On a finite-strain plate theory for growth-induced plane-strain deformations and instabilities of multi-layered hyperelastic plates

In this paper, we propose a finite-strain plate theory to study the growth-induced deformations and instabilities of multi-layered hyperelastic plates. First, under the assumption of plane-strain deformation, we formulate the governing system for a multi-layered hyperelastic (neo-Hookean) plate. The layers in the plate have different geometrical, material and growth parameters. Starting from the governing system, we propose a scheme to establish the plate equation system, where one arbitrary layer is selected as the base layer and a series-expansion approach is adopted. Through a complicated derivation procedure, the iteration relations of the unknowns in the different layers are obtained, and then the plate equation system containing the unknowns in the base layer are derived, which can attain the accuracy of O(h(2)). To show the efficiency of the plate equations, we first study the growth-induced bending deformations of a multi-layered hyperelastic plate. By solving the plate equations, some analytical results are obtained, which provide accurate predictions on the deformations of the plate. The influences of the geometrical and material parameters on the response of the plate are also investigated. Besides that, the plate equation system is applied to study the growth-induced instabilities of a 3-layered hyperelastic plate. Through some conventional analyses, the critical values of the growth parameters corresponding to the different bifurcation modes are determined, which can fit the numerical results very well. In our opinion, the plate theory proposed in the current work is helpful for the design of intelligent soft devices with multi-layered plate forms.