Hyperelastic cloaking theory: Transformation elasticity with pre-stressed solids

Abstract

Transformation elasticity, by analogy with transformation acoustics and optics, converts material domains without altering wave properties, thereby enabling cloaking and related effects. By noting the similarity between transformation elasticity and the theory of incremental motion superimposed on finite pre-strain it is shown that the constitutive parameters of transformation elasticity correspond to the density and moduli of small-on-large theory. The formal equivalence indicates that transformation elasticity can be achieved by selecting a particular finite (hyperelastic) strain energy function, which for isotropic elasticity is semilinear strain energy. The associated elastic transformation is restricted by the requirement of statically equilibrated pre-stress. This constraint can be cast as $\tr {\mathbf F} =$ constant, where $\mathbf{F}$ is the deformation gradient, subject to symmetry constraints, and its consequences are explored both analytically and through numerical examples of cloaking of anti-plane and in-plane wave motion.

arXiv comment20 pages, 5 figures

References in corpus (6)

• Acoustic cloaking theory
• Achieving control of in-plane elastic waves
• Full-wave invisibility of active devices at all frequencies
• Acoustic metafluids
• Employing pre-stress to generate finite cloaks for antiplane elastic waves
• Nonlinear pre-stress for cloaking from antiplane elastic waves

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• Elastic wave cloaking via symmetrized transformation media
• Scattering problems in elastodynamics
• Design of arbitrarily shaped acoustic cloaks through PDE-constrained optimization satisfying sonic-metamaterial design requirements
• Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields. Part II: Singularities, annihilation and invisibility
• Realizing the Willis equations with pre-stresses
• Omnidirectional flexural invisibility of multiple interacting voids in vibrating elastic plates
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• Hyperelastic antiplane ground cloaking
• Cloaking and anamorphism for light and mass diffusion
• The Atomistic Green's Function method for acoustic and elastic wave-scattering problems
• Longitudinal elastic wave control by pre-deforming semi-linear materials
• Modified 1D elastic wave equations that retain time synchronization under spatial coordinate transformations
• Extremal micropolar materials for elastic wave cloaking