This thesis investigates two-dimensional theory of elasticity, exploring the effects of thermal fluctuations, non-equilibrium odd elasticity, and disorder.

We use molecular dynamics to study the vibrations of a thermally fluctuating elastic sheet with one end clamped at its zero-temperature length. We uncover a tilted phase in which the sheet fluctuates about an inclined mean configuration, thus breaking reflection symmetry. We determine the phase behavior as a function of the aspect ratio of the sheet and the temperature. We show that tilt may be viewed as a type of transverse buckling instability induced by clamping coupled to thermal fluctuations, and develop an analytic model that predicts the tilted regions of the phase diagram. Unusual responses, as exemplified by the tilted phase, driven by control of purely geometric quantities like the aspect ratio, as opposed to external fields, provide a rich playground for two-dimensional mechanical metamaterials.

We also investigate the impact of disorder on the elastic moduli of an odd elastic material, defined by a non-symmetric elastic tensor. Using an effective medium theory and numerical simulations, we reveal the behavior of effective odd elastic moduli in the presence of disorder, interpreting it as a crossover between the affine response of the passive elastic backbone and a rigidity percolation transition in the odd elastic components. We find that odd elasticity is generally robust against disorder, although certain finely-tuned features may be affected.