Origami has recently received significant interest from the scientific and engineering communities as a method for designing building blocks of engineered structures to enhance their mechanical properties. However, the primary focus has been placed on their kinematic applications by leveraging the compactness and auxeticity of planar origami platforms. In this thesis, we study two different types of volumetric origami structures, Tachi-Miura Polyhedron (TMP) and Triangulated Cylindrical Origami (TCO), hierarchically from a single unit cell level to an assembly of multi-origami cells. We strategically assemble these origami cells into mechanical metamaterials and demonstrate their unique static/dynamic mechanical responses. In particular, these origami structures exhibit tailorable stiffness and strain softening/hardening behaviors, which leads to rich wave dynamics in origami-based architectures such as tunable frequency bands and new types of nonlinear wave propagations. One of the novel waveforms investigated in this thesis is the rarefaction solitary wave arising from strain-softening nature of origami unit cell. This unique wave dynamic mechanism is analyzed in numerical, analytical, and experimental approaches. By leveraging their tailorable folding mechanisms, the origami-based mechanical metamaterials can be used for designing new types of engineering devices and structures, not only for deployable space and disaster relief applications, but also for vibration filtering, impact mitigation, and energy harvesting.
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