Some elastic materials exhibit strong coupling between mechanical deformation and local degrees of freedom in the material. In materials such as liquid crystal elastomers, these degrees of freedom can be extrinsically controlled by various ambient fields, giving rise to a class of programmable shape-shifting objects. We study the relation between local degrees of freedom in a material, its intrinsic geometry, and its mechanical state, and map the ways in which a material can be deformed given its local symmetries. We explore new theoretical questions that are being rendered applicable by major recent advances in material and metamaterial research, such as shape pluripotency in complex hierarchical materials, and anomalously soft deformation modes in materials with spontaneously broken local symmetries.
Liquid crystals are found midway between liquids and solids in terms of their local symmetry and order. While they cannot be attributed a complete reference metric tensor as in solids, certain aspects of the local geometry must be maintained implying a reference geometry in some weaker sense. This gives rise to geometric frustration between local geometric constraints and the geometry of ambient space or topological constraints on defects in the liquid crystalline structure, which may lead to complex solutions and patterns. We study the topological rules of defects and defect interactions, as well as the equilibrium geometries, elasticity, dynamics, and stability of these topologically distinct states. These questions are essential in understanding phase transitions and bulk properties of materials, and may allow predicting and tuning material properties or designing exotic structures with unique topological attributes.
Liquid crystal theories are general mathematical descriptions of continua with some partial translational and rotational order. As such, they capture essential elements of patterns formed in various continuum systems. One example are wrinkles formed in a thin elastic sheet that is lying on a soft substrate, e.g. skin on flesh, when subject to an external forcing or as a result of geometric incompatibility. We study these systems from a smectic liquid crystal perspective, in order to make sense of the rich zoo of observed patterns. We develop methods to accurately predict, efficiently simulate, and universally design wrinkling patterns over many length scales.