An incremental plate theory for polymer gels in equilibrium

Based on the finite-strain plate theory derived by one of the authors, an incremental plate theory for deformations superimposed on a general base state is formulated in this paper. The unknown functions of the incremental deformation are first expanded into Taylor series in terms of the thickness variable and then expanded around the base state by retaining third-order nonlinearity. From the field equations and the boundary conditions at the top and bottom surfaces, the recursive relations of the expansion coefficients as well as the incremental balance equations are derived. With the constitutive relation for swollen polymer gels in equilibrium, an incremental plate theory is obtained. As an application, this theory is used to study the incremental deformation of a polymer gel layer with a homogeneous base state in a plane strain setting. A linear bifurcation analysis is carried out, which gives the critical values of the external chemical potential and the mode number. The results agree with those obtained directly from a full two-dimensional analysis. Post-bifurcation is also conducted through a perturbation procedure, which reveals that the bifurcation is of supercritical type.