3-DIMENSIONAL PERTURBATION SOLUTION FOR A DYNAMIC PLANAR CRACK MOVINGUNSTEADILY IN A MODEL ELASTIC SOLID

A HALF-PLANE CRACK propagates dynamically, nominally in the x directio n, along the plane y = 0 in an unbounded solid subjected to remote loa ding equivalent to a static stress intensity factor K. The crack fron t at time t lies along the arc x = upsilon0t + epsilonf(z, t) where up silon0 is a constant velocity, f(z, t) is an arbitrary function, and e psilon is a small parameter. The crack front speed thus varies along t he z axis and its shape deviates from straightness. We address this pr oblem within a model 3D elastodynamic theory involving a single displa cement variable u, satisfying a scalar wave equation, and representing tensile opening or shear slippage, with associated tensile or shear s tress sigma = M partial derivative u/partial derivative y across plane s 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 sigma = 0 on gamma = 0 within the rupture. When epsilon = 0 the solut ions for u, sigma, dynamic stress intensity factor K and energy releas e rate G are familiar 2D results. We develop corresponding 3D solution s to first order in epsilon for arbitrary f(z, t). The solutions are u sed to address in some elementary cases how a crack front moves unstea dily through regions of locally variable fracture resistance. When a s traight crack front approaches a slightly heterogeneous strip, lying p arallel to the crack tip along an otherwise homogeneous fracture plane , it may be blocked by asperities after some advancement into the hete rogeneous region if it has a relatively small incoming velocity. If, h owever, the incoming crack velocity is relatively high, the asperities give way and the, now curved, crack front propagates into the borderi ng homogeneous region. There, the moving crack front recovers a straig ht configuration through slowly damped space-time oscillations. The os cillatory crack tip motion results from constructive-destructive inter ferences of stress intensity waves. initiated by encounters of the cra ck front with asperities, and then propagating along the front. Oscill ations in response to a heterogeneity that is spatially periodic in th e direction along the crack front decay as t-1/2 at large t. The slown ess of the decay suggests that the straight crack front configuration may be sensitive to small sustained heterogeneity of the fracture resi stance. 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 t ip motion beyond the immediate region of heterogeneities provides an e xplanation for high frequency seismic radiation, using a lesser amount of heterogeneity than what might be naively assumed by strict corresp ondence of all curved and variable velocity portions of a propagating rupture front to asperities. Also, oscillations of crack lip velocity in the presence of sustained small heterogeneities, suggested by featu res of our 3D results for the model theory, may provide a mechanism fo r the generation of rough tensile fracture surfaces when the average ( macroscopic) propagation speed of the crack is relatively small.