A multi‐scale spectral stochastic method for homogenization of multi‐phase periodic composites with random material properties

Abstract

SUMMARYIn this work a spectral stochastic computational scheme is proposed that links the global properties of multi-phase periodic composites to the geometry and random material properties of their microstructural components. To propagate the uncertainties associated with the material properties to the microstructural response the scheme benefits from a combination of homogenization theory built into a finite element framework and the spectral representation of uncertainty based on Hermite Chaos where a probabilistic characterization of the solutions to a set of local problems defined on the period cell is first sought. A full stochastic description of the global (effective) properties is then obtained by averaging the solutions to the forgoing set of local problems over the unit cell. A representative subset of results is compared with the results obtained using Monte Carlo simulation to demonstrate the accuracy of the proposed procedure.

A hysteretic multiscale formulation for nonlinear dynamic analysis of composite materials

“…These are then used to derive the next increment of the perturbed micro-displacement vector 2 i d m(i) , using relation (49) as well as the increment of the macro equivalent nodal forces using relation (55). Assembling at the coarse element level the increment of the internal forces, defined at the coarse level is readily derived as:…”

“…Thus, a continuous mathematical model that is problem dependent replaces the fine scale information. On the other hand, multiscale methods use the fine scale information to formulate a numerically equivalent problem that can be solved in a coarser scale, usually through the finite element method [2,55]. An extensive review on the subject can be found in [33].…”

On semi‐analytical probabilistic finite element method for homogenization of the periodic fiber‐reinforced composites

“…There are various non-linear models [6,7] accounting for viscoelastic [8], elastoplastic [9], or viscoplastic [10] components, for the overall strength prediction [11] and for prediction of effective properties in really complex engineering systems [12]; the area of homogenized thermal (or electromagnetic) characteristics of composites has also been deeply explored [13,14]. Homogenization of random media in a broad sense has also many well-described realizations [15,16] and recently focuses on some upper and lower bounds for random structures [17], lower [18] and higher order [15] stochastic perturbation techniques as well as some spectral methods [19,20]. As one may realize, the progress in this area may be driven by both new composite models as well as by new computational techniques, which is the issue of this work.…”

A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures

“…Historically, most of the attempts were derived having recourse to the numerical simulations of random microstructures (relying on the experimental identification of some morphological characteristics, or assuming such properties; see [38]), coupled with a (Stochastic) finite elements analysis [5] (see for instance [12] [37]) and a statistical study performed on any quantity of interest (local strain or stress random fields, etc. ).…”