Multipoles, symmetry representations and thermal fluctuations in elastic systems

Abstract

In recent years, we have seen exciting new developments in research on mechanical metamaterials, topological phononics, and mechanics of atomically thin 2D materials. In this thesis, I present how methods from physics can help us in understanding the mechanical properties of these systems as well as gaining further intuition. First, we develop a multipole expansion method to describe the deformation of infinite as well as finite solid structures with cylindrical holes and inclusions by borrowing concepts from electrostatics, such as induction and method of image charges. Our method shows excellent agreement with finite element simulations and experiments. Next, using representation theory, I show how symmetries of phononic crystals affect the degeneracies in their phononic band structures. Deformation of phononic crystals under external load that causes breaking of some symmetries can lead to the lifting of degeneracies for bands and creating gaps such that waves of certain frequencies become disallowed. Finally, using methods from statistical physics, I present how the mechanical properties of atomically thin 2D sheets and shells get modified due to thermal fluctuations. Freely suspended sheets subject to such fluctuations are much harder to bend, but easier to stretch, compress and shear, beyond a characteristic thermal length scale, which is on the order of nanometers for graphene at room temperatures. Just like in critical phenomena, these renormalized elastic constants become scale dependent with universal power-law exponents. In nanotubes, competition between stretching and bending costs associated with radial fluctuations introduces another characteristic elastic length scale, which is proportional to the geometric mean of the radius and effective thickness. Beyond this elastic length scale, bending rigidities and in-plane elastic constants of nanotubes become anisotropic.