  We consider the Cauchy problem for 2-D incompressible isotropic
elastodynamics. Standard energy methods yield local solutions on a time
interval $[0,{T}/{\epsilon}]$, for initial data of the form $\epsilon U_0$,
where $T$ depends only on some Sobolev norm of $U_0$. We show that for such
data there exists a unique solution on a time interval $[0,
\exp{T}/{\epsilon}]$, provided that $\epsilon$ is sufficiently small. This is
achieved by careful consideration of the structure of the nonlinearity. The
incompressible elasticity equation is inherently linearly degenerate in the
isotropic case; in other words, the equation satisfies a null condition. This
is essential for time decay estimates. The pressure, which arises as a Lagrange
multiplier to enforce the incompressibility constraint, is estimated in a novel
way as a nonlocal nonlinear term with null structure. The proof employs the
generalized energy method of Klainerman, enhanced by weighted $L^2$ estimates
and the ghost weight introduced by Alinhac.
