Buckling and imperfection sensitivity of fluctuating one and two dimensional nanostructures

Thermal fluctuations significantly influence the mechanical behavior of low-dimensional elastic nanostructures due to their small bending stiffness. In this work, we develop a theoretical framework to investigate the buckling behavior of one- and two-dimensional flexible structures, namely, elastic rods and crystalline membranes, particularly when they experience large thermal fluctuations. Beginning with a thermally fluctuating elastic rod, we show that classical Euler buckling is recovered when geometric nonlinearities are neglected. Incorporating nonlinearities reveals substantial deviations in force–extension behavior, especially for rods with low bending stiffness. Extending the analysis to crystalline membranes, modeled through a nonlinear von Kármán elasticity of plate, we derive scaling laws for the critical buckling strain as functions of temperature, system size, and further explore their imperfection sensitivity. Our findings show that although imperfections can substantially alter the buckling threshold at zero Kelvin, their influence could be diminished at finite temperatures due to the presence of thermal fluctuations. Further, our results highlight the essential interplay between entropy-driven fluctuations and mechanical instabilities in low-dimensional systems, offering insights relevant to the design of thermally robust nanoscale materials and devices.