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Summary
In this paper, we propose a general multi-layered hyperelastic plate theory to study the growth-induced deformations of soft material samples. First, by considering a multilayered plate sample with an arbitrary geometrical shape and adopting a general hyperelastic constitutive model, the 3D governing system is established, which incorporates the growth effects of the different layers in the plate. Then, a series expansion-truncation approach is adopted to eliminate the thickness variables in the 3D governing system. An elaborate calculation scheme is applied to derive the iteration relations of the coefficient functions in the series expansions. Through some further manipulations, a 2D vector plate equation system with the associated boundary conditions is established. To show the efficiency of the plate theory, three typical examples regarding the growth-induced deformations and instabilities of multi-layered plate samples are studied, where the incompressible neo-Hookean constitutive model is adopted. Some analytical and numerical solutions to the plate equation system are obtained, which can provide accurate predictions on the growth behaviors of the plates. Furthermore, the problem of `shape-programming' of multi-layered plates through differential growth is studied and the explicit formulas for some typical examples are derived. By using these formulas, the shape evolutions of the plates during the growing processes can be controlled accurately. The results obtained in the current work are applicable for the design of intelligent soft devices with multi-layered plate structures.
