  This work comprises a detailed theoretical and computational study of the
boundary value problem for transversely isotropic linear elastic bodies.
General conditions for well-posedness are derived in terms of the material
parameters. The discrete form of the displacement problem is formulated for
conforming finite element approximations. The error estimate reveals that
anisotropy can play a role in minimizing or even eliminating locking behaviour,
for moderate values of the ratio of Young's moduli in the fibre and transverse
directions. In addition to the standard conforming approximation an alternative
formulation, involving under-integration of the volumetric and extensional
terms in the weak formulation, is considered. The latter is equivalent to
either a mixed or a perturbed Lagrangian formulation, analogously to the
well-known situation for the volumetric term. A set of numerical examples
confirms the locking-free behaviour in the near-incompressible limit of the
standard formulation with moderate anisotropy, with locking behaviour being
clearly evident in the case of near-inextensibility. On the other hand,
under-integration of the extensional term leads to extensional locking-free
behaviour, with convergence at superlinear rates.
