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.

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Fig. 1

Notes

  1. 1.

    They, however, do not provide a clear mathematical reasoning as to why Colquitt et al. (2014)’s formulation is incorrect.

  2. 2.

    One should note that in formulating a cloaking problem one can either transform the action and then use Hamilton’s principle for the transformed action or simply transform the Euler–Lagrange equations if they are derived covariantly, i.e., the tensorial form of all quantities are retained and there is a clear distinction between the referential and spatial coordinates.

  3. 3.

    Note the typo in the expression for \(S_{I}\) in their Eq. (14).

  4. 4.

    On a first reading, the reader can skip Sects. 2 and 3 and go directly to Sect. 4.

  5. 5.

    A linear connection is said to be compatible with a metric \(\bar{{\mathbf {G}}}\) on the manifold provided that

    figurea

    where \(\left\langle \left\langle .,. \right\rangle \right\rangle _{\bar{{\mathbf {G}}}}\) is the inner product induced by the metric \(\bar{{\mathbf {G}}}\). It is straightforward to show that \({\bar{\nabla }}\) is compatible with \(\bar{{\mathbf {G}}}\) if and only if \({\bar{\nabla }}\bar{{\mathbf {G}}}={\mathbf {0}}\), or, in components

    $$\begin{aligned} {\bar{G}}_{AB|C}=\frac{\partial {\bar{G}}_{AB}}{\partial X^C}-{\bar{\Gamma }}^S{}_{CA}{\bar{G}}_{SB}-{\bar{\Gamma }}^S{}_{CB}{\bar{G}}_{AS}=0. \end{aligned}$$

    On any Riemannian manifold \(({\mathcal {B}},\bar{{\mathbf {G}}})\) the Levi–Civita connection is the unique linear connection \({\bar{\nabla }}^{\bar{{\mathbf {G}}}}\) that is compatible with \(\bar{{\mathbf {G}}}\) and is symmetric (torsion-free). Note that the metric compatibility of \({\bar{\nabla }}\) (see (2.4)) and the fact that \(\bar{{\mathbf {G}}}(\bar{{\mathbf {N}}},\bar{{\mathbf {Y}}})=0\) are used in deriving the second equality in (2.5).

  6. 6.

    Note that for \(X\in {\mathcal {H}}\), the metric (2.13) has the following representation

    $$\begin{aligned} \bar{{\mathbf {G}}}(X)=\begin{bmatrix} {\bar{G}}_{11}(X) &{} {\bar{G}}_{12}(X) &{} 0 \\ {\bar{G}}_{12}(X) &{} {\bar{G}}_{22}(X) &{} 0 \\ 0 &{} 0 &{} 1 \end{bmatrix}, \end{aligned}$$

    which using (2.10) and (2.16), implies (2.17).

  7. 7.

    Note that the first and the second fundamental forms of \({\mathcal {H}}\) can be expressed in terms of the metric of the embedding space \({\mathcal {B}}\) given by \(\left[ \begin{array}{cc}{\bar{G}}_{11}&{}{\bar{G}}_{12}\\ {\bar{G}}_{12}&{}{\bar{G}}_{22}\end{array} \right] (X)\), \(X\in {\mathcal {B}}\), which in turn, fully characterizes the geometry of \({\mathcal {H}}\).

  8. 8.

    See (Simo et al. 1988) for another equivalent way of characterizing the configuration space of a plate.

  9. 9.

    Note that in coordinates \((x^1,x^2,x^3)\), for which \(x^3\) is the outward normal direction, the second fundamental form of the deformed shell is expressed as

    $$\begin{aligned} \theta _{ab}=-\frac{1}{2}\frac{\partial {\bar{g}}_{ab}}{\partial x^3}\Big |_{\varphi ({\mathcal {H}})}(x), \quad a,b=1,2,\quad \forall x\in \varphi ({\mathcal {H}}). \end{aligned}$$
  10. 10.

    See (Ericksen and Truesdell 1957, P.313) for a discussion on the compatibility equations of a Cosserat shell with deformable directors.

  11. 11.

    Note that \(T_{(X,t)}\varphi \) (cf. (2.25)) is injective, and hence, the local extension vector field always exists, unless \({\mathbf {V}}\) is purely tangential, i.e., \({{\mathbf {V}}}^\perp ={\mathbf {0}}\). In this case, however, one does not need a local extension to compute the acceleration unambiguously as \({\mathbf {V}}={\mathbf {V}}^\top \), and hence, \({\mathbf {A}}(X,t)={\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{{\mathbf {V}}}{\mathbf {V}}={\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{{{\mathbf {V}}}^\top }{\mathbf {V}}^\top \).

  12. 12.

    Let \({\mathbf {W}}\in {\mathcal {X}}(\varphi _t({\mathcal {H}}))\) be an arbitrary vector field defined in a neighborhood containing \((X_o,t_o)\). Thus, \(\bar{{\mathbf {g}}}(\varvec{{\mathcal {V}}}^\perp ,{\mathbf {W}})=0\), and from (2.4), \(\bar{{\mathbf {g}}}({\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{\varvec{{\mathcal {V}}}}{\varvec{{\mathcal {V}}}}^\perp ,{\mathbf {W}})=-\bar{{\mathbf {g}}}({\varvec{{\mathcal {V}}}}^\perp ,{\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{\varvec{{\mathcal {V}}}}{\mathbf {W}})\). Note that

    $$\begin{aligned} \begin{aligned} {\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{\varvec{{\mathcal {V}}}}{\mathbf {W}}&=[\varvec{{\mathcal {V}}},{\mathbf {W}}]+{\bar{\nabla }}^{\bar{{\mathbf {g}}}} _{{\mathbf {W}}}\varvec{{\mathcal {V}}}=[\varvec{{\mathcal {V}}},{\mathbf {W}}] +{\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{{\mathbf {W}}}{\varvec{{\mathcal {V}}}}^\top +{\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{{\mathbf {W}}}{\varvec{{\mathcal {V}}}}^\perp \\&=[\varvec{{\mathcal {V}}},{\mathbf {W}}]+{\nabla }^{{\mathbf {g}}}_{{\mathbf {W}}}{{\varvec{{\mathcal {V}}}}}^\top +\varvec{\theta }({{\varvec{{\mathcal {V}}}}}^\top ,{\mathbf {W}}){\mathbf {n}}+(d{\mathcal {V}}^\perp \cdot {\mathbf {W}}){\mathbf {n}}+{\mathcal {V}}^\perp {{\bar{\nabla }}}^{\bar{{\mathbf {g}}}}_{{\mathbf {W}}}{\mathbf {n}}. \end{aligned} \end{aligned}$$

    Thus, noting that \([\varvec{{\mathcal {V}}},{\mathbf {W}}]\), \({\nabla }^{{\mathbf {g}}}_{{\mathbf {W}}}{{\varvec{{\mathcal {V}}}}}^\top \), \({\mathcal {V}}^\perp {{\bar{\nabla }}}^{\bar{{\mathbf {g}}}}_{{\mathbf {W}}}{\mathbf {n}}\in {\mathcal {X}}(\varphi _t({\mathcal {H}}))\) one concludes that \(\bar{{\mathbf {g}}}(\varvec{{\mathcal {V}}}^\perp ,{\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{\varvec{{\mathcal {V}}}}{\mathbf {W}})={\mathcal {V}}^\perp \varvec{\theta }({{\varvec{{\mathcal {V}}}}}^\top ,{\mathbf {W}})+{\mathcal {V}}^\perp (d{\mathcal {V}}^\perp \cdot {\mathbf {W}})\), which by arbitrariness of \({\mathbf {W}}\) together with \(\bar{{\mathbf {g}}}({\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{\varvec{{\mathcal {V}}}}{\varvec{{\mathcal {V}}}}^\perp ,{\mathbf {W}})=-\bar{{\mathbf {g}}}({\varvec{{\mathcal {V}}}}^\perp ,{\bar{\nabla }}^{\bar{{\mathbf {g}}}}_{\varvec{{\mathcal {V}}}}{\mathbf {W}})\) implies (2.34).

  13. 13.

    Consider a surface embedded in the ambient space such that the embedding is given as \(\varphi : {\mathcal {H}}\rightarrow {\mathcal {S}}\), where for the sake of simplicity one can assume that \({\mathcal {S}}={\mathbb {R}}^3\). The fundamental theorem of surface theory proved by Bonnet (1867) implies that the surface geometry (up to rigid body motions) is completely characterized by the induced first and second fundamental forms \({\mathbf {C}}\) and \({\varvec{\Theta }}\). Therefore, the surface energy density must depend on \({\mathbf {C}}\) and \(\varvec{\Theta }\).

  14. 14.

    \(D_{\varphi _{\epsilon }(X,t)}\) denotes the covariant derivative along the curve \(\varphi _{\epsilon }(X,t)\).

  15. 15.

    Note that (3.31) is equivalent to \(\sigma ^{ab}\) and \({\mathcal {M}}^{ab}\) being symmetric.

  16. 16.

    Note that in the local coordinate chart \(\{x^1,x^2,x^3\}\), for which \(x^3\) is the outward normal direction

    $$\begin{aligned} \theta _{ab}=-\frac{1}{2}\frac{\partial {\bar{g}}_{ab}}{\partial x^3}\Big |_{\varphi ({\mathcal {H}})},\quad a,b=1,2. \end{aligned}$$

    Thus,

    $$\begin{aligned} (\psi _{s*}\theta )_{ a' b'}= & {} -\frac{1}{2}\frac{\partial \left( (T\psi _s^{-1})^a{}_{ a'}(T\psi _s^{-1})^b{}_{ b'}{\bar{g}}_{ab}\right) }{\partial x^3}=(T\psi _s^{-1})^a{}_{ a'}(T\psi _s^{-1})^b{}_{ b'}\left( -\frac{1}{2}\frac{\partial {\bar{g}}_{ab}}{\partial x^3}\right) \\= & {} (T\psi _s^{-1})^a{}_{ a'}(T\psi _s^{-1})^b{}_{ b'}\theta _{ab},\quad a',b'=1,2. \end{aligned}$$

    Note also that \((T\psi ^{-1}_{s})^{a}{}_{a'}=-\delta ^b{}_{a'}\delta ^a{}_{b'}(T\psi _{s})^{b'}{}_b\).

  17. 17.

    Note that \({\dot{\varrho }}=\frac{\partial \varrho }{\partial t}+\nabla \varrho \cdot {\mathbf {v}}\).

  18. 18.

    Note that we do not explicitly specify the source of the initial stress or couple-stress. If the initial stress and couple-stress are due to elastic deformations, and the body has an energy function W with respect to its stress-free configuration, then one may express \(\mathring{{\mathbf {P}}}\) and \(\mathring{{{{\mathbf {\mathsf{{M}}}}}}}\) as

    $$\begin{aligned} \mathring{{\mathbf {P}}}=2\mathring{{\mathbf {F}}}\frac{\partial W}{\partial {\mathbf {C}}}\Big |_{\mathring{{\mathbf {F}}}}, \quad \mathring{{{{\mathbf {\mathsf{{M}}}}}}}=\mathring{{\mathbf {F}}} \frac{\partial W}{\partial {\varvec{\Theta }}}\Big |_{\mathring{{\mathbf {F}}}}. \end{aligned}$$
  19. 19.

    Note that the variation of the normal vector \(\delta \varvec{{\mathcal {N}}}\) is purely tangential, and hence, so is the term \({\bar{{\mathbf {g}}}}(\mathring{\varvec{{\mathfrak {B}}}},\varvec{{\mathcal {N}}})\delta \varvec{{\mathcal {N}}}\) in (3.76). In (3.77), with an abuse of notation we only consider the term in the normal direction.

  20. 20.

    Note that for cloaking applications considered in this paper \({\tilde{\varphi }}_t=id\), i.e., the identity map, and \(\varphi _t\) is time-independent but is not necessarily the identity map as we assume a time-independent pre-stress distribution for the cloak.

  21. 21.

    The Piola transformation of a vector (field) \({\mathbf {w}}\in T_{\varphi (X)}{\mathcal {S}}\) is a vector \({\mathbf {W}}\in T_X{\mathcal {B}}\) given by \({\mathbf {W}}=J\varphi ^*{\mathbf {w}}=J{\mathbf {F}}^{-1}{\mathbf {w}}\). In coordinates, one has \(W^A=J(F^{-1})^A{}_bw^b\), where \(J=\sqrt{\frac{\det {\mathbf {g}}}{\det {\mathbf {G}}}}\det {\mathbf {F}}\) is the Jacobian of \(\varphi \) with \({\mathbf {G}}\) and \({\mathbf {g}}\) the Riemannian metrics of \({\mathcal {B}}\) and \({\mathcal {S}}\), respectively. Note that a Piola transformation can be performed on any index of a given tensor. One can show that \({\text {Div}}{\mathbf {W}}=J({\text {div}}{\mathbf {w}})\circ \varphi \), which in coordinates is written as \(W^A{}_{|A}=Jw^a{}_{|a}\). This is also known as the Piola identity. Another way of writing the Piola identity is in terms of the unit normal vectors of a surface in \({\mathcal {B}}\) and its corresponding surface in \({\mathcal {S}}\), along with the area elements. It is written as \(\hat{{\mathbf {n}}}da=J{\mathbf {F}}^{-\star }\hat{{\mathbf {N}}}dA\), or in components, one writes \(n_ada=J(F^{-1})^A{}_aN_AdA\). In the literature of continuum mechanics, this is known as Nanson’s formula.

  22. 22.

    Note that for both plates we may use two global collinear Cartesian coordinates \(\{z^1,z^2,z^3\}\) and \(\{{\tilde{z}}^1,{\tilde{z}}^2,{\tilde{z}}^3\}\) such that \(z^3\) and \({\tilde{z}}^3\) are the outward normal directions to the physical and virtual plates, respectively. Therefore, \(\{z^1,z^2\}\) and \(\{{\tilde{z}}^1,{\tilde{z}}^2\}\) are two global collinear Cartesian coordinates for \(\varphi ({\mathcal {H}})\) and \({\tilde{\varphi }}(\tilde{{\mathcal {H}}})\), respectively, where \(\Xi :{\mathcal {H}}\rightarrow \tilde{{\mathcal {H}}}\), \(\varphi :{\mathcal {H}}\rightarrow \varphi ({\mathcal {H}})\), \({\tilde{\varphi }}:\tilde{{\mathcal {H}}}\rightarrow {\tilde{\varphi }}(\tilde{{\mathcal {H}}})\), and \(\xi :\varphi ({\mathcal {H}})\rightarrow {\tilde{\varphi }}(\tilde{{\mathcal {H}}})\), and \({{{\mathbf {\mathsf{{s}}}}}}\) is defined as

    $$\begin{aligned} {\mathsf {s}}^{{\tilde{a}}}{}_a(x)=\frac{\partial {\tilde{x}}^{{\tilde{a}}}}{\partial {\tilde{z}}^{{\tilde{i}}}}({\tilde{x}}) \frac{\partial z^i}{\partial x^a}(x) \delta ^{{\tilde{i}}}_i,~~~a,{\tilde{a}},i,{\tilde{i}}=1,2 , \end{aligned}$$

    where \(\{x^a\}\) and \(\{{\tilde{x}}^{{{\tilde{a}}}}\}\) are local coordinate charts for \(\varphi ({\mathcal {H}})\) and \({\tilde{\varphi }}(\tilde{{\mathcal {H}}})\), respectively.

  23. 23.

    See Remark. 3.1 for a discussion on how the boundary surface traction, boundary shear force, and boundary moment as well as their corresponding Dirichlet boundary conditions are prescribed in the boundary-value problem.

  24. 24.

    See (Fox et al. 1993; Le Dret and Raoult 1993; Lods and Miara 1998; Friesecke et al. 2002) for details on the derivation of the Saint Venant–Kirchhoff shell constitutive model.

  25. 25.

    Note that introducing the scalar field \(k=k(X)\) provides an extra degree of freedom in the cloaking problem. It has to be positive because \(\rho =k{\tilde{\rho }}\circ \Xi \).

  26. 26.

    The covariant derivative of a two-point tensor \({\mathbf {T}}\) is given by

    $$\begin{aligned} \begin{aligned} T^{AB\cdots F}{}_{G\cdots Q}{}^{ab\cdots f}{}_{g\cdots q|K}&=\frac{\partial }{\partial X^K}T^{AB\cdots F}{}_{G\cdots Q}{}^{ab\cdots f}{}_{g\cdots q}\\&\quad +T^{RB\cdots F}{}_{G\cdots Q}{}^{ab\cdots f}{}_{g\cdots q}\Gamma ^A{}_{RK}+\mathrm {(all\,\,upper\,\, referential\,\, indices)}\\&\quad -T^{AB\cdots F}{}_{R\cdots Q}{}^{ab\cdots f}{}_{g\cdots q}\Gamma ^R{}_{GK}-\mathrm {(all\,\,lower\,\, referential\,\, indices)}\\&\quad +T^{AB\cdots F}{}_{G\cdots Q}{}^{lb\cdots f}{}_{g\cdots q}\gamma ^a{}_{lr}F^r{}_K+\mathrm {(all\,\,upper\,\, spatial\,\, indices)}\\&\quad -T^{AB\cdots F}{}_{G\cdots Q}{}^{ab\cdots f}{}_{l\cdots q}\gamma ^l{}_{gr}F^r{}_K-\mathrm {(all\,\,lower\,\, spatial\,\, indices)}. \end{aligned} \end{aligned}$$
  27. 27.

    Pomot et al. (2019) used a linear cloaking transformation, which has a covarianlty constant tangent map. However, a linear cloaking map does not satisfy the required traction boundary condition on \(\partial _o{\mathcal {C}}\), i.e., , and therefore, using a linear cloaking map is not acceptable (see Brun et al. 2009 for another improper use of this type of mapping, which does not fix the outer boundary of the cloak to the first order as is required in elastodynamic cloaking.).

  28. 28.

    Note that the physical components of the flexural rigidity tensor are given by \(\hat{\tilde{{\mathbb {C}}}}^{{\tilde{a}}{\tilde{A}}{\tilde{b}}{\tilde{B}}}=\sqrt{{\tilde{g}}_{{{\tilde{a}}}{{\tilde{a}}}}}\sqrt{{\tilde{G}}_{{{\tilde{A}}}{{\tilde{A}}}}}\sqrt{{\tilde{g}}_{{{\tilde{b}}}{{\tilde{b}}}}}\sqrt{{\tilde{G}}_{{{\tilde{B}}}{{\tilde{B}}}}}\tilde{{\mathbb {C}}}^{{{\tilde{a}}}{{\tilde{A}}}{{\tilde{b}}}{{\tilde{B}}}}\) (no summation).

  29. 29.

    Note that

    $$\begin{aligned} \widetilde{{\mathsf {W}}}_{{{\tilde{A}}}{{\tilde{B}}}{{\tilde{C}}} {{\tilde{D}}}}=\frac{\partial ^4\widetilde{{\mathsf {W}}}}{\partial {\tilde{X}}^{{{\tilde{A}}}}\partial {\tilde{X}}^{{{\tilde{B}}}} \partial {\tilde{X}}^{{{\tilde{C}}}}\partial {\tilde{X}}^{{{\tilde{D}}}}}. \end{aligned}$$
  30. 30.

    In a general curvilinear coordinate system, the biharmonic term is given by

    $$\begin{aligned} {\widetilde{\nabla }}^4\widetilde{{\mathsf {W}}}= \frac{1}{\sqrt{\det \tilde{{\mathbf {G}}}}}\frac{\partial }{\partial {\tilde{X}}^{{{\tilde{A}}}}}\left[ \sqrt{\det \tilde{{\mathbf {G}}}} \frac{\partial }{\partial {\tilde{X}}^{{{\tilde{B}}}}} \left( \frac{1}{\sqrt{\det \tilde{{\mathbf {G}}}}}\frac{\partial }{\partial {\tilde{X}}^{{{\tilde{C}}}}}\left[ \sqrt{\det \tilde{{\mathbf {G}}}} \frac{\partial \widetilde{{\mathsf {W}}}}{\partial {\tilde{X}}^{{{\tilde{D}}}}}{\tilde{G}}^{{{\tilde{C}}}{{\tilde{D}}}}\right] \right) {\tilde{G}}^{{{\tilde{A}}}{{\tilde{B}}}}\right] . \end{aligned}$$
  31. 31.

    This assumption ensures that the second term in (4.47) remains form invariant under the cloaking map.

  32. 32.

    Note that (4.62) implies that \(\left( J_{\Xi }-1\right) {\widetilde{\nabla }}^2\widetilde{{\mathsf {W}}}=C\), where C is a constant. Recalling that on the outer boundary of the cloak , and thus, \(J_{\Xi }|_{\partial _o{\mathcal {C}}}=1\), and given that \(\widetilde{{\mathsf {W}}}\) is smooth, one concludes that \(C=0\). Therefore, \(J_{\Xi }=1\).

  33. 33.

    Notice that (4.69) can be represented as a rotation matrix multiplied by the scalar \((\alpha ^2+\beta ^2)\).

  34. 34.

    Note that if one defines the complex function \(f({\tilde{X}}+i{\tilde{Y}})=\beta ({\tilde{X}},{\tilde{Y}})+i\alpha ({\tilde{X}},{\tilde{Y}})\), then (4.70) are the Cauchy-Riemann equations, and hence, the complex function f is holomorphic.

  35. 35.

    Note that \(\mathring{\tilde{{\mathbf {F}}}}=id\), while \(\mathring{{\mathbf {F}}}\) is not the identity, in general. However, for thin plates (due to the inextensibility constraint) \(\mathring{{\mathbf {F}}}=id\), and thus, the mappings \(\xi \) and \(\Xi \) are identical, whence (4.80) reduces to (4.17) with a slight abuse of notation.

  36. 36.

    Conservation of mass for the physical and virtual plates implies that \(\rho _0=\varrho \mathring{J}\) and \({\tilde{\rho }}_0={\tilde{\varrho }}\mathring{{\tilde{J}}}\). Noting that \(\mathring{{\tilde{\varphi }}}=id\), and \(J_{\xi }=\mathring{{\tilde{J}}}J_{\Xi }\mathring{J}^{-1}\), the spatial mass density of the cloak is given by \(\varrho =J_{\xi }{\tilde{\rho }}_0\).

  37. 37.

    Note that

    where \({\tilde{\gamma }}^{{{\tilde{c}}}}{}_{{{\tilde{a}}}{{\tilde{b}}}}\) are the (induced) Christoffel symbols corresponding to the virtual plate in its current configuration.

  38. 38.

    One starts from the governing equations of the virtual plate and substitutes the derivatives with their corresponding transformed derivatives and compares the coefficients of the different derivatives in the transformed governing equations with those in the physical plate. This overdetermined system of equations gives all the transformed fields, and a set of cloaking compatibility equations.

  39. 39.

    Note that , implies that \(\mathring{\varvec{{\mathfrak {L}}}}|_{\partial _o{\mathcal {C}}}={\mathbf {0}}\), and \(\delta \varvec{{\mathfrak {L}}}|_{\partial _o{\mathcal {C}}}={\mathbf {0}}\), (see (4.13) and (4.82d)), and thus,

    where , and the fact that \(\Xi \) (and \(\xi \)) fixes the boundary of the cloak \(\partial _o{\mathcal {C}}\) to the third order were used.

  40. 40.

    Note that in the case of a general isotropic energy function for the virtual plate \({\pmb {{\mathbb {B}}}}\), \(\mathring{\varvec{{\mathfrak {L}}}}\), and \(\mathring{\varvec{{\mathfrak {B}}}}^\perp \) do not vanish and their expressions are given in Remark. 4.16.

  41. 41.

    Note that (4.108) is identical to (4.37).

  42. 42.

    Note that \({\mathsf {F}}_{12}\ne 0\), because otherwise, \(\tilde{{\mathsf {b}}}_1=-2\tilde{{\mathsf {b}}}_2\) (from (4.121)), contradicting the positive definiteness of \(\tilde{\pmb {{\mathbb {B}}}}\).

  43. 43.

    Note that \(\varvec{\eta }\) and \(\varvec{\Lambda }\), respectively, correspond to \({\mathbf {C}}\) and \({\varvec{\Theta }}\) defined previously.

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Acknowledgements

This research was supported by ARO W911NF-18-1-0003 (Drs. Daniel P. Cole and David Stepp). A.G. benefited from discussions with Fabio Sozio, Arzhang Angoshtari, Amirhossein Tajdini, and Souhayl Sadik.

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Dedicated to Professor J.N. Reddy on the occasion of his 75th birthday.

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Communicated by Robert V. Kohn.

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 \(\Lambda _{\alpha \beta }\).Footnote 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.

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Golgoon, A., Yavari, A. Transformation Cloaking in Elastic Plates. J Nonlinear Sci 31, 17 (2021). https://doi.org/10.1007/s00332-020-09660-7

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Keywords

  • Cloaking
  • Nonlinear elasticity
  • Elastic plates
  • Elastic waves

Mathematics Subject Classification

  • 74B20
  • 74K20
  • 74J25
  • 35R30
  • 70G45
  • 53Z30