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.

Growth, boundedness (msc 34C11), Periodic solutions (msc 34C25), Homoclinic and heteroclinic solutions (msc 34C37), None of the above, but in MSC2010 section 49Nxx (msc 49N99), None of the above, but in MSC2010 section 49Rxx (msc 49R99), Nonlinear elasticity (msc 74B20), None of the above, but in MSC2010 section 74Gxx (msc 74G99), None of the above, but in MSC2010 section 74Hxx (msc 74H99), Bifurcation and buckling (msc 74G60), Stability (msc 74H55), Rods (beams, columns, shafts, arches, rings, etc.) (msc 74K10), Random materials and composite materials (msc 74A40), Composite and mixture properties (msc 74E30), Effective constitutive equations (msc 74Q15), Bounds on effective properties (msc 74Q20), Geophysical solid mechanics (msc 74L05), Miscellaneous topics in MSC2010 section 86Axx (msc 86A99), Soil and rock mechanics (msc 74L10), None of the above, but in MSC2010 section 74Gxx (msc 74G99), None of the above, but in MSC2010 section 74Hxx (msc 74H99), Optimization (msc 74Pxx), Geological problems (msc 86A60)
CWI
Modelling, Analysis and Simulation [MAS]

Peletier, M.A. (1999). Sequential buckling : a variational analysis. Modelling, Analysis and Simulation [MAS]. CWI.