Theoretical Solution for Stress and Strain Distributions Induced by Generalized Elastic Constant Variation in Rock Mass

Rock formations experience strength and stiffness degradation or enhancement, which is often reflected by an increase or decrease in the elastic constant. Such elastic constant variation in rock mass often causes the redistributions of the stress and strain; therefore, the geostructure stability is affected. However, to the best of the authors’ knowledge, no studies have been conducted to investigate this problem analytically. Therefore, based on Hooke’s law, this paper analyzed the stress, strain, and energy density distributions in a rock formation with such a variation in the elastic constant. Then, a general analytical solution was presented for two scenarios. One scenario was the elastic constant variation over the whole domain of rock mass, and the other was the elastic constant variation over the partial domain of interest. According to the law of thermodynamics, the elastic constant and stress components could be expressed as a function of the elastic strain energy density (ψ). Then, the stress and strain components distributions after the elastic constant variation could be directly correlated to those that developed before the elastic constant variation. The analysis showed that the stress components before and after the elastic constant variations were unchanged for the case with a variation in the elastic constant over the whole domain of interest. In contrast, in this case, the strain components after the elastic constant change were equal to 1/R times the strain components before the elastic constant change. In this paper, the ratio of elastic constants after the change to before the change was R. The analysis showed that the strain components before and after the elastic constant changes were the same for cases where the elastic constant varied over the partial domain of interest. However, in this case, the stress components after the elastic constant change were equal to R times the stress components that developed before the elastic constant change in this partial domain. The analytical model was verified through numerical simulation with the help of ABAQUS.