Realization of planar and surface conformal mappings through stress-free growth of hyperelastic plates: Analytical formulas and numerical calculations

Abstract

Conformal mapping is a well-known concept and has a long research history in mathematics. Accompanying the developments of computational science and 3D digital scanning technology, conformal mapping has also found wide applications in geometric modeling, computer graphics, medical imaging and other fields. In the virtual image or animation world, conformal mapping offers a convenient approach to achieve various shape changes of planar regions and 3D surfaces. In this work, we propose a promising approach, i.e., through the stress-free growth of hyperelastic plates, to realize arbitrary planar and surface conformal mappings in the physical world. The growth field in a hyperelastic plate can be represented by a symmetric tensor field. For any given planar or surface conformal mappings with explicit analytical expressions, the formulas of growth functions in the growth tensor are derived. In the case that the target surfaces of conformal mappings have no explicit analytical expressions, we further propose a numerical scheme to determine the growth data in the hyperelastic plates. The efficiency of the approach proposed in the current work is validated through several typical examples, where the 3D finite element simulations are conducted. The results demonstrate that, by incorporating the obtained growth functions or growth data, the target surfaces can be generated accurately through growth deformations of hyperelastic plates. As an application of the current work, we further propose an approach of shape-control for double-layer grid structures with plate forms. According to any given target shapes, the values of elongation or contraction of the bar elements are determined, then the desired deformations of the grid structures can be realized.