Pressurized fluid-filled vesicles injected in soft materials

We develop a reduced variational model for a pressurized fluid-filled vesicle opened from a slender precursor in an incompressible neo-Hookean solid. A strictly line-like precursor acquires logarithmically divergent elastic energy when opened to finite transverse size, so the finite-energy problem is posed outside a small prolate-spheroidal cutoff. Within an axisymmetric prolate-spheroidal ansatz, incompressibility reduces the problem exactly to a one-parameter family indexed by an added-volume coordinate. We establish existence, uniqueness, smoothness, and injectivity of the resulting reduced maps. The vesicle geometry and injected volume are explicit, whereas the regularized elastic energy and pressure–volume relation are evaluated by convergent two-dimensional quadrature. Stationarity and energetic stability are assessed with respect to variations along the restricted deformation class. The analysis distinguishes the smooth finite-cutoff response near the precursor from the slit-like line-track regime and shows that large vesicles become asymptotically spherical. The precursor half-length, cutoff radius, shear modulus, and reduced pressure govern the predicted morphology and deployment response.