Shell finite elements, with applications in biomechanics - Bartle, Samantha (Department of Mechanical Engineering, 2014)
This thesis gives a detailed presentation of a formulation for thin shells, and its finite element approximation, with the goal of modelling soft, thin biological tissues. The rigorous but complex theory due to Simo and Fox (1986) is presented in an accessible manner, with detailed derivations where appropriate. The presentation is confined to small strains and linear elasticity, with the constitutive theory extended to take account of transverse isotropy. The finite element formulation is given in such a way as to make various implementational aspects clear. Implementation has been carried out in deal.II, an open source library of finite element code. Substantial detail is given about how the shell formulation was implemented; this includes preprocessing, programming of the solution algorithm, and post-processing of results. The formulation is tested against a series of benchmark problems for flat plates and cylindrical shells, under a variety of loading conditions, and compared with results in the literature. II Two example problems in biomechanics are considered: the problem of arterial clamping, and the modelling of a prosthetic aortic valve. In the case of the clamped artery, the deformed shape for a range of clamp depths compares well with results in the literature obtained using a three-dimensional formulation. The addition of helical fibre families orientated in the same manner as two different arterial layers significantly altered the resulting deformations and agreed qualitatively with those in the literature. Using the geometric and material parameters given in earlier studies of prosthetic aortic valve leaflets, the shell solution algorithm was used to simulate a leaflet with and without transverse isotropy. The deformed leaflet behaved as expected for a diastolic state and showed a significant increase in load carried by the aortic wall with the inclusion of fibres. The work concludes with suggestions for extensions to include, for example, large strains and nonlinear material models.