On cloaking for elasticity and physical equations with a transformation invariant form

By Graeme W. Milton, Marc Briane, J. Willis. 2006

991 citations recorded by the source corpus.

In this paper, we investigate how the form of the conventional elastodynamic equations changes under curvilinear transformations. The equations get mapped to a more general form in which the density is anisotropic and additional terms appear which couple the stress not only with the strain but also with the velocity, and the momentum gets coupled not only with the velocity but also with the strain. These are a special case of equations which describe the elastodynamic response of composite materials, and which it has been argued should apply to any material which has microstructure below the scale of continuum modelling. If composites could be designed with the required moduli then it could be possible to design elastic cloaking devices where an object is cloaked from elastic waves of a given frequency. To an outside observer it would appear as though the waves were propagating in a homogeneous medium, with the object and surrounding cloaking shell invisible. Other new elastodynamic equations also retain their form under curvilinear transformations. The question is raised as to whether all equations of microstructured continua have a form which is invariant under curvilinear space or space-time coordinate transformations. We show that the non-local bianisotropic electrodynamic equations have this invariance under space-time transformations and that the standard non-local, time-harmonic, electromagnetic equations are invariant under space transformations.

Subfields: Biomedical Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/32fde349e701a713794007a001009449b840c2c9

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In Biomedical Engineering

In Mechanics of Materials

Citations in the corpus

Mechanical metamaterials associated with stiffness, rigidity and compressibility: a brief review (2017, background): “Mathematically speaking, the mechanical metamaterials associated with shear/bulk moduli include three regions of the Milton map (Fig.”
Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials (2015, background): “The damping effects can be also suitably designed using special selection of the material microstructure as reported in [3, 4], or the acoustic and optical effects as negative refraction, lensing and cloaking [6, 21].”
Elastic metamaterials and dynamic homogenization: a review (2014, methodology): “For elastodynamics, however, it was found [18] that the equations change form under coordinate transformation and the cloaking ideas from electromagnetism and acoustics cannot be directly applied to elastic waves in solids.”
Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior (2020, background): “Non-standard elastodynamics properties such as, for instance, the cloaking effect, are at the very beginning of the studies of what is presently known as metamaterials [43,47,41].”