考虑尺度效应的微梁静态弯曲数值分析

基于修正偶应力理论及表面弹性理论,本文提出了一种新的双曲线剪切变形梁模型,用于均匀微尺度梁的静态弯曲分析。该理论可以直接利用本构关系获得横向剪切应力,满足梁顶部和底部的无应力边界条件,避免了引入剪切修正因子。根据广义Young-Laplace方程建立了梁的内部与表面层的应力连续性条件,单一的变量场可以描述梁的位移模式。通过在位移场中考虑表面层厚度以及表面层的应力连续条件,可以使新模型能够更准确地预测微尺寸和表面能相关的尺度效应。通过Hamilton原理推导出了梁的控制方程和边界条件。应变能除了考虑经典弹性理论,还要考虑微结构效应和表面能。Navier-type的解析解适用于简支边界条件,而基于拉格朗日插值的微分求积法(DQEM)可以研究在不同边界条件下的力学响应。把该数值解与Navier方法得出的解析解作了对比,得出:微尺度梁在考虑表面能或微尺寸效应、不同载荷和梁高变化下的响应一致;当不考虑微结构相关性和表面能效应时,该模型退化为经典的欧拉梁模型。

Static bending numerical analysis of microbeams considering size effect

In this paper,based on the modified couple stress theory and surface elasticity theory,a new hyperbolic shear deformation beam model is proposed for static bending analysis of uniform micro-scale beams.The theory can directly use the constitutive relation to obtain the transverse shear stress and meet the stress-free boundary conditions at the top and bottom of the beam.It avoids the need of shear correction factor.According to the generalized Young-Laplace equation,the stress continuity conditions of the inner and outer layers of the beam are established.A single variable field can describe the displacement mode of the beam.By considering the thickness of the surface layer and the continuous stress conditions of the surface layer in the displacement field,the new model can more accurately predict the scale effects related to micro-scale and surface energy.The governing equations and boundary conditions of the beam are derived by the Hamilton principle.In addition to the classical elastic theory,strain energy also includes microstructure effects and surface energy.The analytical solution of Navier-type is suitable for simple boundary conditions,and the differential quadrature method based on Lagrangian interpolation(DQEM)can study the mechanical response under different boundary conditions.Comparing the numerical solution with the analytical solution obtained by the Navier method,it is concluded that the micro-scale beam has the same response considering the surface energy or micro-size effect,different loads and beam height variations.When the microstructure correlation and surface energy effects are not considered,the model degenerates into a classical Euler beam model.