Probabilistic analysis of non-local random media

Computational stochastic methods have been devoted over the last years to analysis andquantification of the mechanical response of engineering systems involving randommedia. Specifically analysis of random, heterogeneous media is getting more and moreimportant with the emergence of new complex materials requiring reliable methods toprovide accurate probabilistic response.Advanced materials, often used at nano or meso-levels possess strong non-localcharacters showing that long-range forces between non-adjacent volume elements playan important role in mechanical response. Moreover long and short-range molecularinteractions may have random nature due to unpredictable fabrication process. Thispeculiar character yields probabilistic analysis of non-local interactions still morecumbersome than in case of local heterogeneous random media usually solved withMonte-Carlo analysis that is very time-consuming.Recently analysis of non-local medium with long-range interactions have beenformulated and solved in deterministic setting with the aid of fractional calculusassuming that the long-range forces are spatially decaying with fractional power law oforder a . In this setting a wide class of non-local models may be obtained withvariations of the parameter a both for static and dynamic setting.In this paper the proposed non-local model for long-range forces will be extended to theanalysis in presence of random fluctuations of the elastic properties of the material. Inthis setting a stochastic fractional differential equation is governing the mechanicalresponse of the random media.Solution in terms of the statistics of the random field of the response will be obtainedintroducing a proper discretization of the governing fractional differential equation andyielding a system of algebraic equations with random coefficients. Solution of theproblem may be provided with the aid of the Virtual Distortion Method (VDM) that has proved to be an efficient tool for the analysis of uncertain systems. Statistics will beprovided to the selected level of accuracy and some comparison with Monte-Carlosimulation will be reported.