GEOMETRY, GROWTH AND PATTERN FORMATION IN THIN ELASTIC STRUCTURES

Publication Year:
2018
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Downloads 41
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Repository URL:
https://scholarworks.umass.edu/dissertations_2/1321
Author(s):
Al-Mosleh, Salem
Tags:
Physics; soft matter; geometry; morphogenesis; thin shells; elasticity; Condensed Matter Physics; Statistical, Nonlinear, and Soft Matter Physics
artifact description
Thin shells are abundant in nature and industry, from atomic to planetary scales. The mechanical behavior of a thin shell depends crucially on its geometry and embedding in 3 dimensions (3D). In fact, the behavior of extremely thin shells becomes scale independent and only depends on geometry. That is why the crumpling of graphene will have similarities to the crumpling of paper. In this thesis, we start by discussing the static behavior of thin shells, highlighting the role of asymptotic curves (curves with zero normal curvature) in determining the possible deformations and in controlling the folding patterns. In particular, we found that the presence of these curves on a surface can lead to more rigidity, and to continuously folding deformations. We then move to problems in growing thin shells, where the material properties change adiabatically with time. We derive here expressions for the quasi-static response of the shape to these changes. We focus in particular on changes in the target metric - analogous to the rest length of a spring. We derive an expression for the possible changes in the metric that are consistent with coordinate invariance, locality and dependent on geometry and applied forces. We apply this general framework, through analytic calculations and simulations, to understanding how rod-like E. coli might be able to generate a stable elongating cylindrical shape. We show that coupling to curvature alone is generically linearly unstable and that additionally coupling to stress can lead to stably elongating cylindrical structures. Our approach can readily be extended to gain insights into the general classes of stable growth laws for different target geometries.