A stochastic damage model for concrete considering strain rate effect

To represent the rate-dependent mechanical behavior of concrete, a stochastic damage model is proposed that integrates Langevin dynamics within the Micro-Meso Stochastic Fracture framework. In this model, the fracturing process of each micro-spring is described as a barrier-crossing event where the effective barrier height depends on the external loading rate. Evolution of the reaction coordinate follows the Langevin dynamics. Solving the corresponding Fokker–Planck–Kolmogorov equation yields the failure probability of each micro-spring under dynamic loading, from which the dynamic fracture strain is derived. Numerical results demonstrate that the model naturally captures the characteristic two-stage strength enhancement of concrete: a gradual increase of dynamic increase factor at low strain rates, followed by a rapid rise beyond a critical threshold. By linking the static fracture strain of each micro-spring to its unique initial energy barrier, the model assigns distinct rate sensitivities to different material constituents. This approach enables the model to reflect different rate sensitivities in tension and compression without separate empirical adjustments. Finally, simulations of reinforced concrete beams under various loading velocities, including high-rate impact, validate the model's capability to predict structural dynamic responses in practical engineering scenarios.