effect of different kernel normalization procedures on vibration behaviour of calibrated nonlocal integral continuum model of nanobeams

Although the nonlocal integral (ni) model circumvents the inconsistenciesassociated with the differential model, it is shown in the present study thatthe way its nonlocal kernel function is normalized noticeably affects thedynamic response of nanobeams. to this aim, a two-phase nonlocal integralnanobeam model with different boundary conditions and kernel functions isconsidered and its natural frequencies are obtained using the rayleigh-ritzmethod. also, the kernel function is normalized via two procedures to see theinfluence of each one on the vibration characteristics of nanobeam. from theresults it is found that kernel normalization has a significant effect onvibration response of nanobeam and therefore must be taken into account.further, it is found that the results from each normalized model arenoticeably different from the other. furthermore, by comparing the resultsof continuum ni models with those from atomistic models, it is revealed thatfor certain normalization schemes a calibrated nonlocal parameter cannotbe found due to twofold hardening-softening behavior. moreover, the effectof kernel type, boundary conditions and mode number is thoroughly studied.the results from current study can shed light on the way of choosing ordeveloping more reliable equivalent continuum ni models for nanostructures.