Geometry and topology have emerged as powerful tools for understanding a wide range of phenomena in condensed matter physics. Often, geometric and topological constraints drive the order and dynamics of soft mechanical systems -- systems in which material behavior is energetically accessible at room temperature in tabletop experiments. Here, we investigate the role of geometry and topology in the mechanics of quasi-two-dimensional elastic materials. In Part I of this thesis, we present substrate curvature as a geometric tool for guiding the behavior of cracks. When a flat elastic sheet conforms to a surface with Gaussian curvature, the geometry of the surface redistributes stresses in the sheet in a tunable fashion. Using this insight, we uncover how curvature can stimulate or suppress the growth of cracks and steer or arrest their propagation. We examine the mechanics of this scenario with and without pinning, in systems on both macroscopic and nanometric scales. Potential applications of the results range from stretchable electronics to filtration using nanoparticle membranes. From elastic responses to geometric frustration, we then turn in Part II to discrete metamaterials for which network geometry and real-space lattice topology interact with the topological structure of elastic waves. In these mechanical materials, topological order in their excitation spectra translates to exotic behaviors at the materials' boundaries, such as chiral edge waves that are unusually robust to disorder. We uncover such topological behavior in a simple system composed of interacting gyroscopes and use this metamaterial to explore broken symmetries and tune through topological phase transitions. We then peel away a canonical ingredient for constructing topological insulators: the ordered underlying lattice. We find topological physics emerging from amorphous networks of gyroscopes and establish the basic building blocks for understanding topology in amorphous systems more generally. The results apply to a broad class of systems, from acoustic and mechanical structures to electronic and photonic materials.




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