Sequential Buckling: A Variational Analysis

Abstract

We examine a variational problem from elastic stability theory: a thin elastic strut on an elastic foundation. The strut has infinite length, and its lateral deflection is represented by $u:\R\to\R$. Deformation takes place under conditions of prescribed total shortening, leading to the variational problem \inf \left\{ \frac12 \int {u'}^2 + \int F(u) : \frac12 \int {u'}^2 = \l \right\}. Solutions of this minimization problem solve the Euler--Lagrange equation u' + pu' + F'(u) = 0, \qquad -\infty < x<\infty. The foundation has a nonlinear stress-strain relationship F', combining a destiffening character for small deformation with subsequent stiffening for large deformation. We prove that for every value of the shortening $\l > 0$ the minimization problem has at least one solution. In the limit $\l\to\infty$ these solutions converge on bounded intervals to a periodic profile that is characterized by a related variational problem. We also examine the relationship with a bifurcation branch of solutions of (0.2) and show numerically that all minimizers of (0.1) lie on this branch This information provides an interesting insight into the structure of the solution set of (0.1).