A thermodynamic framework for diffusive shells with finite deformation: application to hydrogel systems

This study establishes a comprehensive thermodynamic theory for analyzing finite deformations in diffusive shells. The geometric description employs covariant tensor analysis with Gaussian local coordinate parametrization to characterize shell kinematics. Building upon Gurtin's classical continuum framework, we derive the force equilibrium equations and orthogonal/lateral moment equilibria through systematic decomposition of the velocity field into three independent kinematic components: in-plane, transverse, and rotational. By integrating through the shell thickness in conjunction with mass conservation principles, we develop an equivalent mid-surface theory specifically tailored for diffusive shells. Notably, the assumption of a uniform chemical potential across the thickness simplifies the diffusive analysis by converting the governing differential equations into a difference form, which significantly reduces computational cost. When particularized for stimuli-responsive hydrogels with an integrated surface free energy density, the developed framework yields closed-form solutions for the diffusion-induced deformation of macromolecule-encapsulating spherical shells immersed in liquid media. The impacts of key physical and geometrical parameters are then investigated in detail, including Flory parameter, environment chemical potential, degree of crosslinking, initial macromolecule percentage, and the shell radius-thickness ratio. Our analysis reveals two fundamental phenomena:i. Counterintuitive inverse correlations between hydrogel and inner solution Flory parameters in deformation modulation;ii. Emergence of discontinuous deformation patterns induced by phase separation mechanisms.The theory demonstrates mathematical consistency through asymptotic recovery of two limit cases: bulk hydrogel behavior when radius-thickness ratio approaches unity, and Kirchhoff plate response under vanishing curvature. These findings provide new insights into the chemo-mechanical coupling of soft active shells, with particular relevance to drug delivery systems and responsive microcapsule design.