  We provide a fully nonlinear port-Hamiltonian formulation for discrete
elastodynamical systems as well as a structure-preserving time discretization.
The governing equations are obtained in a variational manner and represent
index-1 differential algebraic equations. Performing an index reduction one
obtains the port-Hamiltonian state space model, which features the nonlinear
strains as an independent state next to position and velocity. Moreover,
hyperelastic material behavior is captured in terms of a nonlinear stored
energy function. The model exhibits passivity and losslessness and has an
underlying symmetry yielding the conservation of angular momentum. We perform
temporal discretization using the midpoint discrete gradient, such that the
beneficial properties are inherited by the developed time stepping scheme in a
discrete sense. The numerical results obtained in a representative example are
demonstrated to validate the findings.
