基于应变梯度理论的纳米悬臂梁的大位移分析

Large deflection analysis of a nanoscaled cantilever beam with strain gradient effect

以Aifantis发展的应变梯度理论为基础,探讨微纳米尺度下线弹性悬臂梁受集中载荷作用下的大变形问题。基于Euler-Bernoulli梁理论,考虑应变梯度的影响,建立悬臂梁发生大变形时的弹性微分方程,并给出相应的边界条件。通过打靶法并借助于Math CAD软件,求得考虑应变梯度时悬臂梁在自由端集中载荷作用下的挠度数值解。结果表明,在微纳米尺度下应变梯度对悬臂梁的变形有较大影响,弹性变形梯度系数对梁发生大变形比发生小变形时的影响更明显,且弹性梯度系数对于梁的变形有抑制作用。

In this study,we discuss the deformation of a micro / nanoscaled cantilever under a concentrated load. The modeling is based on the strain gradient theory developed by Aifantis. Based on the Euler-Bernoulli beam theory,and considering the effect of strain gradient,the governing equation of the large deformation of a cantilever was built,and the boundary conditions were given. Using the shooting method and the Math CAD software,we obtain the numerical solution of a cantilever with strain gradient,under a concentrated load at the free end. This solution is compared with that of a beam with infinitesimal deformation. The result shows that at the micro / nano scale,the strain gradient has a great effect on the cantilevers deformation. In this case,the gradient coefficient affects a beam more significantly when it is in the finite deformation than in the infinitesimal deformation. Moreover,the gradient coefficient will restrain the deformation of the beam.