The strength of a brittle material is determined by the extreme tail of its crack-size distribution. The Weibull ansatz of algebraic distributions has been widely used but is not derived from any physical mechanism, and recent results on randomly depleted networks exhibiting exponential distributions have called this ansatz into question. In this paper a simple model for the formation and subsequent time-dependent growth of cracklike defects in brittle materials, presumed to occur during processing, is introduced to study possible crack-size distributions. The key aspect of the model is that the crack growth rate has a nonlinear dependence on the local stress at the crack tip, and hence on the crack size. The model predicts evolving crack-size distributions showing a wide range of behaviors: For sufficient non-linearity the crack distribution rapidly becomes nearly algebraic (Weibull) in form and then evolves with a time-dependent power law, or Weibull modulus; in the absence of any nonlinearity the model exactly reproduces the exponential distribution of the one-dimensional random depletion problem. Both algebraic and exponential crack distributions can thus be considered as manifestations of an underlying nonlinear crack growth process. Moreover, the ubiquity of Weibull-like strength distributions observed in brittle materials, with a wide range of Weibull moduli, may be due to the physically expected nonlinearity of damage formation in real materials.