On spherical near fields and far fields in elastic and visco-elastic solids

Abstract

Like divergent fluid pressure waves or electromagnetic radiations, divergent spherical stress waves produced by periodic excitation in infinite elastic and linearly visco-elastic homogeneous isotropic solids can exhibit near fields dominated by stationary vibrations decaying with r −3 and far fields dominated by progressive waves decaying with r −1, if r is the radial distance from the centre of the source. On a divergent motion the zone of transition between these fields, where the length of a peripheral line is of the order of the wave lengths, acts like a high-pass filter in preferentially attenuating the lower frequencies, whereas a convergent motion would have its lower frequencies reinforced preferentially. Due to phase changes in the stresses, the maximum shear stress in the transitional region is no longer half the arithmetical but half the time-vector difference between the principal, radial and circumferential stresses. If a source of excitation with a boundary in the transitional region is compared in its effect with one having its boundary in the near or far field, a damped resonance-like phenomenon is evident which increases in intensity with Poisson's ratio. In materials of high Poisson's ratios, a stress transition giving rise to such a resonance can be distinguished from a subsequent strain transition without resonance. A source of given size and pressure amplitude radiates a maximum of energy into the solid at frequencies that place the boundary in the transitional zone. A superimposed static state of stress does not alter the vibrational phenomena. A weak viscous effect does not influence the stress distribution and amplitudes in the near field but reduces the resonant amplitudes in the zone of transition and causes the known exponential decay in the far field.