A Constrained Cosserat Shell Model up to Order $O(h^{5})$: Modelling, Existence of Minimizers, Relations to Classical Shell Models and Scaling Invariance of the Bending Tensor

Abstract

We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order $O(h^{5})$ in the shell thickness $h$. We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the $O(h^{5})$ and $O(h^{3})$ case and we draw some connections to existing models and classical shell strain measures. Notably, the role of the appearing new bending tensor is highlighted and investigated with respect to an invariance condition of Acharya (Int. J. Solids Struct. 37(39):5517–5528, 2000) which will be further strengthened.

Appendix

1.1 A.1 Useful Identities

We provide some properties of the tensors in the variational formulation of the shell models from [20, 21]:

Remark A.1

The following identities are satisfied:

• (i) $\mathrm{tr}[{\mathrm{A}}_{y_{0}}]\,=\,2$,  ${\det }[{\mathrm{A}}_{y_{0}}]\,=\,0;\quad$ $\mathrm{tr}[{\mathrm{B}}_{y_{0}}]\,=\,2\,\mathrm{H}$, ${\det }[{\mathrm{B}}_{y_{0}}]\,=\,0$,

  ,

  $\mathrm{B}_{y_{0}} = [\nabla \Theta ]^{-T}\; \mathrm{II}_{y_{0}}^{\flat }\; [ \nabla \Theta ]^{-1}$

• (ii) $\mathrm{B}_{y_{0}}$ satisfies the equation of Cayley-Hamilton type $\mathrm{B}_{y_{0}}^{2}-2\,\mathrm{H}\, \mathrm{B}_{y_{0}}+\mathrm{K}\, \mathrm{A}_{y_{0}} \,=\,0_{3}$;

• (iii) $\mathrm{A}_{y_{0}}{\mathrm{B}}_{y_{0}}\,=\,\mathrm{B}_{y_{0}}{\mathrm{A}}_{y_{0}}\,= \,\mathrm{B}_{y_{0}}$,  $\mathrm{A}_{y_{0}}^{2}\,=\,\mathrm{A}_{y_{0}}$,  $\mathrm{C}_{y_{0}}\in \mathfrak{so}(3)$, $\quad \mathrm{C}_{y_{0}}^{2}\,=\,-\mathrm{A}_{y_{0}}$,  $\lVert {\mathrm{C}}_{y_{0}}\rVert ^{2}=2$;

• (iv) ;

• (v) $\mathrm{C}_{y_{0}} \mathcal{K}_{e,s} {\mathrm{A}}_{y_{0}}\,\,=\,\,\mathrm{C}_{y_{0}} \mathcal{K}_{e,s}$, $\mathcal{E}_{m,s} {\mathrm{A}}_{y_{0}}\,\,=\,\,\mathcal{E}_{m,s}$.

Further, in view of $\mathrm{I}_{y_{0}}^{-1} {\mathrm{II}}_{y_{0}}= \mathrm{L}_{y_{0}}$ and considering also the third fundamental form defined by $\mathrm{III}_{y_{0}}= \mathrm{II}_{y_{0}} {\mathrm{L}}_{y_{0}}$, we note the relations

$$ [\nabla \Theta ]^{-1}\;\mathrm{B}_{y_{0}} = \mathrm{L}_{y_{0}}^{\flat }\; [ \nabla \Theta ]^{-1}\qquad \mbox{and}\qquad \mathrm{B}^{2}_{y_{0}} = [ \nabla \Theta ]^{-T}\; \mathrm{III}_{y_{0}}^{\flat }\; [\nabla \Theta ]^{-1}. $$

(A.1)

1.2 A.2 The Classical Nonlinear Koiter Shell Model in Cartesian Matrix Notation

In this subsection, we consider the variational problem for the geometrically nonlinear Koiter energy for a nonlinear elastic shell [15, page 147] and we rewrite it in matrix format. The problem written in tensor format [15, page 147], [40, Eq. (1) and Eq. (101)] is to find a deformation of the midsurface $m:\omega \subset \mathbb{R}^{2}\to \mathbb{R}^{3}$ minimizing on $\omega$:

(A.2)

where the fourth order constitutive tensor $\mathbb{C}_{\mathrm{shell}}^{\mathrm{iso}}:\mathrm{Sym}(2)\to {\mathrm{Sym}}(2)$ for isotropic elastic shells in the Koiter model is given by [14]

$$ \mathbb{C}_{\mathrm{shell}}^{\mathrm{iso}}\,=\, \Big[ \mu \big( a^{\alpha \gamma }a^{\beta \tau }+ a^{\alpha \tau }a^{\beta \gamma }\big)+ \dfrac{2\,\lambda \,\mu }{ \,\lambda \,+ 2\, \mu } \, a^{\alpha \beta }a^{ \gamma \tau }\Big] e _{\alpha }\otimes e _{\beta }\otimes e _{\gamma }\otimes e _{\tau }\,.$$

(A.3)

Since our model is completely written in matrix format, we also transform the above classical minimization problem in matrix format. To this aim, let us remark that in (A.2), for a second order symmetric tensor $X \,=\,X_{\alpha \beta } e _{\alpha }\otimes e _{\beta }$, we have

$$\begin{aligned} \bigl\langle \mathbb{C}_{\mathrm{shell}}^{\mathrm{iso}}.X &,\,X \bigr\rangle \,= \, \Big[ \mu \,\big( a^{\alpha \gamma }a^{\beta \tau }+ a^{\alpha \tau }a^{ \beta \gamma }\big)+\dfrac{2\,\mu \,\lambda \,}{2\,\mu +\lambda } \, a^{ \alpha \beta }a^{ \gamma \tau }\Big]\,X_{\alpha \beta }X_{\gamma \tau } \\ =& \, \,\mu \big( a^{\alpha \gamma }a^{\beta \tau }X_{\alpha \beta }X_{ \gamma \tau }+ a^{\alpha \tau }a^{\beta \gamma }X_{\alpha \beta }X_{ \gamma \tau }\big)+\dfrac{2\,\mu \,\lambda \,}{2\,\mu +\lambda } \, \big(a^{\alpha \beta }X_{\alpha \beta }\big)\big(a^{ \gamma \tau }X_{ \gamma \tau } \big) \\ =&\,\, 2\,\mu \big( a^{\alpha \gamma }a^{\beta \tau }X_{ \alpha \beta }X_{\gamma \tau }\big)+ \dfrac{2\,\mu \,\lambda \,}{2\,\mu +\lambda } \, \big(a^{\alpha \beta }X_{ \alpha \beta }\big)^{2}. \end{aligned}$$

(A.4)

A little calculation shows

$$ \textstyle\begin{array}{l} \lVert [\nabla \Theta ]^{-T} \,{X}^{\flat }\, [\nabla \Theta ]^{-1} \rVert ^{2} \,=\, \lVert P^{-T}\widehat{X}\, P^{-1}\rVert ^{2}\\ \qquad\qquad\,=\, \lVert (a ^{i}\otimes e _{i})\, ( X_{\alpha \beta } \, e _{\alpha }\otimes e _{\beta })\, ( e _{j}\otimes a ^{j})\rVert ^{2} =\rVert X_{ \alpha \beta } \,a ^{\alpha }\otimes a ^{\beta }\rVert ^{2} \\ \qquad \qquad \,=\, \mathrm{tr} \Big[\big(X_{\alpha \beta } \,a ^{\alpha }\otimes a ^{\beta }\big)\, \big(X_{\gamma \delta } \,a ^{\gamma }\otimes a ^{\delta }\big)^{T}\Big] \,=\, \mathrm{tr} \Big[X_{\alpha \beta } X_{\delta \gamma } a^{\beta \gamma } \,\big(a ^{\alpha }\otimes a ^{\delta }\big)\Big]\\ \qquad\qquad\,= \, X_{\alpha \beta } X_{\gamma \delta }\, a^{ \beta \gamma } a^{\alpha \delta } \,=\, \,a^{\alpha \gamma } a^{\beta \tau } X_{\alpha \beta } X_{\gamma \tau }\;, \end{array} $$

(A.5)

where $P\,=\, \nabla \Theta$, and similarly

$$\begin{aligned} \mathrm{tr} \Big[ [\nabla \Theta ]^{-T} \,{X}^{\flat }\, [\nabla \Theta ]^{-1}\Big] &=\, \mathrm{tr} \Big[P^{-T}{X}^{\flat }\, P^{-1} \Big]\,=\, \mathrm{tr} \Big[(a ^{i}\otimes e _{i})\, ( X_{\alpha \beta } \, e _{\alpha }\otimes e _{\beta })\, ( e _{j}\otimes a ^{j})\Big] \vspace{6pt} \\ &=\,\mathrm{tr} \Big[ X_{\alpha \beta } \,a ^{\alpha }\otimes a ^{\beta }\Big] \,=\, X_{\alpha \beta } \,\bigl\langle a ^{\alpha }, a ^{\beta }\bigr\rangle \,=\, a^{\alpha \beta }\,X_{\alpha \beta } \,. \end{aligned}$$

(A.6)

If we substitute (A.5) and (A.6) into (A.4), we obtain

$$ \textstyle\begin{array}{l} \bigl\langle \mathbb{C}_{\mathrm{shell}}^{\mathrm{iso}}.X ,\,X \bigr\rangle \,= \, 2\,\mu \rVert [\nabla \Theta ]^{-T} \,{X}^{\flat }\, [\nabla \Theta ]^{-1} \rVert ^{2} +\dfrac{2\,\lambda \,\mu }{ \lambda + 2 \,\mu } \, \mathrm{tr} \Big[ [\nabla \Theta ]^{-T} \,{X}^{\flat }\, [\nabla \Theta ]^{-1}\Big]^{2}, \end{array} $$

(A.7)

which holds for any symmetric tensor $X \,=\,X_{\alpha \beta } e _{\alpha }\otimes e _{\beta }$.

Writing the equation (A.7) for the symmetric matrix $X\,=\, \frac{1}{2}(\mathrm{I}_{m}-\mathrm{I}_{y_{0}})$ and respectively the matrix $X\,=\, \mathrm{II}_{m}-\mathrm{II}_{y_{0}}$, then we obtain the following relations

(A.8)

Hence, putting all together, in matrix format and for a nonlinear elastic shell, the variational problem for the Koiter energy is to find a deformation of the midsurface $m:\omega \subset \mathbb{R}^{2}\to \mathbb{R}^{3}$ minimizing on $\omega$

$$\begin{aligned} & \int _{\omega }\bigg\{ h\,\bigg( \mu \rVert [\nabla \Theta ]^{-T} \, \underbrace{\frac{1}{2}\big(\mathrm{I}_{m}^{\flat }-\mathrm{I}_{y_{0}}^{\flat }\big)}_{ \mathcal{G}_{\mathrm{Koiter}}} \, [\nabla \Theta ]^{-1}\rVert ^{2} + \dfrac{\,\lambda \,\mu }{\lambda +2\,\mu } \, \mathrm{tr} \Big[ [ \nabla \Theta ]^{-T} \,\big(\mathrm{I}_{m}^{\flat }-\mathrm{I}_{y_{0}}^{\flat }\big) \, [\nabla \Theta ]^{-1}\Big]^{2}\bigg) \\ &\quad \quad +\displaystyle \frac{h^{3}}{12}\bigg( \mu \rVert [\nabla \Theta ]^{-T} \, \underbrace{\big(\mathrm{II}_{m}^{\flat }-\mathrm{II}_{y_{0}}^{\flat }\big)}_{ \mathcal{R}_{\mathrm{Koiter}}} \, [\nabla \Theta ]^{-1}\rVert ^{2} \\ &\quad \quad +\displaystyle \dfrac{\,\lambda \,\mu }{\lambda +2\,\mu } \, \mathrm{tr} \Big[ [ \nabla \Theta ]^{-T} \,\big(\mathrm{II}_{m}^{\flat }-\mathrm{II}_{y_{0}}^{\flat }\big) \, [\nabla \Theta ]^{-1}\Big]^{2}\bigg)\bigg\} \,\mathrm{det} \nabla \Theta \,\, \mathrm{d}a. \end{aligned}$$

(A.9)

The main feature of the classical Koiter model is that it is just the sum of the correctly identified membrane term and bending terms (under inextensional deformation).

1.3 A.3 Thickness Versus Invertibility and Coercivity

1.3.1 A.3.1 Invertibility Conditions for the Parametrized Initial Surface $\Theta$

We note that $\det \nabla \Theta (x_{3})= 1-2\, H\,x_{3}+K\, x_{3}^{2}=(1-\kappa _{1} \,x_{3})(1-\kappa _{2}\, x_{3})>0$ if and only if $1-\kappa _{1}\,x_{3}$ and $1-\kappa _{2}\, x_{3}$ have the same sign. However, $1-\kappa _{1}\,x_{3}$ cannot be negative, since for $\kappa _{1}<0$ this will imply that $1<\kappa _{1}\,x_{3}$ which is not true if $x_{3}>0$, while for $\kappa _{1}\geq 0$ this will imply that $1<\kappa _{1}\,x_{3}$ which is not true if $x_{3}<0$. Therefore, $1-2\, H\,x_{3}+K\, x_{3}^{2}>0$ if and only if $1>\kappa _{1}\,x_{3}$ and $1>\kappa _{2}\, x_{3}$ . These conditions are equivalent with and , i.e., equivalent with (1.18).

1.3.2 A.3.2 Coercivity for the $O(h^{5})$ Model

The decisive point in the proof of the existence where the condition on the thickness is used is only in the proof of the coercivity of the internal energy density. Therefore, in this appendix, we extend the result regarding the coercivity of the internal energy to the following result:

Proposition A.2

Coercivity in the theory including terms up to order $O(h^{5})$

For sufficiently small values of the thickness $h$ such that

$$\begin{aligned} h\max \{\sup _{x\in \omega }|\kappa _{1}|, \sup _{x\in \omega }|\kappa _{2}| \}< \alpha \qquad \textit{with}\qquad \alpha < \sqrt{\frac{2}{3}(29-\sqrt{761})} \simeq 0.97083 \end{aligned}$$

(A.10)

and for constitutive coefficients satisfying $\mu >0, \,\mu _{\mathrm{c}}>0$, $2\,\lambda +\mu > 0$, $b_{1}>0$, $b_{2}>0$ and $b_{3}>0$, the energy density

$$\begin{aligned} W(\mathcal{E}_{m,s}, \mathcal{K}_{e,s})=W_{\mathrm{memb}}\big( \mathcal{E}_{m,s} \big)+W_{\mathrm{memb,bend}}\big( \mathcal{E}_{m,s} , \, \mathcal{K}_{e,s} \big)+W_{\mathrm{bend,curv}}\big( \mathcal{K}_{e,s} \big) \end{aligned}$$

(A.11)

is coercive in the sense that there exists a constant $a_{1}^{+}>0$ such that $W(\mathcal{E}_{m,s}, \mathcal{K}_{e,s})\,\geq \, a_{1}^{+}\, \big ( \lVert \mathcal{E}_{m,s}\rVert ^{2} + \lVert \mathcal{K}_{e,s}\rVert ^{2} \,\big )$, where $a_{1}^{+}$ depends on the constitutive coefficients.

Proof

From the assumptions $h\, |\kappa _{1}|<\alpha$, $\ h\, |\kappa _{2}|<\alpha$, it follows that

$$\begin{aligned} h^{2}|K|=h^{2}\, |\kappa _{1}|\,|\kappa _{2}|< \alpha ^{2}\qquad \text{and}\qquad 2\,h\, |H|=h\, |\kappa _{1}+\kappa _{2}|< 2\, \alpha . \end{aligned}$$

(A.12)

Therefore, for $\alpha <\sqrt{\frac{20}{3}}$ it follows $h-\mathrm{K}\,\frac{h^{3}}{12}> h-\,\frac{h}{12}\alpha ^{2}>0 \ \ \textrm{and} \ \ \frac{h^{3}}{12}\,-\frac{h^{3}}{80}\alpha ^{2}>0$. On the other hand, from [21, Proposition 3.1.] we have

$$\begin{aligned} W(\mathcal{E}_{m,s}, \mathcal{K}_{e,s}) \geq \,& \Big(h-\dfrac{h}{12} \delta +\mathrm{K}\,\dfrac{h^{3}}{12}- \dfrac{h^{2}}{6}\varepsilon \,| \mathrm{H}|\Big)\,{W}_{\mathrm{shell}} \big( \mathcal{E}_{m,s}\big)\\ &+ \Big(\dfrac{h^{3}}{12}\,-\mathrm{K}\,\dfrac{h^{5}}{80}- \dfrac{h^{4}}{6\, \varepsilon }\,\,|\mathrm{H}|\Big)\, {W}_{ \mathrm{shell}} \big( \mathcal{E}_{m,s}{\mathrm{B}}_{y_{0}}+\mathrm{C}_{y_{0}} \, \mathcal{K}_{e,s} \big) \\ &+ \Big(\,\dfrac{h^{5}}{80}- \dfrac{h^{5}}{12\, \delta }\Big)\,\, W_{ \mathrm{shell}} \big(( \mathcal{E}_{m,s} \, \mathrm{B}_{y_{0}} + \mathrm{C}_{y_{0}} \mathcal{K}_{e,s} ) \mathrm{B}_{y_{0}} \,\big) \\ &+\Big(h-\mathrm{K}\, \dfrac{h^{3}}{12}\Big)\, W_{\mathrm{curv}}\big( \mathcal{K}_{e,s} \big)\qquad \ \forall \, \varepsilon >0\ \text{ and } \ \ \delta >0 . \end{aligned}$$

(A.13)

Using the inequalities (A.12), we deduce

$$\begin{aligned} W(\mathcal{E}_{m,s}, \mathcal{K}_{e,s}) \geq \,& \dfrac{h}{12}\Big(12- \delta -\alpha ^{2}- 2\,\varepsilon \,\alpha \Big)\,{W}_{ \mathrm{shell}} \big( \mathcal{E}_{m,s}\big)\\ &+\dfrac{h^{3}}{12}\,\Big(1- \dfrac{3}{20}\,\alpha ^{2}-\dfrac{1}{3\, \varepsilon }\,\alpha \Big)\, {W}_{ \mathrm{shell}} \big( \mathcal{E}_{m,s}{\mathrm{B}}_{y_{0}}+\mathrm{C}_{y_{0}} \, \mathcal{K}_{e,s} \big) \\ &+ \dfrac{h^{5}}{80}\,\Big(1- \dfrac{20 }{3\, \delta }\Big)\,\, W_{ \mathrm{shell}} \big(( \mathcal{E}_{m,s} \, \mathrm{B}_{y_{0}} + \mathrm{C}_{y_{0}} \mathcal{K}_{e,s} ) \mathrm{B}_{y_{0}} \,\big) \\ &+\Big(h-\mathrm{K}\, \dfrac{h^{3}}{12}\Big)\, W_{\mathrm{curv}}\big( \mathcal{K}_{e,s} \big)\qquad \ \forall \, \varepsilon >0\ \text{ and } \ \ \delta >0 . \end{aligned}$$

(A.14)

Let us remark for $0<\alpha <\sqrt{\frac{20}{3}}$, if there exist $\delta >\frac{20}{3}$ and $\varepsilon >\frac{40\, \alpha }{20-3\, \alpha ^{2}}$ such that

$$\begin{aligned} 12-\delta -\alpha ^{2}- 2\,\varepsilon \,\alpha >0 \ \ \Leftrightarrow \ \ 12-\alpha ^{2}>\delta + 2\,\varepsilon \,\alpha > \frac{20}{3}+ 2\,\frac{40\,\alpha ^{2}}{20-3\, \alpha ^{2}} \end{aligned}$$

(A.15)

then $\alpha <\sqrt{\frac{2}{3}(29-\sqrt{761})}\simeq 0.97083$. Since the map $(\delta , \varepsilon )\mapsto \delta + 2\,\varepsilon \,\alpha$ is linear, it increases in gradient direction and hence, the condition $\alpha <\sqrt{\frac{2}{3}(29-\sqrt{761})}\simeq 0.97083$ is also sufficient for the existence of $\delta >\frac{20}{3}$ and $\varepsilon >\frac{40\, \alpha }{20-3\, \alpha ^{2}}$ such that (A.15) is satisfied.

(A.16)

In view of (A.14) and (2.10), we see that there exist positive constants $b_{1}^{+}, b_{2}^{+},b_{3}^{+}>0$ such that

(A.17)

The desired constant $a_{1}^{+}$ from the conclusion can be chosen as $a_{1}^{+}\,{=}\,\min \big \{\dfrac{h}{12}\,b_{1}^{+}\, \min \{c_{1}^{+}, \mu _{\mathrm{c}}\}, \dfrac{h}{12}\,b_{2}^{+}\,\min \{c_{1}^{+},\mu _{\mathrm{c}} \}, h\,b_{3}^{+}\, { c_{2}^{+}} \big \}$. □

1.3.3 A.3.3 Coercivity for the $O(h^{3})$ Model

In this subsection we investigate if the conditions which assure the existence of the solution may be relaxed in the Cosserat shell model up to $O(h^{3})$, too. In fact, as in the previous subsection, it is enough to prove some new coercivity results under weakened conditions on the thickness. We recall that in the Cosserat shell model up to $O(h^{3})$ the shell energy density $W^{(h^{3})}(\mathcal{E}_{m,s}, \mathcal{K}_{e,s})$ is given by (3.65).

Proposition A.3

The first coercivity result in the theory including terms up to order $O(h^{3})$

For sufficiently small values of the thickness $h$ such that

$$\begin{aligned} \begin{aligned} &h\max \{\sup _{x\in \omega }|\kappa _{1}|, \sup _{x\in \omega }|\kappa _{2}| \}< \alpha \qquad \textit{and}\\ &h^{2}< \frac{(5-2\sqrt{6})(\alpha ^{2}-12)^{2}}{4\, \alpha ^{2}} \frac{ {c_{2}^{+}}}{\max \{C_{1}^{+},\mu _{\mathrm{c}}\}}\qquad \textit{with} \quad 0< \alpha < 2\sqrt{3} \end{aligned} \end{aligned}$$

(A.18)

and for constitutive coefficients satisfying the constitutive coefficients are such that $\mu >0, \,\mu _{\mathrm{c}}>0$, $2\,\lambda +\mu > 0$, $b_{1}>0$, $b_{2}>0$ and $b_{3}>0$ and where $c_{2}^{+}$ denotes the smallest eigenvalue of $W_{\mathrm{curv}}( S )$, and $C_{1}^{+}>0$ denotes the largest eigenvalues of the quadratic form $W_{\mathrm{shell}}^{\infty }( S)$, the total energy density $W^{(h^{3})}(\mathcal{E}_{m,s}, \mathcal{K}_{e,s})$ is coercive, in the sense that there exists a constant $a_{1}^{+}>0$ such that $W^{(h^{3})}(\mathcal{E}_{m,s}, \mathcal{K}_{e,s})\,\geq \, a_{1}^{+} \, \big ( \lVert \mathcal{E}_{m,s}\rVert ^{2} + \lVert \mathcal{K}_{e,s} \rVert ^{2}\,\big )$, where $a_{1}^{+}$ depends on the constitutive coefficients.

Proof

Similarly as in the proof presented in [21, Proposition 4.1], we obtain the estimate

(A.19)

which after imposing the conditions $-h^{2} |K|>-\alpha ^{2}$ and $-h\, |H|>-\alpha$, using (A.12), (2.10) and since the Frobenius norm is sub-multiplicative and $\lVert C_{y_{0}}\rVert ^{2}=2$, leads to

(A.20)

for all $\alpha , \delta , \varepsilon >0$ such that $12-\alpha ^{2}- \varepsilon -2\,\alpha \, \delta >0$ and $12-\alpha ^{2}>0$. Since $\lVert B_{y_{0}}\rVert ^{2}=4\, \mathrm{H}^{2}-2\,\mathrm{K}$, it follows

$$\begin{aligned} W^{(h^{3})}(\mathcal{E}_{m,s}, \mathcal{K}_{e,s})\geq \, &\, \dfrac{h}{12} \Big(12-\alpha ^{2}-\varepsilon -2\,\alpha \, \delta \Big)\, \min \{c_{1}^{+},\mu _{\mathrm{c}}\} \lVert \mathcal{E}_{m,s} \rVert ^{2} \\ &+\dfrac{h^{3}}{12}\, \min \{c_{1}^{+},\mu _{\mathrm{c}}\} \lVert \mathcal{E}_{m,s}{\mathrm{B}}_{y_{0}}+\mathrm{C}_{y_{0}}\, \mathcal{K}_{e,s} \rVert ^{2} \\ &+\dfrac{h}{12}\Big[(12-\alpha ^{2})\,\,c_{2}^{+}-\dfrac{4}{\delta } \,\alpha \, \max \{C_{1}^{+},\mu _{\mathrm{c}}\} \,h^{2} \\ &- \dfrac{2}{\varepsilon }\, h^{4}\,\,\max \{C_{1}^{+},\mu _{\mathrm{c}}\} \, (4 \, \mathrm{H}^{2}-2\,\mathrm{K})\Big]\lVert \mathcal{K}_{e,s} \rVert ^{2} + \dfrac{h^{3}}{12}\, c_{2}^{+}\lVert \mathcal{K}_{e,s} {\mathrm{B}}_{y_{0}} \rVert ^{2} \end{aligned}$$

Using again that $h$ is small, we obtain $-h^{2}\,(4\, \mathrm{H}^{2}-2\,\mathrm{K})\geq -h^{2}\,(4\, \mathrm{H}^{2}+2 \,|{\mathrm{K}}|)\geq -6\,\alpha ^{2}$ and

$$\begin{aligned} W^{(h^{3})}(\mathcal{E}_{m,s}, \mathcal{K}_{e,s})\geq \, &\, \dfrac{h}{12} \Big(12-\alpha ^{2}-\varepsilon -2\,\alpha \, \delta \Big)\, \min \{c_{1}^{+},\mu _{\mathrm{c}}\} \lVert \mathcal{E}_{m,s} \rVert ^{2} \\ &+\dfrac{h}{12}\Big[(12-\alpha ^{2})\,\,c_{2}^{+}-\dfrac{4}{\delta } \,\alpha \, \max \{C_{1}^{+},\mu _{\mathrm{c}}\} \,h^{2} \\ &- \dfrac{2}{\varepsilon }\, h^{2}\,\,\max \{C_{1}^{+},\mu _{\mathrm{c}}\} \, 6 \,\alpha ^{2}\Big]\lVert \mathcal{K}_{e,s} \rVert ^{2}. \end{aligned}$$

(A.21)

We consider $\delta =\gamma \, \varepsilon$ and we choose $\epsilon >0$ and $\gamma >0$ such that

$$\begin{aligned} \frac{12-\alpha ^{2}}{1+2\,\alpha \gamma }>\varepsilon >4 \frac{\alpha +3\, \alpha ^{2}\gamma }{\gamma (12-\alpha ^{2})}\, \frac{\max \{C_{1}^{+},\mu _{\mathrm{c}}\} }{c_{2}^{+}}\,h^{2}. \end{aligned}$$

(A.22)

This choice of the variable $\varepsilon >0$ is possible if and only if $\frac{(12-\alpha ^{2})^{2}\gamma }{4\,(a+2\,\alpha \gamma )\,(\alpha +3\, \alpha ^{2}\gamma )}> \frac{h^{2}\,\max \{C_{1}^{+},\mu _{\mathrm{c}}\} }{c_{2}^{+}}$. At this point we use that

$$ \max _{\gamma >0} \frac{(12-\alpha ^{2})^{2}\gamma }{4\,(a+2\,\alpha \gamma )\,(\alpha +3\, \alpha ^{2}\gamma )}= \frac{(5-2\sqrt{6})(\alpha ^{2}-12)^{2}}{4\, \alpha ^{2}}, $$

and this maximum value is attained for $\gamma =\frac{1}{\sqrt{6}\,a}$. Hence, we arrive at the following condition on the thickness $h$:

$$\begin{aligned} h^{2}< \frac{(5-2\sqrt{6})(\alpha ^{2}-12)^{2}}{4\, \alpha ^{2}} \frac{ {c_{2}^{+}}}{\max \{C_{1}^{+},\mu _{\mathrm{c}}\} }, \end{aligned}$$

(A.23)

which proves the coercivity if the condition from the hypothesis is satisfied. □

The condition (A.18) on the thickness does not represent a relaxation of the condition imposed in [21, Proposition 4.1], i.e., not in the sense of the relaxed condition (A.10) which is found for the $O(h^{5})$ Cosserat shell model. Indeed, since the map $\alpha \mapsto \frac{(5-2\sqrt{6})(\alpha ^{2}-12)^{2}}{4\, \alpha ^{2}} \frac{ {c_{2}^{+}}}{\max \{C_{1}^{+},\mu _{\mathrm{c}}\} }$ is monotone decreasing on $[0,2]$ (the interval of the values of the parameter $\alpha$ for which the construction of the model has sense), a large value for $\alpha$ will relax the first condition (A.18)₁ while the other condition (A.18)₂ on the thickness will become more restrictive. Since the second condition (A.18)₂ is expressed in terms of all constitutive parameters, through $c_{2}^{+}$ and $\max \{C_{1}^{+},\mu _{\mathrm{c}}\}$, while the first condition (A.18)₁ depends on the curvatures of the referential configuration, the largest value of the parameter $\alpha$ in the coercivity result would be chosen as the best compromise between the conditions (A.18)₁ and (A.18)₂. In conclusion, in comparison to the conditions imposed in [21, Proposition 4.1], for a specific material and a specific referential configuration the new condition (A.18) would offer a largest interval of values for the upper bound of the thickness.

We also note that in [21, Proposition 4.1], because the condition of the form (A.18)₂ from the hypothesis of Proposition A.3 depends on the length scale $L_{c}$, we have proved another coercivity result which avoids this aspect:

Proposition A.4

The second coercivity result in the theory including terms up to order $O(h^{3})$

For sufficiently small values of the thickness $h$ such that

$$\begin{aligned} h\max \{\sup _{x\in \omega }|\kappa _{1}|, \sup _{x\in \omega }|\kappa _{2}| \}< \frac{1}{a}\qquad \qquad \textit{with} \quad a>\max \Big\{ 1 + \frac{\sqrt{2}}{2}, \frac{1+\sqrt{1+3\frac{\max \{C_{1}^{+},\mu _{\mathrm{c}}\} }{\min \{c_{1}^{+},\mu _{\mathrm{c}}\} }}}{2} \Big\} , \end{aligned}$$

(A.24)

and for constitutive coefficients satisfying the constitutive coefficients are such that $\mu >0, \,\mu _{\mathrm{c}}>0$, $2\,\lambda +\mu > 0$, $b_{1}>0$, $b_{2}>0$ and $b_{3}>0$ and let $c_{1}^{+}$ and $\max \{C_{1}^{+},\mu _{\mathrm{c}}\} >0$ denote the smallest and the largest eigenvalues of the quadratic form $W_{\mathrm{shell}}^{\infty }( S)$, the total energy density $W^{(h^{3})}(\mathcal{E}_{m,s}, \mathcal{K}_{e,s})$ is coercive, in the sense that there exists a constant $a_{1}^{+}>0$ such that $W^{(h^{3})}(\mathcal{E}_{m,s}, \mathcal{K}_{e,s})\,\geq \, a_{1}^{+} \, \big ( \lVert \mathcal{E}_{m,s}\rVert ^{2} + \lVert \mathcal{K}_{e,s} \rVert ^{2}\,\big )$, where $a_{1}^{+}$ depends on the constitutive coefficients.

1.4 A.4 Acharya’s Bending Tensor in Cartesian Matrix Notation

For comparison purpose, let us mark all the vectors and tensors defined by Acharya [1, Page 5520] with the subscript $A$ (e.g., $\boldsymbol{f}_{A}$, $\boldsymbol{b}^{*}_{A}$, $(\boldsymbol{E}_{\alpha })_{A}$). We express them using our notations as follows:

$$\begin{aligned} \begin{aligned} &\boldsymbol{X}_{A} = y_{0}\,,\qquad \boldsymbol{N}_{A} = n_{0}\,, \qquad \boldsymbol{x}_{A} = m\,,\qquad \boldsymbol{n}_{A}= n\,,\\ &(\boldsymbol{E}_{\alpha })_{A} = a_{\alpha }=\partial _{x_{\alpha }} y,\qquad ( \boldsymbol{E}^{\alpha })_{A} = a^{\alpha }\;\;\mbox{etc.} \end{aligned} \end{aligned}$$

(A.25)

Let us denote by $[ \boldsymbol{T}_{A}]$ the $3\times 3$ matrix of the components in the basis $\{ e_{i}\otimes e_{j}\}$ for any tensor $\boldsymbol{T}$ defined by Acharya [1, Page 5519]. Then, we have

$$ [ \boldsymbol{f}_{A} ] = \Big[\Big( \frac{\partial \boldsymbol{x}}{\partial \boldsymbol{X}} \Big)_{A}\, \Big] = [ m_{,\alpha } \otimes a^{\alpha }] = [ (m_{,\alpha } \otimes e_{\alpha })( e_{i} \otimes a^{i}) ] = (\nabla m | 0)\, [ \nabla _{x} \Theta ]^{-1} $$

(A.26)

and

$$ [ \boldsymbol{B}_{A} ] = \Big[\Big( \frac{\partial \boldsymbol{N}}{\partial \boldsymbol{X}} \Big)_{A}\, \Big] = [ n_{0,\alpha } \otimes a^{\alpha }] = [ (n_{0,\alpha } \otimes e_{\alpha })( e_{i} \otimes a^{i}) ] = (\nabla n_{0} | 0)\, [ \nabla _{x} \Theta ]^{-1}, $$

(A.27)

so that

$$ [ \boldsymbol{B}_{A} ] = - \mathrm{B}_{y_{0}} = - [ \nabla _{x} \Theta ]^{-T} \; \mathrm{II}_{y_{0}}^{\flat }\; [ \nabla _{x} \Theta ]^{-1}. $$

(A.28)

Furthermore, we have

$$ [ \boldsymbol{b}^{*}_{A} ] = \Big[\boldsymbol{f}_{A}^{T}\; \Big( \frac{\partial \boldsymbol{n}}{\partial \boldsymbol{x}} \Big)_{A}\; \boldsymbol{f}_{A}\, \Big] = \Big[\boldsymbol{f}_{A}^{T}\; \Big( \frac{\partial \boldsymbol{n}}{\partial \boldsymbol{X}} \Big)_{A} \, \Big] = [\boldsymbol{f}_{A} ]^{T}\;\Big[ \Big( \frac{\partial \boldsymbol{n}}{\partial \boldsymbol{X}} \Big)_{A} \, \Big] $$

and using (A.41) we get

$$ [ \boldsymbol{b}^{*}_{A} ] = [ \nabla _{x} \Theta ]^{-T}\; (\nabla m | 0)^{T} \, (\nabla n | 0)\, [ \nabla _{x} \Theta ]^{-1} = - [ \nabla _{x} \Theta ]^{-T}\; \mathrm{II}_{m}^{\flat }\; [ \nabla _{x} \Theta ]^{-1}. $$

(A.29)

By virtue of (A.28) and (A.29), the classical bending strain measure can be written as [1, Eq. 3]

$$ [\boldsymbol{K}_{A}] = [ \boldsymbol{b}^{*}_{A} ] - [ \boldsymbol{B}_{A} ] = - [ \nabla _{x} \Theta ]^{-T}\; (\mathrm{II}_{m} - \mathrm{II}_{y_{0}} )^{\flat }\; [ \nabla _{x} \Theta ]^{-1}. $$

(A.30)

For the tensor $\boldsymbol{U}_{A}$ we obtain using equation (A.28) the expression

$$ \textstyle\begin{array}{r@{\quad }l} [\boldsymbol{U}_{A}] & = \sqrt{ [ \boldsymbol{f}_{A}^{T}\, \boldsymbol{f}_{A}\, ]} = \sqrt{ [ \boldsymbol{f}_{A}]^{T}\, [ \boldsymbol{f}_{A}\, ]} = \sqrt{[ \nabla _{x} \Theta ]^{-T}\; (\nabla m | 0)^{T}\, (\nabla m | 0)\, [ \nabla _{x} \Theta ]^{-1} } \\ &= \sqrt{[ \nabla _{x} \Theta ]^{-T}\; \mathrm{I}_{m}^{\flat }\; [ \nabla _{x} \Theta ]^{-1} }\,. \end{array} $$

(A.31)

Thus, the bending tensor defined by Acharya [1, Eq. 8] is given by

$$ [\widetilde{\boldsymbol{K}}_{A}] = [ \boldsymbol{b}^{*}_{A} ] - \Big[ \boldsymbol{U}_{A}\;\Big( \frac{\partial \boldsymbol{N}}{\partial \boldsymbol{X}} \Big)_{A}\, \Big] = [ \boldsymbol{b}^{*}_{A} ] - [\boldsymbol{U}_{A}]\,[ \boldsymbol{B}_{A}]\,. $$

Inserting here the relations (A.28), (A.29) and (A.31), we obtain that the bending tensor $\widetilde{\boldsymbol{K}}_{A}$ defined in the paper by Acharya [1, Eq. 7 and 8] can be written in our matrix notation as follows:

(A.32)

Since and , the following relation between the nonlinear bending tensor $\widetilde{\mathcal{R}}_{\mathrm{Acharya}}$ introduced by Acharya and our nonlinear bending tensor $\mathcal{R}_{\infty }^{\flat }$ holds

(A.33)

where we have used and (6.2).

1.5 A.5 A Direct Proof of the Fact That $\mathcal{R}_{\infty }^{\flat }$ Satisfies AR1, AR2 and AR3 ^∗

It is obvious that the tensor (6.1) satisfies condition AR2, see Sect. 6.2. Moreover, using similar calculations as in Sect. 6.2, for a rigid deformation, we have

(A.34)

so that we conclude

(A.35)

which means that $\mathcal{R}_{\infty }^{\flat }$ satisfies AR1.

In order to show that $\mathcal{R}_{\infty }^{\flat }$ satisfies AR3^∗, using

$$ (\mathrm{U_{e}}- n_{0}\otimes n_{0})^{2}=\mathrm{U}_{\mathrm{e}}^{2} - \mathrm{U_{e}}\,n_{0}\otimes n_{0} - n_{0}\otimes n_{0} \, \mathrm{U_{e}}+ n_{0}\otimes n_{0} \overset{\text{(6.20)}}{=} \mathrm{U}_{\mathrm{e}}^{2}-n_{0}\otimes n_{0},$$

(A.36)

we deduce that

(A.37)

where we have used that $\mathrm{U_{e}}$ is positive-definite. Together with the definition of the symmetric matrix $\mathrm{U_{e}}$ we get

(A.38)

Again, using the intermediate steps (6.16) and (A.38) in the expression (6.2) we obtain $\mathcal{R}_{\infty }^{\flat }=0_{3}$ since

(A.39)

so that also $\mathcal{R}_{\infty }^{\flat }$ satisfies AR1, AR2 and AR3^∗.

1.6 A.6 Alternative Representation of Energy in Terms of the New Strain Tensors

We present in the following an alternative formulation of the unconstrained Cosserat-shell model from (3.21). In this section, for a matrix of the form $X\,=\,(X^{S}\, |\, 0)\; [\nabla \Theta \,]^{-1}\in \mathbb{R}^{3 \times 3}$, we consider the matrix $X^{C} \,=\, [\nabla \Theta \,]^{T} X^{S}\in \mathbb{R}^{3\times 2}$, i.e.,

$$\begin{aligned} X\,= \, [\nabla \Theta \,]^{-T} (X^{C} \,|\, 0)\; [\nabla \Theta \,]^{-1}. \end{aligned}$$

(A.40)

For the bilinear form $W_{\mathrm{shell}}( X, Y)$ given in (2.6) we have then the transformations

$$ W_{\mathrm{shell}}( X, Y) = W^{S}_{\mathrm{shell}}( X^{S}, Y^{S}) = W^{C}_{ \mathrm{shell}}( X^{C}, Y^{C}) , $$

(A.41)

where, e.g.,

$$\begin{aligned} W^{C}_{\mathrm{shell}}( X^{C}, Y^{C}) \,=\,& \mu \,\langle \, \mathrm{sym}\, \big( [\nabla \Theta \,]^{-T} (X^{C} \,| \,0)\; [ \nabla \Theta \,]^{-1}\big),\mathrm{sym}\, \big([\nabla \Theta \,]^{-T} (Y^{C} \,| \,0)\; [\nabla \Theta \,]^{-1}\big)\,\rangle \\ & + \mu _{c}\langle \,\mathrm{skew}\, \big( [\nabla \Theta \,]^{-T} (X^{C} \,| \,0)\; [\nabla \Theta \,]^{-1}\big),\mathrm{skew}\, \big([ \nabla \Theta \,]^{-T} (Y^{C} \,| \,0)\; [\nabla \Theta \,]^{-1}\big) \,\rangle \\ & + \,\dfrac{\lambda \,\mu }{\lambda +2\mu }\,\mathrm{tr} ([\nabla \Theta \,]^{-T} (X^{C} \,| \,0)\; [\nabla \Theta \,]^{-1})\, \mathrm{tr} ([\nabla \Theta \,]^{-T} (Y^{C} \,| \,0)\; [\nabla \Theta \,]^{-1}). \end{aligned}$$

(A.42)

We note that the last relation can be written using the first fundamental form $\mathrm{I}_{y_{0}}$ and its square root in the following alternative ways:

$$\begin{aligned} W^{C}_{\mathrm{shell}}( X^{C}, Y^{C}) \,=\,& \, \mu \,\langle \, \mathrm{sym}\, (X^{C} \,| \,0),\,\mathrm{sym}\, \big(\hat{\mathrm{I}}_{y_{0}}^{-1} (Y^{C} \,| \,0)\, \hat{\mathrm{I}}_{y_{0}}^{-1}\big)\,\rangle \\ &+ \mu _{c} \langle \,\mathrm{skew}\, (X^{C} \,| \,0),\,\mathrm{skew}\, \big( \hat{\mathrm{I}}_{y_{0}}^{-1} (Y^{C} \,| \,0)\, \hat{\mathrm{I}}_{y_{0}}^{-1} \big)\,\rangle \\ & + \,\dfrac{\lambda \,\mu }{\lambda +2\mu }\,\mathrm{tr} \big( (X^{C} \,| \,0)\hat{\mathrm{I}}_{y_{0}}^{-1}\big)\,\mathrm{tr} \big( (Y^{C} \,| \,0)\, \hat{\mathrm{I}}_{y_{0}}^{-1}\big) \\ \,=\,& \, \mu \,\langle \, \mathrm{sym}\, \big( \hat{\mathrm{I}}_{y_{0}}^{-1/2} (X^{C} \,| \,0)\hat{\mathrm{I}}_{y_{0}}^{-1/2}\big) ,\,\mathrm{sym}\, \big(\hat{\mathrm{I}}_{y_{0}}^{-1/2} (Y^{C} \,| \,0)\, \hat{\mathrm{I}}_{y_{0}}^{-1/2} \big)\,\rangle \\ & + \mu _{c}\;\langle \,\mathrm{skew}\, \big(\hat{\mathrm{I}}_{y_{0}}^{-1/2} (X^{C} \,| \,0)\hat{\mathrm{I}}_{y_{0}}^{-1/2}\big) ,\,\mathrm{skew}\, \big(\hat{\mathrm{I}}_{y_{0}}^{-1/2} (Y^{C} \,| \,0)\, \hat{\mathrm{I}}_{y_{0}}^{-1/2} \big)\,\rangle \\ & + \,\dfrac{\lambda \,\mu }{\lambda +2\mu }\,\mathrm{tr} \big( \hat{\mathrm{I}}_{y_{0}}^{-1/2} (X^{C} \,| \,0)\hat{\mathrm{I}}_{y_{0}}^{-1/2} \big)\,\mathrm{tr} \big( \hat{\mathrm{I}}_{y_{0}}^{-1/2} (YX^{C} \,| \,0) \, \hat{\mathrm{I}}_{y_{0}}^{-1/2}\big). \end{aligned}$$

(A.43)

Moreover, if we decompose the matrix $X^{C} \in \mathbb{R}^{3\times 2}$ in two block matrices (the matrix and the matrix $e_{3}^{T}\, X^{C} \in \mathbb{R}^{1\times 2}$), then we can write the bilinear form as

(A.44)

where we define for any $X, Y\in \mathbb{R}^{2\times 2}$ the bilinear form

$$\begin{aligned} W_{\mathrm{inplane}}( X, Y) \,=\,& \mu \,\langle \, \mathrm{sym}\, X, \,\mathrm{sym}\, \big(\mathrm{I}_{y_{0}}^{-1}\, Y\, \mathrm{I}_{y_{0}}^{-1} \big)\,\rangle + \mu _{c}\langle \,\mathrm{skew}\, X,\,\mathrm{skew} \, \big(\mathrm{I}_{y_{0}}^{-1}\, Y\, \mathrm{I}_{y_{0}}^{-1}\big)\, \rangle \\ &+ \,\dfrac{\lambda \,\mu }{\lambda +2\mu }\,\mathrm{tr} \big( X \,\mathrm{I}_{y_{0}}^{-1}\big)\,\mathrm{tr} \big( Y\, \mathrm{I}_{y_{0}}^{-1} \big) \\ =\,& \mu \,\langle \, \mathrm{sym}\, \big( \mathrm{I}_{y_{0}}^{-1/2}\, X \, \mathrm{I}_{y_{0}}^{-1/2}\big) ,\,\mathrm{sym}\, \big(\mathrm{I}_{y_{0}}^{-1/2} \, Y\, \mathrm{I}_{y_{0}}^{-1/2}\big)\,\rangle \\ &+ \mu _{c}\;\langle \, \mathrm{skew}\, \big( \mathrm{I}_{y_{0}}^{-1/2}\, X\, \mathrm{I}_{y_{0}}^{-1/2} \big) ,\,\mathrm{skew}\, \big( \mathrm{I}_{y_{0}}^{-1/2}\, Y\, \mathrm{I}_{y_{0}}^{-1/2} \big)\,\rangle \\ & + \,\dfrac{\lambda \,\mu }{\lambda +2\mu }\,\mathrm{tr} \big( \mathrm{I}_{y_{0}}^{-1/2} \, X\, \mathrm{I}_{y_{0}}^{-1/2}\big)\,\mathrm{tr} \big( \mathrm{I}_{y_{0}}^{-1/2} \, Y\, \mathrm{I}_{y_{0}}^{-1/2}\big). \end{aligned}$$

(A.45)

Analogous results hold for the quadratic form $W_{\mathrm{curv}}( X, X)$.

We can decompose the $3\times 2$ matrices in two block matrices ($2 \times 2$ and $1\times 2$) and express the strain energy densities $W_{\mathrm{memb}}$ and $W_{\mathrm{memb,bend}}$ as functions of the matrices $\mathcal{G}_{\infty }$ and $\mathcal{R}_{\infty }$ (we have used that $\mathcal{T}_{\infty }$ vanishes) in the following form

$$\begin{aligned} & W_{\mathrm{memb}}\big( \mathcal{E}_{m,s} \big)+ W_{ \mathrm{memb,bend}}\big( \mathcal{E}_{m,s} ,\, \mathcal{K}_{e,s} \big)= \widetilde{W}_{\mathrm{memb}}\big( \mathcal{G}_{\infty }\big)+ \widetilde{W}_{\mathrm{memb,bend}}\big( \mathcal{G}_{\infty }, \mathcal{R}_{\infty }\big) \\ &= \underbrace{\Big(h+\mathrm{K}\,\dfrac{h^{3}}{12}\Big)\, W_{\mathrm{inplane}}\big( 2\, \mathcal{G}_{\infty }\big)}_{ \textrm{in-plane deformation}} \\ &\ \ \ + \underbrace{\Big(\dfrac{h^{3}}{12}\,-\mathrm{K}\,\dfrac{h^{5}}{80}\Big)\, W_{\mathrm{inplane}}\big( 2 \,\mathcal{G}_{\infty }\, \mathrm{L}_{y_{0}} - \mathcal{R}_{\infty }\big) - \dfrac{h^{3}}{12}\,\,2\, W_{\mathrm{inplane}}\big( 2\, \mathcal{G}_{\infty }, (2 \,\mathcal{G}_{\infty }\, \mathrm{L}_{y_{0}} - \mathcal{R}_{\infty })\,\mathrm{L}_{y_{0}}^{*}\big) + \,\dfrac{h^{5}}{80}\;W_{\mathrm{inp\lambda }}\big( 2 \,\mathcal{G}_{\infty }\,, (\mathrm{L}_{y_{0}} - \mathcal{R}_{\infty })\,\mathrm{L}_{y_{0}}\big)}_{ \textrm{in-plane deformation-bendings coupling terms}}, \end{aligned}$$

(A.46)

where , $W_{\mathrm{inplane}}$ is given by (A.45) and we have denoted

$$\begin{aligned} W_{\mathrm{inp\lambda }}( X, Y) :=& \,\mu \,\bigl\langle \mathrm{sym} \, X,\,\mathrm{sym}\, \big(\mathrm{I}_{y_{0}}^{-1}\, Y\, \mathrm{I}_{y_{0}}^{-1} \big)\bigr\rangle + \mu _{c}\bigl\langle \mathrm{skew}\, X,\, \mathrm{skew}\, \big(\mathrm{I}_{y_{0}}^{-1}\, Y \,\mathrm{I}_{y_{0}}^{-1} \big)\bigr\rangle \\ &+ \,\dfrac{\lambda }{2}\,\mathrm{tr} \big( X\,\mathrm{I}_{y_{0}}^{-1} \big)\,\mathrm{tr} \big( Y\, \mathrm{I}_{y_{0}}^{-1}\big) \\ =\,& \, \mu \,\langle \, \mathrm{sym}\, \big( \mathrm{I}_{y_{0}}^{-1/2} \, X\, \mathrm{I}_{y_{0}}^{-1/2}\big) ,\,\mathrm{sym}\, \big(\mathrm{I}_{y_{0}}^{-1/2} \, Y\, \mathrm{I}_{y_{0}}^{-1/2}\big)\,\rangle \\ &+ \mu _{c}\;\langle \, \mathrm{skew}\, \big( \mathrm{I}_{y_{0}}^{-1/2}\, X\, \mathrm{I}_{y_{0}}^{-1/2} \big) ,\,\mathrm{skew}\, \big( \mathrm{I}_{y_{0}}^{-1/2}\, Y\, \mathrm{I}_{y_{0}}^{-1/2} \big)\,\rangle \\ & + \,\dfrac{\lambda }{2}\,\mathrm{tr} \big( \mathrm{I}_{y_{0}}^{-1/2}\, X \, \mathrm{I}_{y_{0}}^{-1/2}\big)\,\mathrm{tr} \big( \mathrm{I}_{y_{0}}^{-1/2} \, Y\, \mathrm{I}_{y_{0}}^{-1/2}\big). \end{aligned}$$

(A.47)

We can see in the expression (A.46) the different parts of the energy corresponding to in-plane deformation, transverse shear, or coupling terms with bending.

Finally, the bending-curvature energy density $W_{\mathrm{bend,curv}}$ can be written as function of $\mathcal{R}_{\infty }$ and $\mathcal{N}_{\infty }$ as follows

$$\begin{aligned} W_{\mathrm{bend,curv}}&\big( \mathcal{K}_{e,s} \big)=\, \widetilde{W}_{ \mathrm{bend,curv}}\big( \mathcal{R}_{\infty }, \mathcal{N}_{\infty }\big) \\ =\, & \underbrace{\Big(h-\mathrm{K}\,\dfrac{h^{3}}{12}\Big)\, W_{\mathrm{curvpls}}\big( \mathcal{R}_{\infty }\big) + \Big(\dfrac{h^{3}}{12}\,-\mathrm{K}\,\dfrac{h^{5}}{80}\Big)\, W_{\mathrm{curvpls}}\big( \mathcal{R}_{\infty }{\mathrm{L}}_{y_{0}} \big) + \,\dfrac{h^{5}}{80}\,\, W_{\mathrm{curvpls}}\big( \mathcal{R}_{\infty }{\mathrm{L}}_{y_{0}} ^{2} \big)}_{ \textrm{bending strain tensor}} \\ &+ \underbrace{\mu \, L_{c}^{2}\,\dfrac{b_{1}+b_{2}}{2}\,\Big[ \Big(h-\mathrm{K}\,\dfrac{h^{3}}{12}\Big) \bigl\langle \mathcal{N}_{\infty }\, \mathrm{I}_{y_{0}}^{-1}, \mathcal{N}_{\infty }\bigr\rangle + \Big(\dfrac{h^{3}}{12}\,-\mathrm{K}\,\dfrac{h^{5}}{80}\Big) \bigl\langle ( \mathcal{N}_{\infty }\, \mathrm{L}_{y_{0}} ) \mathrm{I}_{y_{0}}^{-1} , ( \mathcal{N}_{\infty }\, \mathrm{L}_{y_{0}} )\bigr\rangle + \dfrac{h^{5}}{80}\, \bigl\langle ( \mathcal{N}_{\infty }\, \mathrm{L}_{y_{0}} ^{2}) \mathrm{I}_{y_{0}}^{-1},( \mathcal{N}_{\infty }\, \mathrm{L}_{y_{0}} ^{2})\bigr\rangle \Big]}_{ \textrm{drilling bendings}}, \end{aligned}$$

(A.48)

where we have denoted for any $2\times 2$ matrix $X$ the quadratic form (positive definite)

(A.49)

Thus, the model can be expressed entirely in terms of the change of metric tensor $\mathcal{G}_{\infty }$, the bending strain tensor $\mathcal{R}_{\infty }$ and the vector of drilling bendings $\mathcal{N}_{\infty }$, through the relations (A.46) and (A.48).