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Acta Mech 219, 291307 (2011)
DOI 10.1007/s00707-011-0461-7
Received: 19 July 2010 / Revised: 17 January 2011 / Published online: 23 February 2011 Springer-Verlag 2011
Abstract When the thickness of metallic cantilever beams reduces to the order of micron, a strong size effect of mechanical behavior has been found. In order to explain the size effect in a micro-cantilever beam, the couple-stress theory (Fleck and Hutchinson, J Mech Phys Solids 41:18251857, 1993) and the C-W strain gradient theory (Chen and Wang, Acta Mater 48:39974005, 2000) are used with the help of the Bernoulli Euler beam model. The cantilever beam is considered as the linear elastic and rigid-plastic one, respectively. Analytical results of the cantilever beam deection under strain gradient effects by applying these two kinds of theories are obtained, from which we nd an explicit relationship between the intrinsic lengths introduced in the two kinds of theories. The theoretical results are further used to analyze the experimental observations, and predictions by both theories are further compared. The results in the present paper should be useful for the design of micro-cantilever beams in MEMS and NEMS.
1 Introduction
Plenty of experiments have shown that metallic material behavior displays a strong size effect when the characteristic length scale is on the order of the micron or submicron scale. In micro-torsion tests, Fleck et al. [3] observed the presence of material hardening of thin copper wires as the wire diameter decreases from 170 to 12 m; Micro-indentation and nano-indentation tests have shown an increasing material hardness with reducing indentation sizes [414]. Experiments for the particle-reinforced composites [15] have shown an obvious increase in the macroscopic ow stress through decreasing the particle diameter from 165 m to 4.5 m, while the volume fraction of the particle is kept constant. Moreover, with the development of MEMS and NEMS [1618], micro-beams have been widely applied. However, the mechanical behavior of the micro-beam cannot be described directly by the classical beam bending theory, due to the existing size effect shown in many experimental observations [1923]. Many other well-known problems also show a strong size effect, such as the increasing yield and ow stresses in polycrystalline materials with decreasing grain diameter [24,25] or the increasing fracture toughness in the micron regions ahead of a...