The variational principle for probabilistic measure, Hashin- Shtrikman bounds and beyond

Abstract eng:
Numerical studies reveal the enormous complexity of local fields in composites with random microstructures. There is no doubt that the local fields can be described adequately only in probabilistic terms. Such description is needed not only for characterization of the state of a composite, but also for prediction of microstructure evolution due to plasticity, fatigue, or fracture, where the path of evolution is controlled by the local fields. At the moment, the probabilistic characteristics of local fields are sought by statistical analysis of a huge number of numerical simulations conducted for different realizations of microstructures. Apparently, a more practical way is desirable. In this talk it is discussed the possibility of using for such purposes the variational principle for probabilistic measure constructed in (Berdichevsky, J. Appl. Math. Mech., 1987). We derive classical results of homogenization theory and obtain statistical characteristics of local fields which were not available previously.

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