Extremal states of energy of a double-layered thick-walled tube - application to residually stressed arteries.

Various biological tissues are designed to optimally support external loads for complex geometries and mechanobiological structures. This results in complex microstructures of such materials. The design of, for instance, (healthy) arteries, which are in the focus of this work, is characterised by a residually stressed fibre-reinforced multi-layered composite with highly non-linear elastic response. The complex interaction of material properties with the geometry and residual stress effects enables the optimal support under different blood pressures, respectively blood flow, within the vessel. The fibres reinforcing the arterial wall, as well as residual stresses present in the vessel, strongly influence its overall behaviour and performance. Turn-over and remodelling processes of the collagenous fibres occurring in the respective layers - either resulting from natural growth phenomena or from artificially induced changes in loading condition such as stent deployment - support the optimisation of the multi-layered composite structure of arteries for the particular loading conditions present in the artery. Within this contribution, the overall energetic properties of an artery are discussed by means of the inflation, bending and extension of a double-layered cylindrical tube. Different states of residual stresses and different fibre orientations are considered so that, for instance, representative fibre angles that result in extremal states of the total potential energy can be identified. In view of turn-over and remodelling processes, these orientations are considered to constitute preferred directions of fibre alignment. In summary, the main goal of this work is to calculate optimal material, structural and loading parameters by concepts of energy-minimisation. Several numerical studies show that the obtained values - such as the fibre orientations, the residual axial stretch and the opening angle - are in good agreement with respective physiological parameters reported in the literature.