Transformation Cloaking in Elastic Plates

Abstract

In this paper, we formulate the problem of elastodynamic transformation cloaking for Kirchoff–Love plates and elastic plates with both in-plane and out-of-plane displacements. A cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic plate (virtual problem) to that of an anisotropic and inhomogeneous elastic plate with a hole surrounded by a cloak that is to be designed (physical problem). For Kirchoff–Love plates, the governing equation of the virtual plate is transformed to that of the physical plate up to an unknown scalar field. In doing so, one finds the initial stress and the initial tangential body force for the physical plate, along with a set of constraints that we call the cloaking compatibility equations. It is noted that the cloaking map needs to satisfy certain boundary and continuity conditions on the outer boundary of the cloak and the surface of the hole. In particular, the cloaking map needs to fix the outer boundary of the cloak up to the third order. Assuming a generic radial cloaking map, we show that cloaking a circular hole in Kirchoff–Love plates is not possible; the cloaking compatibility equations and the boundary conditions are the obstruction to cloaking. Next, relaxing the pure bending assumption, the transformation cloaking problem of an elastic plate in the presence of in-plane and out-of-plane displacements is formulated. In this case, there are two sets of governing equations that need to be simultaneously transformed under a cloaking map. We show that cloaking a circular hole is not possible for a general radial cloaking map; similar to Kirchoff–Love plates, the cloaking compatibility equations and the boundary conditions obstruct transformation cloaking. Our analysis suggests that the path forward for cloaking flexural waves in plates is approximate cloaking formulated as an optimal design problem.

Appendices

Appendix A: Variations of the Right Cauchy–Green Deformation Tensor, the Unit Normal Vector Field, and the Second Fundamental Form

In this appendix, we discuss the derivation of the variations of the right Cauchy–Green deformation tensor ${\mathbf {C}}$, the unit normal vector field (of the deformed shell) $\varvec{{\mathcal {N}}}$, and ${\varvec{\Theta }}$ used in obtaining the Euler–Lagrange equations in Sect. 3 (see also Capovilla and Guven 2002; Kadianakis and Travlopanos 2013, 2018).

Lie derivative. Let ${\mathbf {w}}:{\mathcal {U}}\rightarrow T{\mathcal {S}}$ be a $C^1$ vector field, where ${\mathcal {U}} \subset {\mathcal {S}}$ is an open neighborhood. A curve $\alpha :I \rightarrow {\mathcal {S}}$, where I is an open interval, is an integral curve of ${\mathbf {w}}$ provided that $\frac{d \alpha (t)}{dt}={\mathbf {w}}(\alpha (t)),~\forall ~t \in I$. Consider a time-dependent vector field ${\mathbf {w}}:{\mathcal {S}}\times I\rightarrow T{\mathcal {S}}$, where I is some open interval. The collection of maps $\psi _{\tau ,t}$ is the flow of ${\mathbf {w}}$ if for each t and x, $\tau \mapsto \psi _{\tau ,t}(x)$ is an integral curve of ${\mathbf {w}}_t$, i.e., $\frac{d}{d\tau }\psi _{\tau ,t}(x)={\mathbf {w}}(\psi _{\tau ,t}(x),\tau )$, and $\psi _{t,t}(x)=x$. Assume that ${\mathbf {t}}$ is a time-dependent tensor field on ${\mathcal {S}}$, i.e., ${\mathbf {t}}_t(x)={\mathbf {t}}(x,t)$ is a tensor. The Lie derivative of ${\mathbf {t}}$ with respect to ${\mathbf {w}}$ is defined as

$$\begin{aligned} {\mathbf {L}}_{{\mathbf {w}}}{\mathbf {t}}=\frac{d}{d \tau }\psi _{\tau ,t}^* {\mathbf {t}}_{\tau } \Big |_{\tau =t}. \end{aligned}$$

(A.1)

Note that $\psi _{\tau ,t}$ maps ${\mathbf {t}}_t$ to ${\mathbf {t}}_{\tau }$. Therefore, to calculate the Lie derivative one drags ${\mathbf {t}}$ along the flow of ${\mathbf {w}}$ from $\tau$ to t and then differentiates the Lie dragged tensor with respect to $\tau$. The autonomous Lie derivative of ${\mathbf {t}}$ with respect to ${\mathbf {w}}$ is defined as

$$\begin{aligned} {\mathfrak {L}}_{{\mathbf {w}}}{\mathbf {t}}=\frac{d}{d \tau }\psi _{\tau ,t}^* {\mathbf {t}}_{t} \Big |_{\tau =t}. \end{aligned}$$

(A.2)

Hence, ${\mathbf {L}}_{{\mathbf {w}}}{\mathbf {t}}=\partial {\mathbf {t}}/\partial t+{\mathfrak {L}}_{{\mathbf {w}}}{\mathbf {t}}$. The Lie derivative for a scalar f is given by ${\mathbf {L}}_{{\mathbf {w}}}f=\partial f/\partial t+{\mathbf {w}}[f]$. In a coordinate chart $\{x^a\}$, this is written as, ${\mathbf {L}}_{{\mathbf {w}}}f=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x^a}w^a$. For a vector ${\mathbf {u}}$, it can be shown that ${\mathbf {L}}_{{\mathbf {w}}}{\mathbf {u}}=\frac{\partial {\mathbf {w}}}{\partial t}+[{\mathbf {w}},{\mathbf {u}}]$. If $\nabla$ is a torsion-free connection, then $[{\mathbf {w}},{\mathbf {u}}]=\nabla _{{\mathbf {w}}}{\mathbf {u}}-\nabla _{{\mathbf {u}}}{\mathbf {w}}$, and thus, ${\mathbf {L}}_{{\mathbf {w}}}{\mathbf {u}}=\frac{\partial {\mathbf {w}}}{\partial t}+\nabla _{{\mathbf {w}}}{\mathbf {u}}-\nabla _{{\mathbf {u}}}{\mathbf {w}}$.

The rate of deformation tensor for shells is defined as (Marsden and Hughes 1983)

$$\begin{aligned} 2{\mathbf {D}}^\flat =\varphi ^*_t\left[ (\nabla ^{{\mathbf {g}}}{\mathbf {v}}^\top )^\flat +[(\nabla ^{{\mathbf {g}}}{\mathbf {v}}^\top )^\flat ]^{{\mathsf {T}}} -2v^\perp \varvec{\theta }\right] , \end{aligned}$$

(A.3)

where the spatial velocity is decomposed into the normal and tangential components as ${\mathbf {v}}={\mathbf {v}}^\top +v^\perp {\mathbf {n}}$. In components

$$\begin{aligned} 2 D_{AB}=(v^\top _{a|b}+v^\top _{b|a})F^a{}_AF^b{}_B-2v^\perp \Theta _{AB}. \end{aligned}$$

(A.4)

Note that

$$\begin{aligned} {\mathbf {L}}_{{\mathbf {v}}^\top }{\mathbf {g}}=(\nabla ^{{\mathbf {g}}}{\mathbf {v}} ^\top )^\flat +[(\nabla ^{{\mathbf {g}}}{\mathbf {v}}^\top )^\flat ]^{{\mathsf {T}}}. \end{aligned}$$

(A.5)

Therefore,

$$\begin{aligned} 2{\mathbf {D}}^\flat =\varphi ^*_t({\mathbf {L}}_{{\mathbf {v}}^\top }{\mathbf {g}}) -2v^\perp {\varvec{\Theta }}. \end{aligned}$$

(A.6)

Knowing that $\varphi _t^*({\mathbf {L}}_{{\mathbf {v}}}{\mathbf {g}})=2{\mathbf {D}}^\flat$ (see, e.g., Marsden and Hughes 1983; Simo and Marsden 1984), one obtains

$$\begin{aligned} \varphi _t^*({\mathbf {L}}_{{\mathbf {v}}}{\mathbf {g}})=\varphi ^*_t ({\mathbf {L}}_{{\mathbf {v}}^\top }{\mathbf {g}})-2v^\perp {\varvec{\Theta }}. \end{aligned}$$

(A.7)

Thus, ${\mathbf {L}}_{\delta \varphi }{\mathbf {g}}={\mathbf {L}}_{\delta \varphi ^\top }{\mathbf {g}}-2\,\delta \varphi ^\perp \,\varvec{\theta }$, and hence, $\delta {\mathbf {C}}^\flat =\varphi ^*_t({\mathbf {L}}_{\delta \varphi }{\mathbf {g}})=\varphi ^*_t\,{\mathbf {L}}_{\delta \varphi ^\top }{\mathbf {g}}-2\,\delta \varphi ^\perp \,{\varvec{\Theta }}$. Also note that

$$\begin{aligned} {\mathbf {L}}_{\delta \varphi ^\top }{\mathbf {g}}=\left[ g_{cb}(\delta \varphi ^\top )^c{}_{|a}+g_{ac}(\delta \varphi ^\top )^c{}_{|b}\right] dx^a\otimes dx^b. \end{aligned}$$

(A.8)

Hence, in components, one obtains

$$\begin{aligned} \delta C_{AB}=F^a{}_A\,\delta {\varphi }^{\top }_{a|B}+F^b{}_B\, \delta \varphi ^{\top }_{b|A}-2\,\delta \varphi ^\perp \,\Theta _{AB}. \end{aligned}$$

(A.9)

Or

$$\begin{aligned} \delta C_{AB}=F^a{}_A\,g_{ac}(\delta {\varphi }^{\top })^c{}_{|B} +F^b{}_B\,g_{bc}(\delta \varphi ^{\top })^c{}_{|A}-2\,\delta \varphi ^\perp \,F^a{}_AF^b{}_B\theta _{ab}. \end{aligned}$$

(A.10)

Therefore, (3.11) is implied.

The covariant derivative of ${\mathbf {v}}$ is computed as

$$\begin{aligned} {\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {v}} ={\bar{\nabla }}^{\bar{{\mathbf {g}}}}({\mathbf {v}}^\top +v^\perp {\mathbf {n}})={\bar{\nabla }}^{\bar{{\mathbf {g}}}} {\mathbf {v}}^\top +v^\perp {\bar{\nabla }}^{\bar{{\mathbf {g}}}} {\mathbf {n}}+{\mathbf {d}}v^\perp \otimes {\mathbf {n}}. \end{aligned}$$

(A.11)

Using the relations (2.3) and (2.5) in the ambient space, one obtains

$$\begin{aligned} {\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {v}}=(\nabla ^{{\mathbf {g}}} {\mathbf {v}}^\top -v^\perp \varvec{\theta })+({\mathbf {d}}v^\perp +\varvec{\theta }\cdot {\mathbf {v}}^\top )\otimes {\mathbf {n}}. \end{aligned}$$

(A.12)

In components

$$\begin{aligned} {\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {v}} =\left[ (v^\top )^a{}_{|b}-v^\perp \theta ^a{}_b\right] \partial _a\otimes dx^b +\left[ v^\perp {}_{,b}+\theta _{bc}(v^\top )^c\right] {\mathbf {n}}\otimes dx^b. \end{aligned}$$

(A.13)

Similarly, one can write

$$\begin{aligned} {\bar{\nabla }}^{\bar{{\mathbf {g}}}}\delta \varphi =\left[ (\delta \varphi ^\top )^a{}_{|b}-\delta \varphi ^\perp \theta ^a{}_b \right] \partial _a\otimes dx^b +\left[ \delta \varphi ^\perp {}_{,b}+\theta _{bc}(\delta \varphi ^\top )^c \right] {\mathbf {n}}\otimes dx^b. \end{aligned}$$

(A.14)

Note that for an arbitrary vector field ${\mathbf {W}}={{\mathbf {W}}}^\top +W^\perp \varvec{{\mathcal {N}}}$ defined on a surface embedded in ${\mathbb {R}}^3$, the tangential and normal components of the covariant derivative with respect to the surface coordinates are similarly given by

$$\begin{aligned} ({\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {W}})^\top =\nabla ^{{\mathbf {g}}}{\mathbf {W}}^\top -W^\perp \varvec{\theta }, \quad \quad ({\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {W}})^\perp ={\mathbf {d}}W^\perp +\varvec{\theta }\cdot {\mathbf {W}}^\top . \end{aligned}$$

(A.15)

Therefore, ${\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {W}}=({\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {W}})^\top +({\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {W}})^\perp$, in components reads

$$\begin{aligned} ({\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {W}})^\top= & {} \left[ (W^\top )^a{}_{|b}-W^\perp \theta ^a{}_b\right] \partial _a\otimes dx^b,\nonumber \\ ({\bar{\nabla }}^{\bar{{\mathbf {g}}}}{\mathbf {W}})^\perp= & {} \left[ W^\perp {}_{,b}+\theta _{bc}(W^\top )^c\right] \varvec{{\mathcal {N}}}\otimes dx^b. \end{aligned}$$

(A.16)

Thus, one can use (A.16)$_1$ to write the variation of the deformation gradient in components as

$$\begin{aligned} \delta F^a{}_A=(\delta \varphi ^\top )^a{}_{|A}-\theta ^a{}_bF^b{}_A\delta \varphi ^\perp . \end{aligned}$$

(A.17)

Therefore, (3.60) follows. At any time t, the deformation map $\varphi _t:{\mathcal {H}}\rightarrow {\mathcal {S}}$ is a smooth embedding of the (undeformed) shell into the ambient space. For each $X\in {\mathcal {H}}$, let $T\varphi _t(X):T_X{\mathcal {H}}\rightarrow T_{\varphi _t(X)}{\mathcal {S}}$ be the tangent of $\varphi _t$ at X. The variation of the unit normal vector $\varvec{{\mathcal {N}}}_{\epsilon }={\mathbf {n}}_{\epsilon }\circ \varphi _{\epsilon ,t}$ is defined as

$$\begin{aligned} \delta \varvec{{\mathcal {N}}}=\frac{d}{d\epsilon } \varvec{{\mathcal {N}}}_{\epsilon }\Big |_{\epsilon =0} ={\bar{\nabla }}_{\frac{\partial }{\partial \epsilon }}^{\bar{{\mathbf {g}}}} \varvec{{\mathcal {N}}}_{\epsilon }\Big |_{\epsilon =0} =D_{\varphi _{\epsilon }(X,t)}\varvec{{\mathcal {N}}}_{\epsilon } \Big |_{\epsilon =0}={{\bar{\nabla }}}^{\bar{{\mathbf {g}}} }_{\delta \varphi }\varvec{{\mathcal {N}}}. \end{aligned}$$

(A.18)

In order to compute the variation, let ${\mathbf {W}}$ be a vector field in ${\mathcal {S}}$ tangent to $\varphi _t({\mathcal {H}})$ and note that

$$\begin{aligned} {{\bar{\nabla }}}^{\bar{{\mathbf {g}}} }_{\delta \varphi }{\mathbf {W}}=[\delta \varphi ,{\mathbf {W}}] +{{\bar{\nabla }}}^{\bar{{\mathbf {g}}} }_{{\mathbf {W}}}\delta \varphi . \end{aligned}$$

(A.19)

From (A.12), one obtains

$$\begin{aligned} {{\bar{\nabla }}}^{\bar{{\mathbf {g}}} }_{{\mathbf {W}}}\delta \varphi =\left( \nabla ^{{\mathbf {g}}}_{{\mathbf {W}}}{\delta \varphi ^\top } -\delta \varphi ^\perp \varvec{\theta }\cdot {\mathbf {W}}\right) +\left( ({\mathbf {d}}\,\delta \varphi ^\perp )\cdot {\mathbf {W}} +\varvec{\theta }\cdot \delta \varphi ^\top \cdot {\mathbf {W}}\right) {\mathbf {n}}. \end{aligned}$$

(A.20)

Therefore,

$$\begin{aligned} \bar{{\mathbf {g}}}\left( \varvec{{\mathcal {N}}}, {{\bar{\nabla }}}^{\bar{{\mathbf {g}}} }_{{\mathbf {W}}} \delta \varphi \right) =({\mathbf {d}}\,\delta \varphi ^\perp ) \cdot {\mathbf {W}}+\varvec{\theta }\cdot \delta \varphi ^\top \cdot {\mathbf {W}}=-\bar{{\mathbf {g}}}\left( {{\bar{\nabla }}} ^{\bar{{\mathbf {g}}} }_{\delta \varphi }\varvec{{\mathcal {N}}},{\mathbf {W}}\right) , \end{aligned}$$

(A.21)

where the second equality is a consequence of the metric compatibility of ${\bar{\nabla }}^{\bar{{\mathbf {g}}}}$ (2.4). By arbitrariness of ${\mathbf {W}}$, we have

$$\begin{aligned} \delta \varvec{{\mathcal {N}}}={{\bar{\nabla }}}^{\bar{{\mathbf {g}}} }_{\delta \varphi }\varvec{{\mathcal {N}}}=-\varvec{\theta }\cdot \delta \varphi ^\top -{\mathbf {d}}\,\delta \varphi ^\perp . \end{aligned}$$

(A.22)

In components

$$\begin{aligned} \delta \varvec{{\mathcal {N}}}=-\left[ (\delta \varphi ^\top )^b\theta ^a{}_b +\delta \varphi ^\perp {}_{,b}g^{ab}\right] \partial _a . \end{aligned}$$

(A.23)

Hence, (3.19) and (3.20) are followed. Note that $\delta {\varvec{\Theta }}^\flat =\varphi ^*_t({\mathbf {L}}_{\delta \varphi } \varvec{\theta })$, where ${\mathbf {L}}_{\delta \varphi }\varvec{\theta }={\mathbf {L}}_{\delta \varphi ^\top }\varvec{\theta }-\delta \varphi ^\perp \mathbf {III}+\mathrm {Hess}_{\delta \varphi ^\perp },$ i.e.,

$$\begin{aligned} \delta {\varvec{\Theta }}^\flat =\varphi ^*_t{\mathbf {L}}_{\delta \varphi ^\top } \varvec{\theta }-\delta \varphi ^\perp \,\varphi _t^*\mathbf {III}\, +\varphi _t^*\mathrm {Hess}_{\delta \varphi ^\perp }, \end{aligned}$$

(A.24)

where for ${\varvec{x}},{\varvec{y}}\in {\mathcal {X}}(\varphi ({\mathcal {H}}))$, the third fundamental form of the deformed hypersurface $\mathbf {III}$ and the Hessian of $\delta \varphi ^\perp$, i.e., $\mathrm {Hess}_{\delta \varphi ^\perp }$ are given by (3.15) and (3.16). The Lie derivative with respect to the tangential component of the variation field is given in components by (see, e.g., Marsden and Hughes 1983, p.97)

$$\begin{aligned} {\mathbf {L}}_{\delta \varphi ^\top }\varvec{\theta }^\flat =\left[ \theta _{ab|c}\,(\delta \varphi ^\top )^c+\theta _{ac}( \delta \varphi ^\top )^c{}_{|b}+\theta _{cb}(\delta \varphi ^\top )^c{}_{|a} \right] dx^a\otimes dx^b. \end{aligned}$$

(A.25)

Therefore, in components, one can write the variation of ${\varvec{\Theta }}^\flat$ as

$$\begin{aligned} \begin{aligned} \delta \Theta _{AB}&= F^a{}_AF^b{}_B\,\theta _{ab|c}\,(\delta \varphi ^\top )^c +F^a{}_A\theta _{ac}(\delta \varphi ^\top )^c{}_{|B}+F^b{}_B\theta _{bc}(\delta \varphi ^\top )^c{}_{|A}\\&\quad -\delta \varphi ^\perp F^a{}_AF^b{}_B\theta _{ac}\theta _{bd}\,g^{cd} +F^b{}_A\left( \frac{\partial \,\delta \varphi ^\perp }{\partial \, x^b}\right) _{|B}. \end{aligned} \end{aligned}$$

(A.26)

Using (A.17) and the fact that $\Theta _{AB}=F^a{}_AF^b{}_B\theta _{ab}$, one obtains the variation of $\varvec{\theta }^\flat$ as

$$\begin{aligned} \delta \theta _{ab}=\theta _{ab|c}(\delta \varphi ^\top )^c +\delta \varphi ^\perp \theta _{ac}\theta _{bd}g^{cd}+(\delta \varphi ^\perp {}_{,a})_{|b}. \end{aligned}$$

(A.27)

Hence, (3.17) and (3.75) are obtained.

Proof of the relation (A.24). Here, we give a proof of the relation (A.24), see also (Capovilla and Guven 2002; Lenz and Lipowsky 2000; Deserno 2004). For the sake of simplicity, we assume that the surface is embedded in three-dimensional Euclidean space. For this proof, we adopt a notation different from the rest of the paper. Let us consider an embedded surface denoted by $\Sigma$ in ${\mathbb {R}}^3$. The surface geometry is locally described by three functions ${\mathbf {x}}(x^1,x^2,x^3)={\mathbf {X}}(\nu ^\alpha )$ in the Cartesian coordinates $\{x^1,x^2,x^3\}$ such that $\{\nu ^\alpha \}$, $\alpha =1,2$, is a local coordinate chart on the surface. Let us define two tangent vectors ${\mathbf {e}}_{\alpha }={\partial }{\mathbf {X}}/{\partial \nu ^\alpha }$ on the surface. We note that the surface geometry is completely described by its induced metric $\eta _{\alpha \beta }$ and its induced second fundamental form $\Lambda _{\alpha \beta }$.⁴³ Note that $\eta _{\alpha \beta }={\mathbf {e}}_{\alpha }\cdot {\mathbf {e}}_{\beta }$, where “$\cdot$” denotes the dot product in ${\mathbb {R}}^3$. Let us denote the surface covariant derivative with $\nabla _\alpha$. Then, one can write the Gauss–Weingarten equations as $\nabla _{\alpha }{\mathbf {e}}_{\beta }=\Lambda _{\alpha \beta }{\mathbf {n}}$, and $\nabla _{\alpha }{\mathbf {n}}=-\Lambda _{\alpha \beta }{\mathbf {e}}^{\beta }$, where ${\mathbf {n}}$ is the unit normal vector to the surface. Let us consider the deformation of the embedding functions of the surface ${\mathbf {X}}(\nu )\rightarrow {\mathbf {X}}(\nu )+\delta {\mathbf {X}}(\nu )$ such that the variation field $\delta {\mathbf {X}}$ is decomposed into the tangential and normal components as $\delta {\mathbf {X}}=(\psi ^\top )^\alpha {\mathbf {e}}_{\alpha }+\psi ^\perp {\mathbf {n}}$. Using the relation $\Lambda _{\alpha \beta }={\mathbf {n}}\cdot \nabla _{\alpha }{\mathbf {e}}_{\beta }$, one may write

$$\begin{aligned} \delta \Lambda _{\alpha \beta }=\delta {\mathbf {n}}\cdot \nabla _{\alpha } \nabla _{\beta }{\mathbf {X}}+{\mathbf {n}}\cdot \nabla _{\alpha } \nabla _{\beta }\delta {\mathbf {X}}. \end{aligned}$$

(A.28)

Knowing that the variation of the unit normal vector is purely tangential, the first term vanishes, i.e., $\delta {\mathbf {n}}\cdot \nabla _{\alpha }\nabla _{\beta }{\mathbf {X}}=\delta {\mathbf {n}}\cdot \nabla _{\alpha }{\mathbf {e}}_{\beta }=\Lambda _{\alpha \beta }(\delta {\mathbf {n}})\cdot {\mathbf {n}}=0$. After some simplification and using the Codazzi–Mainardi equation $\nabla _{\alpha }\Lambda _{\beta \gamma }=\nabla _{\beta }\Lambda _{\alpha \gamma }$, one obtains

$$\begin{aligned} \delta \Lambda _{\alpha \beta }= & {} \Lambda _{\beta \gamma }\nabla _{\alpha } (\psi ^\top )^\gamma +\Lambda _{\alpha \gamma }\nabla _{\beta } (\psi ^\top )^\gamma +(\psi ^\top )^{\gamma }\nabla _{\gamma } \Lambda _{\alpha \beta }\nonumber \\&-\Lambda _{\alpha \gamma }\Lambda ^{\gamma }{}_{\beta } \psi ^\perp +\nabla _{\alpha }\nabla _{\beta }\psi ^\perp . \end{aligned}$$

(A.29)

Notice that the first three terms correspond to the Lie derivative of the induced second fundamental form with respect to the tangential component of the variation field, and thus, (A.29) can be rewritten as

$$\begin{aligned} \Lambda _{\alpha \beta }=({\mathbf {L}}_{\psi ^\top }\Lambda )_{\alpha \beta } -\Lambda _{\alpha \gamma }\Lambda ^{\gamma }{}_{\beta }\psi ^\perp +\nabla _{\alpha }\nabla _{\beta }\psi ^\perp . \end{aligned}$$

(A.30)

Therefore, (A.24) follows.

Appendix B: The Euler–Lagrange Equations of Elastic Shells

In this appendix, we discuss the derivation of the Euler–Lagrange equations. Substituting (3.8), (3.12), (3.18), and (3.20) into (3.6), one obtains

$$\begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\Bigg \{\rho ({\mathfrak {B}}^\top )_a (\delta \varphi ^\top )^a+\rho {\mathfrak {B}}^\perp \cdot \delta \varphi ^\perp \nonumber \\&\quad +\frac{\partial {\mathcal {L}}}{\partial {\dot{\varphi }}}\cdot \frac{D\,\delta \varphi }{dt}-\rho {\mathfrak {L}}_b\left( (\delta \varphi ^\top )^a\theta ^b{}_a +\frac{\partial \,\delta \varphi ^\perp }{\partial x^a}g^{ab}\right) \nonumber \\&\quad +\frac{\partial {\mathcal {L}}}{\partial C_{AB}}\left( F^a{}_A \delta \varphi ^\top _{a|B}+F^b{}_B\delta \varphi ^\top _{b|A}-2\delta \varphi ^\perp F^a{}_AF^b{}_B\theta _{ab}\right) \nonumber \\&\quad +\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}} \bigg [F^a{}_AF^b{}_B\,\theta _{ab|c}\,(\delta \varphi ^\top )^c +F^a{}_A\theta _{ac}(\delta \varphi ^\top )^c{}_{|B}+F^b{}_B \theta _{bc}(\delta \varphi ^\top )^c{}_{|A}\nonumber \\&\quad -\delta \varphi ^\perp F^a{}_AF^b{}_B\theta _{ac}\theta _{bd} \,g^{cd}+F^b{}_A\left( \frac{\partial \,\delta \varphi ^\perp }{\partial \, x^b}\right) _{|B}\bigg ]\Bigg \}dAdt=0. \end{aligned}$$

(B.1)

After some simplification, we have

$$\begin{aligned} \begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\Bigg \{\rho ({\mathfrak {B}}^\top )_a(\delta \varphi ^\top )^a+\rho {\mathfrak {B}}^\perp \cdot \delta \varphi ^\perp +\frac{d}{dt}\left( \frac{\partial {\mathcal {L}}}{\partial {\dot{\varphi }}}\cdot \,\delta \varphi \right) \\&\quad -\frac{d}{dt}\left( \frac{\partial {\mathcal {L}}}{\partial {\dot{\varphi }}}\right) \cdot \,\delta \varphi -\rho {\mathfrak {L}}_b(\delta \varphi ^\top )^a\theta ^b{}_a\\&\quad -\left( \rho {\mathfrak {L}}_b\delta \varphi ^\perp \, g^{ab}(F^{-1})^A{}_a\right) _{|A}+\left( \rho {\mathfrak {L}}_b g^{ab}(F^{-1})^A{}_a\right) _{|A}\delta \varphi ^\perp \\&\quad -2\frac{\partial {\mathcal {L}}}{\partial C_{AB}} F^a{}_AF^b{}_B\theta _{ab}\delta \varphi ^\perp +2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^b{}_Bg_{cb}(\delta \varphi ^\top )^c\right) _{|A}\\&\quad -2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^b{}_Bg_{cb}\right) _{|A}(\delta \varphi ^\top )^c\\&\quad +\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^a{}_AF^b{}_B\,\theta _{ab|c}\,(\delta \varphi ^\top )^c+2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^a{}_A\theta _{ac}(\delta \varphi ^\top )^c\right) _{|B}\\&\quad -2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^a{}_A\theta _{ac}\right) _{|B}(\delta \varphi ^\top )^c-\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}} F^a{}_AF^b{}_B\theta _{ac}\theta _{bd}\,g^{cd}\delta \varphi ^\perp \\&\quad +\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\frac{\partial \,\delta \varphi ^\perp }{\partial \, x^b}\right) _{|B}-\left[ \left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}\delta \varphi ^\perp (F^{-1})^D{}_b\right] _{|D}\\&\quad +\left[ \left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}(F^{-1})^D{}_b\right] _{|D}\delta \varphi ^\perp \Bigg \}dAdt=0. \end{aligned} \end{aligned}$$

(B.2)

This can further be simplified to read

$$\begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\Bigg \{\Bigg [\rho ({\mathfrak {B}}^\top )_a-\frac{d}{dt}\left( \frac{\partial {\mathcal {L}}}{\partial ({\dot{\varphi }}^\top )^a}\right) -\rho {\mathfrak {L}}_b\theta ^b{}_a\nonumber \\&\quad -2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^b{}_Bg_{ab}\right) _{|A}+\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B\,\theta _{cb|a}\nonumber \\&\quad -2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\theta _{ba}\right) _{|B}\Bigg ](\delta \varphi ^\top )^a+\Bigg [\rho {\mathfrak {B}}^\perp -\frac{d}{dt}\left( \frac{\partial {\mathcal {L}}}{\partial {\dot{\varphi }}^\perp }\right) +\left( \rho {\mathfrak {L}}_b g^{ab}(F^{-1})^A{}_a\right) _{|A}\nonumber \\&\quad -2\frac{\partial {\mathcal {L}}}{\partial C_{AB}} F^a{}_AF^b{}_B\theta _{ab}-\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}} F^a{}_AF^b{}_B\theta _{ac}\theta _{bd}\,g^{cd}\nonumber \\&\quad +\left( \left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}(F^{-1})^D{}_b\right) _{|D}\Bigg ]\delta \varphi ^\perp \nonumber \\&\quad +\frac{d}{dt}\left( \frac{\partial {\mathcal {L}}}{\partial {\dot{\varphi }}}\cdot \,\delta \varphi \right) -\left( \rho {\mathfrak {L}}_b\delta \varphi ^\perp \, g^{ab}(F^{-1})^A{}_a\right) _{|A}+2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^b{}_Bg_{cb}(\delta \varphi ^\top )^c\right) _{|A}\nonumber \\&\quad +2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^a{}_A\theta _{ac}(\delta \varphi ^\top )^c\right) _{|B}+\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}\,\frac{\partial \,\delta \varphi ^\perp }{\partial X^A}\right) _{|B}\nonumber \\&\quad -\left[ \left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}\delta \varphi ^\perp (F^{-1})^D{}_b\right] _{|D}\Bigg \}dAdt=0. \end{aligned}$$

(B.3)

We assume that $\delta \varphi (X,t_0)=\delta \varphi (X,t_1)=0$. Using Stokes’ theorem, we have

$$\begin{aligned} \begin{aligned}&\int _{t_0}^{t_1}\int _{{\mathcal {H}}}\Bigg \{\Bigg [\rho ({\mathfrak {B}}^\top )_a-\frac{d}{dt}\left( \frac{\partial {\mathcal {L}}}{\partial ({\dot{\varphi }}^\top )^a}\right) -\rho {\mathfrak {L}}_b\theta ^b{}_a\\&\quad -2\left( \frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^b{}_Bg_{ab}\right) _{|A}+\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^c{}_AF^b{}_B\,\theta _{cb|a}\\&\quad -2\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\theta _{ba}\right) _{|B}\Bigg ](\delta \varphi ^\top )^a+\Bigg [\rho {\mathfrak {B}}^\perp -\frac{d}{dt}\left( \frac{\partial {\mathcal {L}}}{\partial {\dot{\varphi }}^\perp }\right) +\left( \rho {\mathfrak {L}}_b g^{ab}(F^{-1})^A{}_a\right) _{|A}\\&\quad -2\frac{\partial {\mathcal {L}}}{\partial C_{AB}} F^a{}_AF^b{}_B\theta _{ab}-\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}} F^a{}_AF^b{}_B\theta _{ac}\theta _{bd}\,g^{cd}\\&\quad +\left( \left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^b{}_A\right) _{|B}(F^{-1})^D{}_b\right) _{|D}\Bigg ]\delta \varphi ^\perp \Bigg \}dAdt+\int _{t_0}^{t_1}\int _{\partial {\mathcal {H}}}\Bigg \{\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}\,\frac{\partial \,\delta \varphi ^\perp }{\partial X^A}{\mathsf {T}}_{B}\\&\quad -\Bigg [\rho {\mathfrak {L}}_b g^{ab}(F^{-1})^A{}_a+\left( \frac{\partial {\mathcal {L}}}{\partial \Theta _{CB}}F^a{}_C\right) _{|B}(F^{-1})^A{}_a\Bigg ]{\mathsf {T}}_{A}\delta \varphi ^\perp \\&\quad +2\Bigg [\frac{\partial {\mathcal {L}}}{\partial C_{AB}}F^a{}_Bg_{ac}+\frac{\partial {\mathcal {L}}}{\partial \Theta _{AB}}F^a{}_B\theta _{ac}\Bigg ]{\mathsf {T}}_{A}(\delta \varphi ^\top )^c\Bigg \}dL\,dt=0, \end{aligned} \end{aligned}$$

(B.4)

where ${{{\mathbf {\mathsf{{T}}}}}}$ is the outward vector field normal to the boundary curve $\partial {\mathcal {H}}$. Knowing that $\delta \varphi ^\top$, $\delta \varphi ^\perp$, and ${\mathbf {d}}(\delta \varphi ^\perp )$ are arbitrary, from (B.4) the Euler–Lagrange equations (3.21) along with the boundary conditions (3.22) are obtained.