On a simplified multi-layered plate model of growth: Asymptotic analyses and numerical implementation

Abstract

In this paper, a simplified multi-layered plate model is proposed, which can be used to study the growth-induced deformations of multi-layered hyperelastic plates. First, we introduce a full-form multi-layered plate theory of growth, which has a relatively complicated plate equation system. To simplify the plate equations, systematic asymptotic analyses are conducted with some specified magnitudes of growth functions and surface tractions. It is found that the terms containing a high-order stress tensor always have higher asymptotic orders. By dropping these higher-order terms from the original plate equations, a simplified multi-layered plate model can be derived. Despite its simplicity, this new plate model is valid in a wide range of growth and loading conditions. To be prepared for practical applications, the associated weak forms of the simplified plate equations are derived and implemented in a finite element software. Some typical examples are further studied to show the efficiency of the plate model. By virtue of the plate model, the growth deformations of multi-layered plate samples are simulated, which show very good consistency with the results of 3D volume model. Especially, the distributions of residual stresses on the interfaces of different layers can be predicted accurately, which has an important meaning for the design of soft devices with multi-layered plate structures.