Generalizations of the Young–Laplace equation for the pressure of a mechanically stable gas bubble in a soft elastic material

The Young-Laplace equation for the pressure of a mechanically stable gas bubble is generalized to include the effects of both surface tension and elastic forces of its surroundings. The latter are taken to be comprised of a soft isotropic material. Generalizations are derived for conditions of constant external pressure and constant system volume. The derived equations are formally exact for a spherical bubble surrounded by a spherical shell of isotropic material, provided that the bubble is sufficiently large for the surface tension to be treated macroscopically, and that the bubble radius is much larger than the thickness of the bubble/soft material interface. The underlying equations are also used to derive a simple expression for the Gibbs free energy of deformation of an elastic medium that surrounds a gas bubble. The possible relevance of this expression to some recently published ideas on decompression sickness ("the bends") is discussed.