Effect of the gradient on the deflection of functionally graded microcantilever beams with surface stress
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O R I G I NA L PA P E R
Xu-Long Peng · Li Zhang · Zi-Xuan Yang · Zhan-Yong Feng · Bing Zhao · Xian-Fang Li
Effect of the gradient on the deflection of functionally graded microcantilever beams with surface stress
Received: 18 March 2020 / Revised: 7 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract The surface stress-induced deflection of a microcantilever beam with arbitrary axial nonhomogeneity and varying cross section is investigated. The surface stresses are assumed to be uniformly distributed on the upper surface of the beam. Based on the small deformation and Euler–Bernoulli beam theory, the second-order integral–differential governing equation is derived. A simple Taylor series expansion method is proposed to calculate the static deformation. The approximate solution of functionally graded microbeams can degenerate into the solution of homogeneous microbeams, and the explicit expressions for the static deflection, slope angle curvature, and surface stress are derived. Particularly, the influence of the gradient parameters on the static deformation of functionally graded rectangular and triangular microbeams is presented by Figures primarily. Obtained results indicate that choosing an appropriate gradient parameter is beneficial for different surface stresses. The proposed method and derived solution can be used as a theoretical benchmark for validating the obtained results of microcantilever beams as micro-mechanical sensors and atomic force microscopy. 1 Introduction With the advantages of low cost, small size, and high performance, microcantilever structures [1–6] have been successfully used in MEMS sensors, such as resonant pressure sensors, force-balanced capacitive microacceleration sensors, and atomic force microscopy. To date, microcantilever beams [3] have become an essential tool to measure the surface properties of various materials at the atomic level [4–6]. Since surface stress plays an important role in the nanomechanical detection of surface science, the measurement of surface stress with a microcantilever beam has become a subject of considerable research [7–10]. To measure the surface stress, it is important to establish the relationship between surface stress and static deformation. The pioneering work can be traced to that of Stoney [11]. Stoney’s formula is a very important theoretical basis for measuring surface stress using microcantilever beams [7,12,13] which is a widely accepted simple quantitative relation linking the surface stress σ and radius at curvature R of a bending beam, that is: σ =
E ∗t 2 , 6R
(1)
where E ∗ = E/(1 − ν) is the biaxial elastic modulus [14] and t is the beam thickness, E and ν are Young’s modulus and Poisson’s ratio of the microcantilever beam, respectively. It is worth mentioning at this point X.-L. Peng · L. Zhang · Z.-X. Yang · Z.-Y. Feng · B. Zhao School of Civil Engineering, Changsha University of Science & Technology, Changsha 410114, Hunan, People’s Republic of China X.-F. Li (B) School of Civil Engineering, Central
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