A novel fractal-statistical scaling model of rocks considering strain rate

The scaling-dependent behaviors of rocks are significant to the stability and safe operation of the structures built in or on rock masses for practical engineering. Currently, many size effect models are employed to connect laboratory measurements at small scales and engineering applications at large scales. However, limited works consider the strain rate effect. In this study, an fractal-statistical scaling model incorporating strain rate is proposed based on a weakest-link approach, fractal theory and dynamic fracture mechanics. The proposed scaling model consists of 8 model parameters with physical meaning, i.e. rate-dependent parameter, intrinsic material parameter, dynamic strain rate, quasi-static strain rate, quasi-static fracture toughness, micro-crack size, micro-crack intensity and fractal dimension, enabling the proposed scaling model to model the scaling behaviors under different external conditions. Theoretical predictions are consistent with experimental data on red sandstone, proving the reliability and effectiveness of the proposed scaling model. Thus, the scaling behaviors of rocks under dynamic loading conditions can be captured by the proposed fractal-statistical scaling model. The sensitivity analysis indicates that the nominal strength difference becomes more obvious with a higher strain rate, larger fractal dimension, smaller micro-crack size or lower micro-crack intensity. Therefore, the proposed scaling model has the potential to capture the scaling behaviors considering the thermal effect, weathering effect, anisotropic characteristic etc., as the proposed scaling model incorporated model parameters with physical meaning. The findings of this study are of fundamental importance to understand the scaling behaviors of rock under dynamic loading condition, and thus would facilitate the appropriate design of rock engineering.