Cyclic loading and unloading strain equations and damage evolution of gypsum specimens considering damping effects

This study aims to establish a strain instanton equation and damage factor evolution law for gypsum specimens by considering damping. First, damping energy is calculated based on the single-degree-of-freedom vibration model, and the instantaneous strain equation is obtained based on the stress balance equation. Second, the dissipation energy is divided into damping and damage energies, and a damage-factor correction algorithm is obtained. Third, cyclic loading and unloading tests were performed at different loading rates and stress amplitudes to verify the accuracy of the strain equation. Finally, the specimens’ magnitude curves and crack characteristics were monitored using moment–tensor acoustic emission simulations. The factors influencing the damping energy and strain equations, energy and damage evolution laws of the specimens, and damage patterns of the specimens at different loading rates were analysed. The results show that the instantaneous strain equation and the modified damage factor considering the damping effect can effectively reflect the deformation law and damage state of the specimens. In contrast, the damage to the specimens in the lower limit of the variable stress experiment was lower than that in the lower limit of the constant stress experiment. As the loading rate increases, the damage energy density of the specimen decreases, and the damage factor within a single cycle gradually decreases. As the loading rate increases, the number of crack events in the model increases significantly, size becomes more uniform, and sequentially exhibits dense and sparse distribution patterns, percentage of shear cracks decreases significantly, number of mixed cracks increases significantly, brittle behaviour of the specimen becomes obvious, and a complete damage state is attained known as the ‘crushed’ state. This study provides a theoretical reference for damage assessments of viscoelastic–plastic materials subjected to perturbing loads.