Deformation of a sheet with coupling between elasticity and concentration

We investigate a simple model for equilibrium deformation of a sheet with concentration-dependent elasticity. The model is motivated by several physical situations where deformation of a sheet is modulated by concentration of a mobile species, for example, a quasi-2D array of carbon nanotubes. Elasticity of the sheet is modeled using a free energy functional that includes concentration-dependent potential energies, and a free energy of mixing. We show that the sheet responds to imposed distortion by rearranging its constituents and can do so in a smooth and stable manner only for sufficiently small distortion. For larger distortions, through phase plane and numerical analysis, we consider how the system can meet boundary conditions through nonsmooth solutions.