The stable shapes and structure formation in prestretched threads of a weak soft polymer gel with permanent cross-links under the action of capillary forces are studied theoretically here. Two cases are considered: when the thread is formed from a drop of gel located between the surfaces after they are moved apart and when it is fixed at the ends. In the first case, we identified three stages of the thread relaxation, characterized by highly different times and associated with the formation of the filament, its lateral compression accompanied by the release of solvent at its surface, and diffusion of the solvent to the terminal droplets. A general linear stability analysis is carried out for threads of any diameter and degree of prestretch, taking into account the effects of elasticity, interactions and inertia, and the corresponding dispersion relations are obtained. The thermodynamically stable beads-on-a-string structures are studied as well and a diagram of the state of the gel thread is presented. It reveals several regimes of different internal structures of gel threads along with phase transitions between them.
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