  Non-Euclidean plates are a subset of the class of elastic bodies having no
stress-free configuration. Such bodies exhibit residual stress when relaxed
from all external constraints, and may assume complicated equilibrium shapes
even in the absence of external forces. In this work we present a mathematical
framework for such bodies in terms of a covariant theory of linear elasticity,
valid for large displacements. We propose the concept of non-Euclidean plates
to approximate many naturally formed thin elastic structures. We derive a thin
plate theory, which is a generalization of existing linear plate theories,
valid for large displacements but small strains, and arbitrary intrinsic
geometry. We study a particular example of a hemispherical plate. We show the
occurrence of a spontaneous buckling transition from a stretching dominated
configuration to bending dominated configurations, under variation of the plate
thickness.
