  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. 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. A cell membrane is simplified as a
composite shell of lipid bilayer and membrane skeleton, which is a little
similar to the solid shell. 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. 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.
