Kinetics and Final Transformations in Self-Similar Processes

We propose a theory for the kinetics and final transformations in two types of self-similar processes, when the operators describing the process at each instant of time are inversely or directly proportional to the difference between the parameters corresponding to the final state and the state achieved. In the first case, the parameter governing the stability of the system reaches a critical value, and the final stage occurs in avalanche fashion. In the second case, there is no critical state, the final stage occurs asymptotically, and the final state is an equilibrium state. As the model processes, we take the fracture process in the composites and their strengthening process. The critical parameter in the fracture process is a measure of the stress tensor. For this process, we obtain an equation relating the initial and final degree of damage in the composite and the operator generating the self-similar process; we construct the theoretical time-to-fracture distribution functions vs. the applied stress, including zero fracture probability. The distribution functions can be constructed without long-term testing. We have obtained an equation describing the kinetics of the strengthening process and an expression defining the equilibrium state. The theory has been confirmed by experiment. The maximum testing time was similar to 20000 h.