A Linear Isotropic Cosserat Shell Model Including Terms up to $O(h^{5})$. Existence and Uniqueness

Abstract

In this paper we derive the linear elastic Cosserat shell model incorporating in the variational problem effects up to order $O(h^{5})$ in the shell thickness $h$ as a particular case of the recently introduced geometrically nonlinear elastic Cosserat shell model. The existence and uniqueness of the solution is proven in suitable admissible sets. To this end, inequalities of Korn-type for shells are established which allow to show coercivity in the Lax-Milgram theorem. We are also showing an existence and uniqueness result for a truncated $O(h^{3})$ model. Main issue is the suitable treatment of the curved reference configuration of the shell. Some connections to the classical Koiter membrane-bending model are highlighted.

Appendix A: The Classical Linear (First) Koiter Membrane-Bending Model

According to [20, page 344], [22, page 154, ] in the linear (first) Koiter model, the variational problem is to find a midsurface displacement vector field $v:\omega \subset \mathbb{R}^{2}\to \mathbb{R}^{3}$ minimizing:

$$\begin{aligned} \displaystyle \int _{\omega }&\bigg\{ h\bigg( \mu \rVert [\nabla \Theta ]^{-T} ( \mathcal{G}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}})^{\flat }[\nabla \Theta ]^{-1} \rVert ^{2} +\dfrac{\,\lambda \,\mu}{\lambda +2\,\mu} \, \mathrm{tr} \Big[ [\nabla \Theta ]^{-T} (\mathcal{G}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}})^{ \flat }[\nabla \Theta ]^{-1}\Big]^{2}\bigg) \vspace{6pt} \\ &+\displaystyle \frac{h^{3}}{12}\bigg( \mu \rVert [\nabla \Theta ]^{-T} \big( \mathcal{R}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}}\big)^{\flat }[\nabla \Theta ]^{-1} \rVert ^{2} +\dfrac{\,\lambda \,\mu}{\lambda +2\,\mu} \, \mathrm{tr} \Big[ [\nabla \Theta ]^{-T} \big(\mathcal{R}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}} \big)^{\flat }[\nabla \Theta ]^{-1}\Big]^{2}\bigg)\bigg\} \\ &\times \,{\mathrm{det}}( \nabla y_{0}|n_{0})\, {\mathrm{d}}a, \end{aligned}$$

(A.1)

where $(\mathcal{G}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}})^{\flat }$ and $\big(\mathcal{R}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}}\big)^{\flat}$ are the lifted quantities of the strain measures [20] given by

$$ \mathcal{G}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}} \,\,:=\frac{1}{2}\big[{\mathrm{I}}_{m} - {\mathrm{I}}_{y_{0}}\big]^{\mathrm{{lin}}}= \,\,\frac{1}{2}\;\big[ (\nabla y_{0})^{T}( \nabla v) + (\nabla v)^{T}(\nabla y_{0})\big] = {\mathrm {sym}}\big[ (\nabla y_{0})^{T}( \nabla v)\big] $$

(A.2)

and

$$\begin{aligned} \mathcal{R}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}} \,\,:&= \,\, \big[{\mathrm{II}}_{m} - { \mathrm{II}}_{y_{0}}\big]^{\mathrm{{lin}}} \,= \Big( \bigl\langle n_{0} , \partial _{x_{\alpha }x_{\beta}}\,v- \displaystyle \sum _{\gamma =1,2}\Gamma ^{ \gamma}_{\alpha \beta}\,\partial _{x_{\gamma}}\,v\bigr\rangle a^{ \alpha}\,\Big)_{\alpha \beta}\in \mathbb{R}^{2\times 2}. \end{aligned}$$

(A.3)

The expression of $\mathcal{R}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}}$ involves the Christoffel symbols $\Gamma ^{\gamma}_{\alpha \beta}$ on the surface given by $\Gamma ^{\gamma}_{\alpha \beta}=\bigl\langle a^{\gamma}, \partial _{x_{ \alpha}} a_{\beta}\bigr\rangle =-\bigl\langle \partial _{x_{\alpha}} a^{ \gamma}, a_{\beta}\bigr\rangle =\Gamma ^{\gamma}_{\beta \alpha}$. Note that, using $m=y_{0}+v$ and $(\nabla m)^{T}\nabla m=(\nabla y_{0})^{T}\nabla y_{0}+(\nabla y_{0})^{T} \nabla v+(\nabla v)^{T}\nabla y_{0}+\text{h.o.t}\in \mathbb{R}^{2 \times 2}$, the linear approximation of the difference $\frac{1}{2}\big[{\mathrm{I}}_{m} - {\mathrm{I}}_{y_{0}}\big]^{\mathrm{{lin}}}$ appearing in the Koiter model can easily be obtained [20, page 92], the linear approximation of the difference $\big[{\mathrm{II}}_{m} - {\mathrm{II}}_{y_{0}}\big]^{\mathrm{{lin}}}$ needs some more insights from differential geometry [20, page 95] and it is based on formulas of Gauß $\partial _{x_{\alpha}} a_{\beta}= \sum _{\gamma =1,2}\Gamma _{\alpha \beta}^{\gamma }\,a_{\gamma}+b_{ \alpha \beta}a_{3}$ and $\partial _{x_{\beta}}a^{\alpha }= - \sum _{\gamma =1,2}\Gamma ^{ \alpha}_{\gamma \beta}\,a^{\gamma }+ b^{\alpha}_{\beta}\, n_{0}$ and the formulas of Weingarten $\partial _{x_{\alpha}} a_{3}=\partial _{x_{\alpha}} a^{3}= -\sum _{ \beta =1,2}b_{\alpha \beta}\, a^{\beta}=-\sum _{\beta =1,2} b^{\gamma}_{ \beta}\, a_{\gamma}$ together with the relations [20, page 76] $b_{\alpha \beta}(m)=-\bigl\langle \partial _{\alpha }a_{3}(m), a_{ \beta}(m)\bigr\rangle =\bigl\langle \partial _{\alpha }a_{\beta}(m), a_{3}(m) \bigr\rangle =b_{\beta \alpha}(m)$, where $b_{\alpha \beta}(m)$ are the components of the second fundamental form corresponding to the map $m$, $b_{\alpha}^{\beta}(m)$ are the components of the matrix associated to the Weingarten map (shape operator), and on the following linear approximation $n=\,n_{0}+\frac{1}{\sqrt{\det ((\nabla y_{0})^{T}\nabla y_{0})}} \left (\partial _{x_{1}} y_{0}\times \partial _{x_{2}} v+\partial _{x_{1}} v\times \partial _{x_{2}} y_{0}+\text{h.o.t}\right ) -{\mathrm {tr}}(((\nabla y_{0})^{T} \nabla y_{0})^{-1}\, {\mathrm {sym}}((\nabla y_{0})^{T}\nabla v) )\,n_{0}$.

We note that other alternative forms of the change of metric tensor and the change of curvature tensor in [20, Page 181] are

$$ \mathcal{G}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}} = \Big( \frac{1}{2}(\partial _{ \beta }v_{\alpha}+\partial _{\alpha }v_{\beta})-\sum _{\gamma =1,2} \Gamma _{\alpha \beta}^{\gamma }v_{\gamma}-b_{\alpha \beta}v_{3} \Big)_{\alpha \beta}\in \mathbb{R}^{2\times 2}, $$

(A.4)

and

$$\begin{aligned} \mathcal{R}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}} &= \Big( \partial _{x_{\alpha }x_{ \beta}}v_{3}-\sum _{\gamma =1,2}\Gamma _{\alpha \beta}^{\gamma } \partial _{x_{\gamma}}v_{3}-\sum _{\gamma =1,2}b_{\alpha}^{\gamma }b_{ \gamma \beta}v_{3} \\ &\quad {}+\sum _{\gamma =1,2}b_{\alpha}^{\gamma}(\partial _{x_{ \beta}}v_{\gamma}-\sum _{\tau =1,2}\Gamma _{\beta \gamma}^{\tau }v_{ \tau}) +\sum _{\gamma =1,2}b_{\beta}^{\gamma}(\partial _{x_{\alpha}}v_{ \gamma}-\sum _{\tau =1,2}\Gamma _{\alpha \tau}^{\gamma }v_{\gamma}) \\ &\quad {}+ \sum _{\tau =1,2}(\partial _{x_{\alpha}}b_{\beta}^{\tau}+\sum _{ \gamma =1,2}\Gamma _{\alpha \gamma}^{\tau }b_{\beta}^{\gamma}-\sum _{ \gamma =1,2}\Gamma _{\alpha \beta}^{\gamma }b_{\gamma}^{\tau})v_{\tau} \Big)_{\alpha \beta}\in \mathbb{R}^{2\times 2}, \end{aligned}$$

(A.5)

respectively. Actually, on one hand, the last form of the curvature tensor will be considered when the admissible set of solutions of the variational problem will be defined. On the other hand, as noticed in [21, Page 175] by considering the form (A.3) of the change of metric tensor, we can impose substantially weaker regularity assumptions on the mapping $y_{0}$. For the linear (first) Koiter model the existence results are given in [20, Theorem 7.1.-1 and Theorem 7.1.-2], see also [16].

While the relation between $\mathcal{G}^{\mathrm{{lin}}}$ and $\mathcal{G}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}}$ holds in the general case, we are able to find a simple explicit relation between $\mathcal{R}^{\mathrm{{lin}}}$ and $\mathcal{R}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}}$ only in the case of the constrained Cosserat-shell model. This is not surprising, since only symmetric stress tensors are taken into account in the classical linear Koiter model, i.e., the internal strain energy does not depend on the skew-symmetric part of the considered strain measures (since it is work conjugate to the skew-symmetric part of the stress tensor). In addition, the linear Koiter model does not consider extra degrees of freedom. In [32] we will discuss the choice of the deformation measures in shell models. We will see that the classical strain measure $\mathcal{R}_{\mathrm{{Koiter}}}^{\mathrm{{lin}}}$ (the classical bending strain measure, also known as the change of curvature tensor) does not represent the unique choice and that some other modified expressions of the classical bending tensor may be more suitable in the modelling of a shell.