kernel function normalization

Although the nonlocal integral (NI) model circumvents the inconsistencies associated with the differential model, it is shown in the present study that the way its nonlocal kernel function is normalized noticeably affects the dynamic response of nanobeams. To this aim, a two-phase nonlocal integral nanobeam model with different boundary conditions and kernel functions is considered and its natural frequencies are obtained using the Rayleigh-Ritz method. Also, the kernel function is normalized via two procedures to see the influence of each one on the vibration characteristics of nanobeam. From the results it is found that kernel normalization has a significant effect on vibration response of nanobeam and therefore must be taken into account. Further, it is found that the results from each normalized model are noticeably different from the other. Furthermore, by comparing the results of continuum NI models with those from atomistic models, it is revealed that for certain normalization schemes a calibrated nonlocal parameter cannot be found due to twofold hardening-softening behavior. Moreover, the effect of kernel type, boundary conditions and mode number is thoroughly studied. The results from current study can shed light on the way of choosing or developing more reliable equivalent continuum NI models for nanostructures.