Investigation of surface instability in an elastic anisotropic cone by the use of surface waves of weak discontinuity

Abstract

The problem of surface instability of a right circular cone with an arbitrary opening made of a hexagonal single crystal (the cone axis coincides with the crystal's axis of isotropy) is investigated. The surface of the cone is free from normal and tangential stresses, but in the layer near the surface initial constant tensile or compressive stresses act in the hoop direction and in the direction of the cone's generators. Surface instability is analyzed by the use of weak nonstationary disturbances which propagate along the surface of the cone in the form of the two types of surface waves: the nonstationary Rayleigh waves polarized in the sagittal plane, and the nonstationary wave of the “whispering gallery” type polarized perpendicular to the sagittal plane. The weak nonstationary surface waves are interpreted as the lines of discontinuity (diverging circles) on which partial derivatives of the stress and strain tensor components with respect to coordinates and time have a discontinuity, but the components of these tensors are continuous. Each of the lines of discontinuity propagates with a constant normal velocity along the cone's surface in the direction of its generators and is obtained as a result of the exit onto the cone surface either of two conic complex wave surfaces of weak discontinuity intersecting along this line (Rayleigh wave) or of one real conic wave surface of weak discontinuity (wave of the “whispering gallery” type). The analysis is carried out within the framework of the theory of discontinuities based on the kinematic, geometric and dynamic conditions of compatibility; using them the velocities of the surface wave propagation and their intensities have been found. It has been shown that the surface wave velocities are dependent only on the initial stress acting in the direction of the propagation of a surface disturbance whereas the damping coefficients for the intensities of the surface waves are dependent not only on this stress but also on the initial stress acting in the hoop direction as well. The relationships for two critical magnitudes of the force compressive in the hoop direction have been obtained, and it has been shown that under the hoop compressive forces in excess of one of these magnitudes the intensity of the Rayleigh wave or the surface wave of the “whispering gallery” type begins to increase without bounds during its propagation, i.e., the surface of the cone loses stability with respect to either of two types of weak nonstationary disturbances.