  We propose a novel approach to the linear viscoelastic problem of
shear-deformable geometrically exact beams. The generalized Maxwell model for
one-dimensional solids is here efficiently extended to the case of arbitrarily
curved beams undergoing finite displacement and rotations. High efficiency is
achieved by combining a series of distinguishing features, that are i) the
formulation is displacement-based, therefore no additional unknowns, other than
incremental displacements and rotations, are needed for the internal variables
associated with the rate-dependent material; ii) the governing equations are
discretized in space using the isogeometric collocation method, meaning that
elements integration is totally bypassed; iii) finite rotations are updated
using the incremental rotation vector, leading to two main benefits: minimum
number of rotation unknowns (the three components of the incremental rotation
vector) and no singularity problems; iv) the same $\rm SO(3)$-consistent
linearization of the governing equations and update procedures as for
non-rate-dependent linear elastic material can be used; v) a standard
second-order accurate time integration scheme is made consistent with the
underlying geometric structure of the kinematic problem. Moreover, taking full
advantage of the isogeometric analysis features, the formulation permits
accurately representing beams and beam structures with highly complex initial
shape and topology, paving the way for a large number of potential applications
in the field of architectured materials, meta-materials, morphing/programmable
objects, topological optimizations, etc. Numerical applications are finally
presented in order to demonstrate attributes and potentialities of the proposed
formulation.
