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# Transformation Cloaking in Elastic Plates

In this paper we formulate the problem of elastodynamic transformation cloaking for Kirchhoff-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 Kirchhoff-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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## Comments

## Plates and Shells

Dear Arash Yavari,

Golgoon is using in your cited paper plate elements and after he call them shell elements. The partial differential equation of plates is shown.

There are two theories for plates elements Kirchoff theory and Mindlin theory which includes the shear deformation energy effect. When the plate element has a great thickness, the Mindlin theory is used. The plate element is considered as a shell element when it has a small thickness and the theory of shell elements can also be used. Shell theory is developed with the midplane of the elements. Is there a relation between these two names in the cited developments ?

## Re: Plates and Shells

Dear Mohammed:

In Kirchhoff-Love plate theory shear deformations are ignored (this is a theory of thin plates). Assuming that energy of a shell depends on the first and the second fundamental forms of its mid surface one is implicitly ignoring shear deformations. To take into account shear deformations one would need to consider director fields at each point of the mid surface. Mindin (or Mindlin-Reissner) plate theory would be a special case of plates with director fields.

Arash

## Shells

Dear Arash,

Thank you for you comment. I am looking for the definition of the p.d.e in (r,s) natural coordinates (or r,s,t) of the curved shell midsurface. You have cited the p.d.e in (x,y) cartesian coordinate of a plane plate which is developed in its plane. There are envelopes theory and cylindric envelopes theory where these domains are known as shells in finite elements. The finite elements is an efficient method which uses interpolations functions and different methods are used to program plates and shells where curved domains can be approximated with several plane plates or with curved shells but this is a numerical method. In this method a plate with membrane effects (quadrilateral element) + bending effects (plate element) is equivalent to a shell elemnt with five to six degrees of freedom.

Mohammed Lamine

## Cloaking in plates

Dear Arash Yavari,

thank you for your interesting work.

Concerning your criticisms on our work concerning your eqs. (1.1) and (1.2), they were already solved in the paper I published in (2018), i.e. https://www.sciencedirect.com/science/article/pii/S0020768317305140?via%3Dihub

Transformed equations and interface/boundary conditions have been reported in Section 4.

In Section 5, we propose the eigenfrequency analysis as a tool in order to check the transformation.

There, higher order polynomials were implemented in order to respect interface conditions between untransformed and transformed domains.

Anyway, concerning the formulation reported in

1) D. J. Colquitt, M. Brun, M. Gei, A. B. Movchan, N. V. Movchan, and I. S. Jones. Transformation elastodynamics and cloaking for flexural waves. Journal of the Mechanics and Physics of Solids, 72:131–143, 2014.

2) M. Brun, D. Colquitt, I. Jones, A. Movchan, and N. Movchan. Transformation cloaking and radial approximations for flexural waves in elastic plates. New Journal of Physics, 16(9):093020, 2014.

and

3) I. Jones, M. Brun, N. Movchan, and A. Movchan. Singular perturbations and cloaking illusions for elastic waves in membranes and kirchhoff plates. International Journal of Solids and Structures, 69:498–506, 2015.

the comparison between untransformed and transformed domain show an excellent agreement apart from the neighbourhood of the interface between untransformed and transformed domains. The difference between solutions in the reference and transformed domains was considered quantitatively in (1) through the scattering measure.

Anyway, the difference is small.

May you guess why?

Best,

Michele Brun

## Re: Cloaking in plates

Dear Michele:

Thank you for your message. I was not aware of your PML paper and just read it. In Eq. (17) of your PML paper the transformed elastic constants are correct. You cited your older papers but I wonder why the discrepancy between the two transformed elastic constants was not discussed? The continuity conditions on the PML boundary (your Eq. (30)) are correct. However, this will not make your cloaking formulation in your previous papers work. There are still some constraints (coming from matching the different terms in the governing equations of the virtual and physical plates) that will ultimately force the cloaking map (or your "transformation" map) to be the identity. Exact cloaking is not possible. I am not claiming that approximate cloaking is not possible. However, it should be formulated properly as an optimization problem.

Regards,

Arash