Joint Modeling of Random Initiation Time and Thermal Damage in Rocks Using a Hybrid Variational Bayesian Approach

Abstract

ABSTRACT Thermal damage significantly influences the reliability and service life of rock materials in many engineering applications, including underground structures, geothermal systems and energy storage. In practice, damage initiates internally at unknown and random times under varying environmental and loading conditions, which complicates degradation assessment and life prediction. This study develops a stochastic modeling framework for thermal damage in rocks that explicitly accounts for both random damage initiation and subsequent degradation progression under covariate effects such as temperature and strain rate. The initiation time is modeled using an exponential distribution with covariate dependent rates, while post initiation damage evolution is described by a Wiener degradation process. A hybrid variational Bayesian method is developed for efficient estimation of model parameters and latent initiation times. The framework enables improved reliability assessment and lifetime prediction and is demonstrated through numerical simulations and experimental sandstone data under thermal loading conditions.