  Topological semimetal antiferromagnets provide a rich source of exotic
topological states which can be controlled by manipulating the orientation of
the N\'eel vector, or by modulating the lattice parameters through strain. We
investigate via ${ab\ initio}$ density functional theory calculations, the
effects of shear strain on the bulk and surface states n two antiferromagnetic
EuCd$_2$As$_2$ phases with out-of-plane and in-plane spin configurations. When
magnetic moments are along the $\textit{c}$-axis, a $3\%$ longitudinal or
diagonal shear strain can tune the Dirac semimetal phase to an axion insulator
phase, characterized by the parity-based invariant $\eta_{4I} = 2$. For an
in-plane magnetic order, the axion insulator phase remains robust under all
shear strains. We further find that for both magnetic orders, the bulk gap
increases and a surface gap opens on the (001) surface up to 16 meV. Because of
a nonzero $\eta_{4I}$ index and gapped states on the (001) surface, hinge modes
are expected to happen on the side surface states between those gapped surface
states. This result can provide a valuable insight in the realization of the
long-sought axion states.
