### Overview

Advanced manufacturing technology has made a large class of complex/architectured materials to be realizable, which, in turn, has provided new avenues for the longstanding quest of finding *better materials*. To fully leverage these technologies, we need theoretical and computational models that can explain the response of materials subject to different conditions, and establish a link between their macroscopic response to their underlying microstructure. In this research thrust, our lab focuses on class of solids that exhibit *some level of translational symmetry, *ranging from architectured lattices/composites with complex unitcells/RVEs, to natural materials. We strive to (i) develop a rigorous understanding of the key foundational features, such as symmetry, on the response of materials, and (ii) capitalize on fundamental mechanisms to design complex functional materials. **The following two projects showcase our efforts along this research thrust:**

### Dynamics of quasiperiodic systems

Quasicrystals are a class of ordered materials that do not have translational symmetry. They are an emerging class of materials with unique properties. Quasicrystalline metals and twistronics are two important examples of such materials. A fundamental understanding of the underlying physics and mathematics will enable discoveries of novel quasiperiodic materials. One class of descriptive theories, with minimal representations of complexity classes, are Frenkel-Kontorova (FK) models. FK models are concerned with a chain of atoms that are simultaneously interacting with the neighboring atoms and with a substrate potential. We have developed a dynamical system framework for studying quasiperiodic FK model, where we aim to establish mathematical tools, as well as computational models, to understand the fundamental properties of quasiperiodic systems.

### Mechanics of architectured tensegrity lattices

Architectured lattices are a novel class of material systems that allow for engineering the effective properties by varying the geometry of the underlying unit cell, as well as the properties of individual elements and their corresponding connections. Among many applications, light-weighted lattices that have high energy absorption properties are highly desirable. Tensegrity (tensional integrity) architectured lattices is one such class, which refers to systems assembled in an arrangement such that the tensile members (cables) belong to a continuous network, while compression members (bars) are either disconnected from each other or belong to isolated compression clusters.

Tensegrity systems also have applications in mechanobiology. Cellular tensegrity theory proposes that living systems use principles of tensegrity to govern how molecules self-assemble to create multi-molecular structures, organelles, cells, tissues, organs and living organisms. We have developed a homogenization framework to compute the effective material properties of a ``large array'' of tensegrities. This approach could also be utilized to homogenize the material properties of general systems with translational symmetries. We have particularly studied how the pre-stretches of the tensile members in the unit-cell level affect the global macro-scale elasticity tensor C.