A virtual element method for transversely isotropic elasticity - Van Huyssteen, Daniel (Department of Mechanical Engineering, 2019)
This work studies the approximation of plane problems concerning transversely isotropic elasticity, using a low-order virtual element method (VEM). The VEM is an alternative finite element method characterised by complete freedom in determining element geometries that are otherwise polygonal in two dimensions, or polyhedral in three. Transversely isotropic materials are characterised by an axis of symmetry perpendicular to a plane of isotropy, and have applications ranging from fibre reinforcement to biological materials. The governing equations of the transversely isotropic elasticity problem are derived and a virtual element formulation of the problem is presented along with a sample implementation of the method. This work focuses on the treatment of near-incompressibility and near-inextensibility. These are explored both for homogeneous problems, in which the plane of isotropy is fixed; and non-homogeneous problems, in which the fibre directions defining the plane of isotropy vary with position. In the latter case various options are explored for approximating the non-homogeneous terms at an element level. The VEM approximations are shown through a range of numerical examples to be robust and locking-free, for a selection of element geometries, and fibre directions corresponding to mild and strong inhomogeneity.