Sequential buckling : a variational analysis

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 $u:RtoR$. Deformation takes place under conditions of prescribed total shortening, leading to the variational problem begin{equation label{abstract:0 inf left{ frac12 int {u''^2 + int F(u) : frac12 int {u'^2 = l right. end{equation Solutions of this minimization problem solve the Euler-Lagrange equation begin{equation label{abstract:1 u'''' + pu'' + F'(u) = 0, qquad -infty<x<infty. end{equation The foundation has a nonlinear stress-strain relationship $F'$, combining a emph{destiffening character for small deformation with subsequent emph{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 $ltoinfty$ 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~pref{abstract:1, and show numerically that all minimizers of~pref{abstract:0 lie on this branch This information provides an interesting insight into the structure of the solution set of~pref{abstract:0.