Three-dimensional perturbation solution for a dynamic planar crack moving unsteadily in a model elastic solid

Abstract

A half-plane crack propagates dynamically, nominally in the x direction, along the plane y = 0 in an unbounded solid subjected to remote loading equivalent to a static stress intensity factor K∗. The crack front at time t lies along the arc x = v 0t + ϵƒ(z, t) where ƒ v 0 is a constant velocity, ( z, t) is an arbitrary function, and ϵ is a small parameter. The crack front speed thus varies along the z axis and its shape deviates from straightness. We address this problem within a model 3D elastodynamic theory involving a single displacement variable u, satisfying a scalar wave equation, and representing tensile opening or shear slippage, with associated tensile or shear stress σ = M δu δy across planes parallel to the crack, where M is an elastic modulus. The problem is then one of finding a solution to the scalar wave equation satisfying σ = 0 on y = 0 within the rupture. When ϵ = 0 the solutions for u, σ, dynamic stress intensity factor K and energy release rate G are familiar 2D results. We develop corresponding 3D solutions to first order in σ, for arbitrary ƒ( z, t). The solutions are used to address in some elementary cases how a crack front moves unsteadily through regions of locally variable fracture resistance. When a straight crack front approaches a slightly heterogeneous strip, lying parallel to the crack tip along an otherwise homogeneous fracture plane, it may be blocked by asperities after some advancement into the heterogeneous region if it has a relatively small incoming velocity. If, however, the incoming crack velocity is relatively high, the asperities give way and the, now curved, crack front propagates into the bordering homogeneous region. There, the moving crack front recovers a straight configuration through slowly damped space-time oscillations. The oscillatory crack tip motion results from constructive-destructive interferences of stress intensity waves, initiated by encounters of the crack front with asperities, and then propagating along the front. Oscillations in response to a heterogeneity that is spatially periodic in the direction along the crack front decay as t t- 1 2 at large t. The slowness of the decay suggests that the straight crack front configuration may be sensitive to small sustained heterogeneity of the fracture resistance. This is consistent with results of a related analysis ( Perrin and Rice, 1994, in press, J. Mech. Phys. Solids) based upon a strictly linearized form of our equations. The persistence of unsteady crack tip motion beyond the immediate region of heterogeneities provides an explanation for high frequency seismic radiation, using a lesser amount of heterogeneity than what might be naively assumed by strict correspondence of all curved and variable velocity portions of a propagating rupture front to asperities. Also, oscillations of crack tip velocity in the presence of sustained small heterogeneities, suggested by features of our 3D results for the model theory, may provide a mechanism for the generation of rough tensile fracture surfaces when the average (macroscopic) propagation speed of the crack is relatively small.