On Structured Surfaces with Defects: Geometry, Strain Incompatibility, Stress Field, and Natural Shapes.

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, aided with a 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 stress field and the deformed shape. We demonstrate this by applying our theory to Kirchhoff-Love shells with a kinematics which allows for small surface strains but moderately large rotations. We discuss implications of our framework in the context of 2-dimensional crystals, growing biological membranes, and isotropic fluid films. [ABSTRACT FROM AUTHOR]