Solution methods for the growth of a repeating imperfection in the line of a strut on a nonlinear foundation

TL;DR

This study investigates the growth of sinusoidal imperfections in a strut on a nonlinear foundation, revealing a maximum imperfection size that influences localization phenomena and providing new analytical insights.

Contribution

It introduces a semi-analytical, Galerkin, and numerical approach to determine the maximum imperfection size, showing its independence from the restoring force and improving previous estimates.

Findings

Existence of a maximum imperfection size leading to a limit point.

The maximum imperfection size is independent of the restoring force.

The exponent of the maximum compressive force decay differs from classical buckling.

Abstract

This paper is a theoretical and numerical study of the uniform growth of a repeating sinusoidal imperfection in the line of a strut on a nonlinear elastic Winkler type foundation. The imperfection is introduced by considering an initially deformed shape which is a sine function with an half wavelength. The restoring force is either a bi-linear or an exponential profile. Periodic solutions of the equilibrium problem are found using three different approaches: a semi-analytical method, an explicit solution of a Galerkin method and a direct numerical resolution. These methods are found in very good agreement and show the existence of a maximum imperfection size which leads to a limit point in the equilibrium curve of the system. The existence of this limit point is very important since it governs the appearance of localization phenomena. Using the Galerkin method, we then establish an exact formula for this maximum imperfection size and we show that it does not depend on the choice of the restoring force. We also show that this method provides a better estimate with respect to previous publications. The decrease of the maximum compressive force supported by the beam as a function of the imperfection magnitude is also determined. We show that the leading term of the development has a different exponent than in subcritical buckling of elastic systems, and that the exponent values depend on the choice of the restoring force.