A two-dimensional triangular spring network model is used to investigate the time-dependent damage evolution and failure of model materials in which the damage formation is a nucleated event. The probability of damage formation r(i)(t) at site i at time t is taken to be proportional to the local stress at site i raised to a power: r(i)(t) = A sigma(i)(t)(eta). As damage evolves in the material, the stress state becomes heterogeneous and drives preferential damage evolution in regions of high stress. As predicted by an analytical model and observed in previous electrical fuse network simulations, there is a transition in the failure behavior at eta = 2: for eta less than or equal to 2, the failure time and damage density are independent of the system size; for eta textgreater 2, the failure time and damage decrease with increasing time and failure occurs by the formation of a finite Critical damage region which rapidly propagates across the remainder of the material. The stress distribution prior to failure exhibits no abrupt changes or scalings that indicate imminent failure. The scalings of the failure time and the failure time distribution are investigated, and compared with analytic predictions. The failure time scales as a power law in In N-T, where N-T is the system size, but the exponent is not the predicted value of 1 - eta/2; this is attributed to a difference in the stress concentration factors (scf) between the discrete lattice and a continuum model. Using the scf values for the lattice lead to predicted scalings consistent with the simulations. Predicted absolute failure times versus size are generally in good agreement with simulation results at larger eta values. The coefficient of variation of the failure time distribution is observed to be nearly constant, in slight contrast to the predicted scaling of (InNT)(-1). Overall, the simulation results quantitatively and qualitatively validate many of the critical predictions of the analytic model.