Bending control and stability of functionally graded dielectric elastomers

A rectangular plate of dielectric elastomer exhibiting gradients of material properties through its thickness will deform inhomogeneously when a potential difference is applied to compliant electrodes on its major surfaces, because each plane parallel to the major surfaces will expand or contract to a different extent. Here we study the voltage-induced bending response of a functionally graded dielectric plate on the basis of the nonlinear theory of electroelasticity, when both the elastic shear modulus and the electric permittivity change with the thickness coordinate. The theory is illustrated for a neo-Hookean electroelastic energy function with the shear modulus and permittivity varying linearly across the thickness. In general the bending angle increases with the potential difference, and this enables the material inhomogeneity to be tuned to control the bending shape. We derive the Hessian criterion that ensures stability of the bent configurations in respect of a general form of electroelastic constitutive law specialized for the considered geometry. This requires that the Hessian remains positive. For the considered model we show that the bent configuration is stable until the voltage reaches the value for which the cross section of the bent configuration forms a complete circle.