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.

中文翻译：

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考虑阻尼效应的石膏试件循环加卸载应变方程及损伤演化


本研究旨在建立考虑阻尼的石膏试件应变瞬子方程和损伤因子演化规律。首先，基于单自由度振动模型计算阻尼能，并基于应力平衡方程得到瞬时应变方程。其次，将耗散能分为阻尼能和损伤能，并得到损伤因子修正算法。第三，在不同加载速率和应力幅值下进行循环加载和卸载试验，以验证应变方程的准确性。最后，使用矩张量声发射模拟监测样本的强度曲线和裂纹特征。分析了影响阻尼能量和应变方程的因素、试件的能量和损伤演化规律以及不同加载速率下试件的损伤模式。结果表明，瞬时应变方程和考虑阻尼效应的修正损伤因子能够有效反映试件的变形规律和损伤状态。相比之下，变应力实验下限对试件的损伤低于恒应力实验下限。随着加载速率的增加，试件的损伤能量密度降低，单次循环内的损伤系数逐渐减小。 随着加载速率的增加，模型中裂纹事件数量显着增加，尺寸变得更加均匀，并依次呈现出密集和稀疏的分布模式，剪切裂纹百分比显着下降，混合裂纹数量显着增加，试件的脆性行为变得明显，并且达到称为“压碎”状态的完全损坏状态。该研究为扰动载荷下粘弹塑性材料的损伤评估提供了理论参考。