The self-similarity of characteristic stages of damage-failure transition has been studied both theoretically and experimentally, with the damage localization kinetics analyzed according to the nonlinearity of free energy release of solids with defects: Free energy in the generalized Ginzburg–Landau form reflecting the specific criticality of solids with defects, and the structural-scaling transition in the presence of two critical points separating qualitatively different material responses corresponds to the types of collective modes of defects. These collective modes have the nature of the self-similar solutions of the damage evolution equation. The final damage stage is associated with the singular self-similar blow-up solutions localized on a set of spatial scales. The presence of two singularities (the stress field at the crack tip and the blow-up damage modes) allows the interpretation of self-similarity of damage localization kinetics to be construed as a criticality sign of damage-failure transition and has been illustrated by the original experimental statements: transition from the steady state to branching crack dynamics, dynamic fragmentation statistics, resonance regimes of failure in shocked materials.

理论上和实验上都研究了破坏-破坏转变特征阶段的自相似性，并根据有缺陷的固体的自由能释放的非线性来分析破坏的局部动力学：广义金茨堡-朗道形式的自由能反映了具有缺陷的固体的特定临界值，以及在存在两个质点分开的质构不同的临界点时，结构缩放比例转换对应于缺陷的集体模式的类型。这些集体模式具有损伤演化方程自相似解的性质。最终的破坏阶段与局部空间集合上的奇异自相似爆炸解决方案相关。