  Motivated by properties-controlling potential of the strain, we investigate
strain dependence of structure, electronic and magnetic properties of
Sr$_2$IrO$_4$ using complementary theoretical tools: {\it ab-initio}
calculations, analytical approaches (rigid octahedra picture, Slater-Koster
integrals), and extended $t-{\mathcal{J}}$ model. We find that strain affects
both Ir-Ir distance and Ir-O-Ir angle, and the rigid octahedra picture is not
relevant. Second, we find fundamentally different behavior for compressive and
tensile strain. One remarkable feature is the formation of two subsets of bond-
and orbital- dependent carriers, a compass-like model, under compression. This
originates from the strain-induced renormalization of the Ir-O-Ir superexchange
and O on-site energy. We also show that under compressive (tensile) strain,
Fermi surface becomes highly dispersive (relatively flat). Already at a tensile
strain of $1.5\%$, we observe spectral weight redistribution, with the
low-energy band acquiring almost purely singlet character. These results can be
directly compared with future experiments.
