两种弹性损伤模型的基本方程与色散关系讨论

ON THE GOVERNING EQUATIONS AND DISPERSION RELATIONS OF TWO ELASTIC DAMAGE MODELS

分析了传统连续介质损伤理论的控制方程在损伤过程中的变化特点,方程类型的改变导致错误预测波不能在损伤区域传播.二阶隐式应变梯度损伤理论是对传统理论的一种改进,严格证明了其控制方程在损伤过程中类型不变,这意味着损伤区域能传播波,同时有助于克服病态的网格相关性.色散分析结果表明:传统连续介质损伤理论不能反映色散现象,隐式梯度模型可以并对波长有上限截断作用.

The governing equations of the conventional continuum damage models have been investigated in this paper. It is found from the analysis that the types of governing equations have been changed from elliptical to hyperbolic during the damage process, which may result in the false appearance of waves impossibly propagating in damaging zone. As an alternative model, the implicit second-order strain gradient theory is powerful in studying damage. It has been rigorously proved that the equation-type of the governing equations of the damaged media is unchangable during the whole damage process. Therefore the false appearance of no wave existing inside the damaged zone has been left. Moreover, the introduction of gradient terms conduces to overcome the pathological mesh-dependence in numerical computation. It is well known that dispersion is always in existence in solids. Further analysis indicates that the implicit second-order strain gradient theory can effectively deal with dispersion which cannot be solved by the continuum damage model. It is also observed that the more serious damage will result in decrease in the critical wave length.