Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams

TL;DR

This paper develops exact analytical solutions for the bending behavior of functionally graded nano-beams using a modified nonlocal strain gradient elasticity theory, providing benchmarks for numerical methods and insights into scale effects in nano-engineering.

Contribution

It introduces an effective analytical approach to derive exact solutions for nonlocal strain gradient nano-beams with various boundary conditions, advancing the understanding of size-dependent nano-mechanics.

Findings

Exact expressions for transverse displacements of nano-beams are derived.

New benchmarks for numerical analysis of nano-beams are established.

The model effectively characterizes scale effects in nano-engineering applications.

Abstract

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.