We present a novel, thermodynamically consistent, model for the charged-fluid flow and the deformation of the morphology of polymer electrolyte membranes (PEM) in hydrogen fuel cells. The solid membrane is assumed to obey linear elasticity, while the pore is completely filled with protonated water, considered as a Stokes flow. The model comprises a system of partial differential equations and boundary conditions including a free boundary between liquid and solid. Our problem generalizes the well-known Nernst-Planck-Poisson-Stokes system by including mechanics. We solve the coupled non-linear equations numerically and examine the equilibrium pore shape. This computationally challenging problem is important in order to better understand material properties of PEM and, hence, the design of hydrogen fuel cells.