Necessary and sufficient conditions for one-dimensional variational problems with applications to elasticity

TL;DR

This paper establishes simplified necessary and sufficient conditions for minimizers of variational problems and applies these results to stability issues in elasticity, notably solving open problems related to the stability of twisted rods and buckling solutions.

Contribution

It derives more straightforward conditions for minimizers of variational functionals and applies them to elasticity, solving previously open stability problems for Kirchhoff rods.

Findings

Simplified conditions for weak and strong minimizers.

Application to stability analysis in elasticity.

Resolution of open stability problems for twisted rods.

Abstract

This paper deals with necessary and sufficient conditions for weak and strong minimizers of functionals $\Phi(u)=\int_a^b f(x,u(x),u'(x))\,dx$, where $u\in C^1([a,b],{\mathbb R}^N)$. We first derive conditions which are simpler than the known ones, and then apply them to several particular problems, including stability problems in the elasticity theory. In particular, we solve some open problems in [A. Majumdar, A. Raisch: Stability of twisted rods, helices and buckling solutions in three dimensions, Nonlinearity 27 (2014), 2841--2867] by finding optimal conditions for the stability of a naturally straight Kirchhoff rod under various types of endpoint constraints.