The time-dependent von K\'arm\'an shell equation as a limit of three-dimensional nonlinear elasticity

The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin shell is considered, as the thickness $h$ of the shell tends to zero. Given the appropriate scalings of the applied force and of the initial data in terms of $h,$ it's verified that three-dimensional solutions of the nonlinear elastodynamic equations converge to solutions of the time-dependent von K\'arm\'an equations or dynamic linear equations for shell of arbitrary geometry.