Title: On structured surfaces with defects: geometry, strain incompatibility, internal stress, and natural shapes,
Abstract: Given a distribution of defects on a structured surface, such as those
represented by 2-dimensional crystalline materials, liquid crystalline
surfaces, and thin sandwiched shells, what is the resulting stress field and
the deformed shape? Motivated by this concern, we first classify, and quantify,
the translational, rotational, and metrical defects allowable over a broad
class of structured surfaces. With an appropriate notion of strain, the defect
densities are then shown to appear as sources of strain incompatibility. The
strain incompatibility relations, with appropriate kinematical assumptions on
the decomposition of strain into elastic and plastic parts, and the stress
equilibrium relations, with a suitable choice of material response, provide the
necessary equations for determining both the internal stress field and the
deformed shape. We demonstrate this by applying our theory to Kirchhoff-Love
shells with a kinematics which allows for small in-surface strains but
moderately large rotations.