Due to the increasing usage of nanostructures in nanotechnology and nanodevice, the following article aims to investigate the free vibration and buckling behaviours of a Timoshenko functionally graded nanobeam under to thermal and magnetic environment. The nanoscale and microstructure effects of FG nanobeam are included to classical mechanics using nonlocal strain gradient theory. The gradation of material properties throughout the beam thickness is described by power-law function. The material properties are assumed to be temperature dependent. Considering the thermal and Lorentz forces, the equations of motion of the functionally graded Timoshenko nanobeam are obtained using the strain gradient and nonlocal elasticity theories. The transverse Lorentz force induced by the horizontal magnetic field vector is derived using Maxwell’s equations. External compressive axial and transverse point loads are included in the formulation and the motion equations are solved using a Navier-type approach. The effects nonlocal size scale, and strain gradient microstructure influence, thermal loadings and magnetic field intensities on the free vibration, transverse bending and buckling behaviours of the functionally graded nanobeam are presented. The following model can be used as benchmark to analyse the nanobeam structure under thermomagnetic field using a finite element or any other numerical method.

由于纳米结构在纳米技术和纳米设备中的使用越来越多，因此本文旨在研究Timoshenko功能梯度纳米束在热和磁环境下的自由振动和屈曲行为。使用非局部应变梯度理论，FG纳米束的纳米级和微观结构效应已包含在经典力学中。整个束厚度的材料特性等级由幂律函数描述。假定材料特性与温度有关。考虑到热力和洛伦兹力，使用应变梯度和非局部弹性理论获得了功能梯度Timoshenko纳米束的运动方程。由水平磁场矢量感应的横向洛伦兹力是使用麦克斯韦方程式导出的。公式中包括了外部压缩轴向和横向点载荷，并且使用Navier型方法求解了运动方程。提出了非局部尺寸尺度的影响，以及应变梯度微结构的影响，热载荷和磁场强度对功能梯度纳米束的自由振动，横向弯曲和屈曲行为的影响。以下模型可以用作基准，以使用有限元或任何其他数值方法分析热磁场下的纳米束结构。提出了功能梯度纳米束的横向弯曲和屈曲行为。以下模型可以用作基准，以使用有限元或任何其他数值方法分析热磁场下的纳米束结构。提出了功能梯度纳米束的横向弯曲和屈曲行为。以下模型可以用作基准，以使用有限元或任何其他数值方法分析热磁场下的纳米束结构。