  In this manuscript we first give the explicit variational structure of the
nonlinear elastic waves for isotropic, homogeneous, hyperelastic materials in
2-D. Based on this variational structure, we suggest a null condition which is
a kind of structural condition on the nonlinearity in order to stop the
formation of finite time singularities of local smooth solutions. In the radial
symmetric case, inspired by Alinhac's work [3] on 2-D quasilinear wave
equations, we show that such null condition can ensure the globalexistence of
smooth solutions with small initial data.
