  We present a stable mixed isogeometric finite element formulation for
geometrically and materially nonlinear beams in transient elastodynamics, where
a Cosserat beam formulation with extensible directors is used. The extensible
directors yields a linear configuration space incorporating constant in-plane
cross-sectional strains. Higher-order (incompatible) strains are introduced to
correct stiffness, whose additional degrees-of-freedom are eliminated by an
element-wise condensation. Further, the present discretization of the initial
director field leads to the objectivity of approximated strain measures,
regardless of the degree of basis functions. For physical stress resultants and
strains, we employ a global patch-wise approxiation using B-spline basis
functions, whose higher-order continuity enables to use much less
degrees-of-freedom, compared to element-wise approximation. For time-stepping,
we employ an implicit energy-momentum consistent scheme, which exhibits
superior numerical stability in comparison to standard trapezoidal and
mid-point rules. Several numerical examples are presented to verify the present
method.
