SummaryWe consider torsional wave propagation in a pre-stressed annular cylinder. Hydrostatic pressure is applied to the inner and outer surfaces of an incompressible hyperelastic annular cylinder, of circular cross-section, whose constitutive behaviour is governed by a Mooney-Rivlin strain energy function. The pressure difference creates an inhomogeneous deformation field, which modifies the inner and outer radii of the annular cylinder. We wish to determine the effect that this pre-stress, and a given axial stretch, has on the propagation of small-amplitude torsional waves through the medium. We use the theory of small-on-large to deduce the linear wave equation that governs incremental torsional waves and then determine the dispersion relation for the prestressed annulus by using an approximation procedure (the Liouville-Green transformation). We show that this scheme compares well to numerical solutions except in regions very close to turning points (of the transformed ordinary differential equation). In particular, we find that the inhomogeneous deformation makes the coefficients of the governing ordinary differential equation spatially dependent and affects the location of the roots of the dispersion relation. We observe that, if the pressure on the outer surface of the annular cylinder is greater (smaller) than that on the inner, then the cut-on frequencies are spaced further apart (closer) than they would be in the stress-free case. This result could potentially be used to tune the propagation characteristics of the cylinder over a range of frequencies.

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We present here a Boundary Element Method (BEM) implementation for SH harmonic waves in a class of inhomogeneous anisotropic media. The inhomogeneity is assumed to be the same not only for the stiffnesses, but also for the density. The implementation is based on a closed form fundamental solution for SH waves derived by Daros [C.H. Daros, A fundamental solution for SH-waves in class of inhomogeneous anisotropic media, Int. J. Eng. Science 46 (2008) 809-817]. We show numerical results obtained by the traditional boundary integral equation. Moreover, the non-hypersingular traction based BEM is also implemented, allowing the modelling of cracks in inhomogeneous anisotropic media. We obtain numerical results for the stress intensity factors (SIF) which are compared to previous published results.

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Based on a monograph by Payton on the wave propagation in transversely isotropic media, some new results for piezoelectric solids are presented here. The dynamic Green's tensors for the displacement fields along the symmetry axis for two-and three-dimensions concerning some piezoelectric solids of the crystal class 6mm are discussed. The central result of this paper is to show that the set of differential equations for linear, quasi-electrostatic piezoelectricity is, when eliminating e.g. the electric potential in a transformed domain, quasi-hyperbolic in the sense defined by Gal'pern. Only when specific constraints are attended, one obtains a hyperbolic operator, guaranteeing the well-posedness and solvability of the Cauchy problem.Mathematics Subject Classification (1991). 73D99, 73R05, 73D10, 35L99, 35L25.