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A nonlinear elasticity approach to modelling the collapse of a shelled microbubble

Cowley, James and Mulholland, Anthony J. and Gachagan, Anthony (2017) A nonlinear elasticity approach to modelling the collapse of a shelled microbubble. IMA Journal of Applied Mathematics. ISSN 1464-3634

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Abstract

There is considerable interest in using shelled microbubbles as a transportation mechanism for localised drug delivery, specifically in the treatment of various cancers. In this paper a theoretical model is proposed which predicts the dynamics of an oscillating shelled microbubble. A neo-Hookean, compressible strain energy density function is used to model the potential energy per unit volume of the shell. The shell is stressed by applying a series of small radially directed stress steps to the inner surface of the shell whilst the outer surface is traction free. Once a certain radial deformation is reached, the stress load at the inner radius is switched off causing the shell to collapse and oscillate about its equilibrium (stress free) position. The inflated shell configuration is used as an initial condition to model the time evolving collapse phase of the shell. The collapse phase is modelled by applying the momentum balance law and mass conservation. The dynamical model which results is then used to predict the collapse time of the shelled microbubble as it oscillates about its equilibrium position. A linear approximation is used in order to gain analytical insight into both the quasistatic inflationary and the oscillating phases of the shelled microbubble. Results from the linearised model are then analysed which show the influence of the shell's thickness, Poisson ratio and shear modulus on the rate of oscillation of the shelled microbubble. The nonlinear model for the quasistatic state is solved numerically and compared to the linearised quasistatic solution. At present, there is no solution to the nonlinear collapsed state. This is a future area of research for the current authors.