Size-Dependent Resonant Frequency and Flexural Sensitivity of Atomic Force Microscope Microcantilevers Based on the Modified Strain Gradient Theory

In the present study, the resonant frequency and flexural sensitivity of atomic force microscope (AFM) microcantilevers are predicted incorporating size effects. To this end, the modified strain gradient elasticity theory is applied to the classical Euler-Bernoulli beam theory to develop a non-classical beam model which has the capability to capture size-dependent behavior of microcantilevers. On the basis of Hamilton's principle, the size-dependent analytical expressions corresponding to the frequency response and sensitivity of AFM cantilevers are derived. It is observed that by increasing the contact stiffness, the resonant frequencies of AFM cantilevers firstly increase and then tend to remain constant at an especial value. Moreover, the resonant frequencies of AFM cantilevers obtained via the developed non-classical model is higher than those of the classical beam theory, especially for the values of beam thickness close to the internal material length scale parameter.