On crack path stability in a layered material

Crack paths in an elastic layer on top of a substrate are considered. Crack growth is initiated from an edge crack in the layer. The plane of the initially straight crack forms an angle to the free surface. The load consists of a pair of forces applied at the crack mouth and parallel to the interface. Crack paths are calculated using a boundary element method. Crack growth is assumed to proceed along a path for which the mode II stress intensity factor vanishes. The inclination and the length of the initial crack are varied. The effect of two different substrates on the crack path evolution is demonstrated. A crack path initially leading perpendicularly to the interface is shown to be directionally unstable for a rigid substrate. Irrespective of its initial angle, the crack does not reach the interface, but reaches the free surface if the layer is infinitely long. At finite layer length the crack reaches the upper free surface if the initial crack inclination to the surface is small enough. For an inextendable flexible substrate, on the other hand, the crack reaches the interface if its initial inclination is large enough. For the flexible substrate an unstable path parallel with the sides of an infinitely long layer is identified. The results are compared with experimental results and discussed in view of characterisation of directionally unstable crack paths. The energy release rate for an inclined edge crack is determined analytically.

Sidewall flow stability in the MHD channel flows

The velocity distribution between two sidewalls is; M-shaped for the MHD channel flows with rectangular cross section and thin conducting walls in a strong transverse magnetic field. Assume that the dimensionless numbers R(m) much less than 1, M, N much greater than 1, and sigma

Distribution, stability, bifurcations and catastrophe of steady rotation of a symmetrical heavy gyroscope with viscous-liquid-filled cavity

The number, the angles of orientation and the stability in Rumyantsev Movchan's sense of oblique steady rotations of a symmetric heavy gyroscope with a cavity completely filled with a uniform viscous liquid, possessing a fixed point 0 on its symmetric axis. are given for various values of the parameters. By taking the square of the upright component of the angular momentum M2 as a control parameter, three types of bifurcation diagrams of the steady rotations, two types of jumps and two kinds of local catastrophes, one being the symmetric reduced cusp type and the other being of the symmetric reduced butterfly type, are obtained. By taking account of the M2-damping owing to the moment of unavoidable faint friction, two different modes for the gyroscope, initially in a stable quasi-steady upright rotation with a nutation angle theta(s) equal to zero, to topple over are found.

Viscosity-induced mode splitting and potential energy criterion for mode stability

This paper points out that viscosity can induce mode splitting in a uniform infinite cylinder of an incompressible fluid with self-gravitation, and that the potential energy criterion cannot be appropriate to all normal modes obtained, i.e., there will be stable modes with negative potential energy (<0). Therefore the condition >0 is not necessary, although sufficient, for the stability of a mode in an incompressible static fluid or magnetohydrodynamics (MHD) system, which is a correction of both Hare's [Philos. Mag. 8, 1305 (1959)] and Chandrasekhar's [Hydrodynamic and Hydromagnetic Stability (Oxford U.P., Oxford, 1961), p. 604] stability criterion for a mode. These results can also be extended to compressible systems with a polytropic exponent.