1 Introduction

Given a surface \(S\), Weingarten’s well-known formulae express the evolution of a unit normal vector \({\mathbf{n}}\) to a two-dimensional surface in terms of the coefficients \(b_{\alpha \beta }\) of second-fundamental form of the surface and the covariant basis vectors \({\mathbf{a}}_{\gamma }\) (cf. Fig. 1). In this paper, we use his formulae to establish new representations for the components of the curvature tensor \({\mathbf{b}} = b_{\alpha \beta }{\mathbf{a}}^{\alpha }\otimes {\mathbf{a}}^{\beta }= b^{ \delta }_{\beta }{\mathbf{a}}_{\delta }\otimes {\mathbf{a}}^{\beta }\), the mean curvature \(H\), and the Gaussian curvature \(K\):

(1.1)

where the angular rate vectors are used to compute the partial derivatives of \({\mathbf{n}}\):

(1.2)

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 \(b_{\alpha \beta }\), \(H\), and \(K\), including those using Cartan’s moving frames that can be found in texts on differential geometry (see, e.g., [1, 5, 12]).

Fig. 1
figure 1

Schematic of a surface \(S\) that is embedded in \(\mathbb{E}^{3}\). The covariant basis vectors \({\mathbf{a}}_{1} = {\mathbf{r}}_{,1}\) and \({\mathbf{a}}_{2} = {\mathbf{r}}_{,2}\), the unit normal \({\mathbf{n}}\) at a point \(P\) of \(S\), the Cartesian basis vectors \(\left \{ {\mathbf{E}}_{1}, {\mathbf{E}}_{2}, {\mathbf{E}}_{3}\right \} \), and the unit normal \({\mathbf{n}}_{0}\) at a point \(Q\) of \(S\) are also shown

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2 Background

Consider a two-dimensional surface \(S\) that is embedded in three-dimensional Euclidean space \(\mathbb{E}^{3}\). The surface is parameterized using a curvilinear coordinate system \(\left \{ \theta ^{1},\theta ^{2}\right \} \) such that every point \(P\) on the surface can be uniquely identified by a position vector (relative to a fixed origin):

$$\begin{gathered} {\mathbf{r}} = {\mathbf{r}}\left (\theta ^{1}, \theta ^{2}\right ). \end{gathered}$$
(2.1)

We now define a covariant basis \(\left \{ {\mathbf{a}}_{1},{\mathbf{a}}_{2} \right \} \) for the tangent space to a point on \(S\):

$$\begin{gathered} {\mathbf{a}}_{\beta }= {\mathbf{r}}_{,\beta } = \frac{\partial {\mathbf{r}}}{\partial \theta ^{\beta }}, \qquad \left (\beta = 1,2\right ). \end{gathered}$$
(2.2)

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 \({\mathbf{n}}\) is assumed to be defined at every point \(P\) on the surface where \(\left ({\mathbf{a}}_{1}\times {\mathbf{a}}_{2}\right )\cdot {\mathbf{n}} > 0\).

We can define a contravariant basis \(\left \{ {\mathbf{a}}^{1}, {\mathbf{a}}^{2}\right \} \) where \({\mathbf{a}}_{\alpha }\cdot {\mathbf{a}}^{\beta }= \delta ^{\beta }_{\alpha }\) and \({\mathbf{n}}\cdot {\mathbf{a}}^{\beta }= 0\). The Kronecker delta \(\delta ^{\beta }_{\alpha }= 0\) if \(\beta \ne \alpha \) and \(= 1\) if \(\alpha = \beta \). Solving \({\mathbf{a}}_{\alpha }\cdot {\mathbf{a}}^{\beta }= \delta ^{\beta }_{\alpha }\) for \({\mathbf{a}}^{\beta }\):

$$\begin{gathered} \sqrt{a} {\mathbf{a}}^{1} = {\mathbf{a}}_{2} \times {\mathbf{n}}, \qquad \sqrt{a} { \mathbf{a}}^{2} = - {\mathbf{a}}_{1} \times {\mathbf{n}}, \end{gathered}$$
(2.3)

where \(\sqrt{a} = \|{\mathbf{a}}_{1}\times {\mathbf{a}}_{2}\| = \left ({\mathbf{a}}_{1} \times {\mathbf{a}}_{2}\right )\cdot {\mathbf{n}}\). We note that

$$\begin{gathered} {\mathbf{a}}_{\beta }= a_{\beta \alpha } {\mathbf{a}}^{\alpha }, \qquad {\mathbf{a}}^{\delta }= a^{\delta \alpha } {\mathbf{a}}_{\alpha }, \end{gathered}$$
(2.4)

where

$$\begin{gathered} a_{\alpha \beta } = {\mathbf{a}}_{\alpha }\cdot {\mathbf{a}}_{\beta }, \qquad a^{ \alpha \beta } = {\mathbf{a}}^{\alpha }\cdot {\mathbf{a}}^{\beta }. \end{gathered}$$
(2.5)

Here, \(a_{\alpha \beta }\) are the coefficients of the first fundamental form of \(S\).

3 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 \({\mathbf{n}}_{0}\). As the normal vector is a unit vector, we can define a rotation that transforms \({\mathbf{n}}_{0}\) to the normal \({\mathbf{n}}\) at any point on the surface. The resulting rotation tensor, which we denote by \({\mathbf{Q}}\), is a function of the coordinates \(\theta ^{\beta }\):

$$\begin{gathered} {\mathbf{n}} = {\mathbf{n}}\left (\theta ^{1}, \theta ^{2}\right ) = {\mathbf{Q}} \left (\theta ^{1}, \theta ^{2}\right ) {\mathbf{n}}_{0}. \end{gathered}$$
(3.1)

As discussed in the sequel, the tensor \({\mathbf{Q}}\) is not uniquely defined. We also note that various representations, including Euler angles, axis-angle, Euler parameters, unit quaternions, and Cayley parameters for \({\mathbf{Q}}\) can be employed but this specification is not needed for present purposes.

Differentiating the identity \({\mathbf{n}} = {\mathbf{Q}}{\mathbf{n}}_{0}\) with respect to \(\theta ^{\beta }\), using the facts that \({\mathbf{Q}}^{T}{\mathbf{Q}} = {\mathbf{I}}\) and \({\mathbf{n}}_{0}\) is constant, we conclude that

$$\begin{gathered} {\mathbf{n}}_{,\beta } = {\mathbf{Q}}_{,\beta }{\mathbf{Q}}^{T} {\mathbf{n}}. \end{gathered}$$
(3.2)

However, as \({\mathbf{Q}}_{,\beta }{\mathbf{Q}}^{T}\) is skew-symmetric, we can define an axial vector such that

(3.3)

for any vector \({\mathbf{a}}\). The vector has the representations

(3.4)

As the vectors associated with \({\mathbf{Q}}\) are identical to those associated with \({\mathbf{Q}}{\mathbf{Q}}_{1}\) where \({\mathbf{Q}}_{1}\) is an arbitrary constant rotation tensor, the choice of \(Q\), as anticipated, does not effect the forthcoming results.

The prescription (3.1) for \({\mathbf{Q}}\) does not define a unique rotation tensor. Indeed, if \({\mathbf{Q}}\) satisfies (3.1) then so does

$$\begin{gathered} {\mathbf{R}} = {\mathbf{L}}\left (\nu , {\mathbf{n}}\right ){\mathbf{Q}}{\mathbf{L}}\left ( \nu _{0}, {\mathbf{n}}_{0}\right ), \end{gathered}$$
(3.5)

where the tensor \({\mathbf{L}}\left (\theta , {\mathbf{p}}\right )\) represents a rotation through an angle \(\theta \) about the unit vector \({\mathbf{p}}\):Footnote 1

$$\begin{aligned} {\mathbf{R}}{\mathbf{n}}_{0} &= {\mathbf{L}}\left (\nu , {\mathbf{n}}\right ){\mathbf{Q}}{\mathbf{L}} \left (\nu _{0}, {\mathbf{n}}_{0}\right ){\mathbf{n}}_{0} \\ &= {\mathbf{L}}\left (\nu , {\mathbf{n}}\right ){\mathbf{Q}}{\mathbf{n}}_{0} \\ &= {\mathbf{L}}\left (\nu , {\mathbf{n}}\right ){\mathbf{n}} \\ &= {\mathbf{n}}. \end{aligned}$$
(3.6)

The angle of rotation \(\nu _{0}\) is a constant. The axial vectors associated with the rotation \({\mathbf{R}}\) can be computed using relative angular velocity vectors [3]:

(3.7)

Thus, the non-uniqueness of \({\mathbf{Q}}\) implies that the \({\mathbf{n}}\) component of are not uniquely prescribed. Examining (1.1), we find that the components of \({\mathbf{b}}\) and \(K\) and \(H\) are independent of . In conclusion, the curvature tensor \({\mathbf{b}}\) is insensitive to the fact that \({\mathbf{Q}}\) is defined modulo an arbitrary rotation about \({\mathbf{n}}\).

4 Formulae for the Curvature Tensor, the Mean Curvature, and the Gaussian Curvature

The curvature tensor \({\mathbf{b}}\) has the representations (see, e.g., [2, 9]):

$$\begin{gathered} {\mathbf{b}} = b^{\alpha }_{\beta }{\mathbf{a}}_{\alpha }\otimes {\mathbf{a}}^{\beta }= b_{ \alpha \beta }{\mathbf{a}}^{\alpha }\otimes {\mathbf{a}}^{\beta }, \end{gathered}$$
(4.1)

where

$$\begin{gathered} b_{\alpha \beta } = {\mathbf{a}}_{\alpha ,\beta }\cdot {\mathbf{n}}, \qquad b^{\gamma }_{\beta }= a^{\gamma \alpha } b_{\alpha \beta }, \end{gathered}$$
(4.2)

and ⊗ is the tensor product: \(\left ({\mathbf{a}}\otimes {\mathbf{b}}\right ){\mathbf{c}} = {\mathbf{a}} \left ({\mathbf{b}} \cdot {\mathbf{c}}\right )\) for any vectors \({\mathbf{a}}\), \({\mathbf{b}}\), and \({\mathbf{c}}\).

Weingarten’s formulae relate the evolution of \({\mathbf{n}}\) to the components of \({\mathbf{b}}\):

$$\begin{gathered} {\mathbf{n}}_{,\beta } = - b_{\alpha \beta }{\mathbf{a}}^{\alpha }. \end{gathered}$$
(4.3)

The standard derivation of this formula starts by differentiating the identities \({\mathbf{n}}\cdot {\mathbf{n}} = 1\) and \({\mathbf{n}}\cdot {\mathbf{a}}_{\beta }= 0\). For our purposes, it is fruitful to consider the components of \({\mathbf{n}}_{,\beta }\):

(4.4)

where \(\epsilon _{11} = \epsilon _{22} = 0\) and \(\epsilon _{12} = - \epsilon _{21} = \sqrt{a}\).

We can use (4.3) along with (4.4) to show that

(4.5)

Thus, we find the sought-after representations (1.1)1,2 for the components \(b_{\alpha \beta }\) and \(b^{\gamma }_{\beta }\):

(4.6)

The identities \(b^{\gamma }_{\delta }= a^{\gamma \beta } b_{\beta \delta }\) and \({\mathbf{a}}^{\alpha }= a^{\alpha \beta } {\mathbf{a}}_{\beta }\) were used to establish the second representation. The identity (4.6)1 along with the symmetry \(b_{12} = b_{21}\) implies that

(4.7)

Thus, the components of and are not all independent.

The representation (1.1)3 for \(H\) can be obtained by first substituting for \(b^{\alpha }_{\beta }\) in the definition of \(H\) (cf. [9]) and then appealing to (2.3) and (4.6):

(4.8)

The representation (1.1)4 for \(K\) can be obtained by first substituting for \(b_{\alpha \beta }\) in the definition of \(K\) (cf. [12]) and then using (2.3):

(4.9)

We observe that the representation for \(K\) provides an elegant representation for \(K dA\) where \(dA\) is the element of area on \(S\):

(4.10)

The integral of \(K dA\) is central to the Gauss-Bonnet theorem.

5 Application to a Sphere

As an example, we consider a sphere of radius \(R\) and use a set of polar angles \(\theta ^{1} = \varphi \) and \(\theta ^{2} = \vartheta \) to parameterize the sphere (cf. Fig. 2). The representation for the position vector \({\mathbf{r}}\) of any point on the sphere is

$$\begin{gathered} {\mathbf{r}} = R {\mathbf{e}}_{R}, \end{gathered}$$
(5.1)

where

$$\begin{gathered} {\mathbf{e}}_{r} = \cos \left (\vartheta \right ) {\mathbf{E}}_{1} + \sin \left (\vartheta \right ) {\mathbf{E}}_{2}, \qquad {\mathbf{e}}_{R} = \sin \left (\varphi \right ) {\mathbf{e}}_{r} + \cos \left (\varphi \right ) { \mathbf{E}}_{3}. \end{gathered}$$
(5.2)

Differentiating \({\mathbf{r}}\), the covariant basis vectors can be computed along with their contravariant counterparts:

$$\begin{gathered} {\mathbf{a}}_{1} = R {\mathbf{e}}_{\varphi }, \qquad {\mathbf{a}}_{2} = R \sin \left ( \varphi \right ) {\mathbf{e}}_{\vartheta }, \qquad {\mathbf{a}}^{1} = \frac{1}{R} { \mathbf{e}}_{\varphi }, \qquad {\mathbf{a}}^{2} = \frac{1}{R \sin \left (\varphi \right )} {\mathbf{e}}_{\vartheta }, \end{gathered}$$
(5.3)

where

$$\begin{gathered} {\mathbf{e}}_{\varphi }= \cos \left (\varphi \right ) {\mathbf{e}}_{r} - \sin \left (\varphi \right ) {\mathbf{E}}_{3}, \qquad {\mathbf{e}}_{\vartheta }= \cos \left (\vartheta \right ) {\mathbf{E}}_{2} - \sin \left (\vartheta \right ) {\mathbf{E}}_{1}. \end{gathered}$$
(5.4)

Clearly, \({\mathbf{n}} = {\mathbf{e}}_{R}\) and \(\sqrt{a} = R^{2} \sin \left (\varphi \right )\).

Fig. 2
figure 2

Schematic of a unit sphere showing the angles \(\varphi \) and \(\vartheta \) and several sets of basis vectors. In light of our comments about the normal components of , it is interesting to note that has a component in the \({\mathbf{n}}\) direction:

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The rotation tensor \({\mathbf{Q}}\) can be defined as the product of two rotations: one about the \({\mathbf{E}}_{3}\) axis through an angle \(\vartheta \) followed by a rotation about \({\mathbf{e}}_{\vartheta }\) through an angle \(\varphi \):

$$\begin{gathered} {\mathbf{Q}} = {\mathbf{L}}\left (\varphi , {\mathbf{e}}_{\vartheta }\right ) {\mathbf{L}} \left (\vartheta , {\mathbf{E}}_{3}\right ). \end{gathered}$$
(5.5)

Thus,

(5.6)

Noting that \({\mathbf{e}}_{\vartheta }\times {\mathbf{e}}_{R} = {\mathbf{e}}_{\varphi }\), we can now compute the components of \({\mathbf{b}}\) using either (1.1)1 or (1.1)2:Footnote 2

$$\begin{gathered} {\mathbf{b}} = - \frac{1}{R} \left ( {\mathbf{e}}_{\varphi }\otimes {\mathbf{e}}_{\varphi }+ {\mathbf{e}}_{\vartheta }\otimes {\mathbf{e}}_{\vartheta }\right ). \end{gathered}$$
(5.7)

The curvatures \(H\) and \(K\) can be computed using (1.1)3,4:

(5.8)

as anticipated.

6 Comments on Applications to Kirchhoff-Love Shell Theory

The representation (1.1)1 is of particular use when demonstrating the equivalence of formulations of Kirchhoff-Love shell theory. The strain energy density function for this shell theory has the representations (cf., e.g., [7, 11]):

$$\begin{gathered} W = \hat{W}\left ( a_{\alpha \beta }, b_{\gamma \delta }, \theta ^{1}, \theta ^{2}\right ) \end{gathered}$$
(6.1)

and (cf., e.g., [4, 6])

(6.2)

The formula (1.1)1 can be used to establish the equivalence of \(\tilde{W}\) and \(\hat{W}\):

(6.3)

It should also be evident from (6.3) that the strain energy function \(\tilde{W}\) should be independent of the components . This restriction is in complete agreement with our earlier remarks that \({\mathbf{Q}}\) is defined modulo a rotation about \({\mathbf{n}}\).

Although the tensor \({\mathbf{Q}}\) 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 \({\mathbf{Q}}\). In some shell theories (cf. [4, 10]) where a rotation tensor is associated with each point of the material surface, a pair of right-handed orthonormal triads (or adapted frames) are chosen: one comprised of \({\mathbf{n}}_{0}\) and two unit tangent vectors \({\mathbf{s}}_{0_{1}}\) and \({\mathbf{s}}_{0_{2}}\) in the reference configuration and the other comprised of \({\mathbf{n}}\) and two unit tangent vectors \({\mathbf{s}}_{1}\) and \({\mathbf{s}}_{2}\) in the present configuration. The rotation tensor , where

(6.4)

is uniquely defined in this case. The fact that there are infinitely many choices of the triads \(\left \{ {\mathbf{n}}_{0}, {\mathbf{s}}_{0_{1}}, {\mathbf{s}}_{0_{2}} \right \} \) and \(\left \{ {\mathbf{n}}, {\mathbf{s}}_{1}, {\mathbf{s}}_{2}\right \} \) provides another explanation for the non-uniqueness of \({\mathbf{Q}}\).Footnote 3