1 Introduction

This brief note discusses the mid-surface scaling invariance of three nonlinear measures of pure bending strain, as introduced in [1] and physically motivated therein more than 20 years ago, in light of the recent work of [3] where the said invariance is introduced.

It is shown that one of the strain measures introduced in [1] possesses scaling invariance, and the other two are easily modified to have the invariance as well. There has been a recent surge of interest in such matters, as can be seen from the works of [3, 5, 6].

We use the notation of [1]: a shell mid-surface is thought of as a 2D surface in ambient 3D space (the qualification ‘mid-surface’ will not be used in all instances; it is hoped that the meaning will be clear from the context). Both the reference and deformed shells are parametrized by the same coordinate system \(((\xi ^{\alpha}), \alpha = 1,2)\) (convected coordinates). Points on the reference geometry are denoted generically by \({\boldsymbol {X}}\) and on the deformed geometry by \({\boldsymbol {x}}\). The reference unit normal is denoted by \({\boldsymbol {N}}\) and the unit normal on the deformed geometry by \({\boldsymbol {n}}\). A subscript comma refers to partial differentiation, e.g. \(\frac{\partial ()}{\partial \xi ^{\alpha}} = ()_{,\alpha}\). Summation over repeated indices will be assumed. The convected coordinate basis vectors in the reference geometry will be referred to by the symbols \(({\boldsymbol {E}}_{\alpha})\) and those in the deformed geometry by \(({\boldsymbol {e}}_{\alpha})\), \(\alpha =1,2\), with corresponding dual bases \(({\boldsymbol {E}}^{\alpha})\), \(({\boldsymbol {e}}^{\alpha})\), respectively. A suitable number of dots placed between two tensors represent the operation of contraction, while the symbol ⊗ will represent a tensor product. The deformation gradient will be denoted by \({\boldsymbol {f}}= {\boldsymbol {e}}_{\alpha }\otimes {\boldsymbol {E}}^{ \alpha}\) and admits the right polar decomposition \({\boldsymbol {f}}= {\boldsymbol {r}}\cdot {\boldsymbol {U}}\), where \({\boldsymbol {U}}({\boldsymbol {X}}): T_{{\boldsymbol {X}}}\to T_{{ \boldsymbol {X}}}\) and \({\boldsymbol {r}}({\boldsymbol {X}}): T_{{\boldsymbol {X}}}\to T_{{ \boldsymbol {x}}}\), where \(T_{{\boldsymbol {c}}}\) represents the tangent space of the shell at the point \({\boldsymbol {c}}\). The curvature tensor on the deformed shell is denoted as \({\boldsymbol {b}} = {\boldsymbol {n}}_{,\beta} \otimes {\boldsymbol {e}}^{\beta}\) and that on the undeformed shell as \({\boldsymbol {B}} = {\boldsymbol {N}}_{,\beta} \otimes {\boldsymbol {E}}^{\beta}\).

2 Some Measures of Pure Bending and Their Invariance Under Mid-Surface Scaling

In [1] three measures of bending strain were proposed, given by

$$\begin{aligned} \widetilde{{\boldsymbol {K}}} & = \left ({\boldsymbol {E}}_{\alpha } \cdot {\boldsymbol {U}}\cdot {\boldsymbol {r}}^{T} \cdot {\boldsymbol {n}}_{, \beta} - {\boldsymbol {E}}_{\alpha }\cdot {\boldsymbol {U}}\cdot { \boldsymbol {N}}_{,\beta} \right ) {\boldsymbol {E}}^{\alpha }\otimes { \boldsymbol {E}}^{\beta } = {\boldsymbol {f}}^{T} \cdot {\boldsymbol {b}} \cdot {\boldsymbol {f}} - {\boldsymbol {U}} \cdot {\boldsymbol {B}}, \end{aligned}$$
(1a)
$$\begin{aligned} \check{{\boldsymbol {K}}} & = \left ( {\boldsymbol {E}}_{\alpha }\cdot { \boldsymbol {U}}\cdot {\boldsymbol {r}}^{T} \cdot {\boldsymbol {n}}_{, \beta} - \frac{1}{2}\left ( {\boldsymbol {E}}_{\alpha }\cdot { \boldsymbol {U}}\cdot {\boldsymbol {N}}_{,\beta} + {\boldsymbol {E}}_{ \beta }\cdot {\boldsymbol {U}}\cdot {\boldsymbol {N}}_{,\alpha}\right ) \right ) {\boldsymbol {E}}^{\alpha }\otimes {\boldsymbol {E}}^{\beta } \\ & = {\boldsymbol {f}}^{T} \cdot {\boldsymbol {b}} \cdot {\boldsymbol {f}} - ({\boldsymbol {U}} \cdot {\boldsymbol {B}})_{sym}, \end{aligned}$$
(1b)
$$\begin{aligned} \overline{{\boldsymbol {K}}} & = \left ({\boldsymbol {E}}_{\alpha }\cdot { \boldsymbol {r}}^{T} \cdot {\boldsymbol {n}}_{, \beta} - {\boldsymbol {E}}_{ \alpha }\cdot {\boldsymbol {N}}_{,\beta} \right ) {\boldsymbol {E}}^{ \alpha }\otimes {\boldsymbol {E}}^{\beta} = {\boldsymbol {r}}^{T} \cdot {\boldsymbol {b}} \cdot {\boldsymbol {f}} - {\boldsymbol {B}}. \end{aligned}$$
(1c)

Equation (1c) was unnumbered in that work, as the main emphasis was to obtain a nonlinear generalization of the Koiter-Sanders-Budiansky bending strain measure [2, 4]; \(\widetilde{{\boldsymbol {K}}}\) is introduced as Equation (8) and \(\check{{\boldsymbol {K}}}\) as Equation (10) in [1].

In [3] a physically natural requirement of invariance of bending strain measure under simple scalings of the form

$$ {\boldsymbol {x}}\to a {\boldsymbol {x}}, \qquad 0 < a \in \mathbb{R} $$

is introduced (for plates, but the requirement is natural for shells as well) and it is shown that the measures \(\widetilde{{\boldsymbol {K}}}\), \(\check{{\boldsymbol {K}}}\) are not invariant under such a scaling. The measure \(\overline{{\boldsymbol {K}}}\) is not discussed in [3].

It is straightforward to see that under the said scaling, the deformation gradient scales as

$$ \frac{\partial {\boldsymbol {x}}}{\partial {\boldsymbol {X}}} = { \boldsymbol {r}}\cdot {\boldsymbol {U}}= {\boldsymbol {f}}\qquad \to \qquad a {\boldsymbol {f}}= {\boldsymbol {r}}\cdot (a {\boldsymbol {U}}) = a \frac{\partial {\boldsymbol {x}}}{\partial {\boldsymbol {X}}}, $$

resulting in the bending measures scaling as

$$ \widetilde{{\boldsymbol {K}}} \to a \widetilde{{\boldsymbol {K}}}; \qquad \check{{\boldsymbol {K}}} \to a \check{{\boldsymbol {K}}}; \qquad \overline{{\boldsymbol {K}}} \to \overline{{\boldsymbol {K}}}. $$

Thus, the bending strain measure \(\overline{{\boldsymbol {K}}}\) from [1], not discussed by [3], is actually invariant under scaling deformations of the deformed shell mid-surface. Furthermore, the simple modifications of the measures \(\widetilde{{\boldsymbol {K}}}\), \(\check{{\boldsymbol {K}}}\) to

$$\begin{aligned} \widetilde{{\boldsymbol {K}}}_{mod} & = \frac{1}{|{\boldsymbol {U}}|} \left ({\boldsymbol {E}}_{\alpha }\cdot {\boldsymbol {U}}\cdot { \boldsymbol {r}}^{T} \cdot {\boldsymbol {n}}_{, \beta} - {\boldsymbol {E}}_{ \alpha }\cdot {\boldsymbol {U}}\cdot {\boldsymbol {N}}_{,\beta} \right ) { \boldsymbol {E}}^{\alpha }\otimes {\boldsymbol {E}}^{\beta } \end{aligned}$$
(2a)
$$\begin{aligned} & = \left (tr\left ({\boldsymbol {f}}^{T} {\boldsymbol {f}}\right ) \right )^{-\frac{1}{2}} \left ({\boldsymbol {x}}_{,\alpha} \cdot { \boldsymbol {n}}_{, \beta} - {\boldsymbol {E}}_{\alpha }\cdot { \boldsymbol {U}}\cdot {\boldsymbol {N}}_{,\beta} \right ) {\boldsymbol {E}}^{ \alpha }\otimes {\boldsymbol {E}}^{\beta }, \\ \check{{\boldsymbol {K}}}_{mod} & = \frac{1}{|{\boldsymbol {U}}|}\left ( { \boldsymbol {E}}_{\alpha }\cdot {\boldsymbol {U}}\cdot {\boldsymbol {r}}^{T} \cdot {\boldsymbol {n}}_{, \beta} - \frac{1}{2}\left ( {\boldsymbol {E}}_{ \alpha }\cdot {\boldsymbol {U}}\cdot {\boldsymbol {N}}_{,\beta} + { \boldsymbol {E}}_{\beta }\cdot {\boldsymbol {U}}\cdot {\boldsymbol {N}}_{, \alpha}\right ) \right ) {\boldsymbol {E}}^{\alpha }\otimes { \boldsymbol {E}}^{\beta } \\ & = \left (tr\left ({\boldsymbol {f}}^{T} {\boldsymbol {f}}\right ) \right )^{-\frac{1}{2}} \left ( {\boldsymbol {x}}_{,\alpha} \cdot { \boldsymbol {n}}_{, \beta} - \frac{1}{2}\left ( {\boldsymbol {E}}_{ \alpha }\cdot {\boldsymbol {U}}\cdot {\boldsymbol {N}}_{,\beta} + { \boldsymbol {E}}_{\beta }\cdot {\boldsymbol {U}}\cdot {\boldsymbol {N}}_{, \alpha}\right ) \right ) {\boldsymbol {E}}^{\alpha }\otimes { \boldsymbol {E}}^{\beta } , \end{aligned}$$
(2b)

where

$$ |{\boldsymbol {U}}| = \sqrt{{\boldsymbol {U}}: {\boldsymbol {U}}} = \sqrt{tr \left ({\boldsymbol {f}}^{T} {\boldsymbol {f}}\right )}, $$

make them mid-surface scaling invariant.