Transformation elastodynamics and cloaking for flexural waves
Graphical abstract
Introduction
Transformation optics, as introduced by Leonhardt (2006) and Pendry et al. (2006), has proven to be a useful tool in the design and fabrication of invisibility cloaks for electromagnetic waves (see Schurig et al., 2006, Landy and Smith, 2013, Ergin et al., 2011, Chen and Zheng, 2012, among many others). Prior to cloaking, the ideas of transformation optics were used as a computational tool (Ward and Pendry, 1996) to aid in simulations involving several length scales or complex geometries. The cornerstone of transformation optics is the invariance of Maxwell׳s equations under coordinate transformations (Post, 1962). In essence, the design of invisibility cloaks via transformation optics involves deforming a region of space such that a disc is mapped to an annulus. Electromagnetic waves then propagate around the annulus as if it were a disc and, in this sense, any object placed inside the inner annulus is said to be invisible. The ideas behind transformation optics have also been successfully applied to systems that are governed by equations isomorphic to Maxwell׳s equations, such as acoustics (Chen and Chan, 2007, Cummer and Schurig, 2007, Norris, 2008, Chen and Chan, 2010), thermodynamics (Guenneau et al., 2012, Schittny et al., 2013, Han et al., 2013), and out-of-plane elastic waves (Parnell et al., 2012, Parnell and Shearer, 2013, Colquitt et al., 2013). Metamaterials have also found broad application in a wide range of physical settings (Kadic et al., 2013).
However, in general, the partial differential equations governing physical systems are not invariant under coordinate transformations. In particular, the elastodynamic wave equation is not invariant under a general mapping (Milton et al., 2006, Norris and Shuvalov, 2011). Norris and Shuvalov (2011) showed that the transformed equation requires either non-symmetric stress or tensorial density. However, Milton et al. (2006) demonstrated that, for a particular choice of gauge, a more general constitutive model (the so-called Willis model Milton and Willis, 2007) remains invariant under an arbitrary coordinate transformation yielding symmetric stress but tensorial density. Brun et al. (2009) applied a cloaking transformation to the Navier equations for isotropic linear elasticity and found that the transformed equations correspond to non-symmetric constitutive relations where only the major symmetry is preserved. More recently Norris and Parnell (2012) showed that it is possible to obtain the constitutive equations required for elastodynamic cloaking by the application of a finite pre-strain. However, using this approach there is a limit on the relative size of the cloaked region and regularisation parameter used to create the near cloaks.
Transformation elastodynamics has also been used in the design of invisibility cloaks for flexural waves in thin elastic plates (Farhat et al., 2009a, Farhat et al., 2009b) with corresponding experiments recently performed by Stenger et al. (2012). The equations governing the motion of flexural waves in thin plates are not, in general, invariant under coordinate transformations. Nevertheless, Farhat et al. (2009a) found that the transformed equation can be identified with a thin plate if one assumes strains of the von-Kármán form (see, for example, Timoshenko and Woinowsky-Krieger, 1959, Lekhnitskii et al., 1968). However, as demonstrated by Ciarlet (1980) and Blanchard and Ciarlet (1983) the theory of von-Kármán corresponds to the leading order behaviour of a thin elastic plate under moderate deformation provided that the applied loads and/or the elastic moduli have a specific dependence on the thickness of the plate.
The present paper is concerned with the construction of a transformation theory for the dynamic equations of flexural deformations in Kirchhoff–Love plates. In contrast to the elastodynamic case (Milton et al., 2006, Norris and Shuvalov, 2011, Norris and Parnell, 2012, Brun et al., 2009) and previous work on plates (Farhat et al., 2009a, Farhat et al., 2009b), it is shown that it is possible to construct an invisibility cloak for flexural waves in thin plates without recourse to non-symmetric stresses, tensorial densities, or non-linear theories. In particular, it is shown that by the application of appropriate in-plane forces it is possible to construct an exact invisibility cloak for flexural waves within the framework of linear Kirchhoff–Love plate theory. This result could lead to a refinement of the experimental implementation of invisibility cloaks for flexural waves yielding an improvement of the experimental results reported in Stenger et al. (2012).
The structure of the paper is as follows. After the introduction, a formal framework for transformation elastodynamics applied to Kirchhoff–Love plates is developed. The material parameters in the transformed system are given explicitly in terms of the Jacobi matrix of the map. In Section 2, a general coordinate transformation is applied to the equation, which governs the flexural displacement of a Kirchhoff–Love plate. It is demonstrated that, in general, the bi-harmonic operator is not invariant under a cloaking transformation. However in Section 2.1, it is shown that a sensible physical interpretation can be given to the transformed equation corresponding to a pre-stressed linear anisotropic inhomogeneous Kirchhoff–Love plate. The natural and essential interface conditions are discussed in Section 2.2. A regularised cloaking push-out transformation (Colquitt et al., 2013) is discussed in Section 3. The material parameters of the cloak and the applied pre-stresses are given explicitly in Section 3.1, where it is shown that it is possible to reduce the fully anisotropic plate to a locally orthotropic plate. A series of illustrative simulations are presented in Section 3.2, along with a comparison with the corresponding problem for the membrane in Section 3.2.1. The efficiency of the cloak is examined in Section 3.2.2 while the quality of the cloaking effect is demonstrated using a delicate interference pattern in Section 3.2.3. The paper concludes with some remarks in Section 4.
Section snippets
Transforming Kirchhoff–Love plates
In the absence of applied in-plane forces, the equation governing the time-harmonic out-of-plane displacement amplitude of an isotropic homogeneous Kirchhoff–Love plate under pure bending is (Timoshenko and Woinowsky-Krieger, 1959, Lekhnitskii et al., 1968)where , , and h are the flexural rigidity, density, and thickness of the plate, respectively, and ω is the radian frequency. Consider an invertible transformation and . By a double application
A push-out transformation: the square cloak
This section is devoted to the construction of a square cloak for flexural waves in a Kirchhoff–Love plate. Square cloaks have already been constructed for electromagnetic (Rahm et al., 2008) and out-of-plane elastic (Colquitt et al., 2013) waves; the cloak presented here is based on the coordinate transformation as in Colquitt et al. (2013). Geometrically, the coordinate transformation deforms a small square together with the surrounding four trapezoids χ(i), into a
Concluding remarks
A formal framework for transformation elastodynamics as applied to Kirchhoff–Love plates has been developed. The material properties of the transformed system and the applied pre-stresses and body loads are given explicitly in terms of the deformation gradient in Eq. (9a), (9b), (9c), (9d). It has been demonstrated that the bi-harmonic equation governing the flexural deformation of a linear homogeneous and isotropic Kirchhoff–Love plate is not invariant under a general coordinate mapping.
Acknowledgements
D.J.C., M.G., A.B.M. and N.V.M. acknowledge the financial support of the European Community׳s Seven Framework Programme under contract number PIAP-GA-2011-286110-INTERCER2. M.B. acknowledges the financial support of the European Community׳s Seven Framework Programme under contract number PIEF-GA-2011-302357-DYNAMETA and of Regione Autonoma della Sardegna (LR7 2010, Grant ‘M4’ CRP-27585). D.J.C. also acknowledges the financial support of EPSRC in the form of a Doctoral Prize Fellowship and grant
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