Breakdown of Classical Kinematic Continuity Conditions at Cracked Cross-Sections of Nanobeams

In the existing literature, the effect of a crack in a nanobeam is commonly modeled by introducing a discontinuity in the slope at the cracked cross-sections, with the magnitude proportional to the bending moment transmitted through the section. The proportionality factor (i.e., the crack compliance) is typically derived from closed-form solutions based on classical linear elastic fracture mechanics, whose validity at micro- and nanoscale dimensions is not well established. At such small scales, atomic interactions across crack surfaces become significant and can strongly influence the kinematic fields at the cracked cross-section. Consequently, the accuracy of the majority of existing models for the mechanical behavior of cracked micro- and nanobeams remains unclear. This study examines size effects on the crack compliance of silicon nanobeams by integrating molecular dynamics simulations with beam formulations derived from a nonlocal theory. Size-dependent bending and free-vibration responses of intact nanobeams are first obtained through molecular dynamics simulations, and these results are used to calibrate the nonlocal parameters of the continuum models. The calibrated models are then employed to study cracked nanobeams. Comparisons between molecular dynamics predictions and theoretical results reveal pronounced size effects: classical formulas substantially underestimate crack-induced flexibility in nanobeams, while the discrepancy decreases with increasing beam length. For sufficiently long nanobeams, crack compliance converges toward classical predictions. These results provide direct atomistic evidence that classical kinematic continuity conditions at cracked cross sections are not generally valid at small scales and must be reformulated to accurately capture the mechanical behavior of nanobeams.