Abstract
We investigate the applicability of the path integral of non-equilibrium statistical mechanics to non-equilibrium damage phenomena. As an example, a fiber-bundle model with a thermal noise and a fiber-bundle model with a decay of fibers are considered. Initially, we develop an analogy with the Gibbs formalism of non-equilibrium states. Later, we switch from the approach of non-equilibrium states to the approach of non-equilibrium paths. Behavior of path fluctuations in the system is described in terms of effective temperature parameters. An equation of path as an analogue of the equation of state and a law of path-balance as an analogue of the law of conservation of energy are developed. Also, a formalism of a free energy potential is developed. For fluctuations of paths in the system, the statistical distribution is found to be Gaussian. Also, we find the ‘true’ order parameters linearizing the matrix of fluctuations. The last question we discuss is the applicability of the phase transition theory to non-equilibrium processes. From near-equilibrium processes to stationary processes (dissipative structures), and then to significantly non-equilibrium processes: Through these steps we generalize the concept of a non-equilibrium phase transition.
Highlights
One of the most important aspects in the modern theory of damage phenomena is the possibility to describe damage occurrence on the base of the formalism of statistical physics
Once the analogy between damage mechanics and statistical physics had been discovered, it immediately became clear that any damage phenomenon can be described as a phase transition, where damage growth represents the appearance of another phase and the point of material failure is the point of the phase transition
In our previous studies [21, 22, 30], we introduced quenched disorder by means of the fiber strength variability and for the ensemble of constant strain we were able to map the fluctuations observed on thermodynamic fluctuations in statistical physics
Summary
One of the most important aspects in the modern theory of damage phenomena is the possibility to describe damage occurrence on the base of the formalism of statistical physics. Our mapping of the NFBM onto the DFBM is introduced only for one time-step In this sense, we will consider only one of these models (the DFBM for the approach of states at Sections The Classical Gibbs Approach of States and Restrictions of the Classical Gibbs Approach of States, which is more convenient for this as it is a model with a well-known solution; the NFBM for the approach of paths at Section Statistical Mechanics of the Path Integral Approach since it is more illustrative of the generalization to more complex model formulations), assuming, that all results are valid for the other model as well. All formulae can be generalized for this case as well, but due to the complexity of the results already present for the simpler model, we leave it for further studies
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