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.
We human beings live in a three-dimensional (3D) space which contains many geometric entities composed of atoms or molecules. The length scale of objects observed with our naked eyes is much larger than the distance between nearest neighbor atoms or molecules in the objects. As a result, the objects can be regarded as continua. If one dimension of an object is much larger than the other two dimensions, such as a rod, we call it a one-dimensional (1D) entity. If one dimension of an object is much smaller than the other two dimensions, such as a thin film, we call it a two-dimensional (2D) entity. In this review, the term "low-dimensional continua" represents 1D and 2D entities.
Elasticity is a property of materials. It means that materials deform under external forces, but return to their original shapes when the forces are removed. Elastic theory, the study on the elasticity of continuum materials, has a long history 1,2 which records many geniuses such as Hooke (1635-1703), Bernoulli (1700-1782), Euler (1707-1783), Lagrange (1736-1813), Young (1773-1829), Poisson (1781-1840), Navier (1785-1836), Cauchy (1789-1857), Green (1793-1841), Lamé (1795-1870), Saint-Venant (1797-1886), Stokes (1819-1903), Kirchhoff (1824-1887), and so on. Now elastic theory has been a mature branch of physics and summarized in several excellent textbooks. 2,3,4 Although the classical elastic theory is applied to macroscopic continuum materials, more and more facts reveal that it can be also available for bio-or nanostructures such as short DNA rings, 5,6,7,8,9,10,11,12,13 α-helical coiled coils, 14 chiral filaments, 15,16,17,18,19,20,21 climbing plants, 22,23 bacterial flagella, 24 viral shells, 25,26,27 bio-membranes, 28,29,30,31,32,33,34,35,36 zinc oxide nanoribbons, 37,38,39 and carbon nanotubes, 40,41,42,43,44,45,46 to some extent. This review presents the elastic theory of lowdimensional continua and its applications in bio-and nano-structures, which is organized as follows: In Sec. II, we briefly introduce the geometric representation and the elastic theory of low-dimensional continua including 1D rod and 2D fluid membrane or solid shell. The free energy density of the continua is constructed on the basis of the symmetry argument. The fundamental equations can be derived from the bottom-up and the top-down viewpoints. Although they have different forms obtained from these two standpoints, several examples reveal that they are, in fact, equivalent to each other. In Sec. III, the application and availability of the elastic theory of lowdimensional continua in bio-structures, including short DNA rings, lipid membranes, and cell membranes, are discussed. We investigate the kink stability of short DNA rings, the elasticity of lipid membranes, and the adhesions between a vesicle and a substrate or another vesicle. A cell membrane is simplified as a composite shell of lipid bilayer and membrane skeleton. The membrane skeleton is shown to enhance highly the mechanical stability of cell membranes. In Sec. IV, the application and availability of the elastic theory of low-dimensional continua in nano-structures, including graphene and carbon nanotubes, are discussed. We propose a revised Lenosky lattice model and fit four parameters in this model through the local density approximation. We derive its continuum form up to the second order terms of curvatures and strains, which 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 by using this continuum form. Sec. V is a brief summary and prospect.
In this section, we describe the mathematical basis and the elastic theory of 1D and 2D continua.
Fig. 1 depicts a curve C embedded in the 3D Euclid space. Each point in the curve can be expressed as a vector r and let s be the arc length parameter. At point r(s), one can take T, N, and B as the tangent, normal and binormal vectors, respectively. {r; T, N, B} is called the Frenet frame which satisfies the Frenet formula:
where the prime represents the derivative with respect to s. κ and τ are the curvature and torsion of the curve, respectively. The fundamental theory of curve 47 tells us that the bending and twist properties of a smooth curve are uniquely determined by the Frenet formula (1).
Fig. 2 depicts a surface M embedded in the 3D Euclid space. Imagine that a mass point moves on the surface in the speed of unit and that a right-handed frame, which consists of three unit orthonormal vectors with two vectors always in the tangent plane of the surface, adheres to the mass point. Assume that the mass point is at position expressed as vector r and the frame superposes three unit orthonormal vectors {e 1 , e 2 , e 3 } with e 3 being the normal vector of surface M at some time s. When the mass point moves to another position r ′ at time s + ∆s, the frame will superpose three unit orthonormal vectors {e ′ 1 , e ′
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