Transformation theory is developed for the equations of linear anisotropic elasticity. The transformed equations correspond to non-unique material properties that can be varied for a given transformation by selection of the matrix relating displacements in the two descriptions. This gauge matrix can be chosen to make the transformed density isotropic for any transformation although the stress in the transformed material is not generally symmetric. Symmetric stress is obtained only if the gauge matrix is identical to the transformation matrix, in agreement with Milton et al. [1]. The elastic transformation theory is applied to the case of cylindrical anisotropy. The equations of motion for the transformed material with isotropic density are expressed in Stroh format, suitable for modeling cylindrical elastic cloaking. It is shown that there is a preferred approximate material with symmetric stress that could be a useful candidate for making cylindrical elastic cloaking devices.

Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.

A theoretical framework is developed which describes wave propagation in in nite homogeneous elastic plates of unrestricted anisotropy. The approach exploits the propagator matrix which is the exponential of the fundamental elasticity matrix underlying Stroh's formalism of anisotropic elastodynamics. The matrices of plate impedance and admittance are introduced, and their analytical properties are established, which appear fruitful for computing and analysing the plate wave spectra. On this basis, the dispersion equation can be cast into the form of a real equation involving the monotonic function, whose zeros and poles are the wave velocities in a given plate subjected to di¬erent boundary conditions (traction-free or clamped faces). It is proved that three fundamental wave branches exist for any orientation of wave propagation in an anisotropic plate with traction-free faces, and that those branches are missing if one or both faces are clamped. The intrinsic symmetry of the wave motion in an arbitrary plate is revealed. The general formalism is applied to elaborate the long-wavelength low-frequency approximation for a (thin) free plate of unrestricted anisotropy. The frequency-dispersive wave velocities, displacements and tractions at the onset of the fundamental wave branches are derived explicitly. The conditions for the extreme velocity values and some other useful universal connections are determined for an arbitrary thin plate, and exempli ed for speci c anisotropy cases.