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Stochastic finite elements: where is the physics? (English) Zbl 1313.74108

Summary: 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.

MSC:

74M25 Micromechanics of solids
65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
60G60 Random fields
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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