Bending and Fracture Properties of Small Scale Elastic Beams – A Nonlocal Analysis

Abstract

Using the nonlocal elasticity theory, this paper presents a static analysis of a microbeam according to the Timoshenko beam model. A fourth-order governing differential equation is derived and a general solution is suggested. For a cantilever beam at nanoscale subjected to uniform distributed loading, explicit expressions for deflection, rotation and strain energy are obtained. The nonlocal effect decreases the deflection and maximum stress distribution. With a double cantilever beam model, the strain energy release rate of a cracked beam is evaluated, and the results obtained show that the strain energy release rate is decreased (hence an increased apparent fracture toughness is measured) when the beam thickness is several times the material characteristic length. However, in the absence of a uniformly distributed loading, the nonlocal beam theory fails to account for the size-dependent properties for static analysis. Particularly, the nonlocal Euler-Bernoulli beam can be analytically obtained from the nonlocal Timoshenko beam if the apparent shear modulus is sufficiently large.