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We present an atomic scale theory of lattice distortions using strain relatedvariables and their constraint equations. Our approach connects constrainedatomic length scale variations to continuum elasticity and describes elasticityat all length scales. We apply the general approach to a two-dimensional squarelattice with a monatomic basis, and find the atomic scale elastic texturesaround a structural domain wall and a single defect, as exemplary textures. Weclarify the microscopic origin of gradient terms, some of which are includedphenomenologically in Landau-Ginzburg theory. The obtained elastic textures areused to investigate the effects of elasticity-driven lattice deformation on thenanoscale electronic structure in superconductor by solving the Bogliubov-deGennes equations with the electronic degrees of freedom coupled to the latticeones. It is shown that the order parameter is depressed in the regions wherethe lattice deformation takes place. The calculated local density of statessuggests the electronic structure is strongly modulated as a response to thelattice deformation-- the elasticity propagates the electronic response overlong distances. In particular, it is possible for the trapping of low-lyingquasiparticle states around the defects. These predictions could be directlytested by STM experiments in superconducting materials.Comment: Proceeding paper for "Conference on Dynamic Inhomogeneities in  Complex Oxides" (to appear in J. Superconductivity

Atomic Scale Elastic Textures Coupled to Electrons in Superconductors