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Elastodynamic transformation cloaking for non-centrosymmetric gradient solids

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Abstract

In this paper, we investigate the possibility of elastodynamic transformation cloaking in bodies made of non-centrosymmetric gradient solids. The goal of transformation cloaking is to hide a hole from elastic disturbances in the sense that the mechanical response of a homogeneous and isotropic body with a hole covered by a cloak would be identical to that of the corresponding homogeneous and isotropic body outside the cloak. It is known that in the case of centrosymmetric gradient solids exact transformation cloaking is not possible; the balance of angular momentum is the obstruction to transformation cloaking. We will show that this no-go theorem holds for non-centrosymmetric gradient solids as well.

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Notes

  1. In Sect. 3.1, in linearized gradient elasticity, we will use \({\mathbf {U}}\) for the linearized displacement instead of \(\delta {\mathbf {U}}\).

  2. Once the balance of angular momentum is enforced, both coupling elasticity tensors \({\mathbb {B}}\) and \({\mathbb {L}}\) have 108 independent components in the most general case. In [31], it was mentioned that \({\mathbb {B}}\) has 90 independent components, which is incorrect. However, this inaccurate statement did not affect any of the results or conclusions of that work.

  3. Note that this tensor does not have any major symmetries; the symmetries claimed in Eq. (3.2)\(_2\) in [7] are incorrect. As a matter of fact, from the representation (3.20) in the isotropic case, the coupling elasticity tensor \({\mathbb {L}}\) has the following major antisymmetry: \({\mathbb {L}}^{IJKLM} = -{\mathbb {L}}^{KLIJM}\). From (3.17), \({\mathbb {B}}\) has the same property in the isotropic case.

  4. This is identical to the corresponding relation for centrosymmetric gradient solids investigated in [31].

  5. Symbolic computations were done with Mathematica Version 12.0.0.0, Wolfram Research, Champaign, IL.

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Acknowledgements

This research was supported by ARO W911NF-18-1-0003 (Dr. Daniel P. Cole) and NSF—Grant No. CMMI 1561578, CMMI 1939901.

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Fabio Sozio & Arash Yavari

  2. Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208, USA

    Ashkan Golgoon

  3. The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Arash Yavari

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  1. Fabio Sozio
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  2. Ashkan Golgoon
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  3. Arash Yavari
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Correspondence to Arash Yavari.

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Sozio, F., Golgoon, A. & Yavari, A. Elastodynamic transformation cloaking for non-centrosymmetric gradient solids. Z. Angew. Math. Phys. 72, 123 (2021). https://doi.org/10.1007/s00033-021-01555-1

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  • DOI: https://doi.org/10.1007/s00033-021-01555-1

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