  We consider the dynamics of an elastic continuum under large deformation but
small strain. Such systems can be described by the equations of geometrically
nonlinear elastodynamics in combination with the St. Venant-Kirchhoff material
law. The velocity-stress formulation of the problem turns out to have a formal
port-Hamiltonian structure. In contrast to the linear case, the operators of
the problem are modulated by the displacement field which can be handled as a
passive variable and integrated along with the velocities. A weak formulation
of the problem is derived and essential boundary conditions are incorporated
via Lagrange multipliers. This variational formulation explicitly encodes the
transfer between kinetic and potential energy in the interior as well as across
the boundary, thus leading to a global power balance and ensuring passivity of
the system. The particular geometric structure of the weak formulation can be
preserved under Galerkin approximation via appropriate mixed finite elements.
In addition, a fully discrete power balance can be obtained by appropriate time
discretization. The main properties of the system and its discretization are
shown theoretically and demonstrated by numerical tests.
