parameters. When the small scale parameter is 1 nm, the stability of the nanoplate varies sharply with the change of magnetic induction intensity, plate length in the range of 200 nm^450 nm and plate thickness in the range of 1 nm^10 nm. Changing the magnetic induction intensity, plate length and plate thickness in this sensitive range can effectively improve the stability of nanoplates.