Nonlocal Buckling of Double‐Walled Carbon Nanotubes on Kerr Foundations: An Explicit Interface–Displacement Formulation

Source abstract

ABSTRACT A double‐walled carbon nanotube resting on a Kerr‐type elastic foundation is analyzed for buckling within Eringen's differential nonlocal framework. The commonly used two‐parameter reduction of this foundation can substantially overestimate the critical buckling load because it overstates the effective shear‐layer stiffness. This study quantifies this discrepancy and identifies its mechanical origin. The foundation interface displacement is retained as an independent field with its own governing equation and natural end condition. A consistent long‐wave reduction shows that the effective shear coefficient scales with the square of the upper‐spring stiffness ratio, whereas the series‐spring reduction retains only its first power. Because the shear layer acts on the interface rather than directly on the nanotube, the explicit‐interface formulation preserves the classical free‐end conditions, whereas the reduced model modifies the shear‐force condition at a free end. The resulting ten‐state system is solved by an initial value formulation with a stabilized approximate transfer matrix and checked against published benchmarks and an independent Chebyshev spectral discretization. Over the parameter range examined, the largest overestimation occurs for the clamped–free configuration. Increasing the nonlocal parameter narrows the admissible load range of the differential nonlocal model, causing progressively fewer higher buckling branches to remain admissible.