Beyond a characteristic value of the negative peak pressure, ultrasound fracture the shell of ultrasonic contrast agents (UCAs). Existing criteria for ascertaining this threshold value exploit the dependence of the amplitude of the UCA acoustic response on the incident pressure. However, under the common experimental conditions used in this work, these criteria appear to be unreliable when they are applied to UCAs that are stabilized by a thick polymeric shell. An alternative criterion for determining the onset of shell fracture is introduced here, which uses variations of the shape of the acoustic time-domain response of an UCA suspension. Experimental evidence is presented that links the changes of the cross-correlation coefficient between consecutive time-domain signals to the fracture of the shells, and consequent release of air microbubbles. In principle, this criterion may be used to characterize similar properties of other types of particles that cannot undergo inertial cavitation.