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Transformation Cloaking in Elastic Plates

Arash_Yavari's picture

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

mohamedlamine's picture

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 ?

Arash_Yavari's picture

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

mohamedlamine's picture

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

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