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Nonlinear and Linear Elastodynamics Transformation Cloaking
Soft Condensed Matter · Submitted 27 Jul 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that transformation cloaking is not possible regardless of the shape of the hole and the cloak.
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The abstract, sentence by sentence
We formulate the problems of nonlinear and linear elastodynamics transformation cloaking in a geometric framework. It is noted that a cloaking transformation is neither a spatial nor a referential change of frame (coordinates); a cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic body (virtual problem) to that of an anisotropic and inhomogeneous elastic body with a hole surrounded by a cloak that is to be designed (physical problem). The virtual body has a desired mechanical response while the physical body is designed to mimic the same response outside the cloak using a cloaking transformation. We show that nonlinear elastodynamics transformation cloaking is not possible while nonlinear elastostatics cloaking may be possible for special deformations, e.g., radial deformations in a body with either a cylindrical or a spherical cavity. For linear elastodynamics, in line with the previous observations in the literature, we show that the elastic constants in the cloak are not fully symmetric; they do not possess the minor symmetries. We prove that transformation cloaking is not possible regardless of the shape of the hole and the cloak. We next show that linear elastodynamics transformation cloaking cannot be achieved for gradient elastic solids either; similar to classical linear elasticity the balance of angular momentum is the obstruction to transformation cloaking. We finally prove that transformation cloaking is not possible for linear elastic generalized Cosserat solids in 2D for any shape of the hole and the cloak. Particularly, in 2D, transformation cloaking cannot be achieved in linear Cosserat elasticity. We also show that transformation cloaking for a spherical cavity covered by a spherical cloak is not possible in the setting of linear elastic generalized Cosserat solids. We conjecture that this result holds for a cavity of any shape.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We formulate the problems of nonlinear and linear elastodynamics transformation cloaking in a geometric framework. | contribution 0.85 | 0.91 | 0.04 | asserted 0.98 |
| s1 | It is noted that a cloaking transformation is neither a spatial nor a referential change of frame (coordinates); a cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic body (virtual problem) to that of an anisotropic and inhomogeneous elastic body with a hole surrounded by a cloak that is to be designed (physical problem). | background 0.34 | 0.51 | 0.06 | asserted 1.00 |
| s2 | The virtual body has a desired mechanical response while the physical body is designed to mimic the same response outside the cloak using a cloaking transformation. | background 0.55 | 0.26 | 0.05 | asserted 0.99 |
| s3 | We show that nonlinear elastodynamics transformation cloaking is not possible while nonlinear elastostatics cloaking may be possible for special deformations, e.g., radial deformations in a body with either a cylindrical or a spherical cavity. | result 1.00 | 0.88 | 0.25 | asserted 0.72 |
| s4 | For linear elastodynamics, in line with the previous observations in the literature, we show that the elastic constants in the cloak are not fully symmetric; they do not possess the minor symmetries. | result 0.99 | 0.79 | 0.08 | asserted 1.00 |
| s5 | We prove that transformation cloaking is not possible regardless of the shape of the hole and the cloak. | result 1.00 | 0.92 | 0.48 | asserted 1.00 |
| s6 | We next show that linear elastodynamics transformation cloaking cannot be achieved for gradient elastic solids either; similar to classical linear elasticity the balance of angular momentum is the obstruction to transformation cloaking. | result 0.98 | 0.89 | 0.12 | asserted 1.00 |
| s7 | We finally prove that transformation cloaking is not possible for linear elastic generalized Cosserat solids in 2D for any shape of the hole and the cloak. | result 0.99 | 0.93 | 0.24 | asserted 1.00 |
| s8 | Particularly, in 2D, transformation cloaking cannot be achieved in linear Cosserat elasticity. | result 0.99 | 0.81 | 0.14 | asserted 1.00 |
| s9 | We also show that transformation cloaking for a spherical cavity covered by a spherical cloak is not possible in the setting of linear elastic generalized Cosserat solids. | result 0.99 | 0.91 | 0.10 | asserted 1.00 |
| s10 | We conjecture that this result holds for a cavity of any shape. | future work 0.56 | 0.58 | 0.09 | speculative 1.00 |
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Original abstract
We formulate the problems of nonlinear and linear elastodynamics transformation cloaking in a geometric framework. It is noted that a cloaking transformation is neither a spatial nor a referential change of frame (coordinates); a cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic body (virtual problem) to that of an anisotropic and inhomogeneous elastic body with a hole surrounded by a cloak that is to be designed (physical problem). The virtual body has a desired mechanical response while the physical body is designed to mimic the same response outside the cloak using a cloaking transformation. We show that nonlinear elastodynamics transformation cloaking is not possible while nonlinear elastostatics cloaking may be possible for special deformations, e.g., radial deformations in a body with either a cylindrical or a spherical cavity. For linear elastodynamics, in line with the previous observations in the literature, we show that the elastic constants in the cloak are not fully symmetric; they do not possess the minor symmetries. We prove that transformation cloaking is not possible regardless of the shape of the hole and the cloak. We next show that linear elastodynamics transformation cloaking cannot be achieved for gradient elastic solids either; similar to classical linear elasticity the balance of angular momentum is the obstruction to transformation cloaking. We finally prove that transformation cloaking is not possible for linear elastic generalized Cosserat solids in 2D for any shape of the hole and the cloak. Particularly, in 2D, transformation cloaking cannot be achieved in linear Cosserat elasticity. We also show that transformation cloaking for a spherical cavity covered by a spherical cloak is not possible in the setting of linear elastic generalized Cosserat solids. We conjecture that this result holds for a cavity of any shape.
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