功能梯度弯曲纳米梁的非局部应变梯度精确解
Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams
摘要：
采用修正的非局部应变梯度弹性理论研究了纳米梁的尺寸相关弯曲行为。根据该模型，弯矩表示为弹性弯曲曲率及其导数的积分卷积与双指数平均核。最近已经证明，这样的关系是等效的微分方程，涉及弯矩和弯曲曲率场，配备了自然的高阶边界条件的本构类型。相关的弹性静力问题的Bernoulli-Euler功能梯度纳米梁制定和解决简单的静态计划的技术利益。提出了一种有效的分析方法，并利用建立精确的表达的非局部应变梯度横向位移的双固支，悬臂，固支销和pinned-pinned纳米梁，检测因此也为数值分析的新基准。与文献结果的比较，相应的选择高阶边界条件提供和讨论。所考虑的非局部应变梯度模型可以有利地采用纳米工程问题中的尺度现象的特征。
英文摘要：
The size-dependent bending behavior of nano-beams is investigated by the modified nonlocal strain gradient elasticity theory. According to this model, the bending moment is expressed by integral convolutions of elastic flexural curvature and of its derivative with a bi-exponential averaging kernel. It has been recently proven that such a relation is equivalent to a differential equation, involving bending moment and flexural curvature fields, equipped with natural higher-order boundary conditions of constitutive type. The associated elastostatic problem of a Bernoulli-Euler functionally graded nanobeam is formulated and solved for simple statical schemes of technical interest. An effective analytical approach is presented and exploited to establish exact expressions of nonlocal strain gradient transverse displacements of doubly clamped, cantilever, clamped-pinned and pinned-pinned nano-beams, detecting thus also new benchmarks for numerical analyses. Comparisons with results of literature, corresponding to selected higher-order boundary conditions are provided and discussed. The considered nonlocal strain gradient model can be advantageously adopted to characterize scale phenomena in nano-engineering problems.
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