  This work studies discontinuous Galerkin (DG) approximations of the boundary
value problem for homogeneous transversely isotropic linear elastic bodies.
Low-order approximations on triangles are adopted, with the use of three
interior penalty DG methods, viz. nonsymmetric, symmetric and incomplete. It is
known that these methods are uniformly convergent in the incompressible limit
for the isotropic case. This work focuses on behaviour in the inextensible
limit for transverse isotropy. An error estimate suggests the possibility of
extensional locking, a feature that is confirmed by numerical experiments.
Under-integration of the extensional edge terms is proposed as a remedy. This
modification is shown to lead to an error estimate that is consistent with
locking-free behaviour. Numerical tests confirm the uniformly convergent
behaviour, at an optimal rate, of the under-integrated scheme.
