Effect of surface stress on the stiffness of thin elastic plates and beams

Nanomechanical doubly-clamped beams and cantilever plates are often used to sense a host of environmental effects, including biomolecular interations, mass measurements, and responses to chemical stimuli. Understanding the effects of surface stress on the stiffness of such nanoscale devices is essential for rigorous analysis of experimental data. Recently, we explored the effects of surface stress on cantilever plates and presented a theoretical framework valid for thin plate structures. Here, we generalize this framework and apply it to cantilever plates and doubly-clamped beams, exploring in detail the relative physical mechanisms causing a stiffness change in each case. 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 relative change in effective spring constant is also examined, and its connection to the relative frequency shift is discussed. Interestingly, this differs from what is naively expected from elementary mechanics. Finally, a discussion of the practical implications of our theoretical findings is presented, which includes an assessment of available experimental results and potential future measurements on nanoscale devices.