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Nonlinear Problems for Thin Elastic Structures and the Ubiquitous Isola Bifurcation

Monday, January 28, 2013 at 4:00pm

Rockefeller Hall, Schwartz Auditorium
Central Campus

Physics Colloquium, Tim Healey, Cornell University. Refreshments at 3:30pm.

Abstract: We begin with a simple 1-dimensional, 2-phase solid under "hard" tensile end loading in the presence of inter-facial effects. This is equivalent to a phase-field model, of the van der Waals-Cahn-Hilliard type, that illustrates well the concept of an isola bifurcation. Most of the talk is then focused on the wrinkling of highly stretched, finely thin rectangular sheets – a problem that has attracted the attention of several investigators in recent years, nearly all of which employ the classical Föppl-von Kármán (F-K) theory of plates. We first propose a rational model that correctly accounts for the large mid-plane strain in highly stretched membranes. We mention some mathematical existence issues, indicating the appropriateness of our model. We then carefully perform a numerical bifurcation/continuation analysis, identifying stable solutions (local energy minimizers). Our results in comparison to those from the F-K theory (also obtained herewith) show:
(i) For a given fine thickness, only a certain range of aspect ratios admit stable wrinkling; for a fixed length (in the highly stretched direction), wrinkling does not occur if the width is too large or too small. In contrast, the F-K model erroneously predicts wrinkling in those very same regimes for sufficiently large applied macroscopic strain.
(ii) When stable wrinkling emerges as the applied macroscopic strain is steadily increased, the amplitude first increases, reaches a maximum, decreases, and then returns to zero again. In contrast, the F-K model predicts an ever-increasing wrinkling amplitude as the macroscopic strain is increased.
We recognize (i), (ii) as global isola bifurcations – in term of both the macroscopic strain parameter and an aspect-ratio parameter.
(iii) When stable wrinkling occurs, for fixed parameters, the transverse pattern admits an entire orbit of neutrally stable (equally likely) possibilities: These include reflection symmetric solutions about the mid-plane, anti-symmetric solutions about the mid-line (a rotation by radians about the mid-line leaves the wrinkled shape unchanged) and a continuously evolving family of shapes “in-between”, say, parametrized by an arbitrary phase angle, each profile of which is neither reflection symmetric nor anti-symmetric. While the rectangular symmetry of the system is not rich enough to support such an orbit of solutions, we argue that the apparent symmetry is asymptotically induced in the limit of vanishing thickness.

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Christine Clay

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Tim Healey

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Cornell University