Role of material microstructure in plate stiffness with relevance to microcantilever sensors

TLDR

Traditional beam theory based on isotropic Hooke’s law is contrasted with a micropolar elasticity model that incorporates microstructural effects. The study investigates how microstructure influences the bending stiffness of microcantilevers. Using a micropolar constitutive relation, polypropylene microcantilevers were fabricated by injection molding, their stiffness measured with a nanoindenter, and the results compared to predictions from classical beam theory. The micropolar model predicts a pronounced stiffening as thickness decreases, and nanoindentation measurements show stiffness at least four times higher than beam theory, underscoring the importance of microstructure for MEMS devices.

Abstract

This work examines the effect of microstructure upon microcantilever bending stiffness. An existing beam theory model, based upon an isotropic Hooke's law constitutive relationship, is compared to a model based upon a micropolar elasticity constitutive model. The micropolar approach introduces a bending stiffness relation which is a function of any two independent elastic constants of the Hooke's law model (e.g., the elastic modulus and the Poisson's ratio), and an additional material constant (called γ). A consequence of the additional material constant is the prediction of an increased bending stiffness as the cantilever thickness decreases—a stiffening due to the material microstructure which becomes measurable at micron-order thicknesses. Polypropylene microcantilevers, which have a non-homogeneous microstructure due to their semi-crystalline nature, were fabricated via injection molding. A nanoindenter was used to measure their stiffness. The nanoindenter-determined stiffness values, which include the effect of the additional micropolar material constant, are compared to stiffness values obtained from beam theory. The nanoindenter stiffness values are seen to be at least four times larger than the beam theory stiffness predictions. This stiffening effect has relevance in future MEMS applications which employ materials with non-homogeneous microstructures instead of the conventional MEMS materials (e.g., silicon, silicon nitride), which have a very uniform microstructure.