  We present an atomic scale theory of lattice distortions using strain related
variables and their constraint equations. Our approach connects constrained
atomic length scale variations to continuum elasticity and describes elasticity
at all length scales. We apply the general approach to a two-dimensional square
lattice with a monatomic basis, and find the atomic scale elastic textures
around a structural domain wall and a single defect, as exemplary textures. We
clarify the microscopic origin of gradient terms, some of which are included
phenomenologically in Landau-Ginzburg theory. The obtained elastic textures are
used to investigate the effects of elasticity-driven lattice deformation on the
nanoscale electronic structure in superconductor by solving the Bogliubov-de
Gennes equations with the electronic degrees of freedom coupled to the lattice
ones. It is shown that the order parameter is depressed in the regions where
the lattice deformation takes place. The calculated local density of states
suggests the electronic structure is strongly modulated as a response to the
lattice deformation-- the elasticity propagates the electronic response over
long distances. In particular, it is possible for the trapping of low-lying
quasiparticle states around the defects. These predictions could be directly
tested by STM experiments in superconducting materials.
