New Representations for the Curvature Tensor of a Surface with Application to Theories of Elastic Shells

Consider two points P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P$\end{document} and Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q$\end{document} on a surface. Modulo rotations about the normal vector to the surface at P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P$\end{document} and the normal vector to the surface at Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q$\end{document}, a rotation can be defined that maps the unit normal vector to the surface at Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q$\end{document} to the corresponding unit normal vector at P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P$\end{document}. With the help of Weingarten’s formulae, new representations are established for the components of the curvature tensor of a surface and the associated mean and Gaussian curvatures in terms of components of a pair of vectors associated with the rotation. The formulae are shown to be helpful in demonstrating how different strain measures for Kirchhoff-Love shell theory are equivalent.


Introduction
Given a surface S, Weingarten's well-known formulae express the evolution of a unit normal vector n to a two-dimensional surface in terms of the coefficients b αβ of second-fundamental form of the surface and the covariant basis vectors a γ (cf. Fig. 1). In this paper, we use his formulae to establish new representations for the components of the curvature tensor b = b αβ a α ⊗ a β = b δ β a δ ⊗ a β , the mean curvature H , and the Gaussian curvature K: where the angular rate vectors ω β are used to compute the partial derivatives of n: n ,β = ω β × n.
(1.2) Fig. 1 Schematic of a surface S that is embedded in E 3 . The covariant basis vectors a 1 = r ,1 and a 2 = r ,2 , the unit normal n at a point P of S, the Cartesian basis vectors {E 1 , E 2 , E 3 }, and the unit normal n 0 at a point Q of S are also shown As an illustrative example, the case of a spherical surface is considered. The representations (1.1) are then shown to be helpful in relating two different formulations for some of the strain measures used in Kirchhoff-Love shell theory. We also note that the discussion in this paper complements alternative representations for b αβ , H , and K, including those using Cartan's moving frames that can be found in texts on differential geometry (see, e.g., [1,5,12]).

Background
Consider a two-dimensional surface S that is embedded in three-dimensional Euclidean space E 3 . The surface is parameterized using a curvilinear coordinate system θ 1 , θ 2 such that every point P on the surface can be uniquely identified by a position vector (relative to a fixed origin): We now define a covariant basis {a 1 , a 2 } for the tangent space to a point on S: Here, and in the sequel, lower-case Greek letters range from 1 to 2, the summation convention on repeated indices is employed, and the comma denotes partial derivative. A unit normal n is assumed to be defined at every point P on the surface where (a 1 × a 2 ) · n > 0. We can define a contravariant basis a 1 , a 2 where a α · a β = δ β α and n · a β = 0. The Kronecker delta δ β α = 0 if β = α and = 1 if α = β. Solving a α · a β = δ β α for a β : where √ a = a 1 × a 2 = (a 1 × a 2 ) · n. We note that a β = a βα a α , a δ = a δα a α , (2.4) where a αβ = a α · a β , a αβ = a α · a β .
Here, a αβ are the coefficients of the first fundamental form of S.

A Rotation of the Unit Normal Vector and a Pair of Axial Vectors
We consider a point Q on the surface and denote the normal to the surface at this point by n 0 . As the normal vector is a unit vector, we can define a rotation that transforms n 0 to the normal n at any point on the surface. The resulting rotation tensor, which we denote by Q, is a function of the coordinates θ β : As discussed in the sequel, the tensor Q is not uniquely defined. We also note that various representations, including Euler angles, axis-angle, Euler parameters, unit quaternions, and Cayley parameters for Q can be employed but this specification is not needed for present purposes.
Differentiating the identity n = Qn 0 with respect to θ β , using the facts that Q T Q = I and n 0 is constant, we conclude that However, as Q ,β Q T is skew-symmetric, we can define an axial vector ω β such that for any vector a. The vector ω β has the representations As the vectors ω δ associated with Q are identical to those associated with QQ 1 where Q 1 is an arbitrary constant rotation tensor, the choice of Q, as anticipated, does not effect the forthcoming results. The prescription (3.1) for Q does not define a unique rotation tensor. Indeed, if Q satisfies (3.1) then so does R = L (ν, n) QL (ν 0 , n 0 ) , (3.5) where the tensor L (θ, p) represents a rotation through an angle θ about the unit vector p: 1 The angle of rotation ν 0 is a constant. The axial vectors associated with the rotation R can be computed using relative angular velocity vectors [3]: Thus, the non-uniqueness of Q implies that the n component of ω α are not uniquely prescribed. Examining (1.1), we find that the components of b and K and H are independent of ω α · n. In conclusion, the curvature tensor b is insensitive to the fact that Q is defined modulo an arbitrary rotation about n.
4 Formulae for the Curvature Tensor, the Mean Curvature, and the Gaussian Curvature The curvature tensor b has the representations (see, e.g., [2,9] and ⊗ is the tensor product: (a ⊗ b) c = a (b · c) for any vectors a, b, and c.
Weingarten's formulae relate the evolution of n to the components of b: The standard derivation of this formula starts by differentiating the identities n · n = 1 and n · a β = 0. For our purposes, it is fruitful to consider the components of n ,β : where 11 = 22 = 0 and 12 = − 21 = √ a. We can use (4.3) along with (4.4) to show that b αβ = −n ,β · a α = − ω β × n · a α = −ω β · (n × a α ) .
The identities b γ δ = a γβ b βδ and a α = a αβ a β were used to establish the second representation. The identity (4.6) 1 along with the symmetry b 12 = b 21 implies that ω 1 · a 1 + ω 2 · a 2 = 0. (4.7) Thus, the components of ω 1 and ω 2 are not all independent. The representation (1.1) 3 for H can be obtained by first substituting for b α β in the definition of H (cf. [9]) and then appealing to (2.3) and (4.6): Fig. 2 Schematic of a unit sphere showing the angles ϕ and ϑ and several sets of basis vectors. In light of our comments about the normal components of ω β , it is interesting to note that ω 2 has a component in the n direction: The representation (1.1) 4 for K can be obtained by first substituting for b αβ in the definition of K (cf. [12]) and then using (2.3): We observe that the representation for K provides an elegant representation for KdA where dA is the element of area on S: The integral of KdA is central to the Gauss-Bonnet theorem.

Application to a Sphere
As an example, we consider a sphere of radius R and use a set of polar angles θ 1 = ϕ and θ 2 = ϑ to parameterize the sphere (cf. Fig. 2). The representation for the position vector r of any point on the sphere is where e r = cos (ϑ) E 1 + sin (ϑ) E 2 , e R = sin (ϕ) e r + cos (ϕ) E 3 .
Differentiating r, the covariant basis vectors can be computed along with their contravariant counterparts: Clearly, n = e R and √ a = R 2 sin (ϕ). The rotation tensor Q can be defined as the product of two rotations: one about the E 3 axis through an angle ϑ followed by a rotation about e ϑ through an angle ϕ: Thus, Noting that e ϑ × e R = e ϕ , we can now compute the components of b using either (1.1) 1 or The curvatures H and K can be computed using (1.1) 3,4 : as anticipated.
The formula (1.1) 1 can be used to establish the equivalence ofW andŴ : It should also be evident from (6.3) that the strain energy functionW should be independent of the components ω β · n. This restriction is in complete agreement with our earlier remarks that Q is defined modulo a rotation about n.
Although the tensor Q plays a central role in some formulations of Kirchhoff-Love shell theory (see [6]), we have not found a previous discussion of the non-uniqueness of Q. In some shell theories (cf. [4,10]) where a rotation tensor Ψ is associated with each point of the material surface, a pair of right-handed orthonormal triads (or adapted frames) are chosen: one comprised of n 0 and two unit tangent vectors s 0 1 and s 0 2 in the reference configuration and the other comprised of n and two unit tangent vectors s 1 and s 2 in the present configuration. The rotation tensor Ψ, where Ψ = s 1 ⊗ s 0 1 + s 2 ⊗ s 0 2 + n ⊗ n 0 , (6.4) is uniquely defined in this case. The fact that there are infinitely many choices of the triads n 0 , s 0 1 , s 0 2 and {n, s 1 , s 2 } provides another explanation for the non-uniqueness of Q. 3