Topological Mechanical Systems Beyond the Limit of Linear Ideal Springs

Abstract

In recent years, topological mechanical metamaterials (TMM) have gradually attracted attention across different disciplines including physics, mechanical engineering, and materials science. It has achieved great success in analyzing mechanical rigidity and programming mechanical responses in Maxwell lattices. The engineering of TMMs opened numerous opportunities for achieving mechanical functionalities in engineering applications such as mechanical impact buffers, waveguides, phonon diodes, etc. Besides the achievements in the theoretical aspect, there have been also numerous progresses made in the manufacturing of TMMs. However, issues such as finite hinge and material composition takes the realization of TMMs away from the limit of ideal springs. The focus of this thesis is to examine systems beyond the limit of ideal spring models and study both the novel physical phenomena and engineering advantage by these considerations. The first project presented in this dissertation expands on a lattice model in which the elastic energy is related to rings of springs. This so-called Vertex Model (VM) is presented as an attempt to understand the geometric effect on epithelial tissues. In a slight variation of the VM, the Active Tension Network model (ATN), topologically polarized zero modes (ZMs) were discovered. This work may provide useful insights into tissue morphogenesis. The second project presents a numerical study on the effect of bending rigidity in Maxwell lattices. Specifically, the study focuses on the stress-focusing effect on a domain wall that carries states of self-stress (SSS) formed by connecting two opposite topologically polarized domains. By including the bending stiffness of the hinges, a masking effect on the stress focusing was observed as the bending stiffness increased, and by designing the lattice geometry, lowering and homogenization of the bending stress were achieved. Furthermore, stress focusing was able to be achieved for shear strain as well. This geometric manipulation could help prevent fracturing at the hinges in response to different strains. The third project shows an engineering effort of designing bistable unit cells in Maxwell lattices. Such a design makes the entire lattice to be multi-stable, and the topological transitions more easily achievable. The multi-stable lattice also shows interesting interface profiles during the transition due to the incompatibility of the lattice spacing in the two topologically distinguished lattice regions. Furthermore, structural assembly and 3D-printing techniques were employed to realize the multi-stable lattices to test the numerically predicted edge stiffness difference between the opposite lattice boundaries. Such an engineering design offers potential application opportunities for impact mitigation, mechenological computation, and flexible robots. The fourth project is an analytical and numerical study of the non-linear effect on the Maxwell lattices. In this study, exact geometric relations were solved to describe the Maxwell lattice with two boundaries prescribed. Under such a condition, a sinusoidal perturbation was given at one boundary of the lattice to study the nonlinear effect on the topological polarization of the lattice. Under such a condition, local regions of topological polarization switching were observed, along with well-known non-linear effects such as harmonic generations. Furthermore, solitary waves were observed as the perturbation is localized, which allows us to make an analogy to the time domain to create an artificial “non-Hermitian” system. The inclusion of non-linearity promotes further understanding of TMMs beyond the small deformation regime, as well as opens opportunities for further applications such as adaptive smart materials.