Higher order asymptotic field at the moving crack tip

The higher order asymptotic field at a smoothly curved dynamic crack tip under mixed loading conditions is investigated by the use of a moving coordinate system and the mathematical framework of the complex potential theory. On the basis of the general representation for stress functions in a moving coordinate system, the recurrence formulae for determining the higher order solutions from lower order ones are derived. The calculation in this paper shows that the higher order asymptotic field can be separated into two parts: the steady state asymptotic field which depends on the crack velocity; and the non-steady state asymptotic field which is determined by the time rate of change of the intensity factor, the crack tip acceleration and rotation speed. The second order solution giving the sress distributions at the moving crack tip are presented. The important result that the experimentally observed crack branching velocities are estimated to be much smaller than Yoffe's prediction might be explained by this second order asymptotic solution.