Elastic Theory of Low-Dimensional Continua and Its Applications in Bio- and Nano-Structures

Abstract

This review presents the elastic theory of low-dimensional (one- and two-dimensional) continua and its applications in bio- and nano-structures. First, the curve and surface theory, as the geometric representation of the low-dimensional continua, is briefly described through Cartan moving frame method. The elastic theory of Kirchhoff rod, Helfrich rod, bending-soften rod, fluid membrane, and solid shell is revisited. The free energy density of the continua, is constructed on the basis of the symmetry argument. The fundamental equations can be derived from two kinds of viewpoints: the bottom-up and the top-down standpoints. In the former case, the force and moment balance equations are obtained from Newton's laws and then some constitute relations are complemented in terms of the free energy density. In the latter case, the fundamental equations are derived directly from the variation of the free energy. Although the fundamental equations have different forms obtained from these two viewpoints, several examples reveal that they are, in fact, equivalent to each other. Secondly, the application and availability of the elastic theory of low-dimensional continua in bio-structures, including short DNA rings, lipid membranes, and cell membranes, are discussed. The kink stability of short DNA rings is addressed by using the theory of Kirchhoff rod, Helfrich rod, and bending-soften rod. The lipid membranes obey the theory of fluid membrane. The shape equation and the stability of closed lipid vesicles; the shape equation and boundary conditions of open lipid vesicles with free edges as well as vesicles with lipid domains, and the adhesions between a vesicle and a substrate or another vesicle are fully investigated. A cell membrane is simplified as a composite shell of lipid bilayer and membrane skeleton, which is a little similar to the solid shell. The equations to describe the in-plane strains and shapes of cell membranes are obtained. It is found that the membrane skeleton enhances highly the mechanical stability of cell membranes. Thirdly, the application and availability of the elastic theory of low-dimensional continua in nano-structures, including graphene and carbon nanotubes, are discussed. A revised Lenosky lattice model is proposed based on the local density approximation. Its continuum form up to the second order terms of curvatures and strains is the same as the free energy of 2D solid shells. The intrinsic roughening of graphene and several typical mechanical properties of carbon nanotubes are revisited and investigated based on this continuum form. It is possible to avoid introducing the controversial concepts, the Young's modulus and thickness of graphene and single-walled carbon nanotubes, with this continuum form.