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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium

Scott Spector
Southern Illinois University
Title: The Stability of Equilibria for Elastic Bars

Abstract: When an elastic bar is subjected to uniaxial compression or tension, one expects it to deform (approximately) homogeneously until a critical load is reached. A bifurcation, such as buckling in compression and necking in tension, may then be observed.

In this seminar, I will model such an experiment as the in-plane contraction or extension of a two-dimensional, homogeneous, isotropic, hyperelastic material in which the length of the bar is prescribed, the ends of the bar are assumed to be free of shear, and the sides are left completely free. I will show that standard, additional, constitutive hypotheses on the stored-energy function imply that:

(1) In compression, in accord with Davies [1989], a supercritical, pitchfork bifurcation to mode-one buckling will normally occur. A theorem of Crandall and Rabinowitz [1971] can then be used to show that, locally, a buckled solution branch does indeed exist. In addition, if one follows the homogeneous solution branch past all such buckling instabilities one reaches a load at which the complementing condition fails for the linearized problem. The remainder of the homogeneous solution branch then shows an even more significant instability: Agmon’s condition fails for the linearized problem.

(2) In tension, this model does not allow the expected bifurcation due to the fact that the homogeneous deformation is the unique absolute minimizer of the elastic energy. Thus, in order for a bifurcation to occur in tension, either the material must cease to be elastic or the stored-energy function must violate the additional constitutive hypotheses. The fact that, under such assumptions, no bifurcation can occur in tension follows from a result of Hill and Hutchinson [1975], since my hypotheses prohibit the load on the bar from reaching a maximum value. However, the fact that the homogeneous deformation is the unique absolute minimizer of the energy appears to be a new result.

Date: Tuesday, March 15, 2011
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer