Mechanical Metamaterials: Nonlinear Beams and Excess Zero Modes

Mechanical metamaterials are man-made materials which derive their unusual properties from their structure rather than their composition. Their structure, or architecture, often consists of periodically arranged building blocks whose mutual interactions realize unusual properties. In this thesis, we investigate the role of two aspects of mechanical metamaterials: The beam ligaments and the microstructures of hinging squares. Both aspects play an important role in mechanical metamaterials, but give rise simultaneously to several open problems. First, although the mechanical behaviour of slender beam ligaments is well understood, the finite-width ligaments that often occur in mechanical metamaterials lead to new physics; wide beams exhibit a negative post-buckling stiffness, characterized by a decreasing force after buckling, which is not well understood. Second, fully filled microstructures of hinging squares constitute an auxetic mechanism, but possible new zero modes derived from (diluted) microstructures with missing squares remain largely unexplored. How does the number of zero modes increases in such diluted systems, can we count these additional modes, and what is their spatial structure? Here we address these open problems, thereby providing the necessary understanding to fully leverage the characteristics of wide beam ligaments and diluted collections of hinging squares for the design of novel mechanical functionalities.