Elastic Stability of a Cantilever Beam (Rod) Supported by an Elastic Foundation, With Application to Nano-Composites

Abstract

A simple analytical (“mathematical”) predictive model is developed with an objective to establish the condition of elastic stability for a compressed cantilever beam (rod) of finite length lying on a continuous elastic foundation. Based on the developed model, practical guidelines are provided for choosing the adequate length of the beam and/or its flexural rigidity and/or the spring constant of the foundation, so that the beam remains elastically stable. The obtained solution can be used, perhaps with some additional assumptions and modifications, for the assessment of the critical force for high-modulus and low-expansion fibers (including nano-fibers) embedded into a low-modulus and high-expansion medium (matrix). Composite systems are often fabricated at elevated temperatures and operated at lower temperature conditions. It is imperative that an embedded fiber remains elastically stable, i.e., does not buckle as a result of the thermal contraction mismatch of its material with the material of the matrix. If buckling occurs, the functional (e.g., thermal) and/or the structural (“physical”) performance of the composite might be compromised.