Abstract
It is known that design of elastic cloaks is much more challenging than that of acoustic cloaks, cloaks of electromagnetic waves or scalar problems of anti-plane shear. In this paper, we address fully the fourth-order problem and develop a model of a broadband invisibility cloak for channelling flexural waves in thin plates around finite inclusions. We also discuss an option to employ efficiently an elastic pre-stress and body forces to achieve such a result. An asymptotic derivation provides a rigorous link between the model in question and elastic wave propagation in thin solids. This is discussed in detail to show connection with non-symmetric formulations in vector elasticity studied in earlier work.
Highlights
There is a theoretical and practical interest in wave cloaking in the context of metamaterials, as outlined in the publications [1,2,3,4,5,6,7,8]
The present paper addresses cloaking for flexural waves in Kirchhoff elastic plates
We present a detailed asymptotic analysis, which establishes a connection between the transformed equations for the fourth-order model of flexural waves and those for a vector problem of elasticity in thin solids
Summary
There is a theoretical and practical interest in wave cloaking in the context of metamaterials, as outlined in the publications [1,2,3,4,5,6,7,8]. We show that the governing equations are not invariant with respect to the radial “push-out” transformation [3, 22] This observation implies that the cloaking design procedure, well developed for acoustics, vibration of elastic membranes and anti-plane shear problems (see, for example, [23, 24]), does not apply to problems of flexural vibrations of elastic plates. The paper [19] has shown, for a model of a square cloak, that a formulation for flexural waves in a Kirchhoff plate, after the cloaking transformation, includes additional terms in the governing equation; these may represent in-plane body forces and pre-stress. The coefficients of the expansion (16), (17) are determined from the boundary and the interface conditions on the contour of the cloak
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