Uncovering size effects in crack compliance in nanobeams using MD simulations

Abstract

The effect of a crack on the structural response of a nanobeam is commonly modeled by introducing a discontinuity in the slope at the cracked cross-section, 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. This study reveals size effects in the crack compliance of silicon nanobeams by integrating large-scale molecular dynamics (MD) simulations involving more than 2.3 million atoms with beam formulations derived from a local/nonlocal stress-driven gradient elasticity theory. Size-dependent bending and free transverse vibration responses of intact nanobeams are obtained through MD simulations and used to calibrate the nonlocal parameters of the continuum models. The calibrated models are then employed to study cracked nanobeams. Comparisons between MD 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. We demonstrate that this size-dependent crack compliance is influenced by both the crystallographic orientation and the imposed boundary conditions. An atomic strain analysis reveals that the near-tip strain distribution in short nanobeams deviates from the classical singular form; however, for sufficiently long nanobeams (e.g., L ≈ 180 nm for a fixed–guided nanobeam under bending), discrete atomic effects become negligible, and strain distribution and crack compliance converge toward classical predictions.