A uniformly-valid asymptotic plate theory of growth with numerical implementation

Abstract

In this paper, we aim to develop a simplified and uniformly-valid finite-strain plate theory of growth. First, starting from a consistent finite-strain plate theory of growth proposed in our previous work, we specify the magnitudes of growth functions in five different cases and conduct systematic asymptotic analyses, from which the original plate theory can be reduced to some classical plate and membrane theories. Based on the asymptotic analyses, it is found that some terms in the original plate equations (which contain the high-order stress tensor S(2)) always have higher asymptotic orders. By dropping these terms, the plate equations can be simplified significantly. Then, a reduced finite-strain plate theory of growth is established, which is valid in a wide range of growth and mechanical loading conditions. The associated weak form of this uniformly-valid plate theory has also been derived for numerical implementation. To demonstrate the efficiency of this plate theory, it is implemented into a finite element software and applied to study four typical examples. To verify the accuracy of the 2D plate models, the corresponding 3D volume models have also been constructed for these examples. Through some comparisons, it is found that the numerical results obtained from the 2D and 3D models can fit each other quite well. Furthermore, the numerical calculations based on the plate models show obvious advantages in the aspects of computational efficiency, convergence rate and stability. In our opinion, the uniformly-valid asymptotic plate theory of growth proposed in the current work is adequate for studying the complex growth behaviors of thin hyperelastic plates.