A generalized cumulative damage approach is presented which yields a large family of accelerated test inverse Gaussian-type models for strength of materials that incorporate the size effect as the acceleration variable. Previous models are generalized here in three aspects: the cumulative damage model is more general and can include damage functions other than the additive and multiplicative damages; the strength reduction function due to initial damage existing in the material is taken to be a very general function; and the initial damage process is a more general stochastic process that includes those previously assumed as special cases. The approach taken here is therefore the most general cumulative damage model obtained to date and yields a large number of potentially more useful accelerated test models for material strength. Estimation of model parameters by maximum likelihood methods is discussed, and two examples using real tensile strength data for carbon micro-composites and single carbon fibers are presented, illustrating the improvement of the new approach over previous models. (c) 2005 Elsevier B.V. All rights reserved.