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