Transformation theory has long been known to be a mechanism for

the design of metamaterials. It gives rise to the required properties of the

material in order to direct waves in the manner desired. This talk will

focus on the mathematical theory underpinning the design of acoustic and

elastodynamic metamaterials based on transformation theory and aspects of

the experimental confirmation of these designs. In the acoustics context it

is well-known that the governing equations are transformation invariant and

therefore a whole range of microstructural options are available for design,

although designing materials that can harness incoming acoustic energy in

air is difficult due to the usual sharp impedance contrast between air and

the metamaterial in question. In the elastodynamic context matters become

even worse in the sense that the governing equations are not transformation

invariant and therefore we generally require a whole new class of materials.

In the acoustics context we will describe a new microstructure that consists

of rigid rods that is (i) closely impedance matched to air and (ii) slows

down sound in air. This is shown to be useful in a number of configurations

and in particular it can be employed to half the resonant frequency of the

standard quarter-wavelength resonator (or alternatively it can half the size

of the resonator for a specified resonant frequency) [1].

In the elastodynamics context we will show that although the equations are

not transformation invariant one can employ the theory of waves in

pre-stressed hyperelastic materials in order to create natural elastodynamic

metamaterials whose inhomogeneous anisotropic material properties are

generated naturally by an appropriate pre-stress. In particular it is shown

that a certain class of hyperelastic materials exhibit this so-called

“invariance property” permitting the creation of e.g. hyperelastic cloaks

[2,3] and invariant metamaterials. This has significant consequences for the

design of e.g. phononic media: it is a well-known and frequently exploited

fact that pre-stress and large deformation of hyperelastic materials

modifies the linear elastic wave speed in the deformed medium. In the

context of periodic materials this renders materials whose dynamic

properties are “tunable” under pre-stress and in particular this permits

tunable band gaps in periodic media [4]. However the invariant hyperelastic

materials described above can be employed in order to design a class of

phononic media whose band-gaps are invariant to deformation [5]. We also

describe the concept of an elastodynamic ground cloak created via pre-stress

[6].

[1] Rowley, W.D., Parnell, W.J., Abrahams, I.D., Voisey, S.R. and Etaix, N.

(2018) “Deepening subwavelength acoustic resonance via metamaterials with

universal broadband elliptical microstructure”. Applied Physics Letters 112,

251902.

[2] Parnell, W.J. (2012) “Nonlinear pre-stress for cloaking from antiplane

elastic waves”. Proc Roy Soc A 468 (2138) 563-580.

[3] Norris, A.N. and Parnell, W.J. (2012) “Hyperelastic cloaking theory:

transformation elasticity with pre-stressed solids”. Proc Roy Soc A 468

(2146) 2881-2903

[4] Bertoldi, K. and Boyce, M.C. (2008) “Mechanically triggered

transformations of phononic band gaps in periodic elastomeric structures”.

Phys Rev B 77, 052105.

[5] Zhang, P. and Parnell, W.J. (2017) “Soft phononic crystals with

deformation-independent band gaps” Proc Roy Soc A 473, 20160865.

[6] Zhang, P. and Parnell, W.J. (2018) “Hyperelastic antiplane ground

cloaking” J Acoust Soc America 143 (5)