Ⅰ阶梯度损伤理论

从热力学基本定律出发，将应变张量、标量损伤变量、损伤梯度作为Helmholtz自由能函数的状态变量，利用本构泛函展开法在自然状态附近作自由能函数的Taylor展开，未引入附加假设，推导出Ⅰ阶梯度损伤本构方程的一般形式．该形式在损伤为0时可退化为线弹性应力-应变本构方程，在损伤梯度为0时可退化为基于应变等效假设给出的线弹性局部损伤本构方程．一维解析解表明，随着应力增大，损伤场逐步由空间非周期解变为关于空间的类周期解，类周期解的峰值区域形成局部化带．局部化带内的损伤变量将不同于局部化带外的损伤变量，由此可以反映出介质的局部化特征．损伤局部化并不是与损伤同时发生，而是在损伤发生后逐渐显现出来，模型的局部化机制开始启动；损伤局部化的宽度同内部特征长度成正比．

First-Order Gradient Damage Theory

Taking the strain tensor, scalar damage variable and damage gradient as the state variables of Helmholtz free energy, the general expressions of first-order gradient damage constitutive equations were derived directly from the basic law of irreversible thermodynamics by constitutive functional expansion method at natural state. When damage variable was equal to zero, the expressions could be simplified to linear elastic constitutive equations; when the damage gradient vanished, the expressions could be smiplified to the classical damage constitutive equations based on the strain equivalence hypothesis. One-dimensional problem is presented to indicate that the damage field changes from non-periodic solutions to the spatial periodic-like solutions with stress in crement. The peak value region developes to a localization band. The onsetm echanism of strain localization is advised. Damage localization emerges after damage occurs for a short tmie. The width of localization band is proportional to the internal characteristic length.