The design of cellular materials with controlled energy dissipation is relevant to a broad range of applications, ranging from cushioning (such as shoes and athletic protection) to vibration control in damped, lightweight structures. Additive manufacturing has dramatically expanded the design space for such structures, enabling a diverse range of topologies. This dissertation establishes analysis techniques to make quantitative links between base properties, strut topology and cellular response; those techniques are then used to conduct case studies of various classes of strut-based structures to generate insight regarding effective strategies for designs that achieve specific types of response. Key contributions include: (i) a detailed numerical study of buckling behaviors that control large deformation response of low-density, elastic structures, (ii) the development of analytical and reduced-order models for buckling behaviors in viscoelastic struts subject to dynamic loading, (iii) a highly efficient framework to predict the damped frequency

response of cellular materials, and (iv) a broad study of the effects of topology in single celled structures, including the use of multiple strut sizes.

These contributions have led to several new, quantitative insights regarding the design of low- density structures to control energy dispersion. (i) For cellular materials that include struts that lie at an angle to the direction of compression, snap-through behaviors influence both the initial softening at small strains and the stiffening behavior observed at moderate strains. The models presented in this work illustrate that novel structures comprising cells with struts at multiple angles create significant opportunities to control the softening regime that falls between initial response and stiffening associated with large deformation. (ii) Cellular structures comprising viscoelastic struts create significant opportunities to improve vibration damping through a combination of materials selection and topological design. The use of high damping materials as the core of composite struts

can improve damping over the base shell material by a factor of 5-10 while maintaining the stiffness of the base structure. The use of non-uniform cells that disrupt standing waves insight structures can further increase damping by a factor of two, with potentially larger gains possible with topology optimization. (iii) The use of internal struts that sub-divide larger cells provides stabilization of buckling events; this can be exploited to improve both the onset of non-linearity (broadly defined, strength) and the energy absorbed by purely elastic structures subjected to compression. The broad topology study illustrates that response is highly sensitive to small variations in internal topology, indicating that topology optimization must rely on direct search algorithms.