Mid-surface scaling invariance of some bending strain measures

The mid-surface scaling invariance of bending strain measures proposed in [Acharya (2000)] is discussed in light of the work of [arXiv:2010.14308].


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 ((ξ α ), α = 1, 2) (convected coordinates).Points on the reference geometry are denoted generically by X and on the deformed geometry by x.The reference unit normal is denoted by N and the unit normal on the deformed geometry by n.A subscript comma refers to partial differentiation, e.g.∂() ∂ξ α = () ,α .Summation over repeated indices will be assumed.The convected coordinate basis vectors in the reference geometry will be referred to by the symbols (E α ) and those in the deformed geometry by (e α ), α = 1, 2, with corresponding dual bases (E α ), (e α ), 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 f = e α ⊗ E α and admits the right polar decomposition f = r • U , where U (X) : T X → T X and r(X) : T X → T x , where T c represents the tangent space of the shell at the point c.The curvature tensor on the deformed shell is denoted as b = n ,β ⊗ e β and that on the undeformed shell as 2 Some measures of pure bending and their invariance under midsurface scaling In [1] three measures of bending strain were proposed, given by 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 [4,2]; K is introduced as Equation ( 8) and Ǩ as Equation ( 10) in [1].
In [3] a physically natural requirement of invariance of bending strain measure under simple scalings of the form x → ax, 0 < a ∈ R is introduced (for plates, but the requirement is natural for shells as well) and it is shown that the measures K, Ǩ are not invariant under such a scaling.The measure K is not discussed in [3].
It is straightforward to see that under the said scaling, the deformation gradient scales as resulting in the bending measures scaling as Thus, the bending strain measure 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 K, Ǩ to where |U | = √ U : U = tr (f T f ), make them mid-surface scaling invariant.