Mesoscopic modelling of membrane-fluid interactions using peridynamics: application to deformable and fracturing beams

Abstract

In the veins, blood flow is controlled by deformable valves; damage to and subsequent deterioration of the tissues of the vein wall and valves of the lower limbs leads to changing material properties, valve remodelling, and serious medical conditions such as varicose veins and hypertension. Damage to the system can be significant, to the point of valve disappearance. Investigation of the damage cascade is difficult in vivo and thus it is not well understood. In this thesis, the bond-based PD-IB-CLBM is applied to this complex problem and two curved interacting beams are simulated, representative of a bicuspid valve in 2-dimensions. The strong coupling of Peridynamics (PD) for the solid material and the cascaded lattice Boltzmann method (CLBM) for the working fluid, via the immersed boundary (IB) method is an efficient technique for modelling fluidstructure interaction (FSI) and can capture the inception and propagation of damage.

Initially, a small number of simulations were carried out to investigate the effects of Young’s modulus (E) and Reynolds number (Re) on fluid and solid dynamics. The respective ranges were 10 ≤ E ≤ 200 kPa and 70 ≤ Re ≤ 110. The results show that a higher E will lead to a reduction in opening area and a higher jet velocity, which agrees with findings in the literature. A higher Re leads to a larger opening aperture and tip displacement for a given E. Furthermore, the length of the fluid recirculation region downstream of the beams increases with both Re and E, as does the rate of growth, but due to limitations in solid displacement, the normalised throat velocity will decrease.

A set of 65 simulations with the bond-based PD-IB-CLBM was then setup to investigate the impact of critical stretch ratio (s꜀) on flow characteristics and beam displacements. When failure is permitted, systems fall in to 1 of 4 failure zones depending on E and s꜀, which is based on the solid material fracture energy. Very low E and s꜀ will result in weak beams which fail due to very small displacements from the initial configuration. High s꜀ results in beams which can move freely with little damage propagation, generally replicating the expected valve motions. Partial failures can occur leading to monocuspid-like behaviour as only one of the leaflets remains operational. The most common mode of failure was upon close, where the beams press together and deform significantly. The damage of PD material points was heavily dependent on the failure mechanism; trends based on E and s꜀ can be seen after this categorisation.

Bond-based PD is only capable of accurately modelling materials with Poisson’s ratio ν = 1/3, and hence the more advanced state-based PD-IB-CLBM model was introduced. This model was validated via comparison with results from COMSOL Multiphysics with the hope of applying it in future studies to more complex PD materials.

This thesis provides an initial look at the damage mechanisms which occur when fluidsolid and solid-solid interaction of this type take place. This can be extrapolated to more complex systems, aid in understanding valve pathology and the illness cascade, and instruct the development of replacement vein valves. Applications for valves such as these may extend to means of passive control in soft-robotics and therefore knowledge of standard operation and failure behaviour can be of importance in industrial applications. More generally, the study of beams in a flow is applicable to many FSI phenomena such as flapping of insect wings, and hair and fibre beds prevalent in biological organisms.