On elastic deformation of radially prestressed incompressible spherical shells

Abstract

We formulate and study inflation of radially prestressed incompressible spheres and annular spherical shells. We find that radially prestressed Neo-Hookean spheres subjected to tensile radial component of normal stress at the outer boundary admit deformations fields such that a cavity forms at the center of the sphere and that for a critical value of this radial component of the normal stress at the outer boundary, \(\mathcal{P}_{o}^{cr}, \) the surface of the cavity becomes traction free. This is the cavitation problem first studied by Ball [Phil Trans Royal Soc London A 306(1496): 557–611 (1982)]. Our studies show that the value of \(\mathcal{P}_{o}^{cr}\) depends on the prestress. While some prestress distributions requires the radial component of the normal stress at the outer surface to increase in value, at least initially, for the cavity to grow, for others, like in the stress free case, requires this normal stress to decrease monotonically. Consistent with the observation of Saravanan [ Int J Non-Linear Mech 46(1): 96–113 (2011)] we also find that the boundary traction required to engender a given deformation field in a prestressed annular spherical shell depends on the prestresses.