Simulations by the finite-difference time-domain method were performed to model the interaction of laser-excited broadband surface acoustic wave (SAW) pulses with a partially closed surface-breaking crack, prepared by stress-induced impulsive fracture. One- and two-parameter models were developed to study the role played by the interfacial stress of the crack. Good agreement with laser-scanning nondestructive evaluation was found for the measured reflected and transmitted SAW velocity profiles. Even the single-parameter reduced-stress model, based on the same softening of all interface stress-tensor components, provided a reasonable description, consistent with the two-parameter simulations treating the tensile and shear interfacial stress contributions separately. In addition, the enhancement of the velocity profile at the front edge of the crack was calculated for the one- and two-parameter stress models and compared with experimentally observed enhancement effects.