微结构非局部效应的多重性模型及在微梁弯曲中的应用

基于Eringen非局部弹性理论，直接利用逐次逼近法推导了非局部应力场的精确表达，该精确的非局部应力可具体表示为一个无穷级数的形式. 然后以微梁的横向弯曲和纯弯曲变形为例，建立平衡方程并求解及分析了挠度受非局部效应的影响. 结果表明：根据所取非局部小尺度参数大小的不同，非局部微梁的弯曲挠度可低于也可以高于经典力学下的挠度，非局部效应的增大可提高亦可降低结构的抗弯刚度. 本文结果证明了Wang以及Lim等人分别提出的两种相反的非局部模型的各自正确性. 同时首次发现，弯曲挠度随着非局部效应的增大而上下波动且存在若干跳跃点，挠度是非局部小尺度参数的非单调函数，研究同时给出了一种确定材料非局部常数的建议途径.

A MULTI-SCALE MODEL FOR NONLOCAL EFFECTS AND ITS APPLICATION ON BENDING ANALYSIS OF MICRO-BEAMS

Exact nonlocal stress field is derived via a successive approximation approach based on Eringen’s nonlocal elasticity theory. The explicit solution of nonlocal stress is provided and it can be expressed as an infinite series. Subsequently, the transverse bending and pure bending deformations of micro-beams are taken as examples to exhibit the effect of nonlocal small scale parameter on static deflection. The equilibrium equations are solved and the results are analyzed and discussed in detail. It shows that with different nonlocal small scale parameters, nonlocal bending deflection is lower or higher than that predicted by the classical continuum mechanics, or nonlocal effect increases or decreases structural stiffness of micro-beams. Two opposite nonlocal models proposed respectively by Wang et al and Lim et al are both existence and acceptable. It is demonstrated firstly that deflection fluctuates up and down with increasing the scale parameters, and the existence of some discontinuity points is also observed. The deflection is a non-monotone function with respect to nonlocal small scale parameter. The present study also suggests a possible approach to determine the nonlocal material parameter.