A fatigue model for the growth of a crack in a discrete lattice proposes that the size of the crack evolves as a pure birth process with parameters dx = (1+k) for k = 1, 2, .... The theory behind the model postulates that the growth rate of the crack is proportional to some power of the stress concentration at its ends and that this stress concentration is itself proportional to some power of 1 +k, where k is the crack length. Use the sojourn time description to deduce that the mean time for the crack to grow to infinite length is finite when p > 1 and that, there- fore, the failure time of the system is a well-defined and finite-valued random variable. A fatigue model for the growth of a crack in a discrete lattice proposes that the size of the crack evolves as a pure birth process with parameters dx = (1+k) for k = 1, 2, .... The theory behind the model postulates that the growth rate of the crack is proportional to some power of the stress concentration at its ends and that this stress concentration is itself proportional to some power of 1 +k, where k is the crack length. Use the sojourn time description to deduce that the mean time for the crack to grow to infinite length is finite when p > 1 and that, there- fore, the failure time of the system is a well-defined and finite-valued random variable.