Date of Award

Summer 8-27-2021

Degree Type


Degree Name

Doctor of Philosophy (PhD)




Schwarz, Jennifer


Compression stiffening, Morphogenesis, Random polygon, Rigidity transitions, Spring networks, Strain-induced transitions

Subject Categories

Physical Sciences and Mathematics | Physics


This thesis reports work in three topics - I) isotropic strain-induced rigidity transitions in under-constrained spring networks II) uniaxial strain-induced stiffening transitions in semiflexible networks with area-conserving inclusions and III) non-linearities in the buckling without bending morphogenesis model for growing cerebella. All models are two dimensional and we employ either discrete and continuum models to study these systems.In I), motivated by the rigidity transitions in isotropically strained disordered spring networks, we study rigidity transitions in isotropically strained area-conserving random polygonal loops. We find a crossover transition in these loops. We also provide arguments towards showing convexity as a necessary condition for the transition and cyclic polygonal configurations, a sufficient condition. In II), towards modeling the uniaxial compression stiffening experiments in mEF cells and composite systems of fibrin and dextran beads, we construct area-conserving regular polygonal loops. These loops demonstrate compression stiffening. We also report the compression softening of on-lattice semiflexible polymer networks. The softening mechanism is independent of Euler-buckling instabilities. Introduction of area-conserving regular polygons as inclusions in the semiflexible network introduces non-affinities in the elastic response of the system. The non-affine bending of filaments leads to compression stiffening of the semiflexible network. In III), we find that by adding non-linearities to the buckling without bending morphogenesis model, we obtain cusped folds which visually resemble the cusped folds of the cerebellum. Introduction of non-linearities in the energy functional of the model robustly develops a quadratic non-linearity in the Euler-Lagrange equations. We study the effect of such a non-linear force for a simple harmonic oscillator like system and see that there too we obtain cusped `folds'. We also discuss steric confinements on the growing cerebellum and a paradigmatic demonstration of hierarchical folds in the cerebellum.


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