  We present a first-principles investigation of structural and elastic
properties of experimentally observed phases of bulk SrRuO$_3$ - namely
orthorhombic, tetragonal, and cubic - by applying density functional theory
(DFT) approximations. At first, we focus our attention on the accuracy of
calculated lattice constants in order to find out DFT approaches that best
represent the crystalline structure of SrRuO$_3$, since many important physical
quantities crucially depend on change in volume. Next, we evaluate
single-crystal elastic constants, mechanical stability, and macroscopic elastic
parameters trying to at least partially compensate for the existing lack of
information about these fundamental features of SrRuO$_3$. Finally, we analyze
the anomalous behavior of low-temperature orthorhombic phase under $C_{44}$
related shear deformation. It turns out that at critical strain values the
system exhibits a distinct deviation from the initial behavior which results in
an isosymmetric phase transition. Moreover, under $C_{44}$ related shear
deformation tetragonal SrRuO3 becomes mechanically unstable raising an open
question of what makes it experimentally observable at high temperatures.
