  This work studies the approximation of plane problems concerning transversely
isotropic elasticity, using a low-order Virtual Element Method (VEM), with a
focus on near-incompressibility and near-inextensibility. Additionally, both
homogeneous problems, in which the plane of isotropy is fixed; and
non-homogeneous problems, in which the fibre direction defining the isotropy
plane varies with position, are explored. In the latter case various options
are considered for approximating the non-homogeneous fibre directions at
element level. Through a range of numerical examples the VEM approximations are
shown to be robust and locking-free for several element geometries and for
fibre directions that correspond to mild and strong non-homogeneity.
