Abstract
A relevant application of transformation elastodynamics has shown that flexural waves in a Kirchhoff-Love plate can be diverted and channeled to cloak a region of the ambient space. To achieve the goal, an orthotropic meta-structural plate should be employed. However, the corresponding mathematical transformation leads to the presence of an unwanted strong compressive prestress, likely beyond the buckling threshold of the structure, with a set of in-plane body forces to warrant equilibrium. In addition, the plate must possess, at the same time, high bending stiffnesses, but a null twisting rigidity. With the aim of estimating the performance of cloaks modelled with approximate parameters, an in-house finite element code, based on a subparametric technique, is implemented to deal with the cloaking of transient waves in orthotropic thin plates. The tool allows us to explore the sensitivity of specific stiffness parameters that may be difficult to match in a real cloak design. In addition, the finite element code is extended to investigate a meta-plate interacting with a Winkler foundation, to confirm how the subgrade modulus should transform in the cloak region.
Highlights
The control of elastic waves to cloak a region of the ambient space has been shown achievable by transformation elastodynamics, which provides the mechanical properties of the material surrounding the region (Milton et al, 2006; Brun et al, 2009; Norris and Shuvalov, 2011)
A relevant application of this broad area is the control of transverse waves in plates, for which solutions have been proposed mainly based on two approaches: a “passive” one, where the features of the cloak are achieved by a given microstructure (Farhat et al, 2009a; Farhat et al, 2009b; Brun et al, 2014; Colquitt et al, 2014; Climente et al, 2016; Zareei and Alam, 2017; Liu and Zhu, 2019; Misseroni et al, 2019) and an “active” one, in which tunable quantities depending on the actual mechanical input are employed for the same goal
The definition of the flexural cloak is based on a transformation, say χ, that maps a two-dimensional subdomain C0 ⊂ K occupied by a homogeneous, isotropic Kirchhoff-Love plate, to the domain C where the meta-plate is to be constructed, i.e., χk : C0 → C, xK0 → xk χk(xK0 ).1. The map is such that the transformed equation in the latter is still the governing equation for flexural waves supported by a Kirchhoff-Love plate
Summary
State of the Art and Research ChallengesThe control of elastic waves to cloak a region of the ambient space has been shown achievable by transformation elastodynamics, which provides the mechanical properties of the material surrounding the region (Milton et al, 2006; Brun et al, 2009; Norris and Shuvalov, 2011). The general framework was specialised to the case of a square cloak composed of four trapezoidal elements (see Figure 1) embedded in an isotropic, homogeneous domain. For this geometry, a set of relationships was established to provide explicit expressions of the quantities concerned in each part of the cloak. The definition of the flexural cloak is based on a transformation, say χ, that maps a two-dimensional subdomain C0 ⊂ K occupied by a homogeneous, isotropic Kirchhoff-Love plate, to the domain C where the meta-plate is to be constructed, i.e., χk : C0 → C, xK0 → xk χk(xK0 ).. When free-standing, the governing equation of the plate under harmonic vibrations takes the form
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