On the theory of wave propagation in anisotropic plates

Abstract

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.

The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides

“…PWD, used extensively [3,10,39,40], provides a very intuitive physical approach; the symmetry and coupling properties of guided waves in isotropic and various anisotropic materials, including monoclinic symmetry, have been investigated using this method [39,40]. More recently, as an alternative, Shuvalov et al [5] used the Stroh formalism to study guided waves in free anisotropic plates and derived a formula for the symmetries of the solutions in arbitrarily anisotropic plates in agreement with [39,40]. To complete the classification of materials and axes configurations according to their symmetry and coupling properties for a single layer, we could extract formulae from [5] for each case.…”

“…Analytical solutions in anisotropy are somewhat harder to find, but none the less for free plates with higher anisotropic symmetries such as cubic [3] and orthorhombic and monoclinic [4], solutions do exist. More recently, and by means of the Stroh formalism, dispersion relations have been obtained for generally anisotropic plates and some general properties of the modes derived analytically as well as the asymptotic behaviour of the first fundamental modes at high and low frequencies, see [5,6]. Some computational aspects of this problem are to be found in [7].…”

“…Section II presents the study, dispersion curves and mode shapes, of new cases in free anisotropic plates; these complement [4,5] and provide the classification of single layer anisotropic waveguides that is necessary to move on to the study of multiple-layer systems. In section III, the properties of the guided waves in 2 and 3 layer systems are studied in detail using partial wave decomposition (PWD).…”

Existence of one-component Rayleigh waves, Stoneley waves, Love waves, slip waves and one-component waves in a plate or layered plate

“…The readers may consult the references listed in the more recent studies by Shuvalov [2000;2004] and Ting [2008].…”

Incremental Equations for Soft Fibrous Materials

“…Finally, it can be shown (e.g., Shuvalov, 2000;Mielke and Fu, 2004) that Z is Hermitian so that it is of the form…”