Phase diagrams in inflating balloon

Volume transition of an inflating cylindrical balloon made of rubber under external force $F$ is studied based on the non-linear elastic theory for rubber. The pressure difference $\Delta P$ between inside and outside of the balloon is calculated as a function of the volume $V$ of the fluid inside the balloon. It is shown that if $F$ is smaller than a critical value $F_c$, the inflation of the balloon takes place with coexistence of strongly inflated region and weakly inflated region, while if $F$ is larger than $F_c$, the inflation takes place uniformly. It is shown that the critical force $F_c$ depends on the material parameter of the rubber, and that the discontinuous transition for finite stretching force can take place only when the rubber has enough stretchability