Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory

Abstract

In this paper, the resonant frequency and sensitivity of atomic force microscope (AFM) microcantilevers are studied using the modified couple stress theory. The classical continuum mechanics is incapable of interpreting micro-structure-dependent size effects when the size of structures is in micron- and sub-micron scales. However, this dependency can be well treated by using non-classical continuum theories. The modified couple stress theory is a non-classic continuum theory which employs additional material parameters besides those appearing in classical continuum theory to treat the size-dependent behavior. In this work, writing differential equations of motion of AFM cantilevers together with appropriate boundary conditions based on the couple stress theory, the analytical expressions are derived for the natural frequency and sensitivity. According to the numerical results, when the ratio of beam thickness to the material length scale parameter is less than 10, the difference between the classical based and the couple stress based results of resonance frequencies and sensitivities is considerable. The results show the significant importance of the size effects in behavior of AFM microcantilevers.