梯度型非局部高阶梁理论与非局部弯曲新解法  

GRADIENT TYPE OF NONLOCAL HIGHER-ORDER BEAM THEORY AND NEW SOLUTION METHODOLOGY OF NONLOCAL BENDING DEFLECTION

作　　者：陈玲[1] 沈纪苹[1] 李成[1] 刘鑫培[1] 

机构地区：[1]苏州大学城市轨道交通学院,江苏苏州215131

出　　处：《力学学报》2016年第1期127-134,共8页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金(11202145);苏州大学"东吴学者计划"项目;江苏省高校自然科学基金(14KJB460026)资助项目

摘　　要：针对文献中关于纳米结构刚度受非局部效应影响趋势的不一致预测,基于梯度型的非局部微分本构模型,利用迭代法及泰勒展开法求得了非局部高阶应力的无穷级数表达,非局部应力相当于经典弯曲应力与非局部挠度的逐阶梯度之和.然后推导了梯度型非局部高阶梁弯曲的挠曲轴微分方程,并结合正则摄动思想,求解了非局部挠度的表达式.最后给出数值算例,具体量化挠度受非局部尺度因子的影响大小.研究表明:相比于其经典值,纳米结构的非局部弯曲挠度可呈现出或增大或减小或不变的趋势,考虑梯度型非局部高阶应力降低或提高或不影响纳米结构的刚度,具体结果依赖于外载和边界约束的类型.算例显示外载形式和边界约束条件均各自独立地影响着纳米结构的非局部弯曲挠度,同时首次观察到非局部最大弯曲挠度的位置可能受非局部尺度因子的影响.研究结论解决了非局部弹性力学应用于纳米结构的若干疑点,并为理论的发展和优化提供支持.Different predictions were found in different literatures about the trend of nonlocal effects on nanostructural stiffness. By employing an iterative method and the Taylor expansion method, an infinite series for nonlocal high-order stress is achieved based on the gradient-type of nonlocal differential constitutive model. The nonlocal stress consists of the classical bending stress and each order gradient of nonlocal deflection. Consequently, the differential equation of the bending deflection curve for nonlocal high-order beam is derived. The nonlocal deflection is determined via the regular perturbation method. Some numerical examples are provided to reveal and quantize the effects of nonlocal scale factor on bending deflection. It is shown that compared with the corresponding classical results, the nonlocal bending deflections of nanostructure may increase, decrease or remain unchanged. The stiffness of nanostructures can be reduced or enhanced or the same as classical structures by considering the gradient-type of nonlocal high-order stress effect, and the trend depends on external loads and boundary constraints which are found to play significant roles independently in nonlocal bending of nanostructures. Moreover, it is observed for the first time that the position of maximum nonlocal deflection may be influenced by nonlocal scale factor. The present studies are expected to solve the problems in the application of nonlocal elasticity theory to nanostructures, and further provide supports for the development and optimization of such theory.

关 键 词：纳米结构 高阶应力 梯度型非局部理论 正则摄动法 

分 类 号：O331[理学—一般力学与力学基础]