A degenerate polar lattice for cloaking in full two-dimensional elastodynamics and statics

Abstract

A lattice design of a cloak for full two-dimensional elasticity is suggested when the background continuum is isotropic with Lamé parameters (λ, μ) satisfying μ ≤ λ. The lattice is polar in the sense that it elastically resists rotations; and is degenerate meaning it admits a stressless collapse mechanism. These characteristics are attained through the use of appropriately distributed restoring torques in conjunction with hinge-like spring-mass contacts. Thus, the lattice is proven to exhibit a rank-3 elasticity tensor lacking the minor symmetries. Accordingly, it rigorously adheres to the form-invariance requirements of the transformation method under the Brun–Guenneau–Movchan gauge. The cloak is numerically tested in statics and in dynamics under pressure and shear incident waves and shows satisfactory performance. Finally, a theoretical generalization extends the design to three dimensions and to arbitrarily anisotropic backgrounds so as to enable cloaking as well as other transformation-based static and dynamic field manipulation techniques in these cases.