Flexural wave dispersion in finitely pre-strained solid and hollow circular cylinders made of compressible materials

Abstract

Flexural wave dispersion in finitely pre-stretched (or pre-compressed) solid and hollow, circular cylinders is investigated with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies. It is assumed that the initial strains in the cylinders are homogeneous and correspond to the uniaxial tension, or compression, along their central axes. The elasticity relations of the cylinders' materials are described by the harmonic potential. The analytical solution of the corresponding field equations is presented and, using these solutions, the dispersion equations for the cases under consideration are obtained. The dispersion equations are solved numerically and based on these solutions, dispersion curves and dispersion diagrams are constructed for various values of the elongation parameter through which the magnitude of the initial strains is determined. The numerical results are obtained for the first and second lowest modes of the solid cylinder and for the first three lowest modes of the hollow cylinder. According to the analyses, in particular, it is established that the finite initial uniaxial stretching, as well the finite initial uniaxial compressing, change the dispersion of the flexural waves in the solid and hollow cylinders not only quantitatively, but also qualitatively.