Theoretical analysis of surface stress for a microcantilever with varying widths

A theoretical model of surface stress is developed in this paper for a microcantilever with varying widths, and a method for calculating the surface stress via static deflection, slope angle or radius at curvature of the cantilever beam is presented. This model assumes that surface stresses are uniformly distributed on one surface of the cantilever beam. Based on this stressor model and using the small deformation Euler-Bernoulli beam theory, a fourth-order ordinary differential governing equation with varying coefficients or an equivalent second-order integro-differential equation is derived. A simple approach is then proposed to determine the solution of the resulting equation, and a closed-form approximate solution with high accuracy can be obtained. For rectangular and V-shaped microfabricated cantilevers, the dependences of transverse deflection, slope and curvature of the beam on the surface stresses are given explicitly. The obtained results indicate that the zeroth order approximation of the stressor model reduces to the end force model with a linear curvature for a rectangular cantilever. For larger surface stresses, the curvature exhibits a non-linear behaviour. The predictions through the stressor model give higher accuracy than those from the end moment and end force models and satisfactorily agree with experimental data. The derived closed-form solution can serve as a theoretical benchmark for verifying numerically obtained results for microcantilevers as atomic force microscopy and micromechanical sensors.