A virtual element method for hyperelasticity

Doctoral Thesis


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This thesis studies the approximation of plane problems of hyperelasticity, using a loworder virtual element method (VEM). The VEM is an extension of the finite element method (FEM). It is characterised by considerable freedom with regard to element geometry, permitting arbitrary polygonal and polyhedral elements in two and three dimensions respectively. Furthermore, the local basis functions are not known explicitly on elements and take the simple form of piecewise-linear Lagrangian functions on element boundaries. All integrations are performed on element edges. The VEM formulation typically involves a consistency term, computed via a projection, and a stabilization term, which must be approximated. Problems concerning isotropic and transversely isotropic hyperelastic material models are considered. Examples of transversely isotropic materials, which are characterised by an axis of symmetry normal to a plane of isotropy, range from simple fibre-reinforced materials to biological tissues. To date, in the context of hyperelasticity, investigation of the performance of VEM has primarily focused on problems involving the isotropic neo-Hookean material model. Furthermore, there has been limited investigation into the behaviour of the VEM in the nearly incompressible and nearly inextensible limits. In this thesis a VEM formulation with a novel approach to the construction of the stabilization term is formulated and implemented for problems involving isotropic and transversely isotropic hyperelastic materials. The governing equations of hyperelasticity are derived and various isotropic and transversely isotropic constitutive models are presented. This is followed by presentation of the virtual element formulation of the hyperelastic problem and a possible approach to its practical implementation. Through a range of numerical examples, the VEM with the proposed stabilization term is found to exhibit robust and accurate behaviour for a variety of mesh types, including those comprising highly non-convex element geometries, and for problems involving severe deformations. Furthermore, the versatility of the proposed VEM formulation is demonstrated through its application to a range of popular isotropic and transversely isotropic material models for a wide variety of material parameters. Through this investigation the VEM is found to exhibit locking-free behaviour in the limiting cases of near-incompressibility and near-inextensibility, both separately and combined.