Stochastic finite elements: Where is the physics?

Abstract

The micromechanics based on the Hill-Mandel condition indicates that the majority of stochastic finite element methods hinge on random field (RF) models of material properties (such as Hooke?s law) having no physical content, or even at odds with physics. At the same time, that condition allows one to set up the RFs of stiffness and compliance tensors in function of the mesoscale and actual random microstructure of the given material. The mesoscale is defined through a Statistical Volume Element (SVE), i.e. a material domain below the Representative Volume Element (RVE) level. The paper outlines a procedure for stochastic scale-dependent homogenization leading to a determination of mesoscale one-point and two-point statistics and, thus, a construction of analytical RF models.Mikromehanika zasnovana na Hill-Mandel-ovom uslovu ukazuje da vecina stohastickih metoda konacnih elemenata pociva na RF (random field) modelima materijalnih svojstava, koji su cesto u neslaganju sa fizikom problema. Istovremeno, ovaj uslov omogucava da se formiraju RF tenzori krutosti i fleksibilnosti u funkciji mikrostrukture analiziranog materijala na mezonivou. Mezonivo je definisan na bazi statistickog zapreminskog elementa (SZE), koji je ispod materijalne skale reprezentativnog zapreminskog elementa (RZE). U radu je formulisana procedura homogenizacije, zavisna od stohastickog nivoa, koja vodi odgovarajucoj na mezonivou statistici i izgradnji analitickih RF modela.