This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional (3-D) potential energy under general loadings with a third-order error. Staring from the 3-D nonlinear elasticity (with both geometrical and
material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the 3-D field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a 2-D virtual work principle. An alternative approach based on a 2-D truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a 2-D energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Comparing with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of the loadings, applicability to finite-strain problems and no involvement of unphysical quantities.
A polymer network can imbibe water, forming an aggregate called hydrogel, and undergo large and inhomogeneous deformation with external mechanical constraint. Due to the large deformation, nonlinearity plays a crucial role, which also causes the math
ematical difficulty for obtaining analytical solutions. Based on an existing model for equilibrium states of a swollen hydrogel with a core-shell structure, this paper seeks analytical solutions of the deformations by perturbation methods for three cases, i.e. free-swelling, nearly free-swelling and general inhomogeneous swelling. Particularly for the general inhomogeneous swelling, we introduce an extended method of matched asymptotics to construct the analytical solution of the governing nonlinear second-order variable-coefficient differential equation. The analytical solution captures the boundary layer behavior of the deformation. Also, analytical formulas for the radial and hoop stretches and stresses are obtained at the two boundary surfaces of the shell, making the influence of the parameters explicit. An interesting finding is that the deformation is characterized by a single material parameter (called the hydrogel deformation constant), although the free-energy function for the hydrogel contains two material parameters. Comparisons with numerical solutions are also made and good agreements are found.
