Effect of surface stress and thermal loads on the stiffness of cantilever beams and plates

© 2013 Dr. Michael J. LachutNanomechanical doubly-clamped beams and cantilever plates are often used to sense a host of environmental processes, including biomolecular interactions, mass measurements and responses to chemical stimuli. Understanding the effects of surface stress on the stiffness of such nanoscale devices is essential for proper analysis of experimental data. In this thesis, we present a rigorous theoretical treatment of this problem, and explore both linear and nonlinear effects. In Chapter 2, we begin by examining effects of surface stress on the stiffness of cantilever plates and doubly-clamped beams, in the linear stiffness regime. The relative physical mechanisms causing a stiffness change in each case is explored. Specifically, Poisson's ratio is found to exert a dramatically different effect in cantilevers than in doubly-clamped beams, and here we explain why. The relationship between the change in spring constants and resonant frequency is also discussed. The relative change in effective spring constant is also examined, and its connection to the relative frequency shift is presented. Interestingly, this differs from what is naively expected from elementary mechanics. A discussion of the practical implications of our theoretical findings is provided, which includes an assessment of available experimental results and potential future measurements on nanoscale devices. Buckling of elastic structures can occur for loads well within the proportionality limit of their constituent materials. Given the ubiquity of beams and plates in engineering design and application, their buckling behaviour has been widely studied. However, buckling of a cantilever plate is yet to be investigated, despite the widespread use of cantilevers in modern technological developments. In chapter 3, a theoretical study into the buckling behaviour of a cantilever plate that is uniformly loaded in its plane is presented. Applications of this fundamental problem include loading due to uniform temperature and surface stress changes. This is achieved using a scaling analysis and full three-dimensional numerical solution, leading to explicit formulas for the buckling loads. Unusually, we observe buckling for both tensile and compressive loads, the physical mechanisms for which are explored. We also examine the practical implications of these results to modern developments in ultra sensitive micro- and nano-cantilever sensors, such as those composed of silicon nitride and graphene. In chapter 4, the influence of the cantilever geometry is investigated and shown to play a critical role, with V-shaped cantilevers displaying greatly enhanced sensitivity to surface stress in comparison to rectangular cantilevers. The physical mechanisms controlling the effect on V-shaped cantilever stiffness are discussed. We also investigate the buckling behaviour of V-shaped cantilevers and demonstrate that they too can buckle under tensile and compressive loads. The analysis concludes with an assessment of the stiffness and stability of V-shaped cantilevers made of silicon and graphene, under surface stress loads