Buckling of a critically tapered rod: properties of some global branches of solutions

Abstract

This paper, which is a continuation of C. A. Stuart & G. Vuillaume (2003 Proc. R. Soc. Lond. A459, 1863–1889), is concerned with the study of the buckling of a tapered rod. This physical phenomenon leads to the nonlinear eigenvalue problem: {A(s)u'(s)}'+μsinu(s)=0s∈(0,1), u(1)= lim s→0 A(s)u'(s)=0 ∫ 0 1 A(s)u' (s) 2 ds <∞, where A(s) ε C([0,1]) is such that A(s) > 0 for all s > 0 and lims→0A(s)/sp = L for some constants p ⩾ 0 and L ε (0,∞). We deal with the critical case p= 2 and study the set of all the solutions of the problem. In Stuart & Vuillaume (2003) and under additional assumptions on A, we found a set of points {μi , i ε I ⫅ N* ={1,2,3,...}} ⊂ R+ such that a global branch of non–trivial solutions emanates from each μi , iε I. In this paper, we provide more detailed information about these global branches of solutions.