School of Physics Thesis Dissertation Defense

Andrew Wu 
Advisor: Dr. Zeb Rocklin, School of Physics, Georgia Institute of Technology

An Investigation into the Mechanics of Parallelogram-based Origami
Date: Thursday, July 17, 2025 
Time:  12:00 p.m.    
Location: Howey N2021/202 
Zoom link:  https://gatech.zoom.us/j/95864766249?pwd=fMdMlhglCE2SAVmsaGrlyhDNAauGqq…
Meeting ID: 958 6476 6249
Passcode: 696219

Committee Members:
Dr. Brian Kennedy, School of Physics, Georgia Institute of Technology
Dr. JC Gumbart, School of Physics, Georgia Institute of Technology
Dr. Kurt Wiesenfeld, School of Physics, Georgia Institute of Technology
Dr. Glaucio Paulino, Department of Civil and Environmental Engineering, Princeton University

Abstract:
Origami sheets, particularly those with parallelogram faces, are archetypal examples of flexible mechanical metamaterials, which possess well-defined uniform deformation modes, but exhibit spatially complex response patterns under generic loads that are relatively less explored. Thus, I explore which nonuniform, low-energy deformations are accessible to one such broad but special class of origami, four-parallelogram origami, and report on three approaches I take with collaborators to address this question. The first approach centers on a reciprocal-space compatibility matrix that uncovers a topological invariant, the sign of the Pfaffian, previously only identified in quantum mechanical systems, which governs the existence of linear, nonuniform modes of deformation of four-parallelogram origami. The second approach applies differential geometric methods to nonlinear, nonuniform deformations of the Morph, a broad subclass within the four-parallelogram class of origami that includes the Miura and eggbox patterns, which results in a numerically validated prediction that the volume in configuration space swept out by the origami sheet adheres closely to a "hyperribbon" with a large extent resulting from the four-parallelogram's planar mode and small extents resulting from its bend and twist modes. The third approach is another nonlinear, continuum theoretical approach, but one which begins from enforcing compatibility and utilizing symmetry of the microscopic structure of the sheet. The main result of this approach is a "master equation" which enforces the compatibility of each vertex in the unit cell, and which we expand in the long-wavelength limit to derive a continuum model for the system. We find strong statistical agreement between theory and simulation for our master equation over a wide range of geometries and applied loads over decades of amplitudes of the applied loads, as well as for the linear part of our continuum theory, which works over the same wide range of geometries and applied loads for small amplitudes of deformation as well as for nonlinear activations of the Miura ori's rigid, planar mechanism, and discuss ongoing work to extend the continuum theory to capture nonlinear departures from uniform configurations of the four-parallelogram.