Bifurcations in the regularized Ericksen bar model. (English) Zbl 1134.74018

Grinfeld, M.; Lord, G. J.

J. Elasticity 90, No. 2, 161-173 (2008).

Summary: We consider the regularized Ericksen model of an elastic bar on elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of S. Müller’s conjecture [Calc. Var. Partial Diff. Equ. 1, No. 2, 169–204 (1993; Zbl 0821.49015)] concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of primary branch connections that occur in this problem. We give a reformulation of Müller conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a “loop” structure that characterizes \((k,3k)\) bifurcations.

MSC:

74G60	Bifurcation and buckling
74K10	Rods (beams, columns, shafts, arches, rings, etc.)

Keywords:

Lyapunov-Schmidt analysis; continuation methods; Müller conjecture

Citations:

Zbl 0821.49015

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