  We study two aspects of the elasticity of smectic-$A$ elastomers that make
these materials genuinely and qualitatively different from conventional
uniaxial rubbers. Under strain applied parallel to the layer normal, monodomain
smectic-$A$ elastomers exhibit a drastic change in Young's modulus above a
threshold strain value of about 3%, as has been measured in experiments by
Nishikawa and Finkelmann [Macromol. Chem. Phys. {\bf 200}, 312 (1999)]. Our
theory predicts that such strains induce a transition to a smectic-$C$-like
state and that it is this transition that causes the change in elastic modulus.
We calculate the stress-strain behavior as well as the tilt of the smectic
layers and the molecular orientation for strain along the layer normal, and we
compare our findings with the experimental data. We also study the
electroclinic effect in chiral smectic-$A^\ast$ elastomers. According to
experiments by Lehmann {\em et al}. [Nature {\bf 410}, 447 (2001)] and
K\"{o}hler {\em et al}. [Applied Physics A {\bf 80}, 381 (2003)], this effect
leads in smectic-$A^\ast$ elastomers to a giant or, respectively, at least very
large lateral electrostriction. Incorporating polarization into our theory, we
calculate the height change of smectic-$A^\ast$ elastomer films in response to
a lateral external electric field, and we compare this result to the
experimental findings.
