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Based on a generalized cumulative damage approach with a stochastic process describing initial damage for a material specimen, a broad class of statistical models for material strength is developed. Plausible choices of stochastic processes for the initial damage include Brownian motion, geometric Brownian motion, and the gamma process; and additive & multiplicative cumulative damage functions are considered. The resulting general statistical model gives an accelerated test form of the inverse Gaussian distribution, special cases of which include some existing models in addition to several new models. Model parameterizations & estimation by maximum likelihood from accelerated test data are discussed, and the applicability of the general model is illustrated for three sets of strength data. The proposed models are compared with the power-law Weibull model, and the inverse Gaussian generalized linear models.