We developed a novel concept, the Embedded Unit Cell (EUC) approach, to calculate local responses in elastic media. The EUC approach is based on a new, multiscale formulation of nonperiodic homogenization that facilitates zooming in on the local/micro domains where high concentrations of stress are expected. The formulation was developed using the appropriate boundary conditions and a homogenization procedure that does not use periodic assumptions for the unit cell response. This approach provides a reduced computational cost model of the macroscopic problem while preserving accuracy. It broadens the applicability of multiscale analysis techniques, which are widely used in engineering problems, to deal with local-global problems through domain decomposition. We conclude the work with a numerical implementation and a verification study.