2D versus 3D probabilistic homogenization of the metallic fiber-reinforced composites by the perturbation-based stochastic Finite Element Method Marcin Kamin ´ ski ⇑ , Marlena Kazimierczak Department of Structural Mechanics, Faculty of Civil Engineering, Architecture and Environmental Engineering Technical University of Lódz ´, Al. Politechniki 6, 90-924 Lódz ´, Poland article info Article history: Available online 30 October 2013 Keywords: Homogenization method Probabilistic analysis Least Squares Method Metallic composites Finite Element Method Representative Volume Element abstract The main purpose of this work is computational simulation of the expectations, standard deviations, skewness and kurtosis of the homogenized tensor for some composites with metallic components. The Representative Volume Element (RVE) of this composite contains a single cylindrical fiber and their com- ponents are treated as statistically homogeneous and isotropic media uniquely defined by the Gaussian elastic modulus. Probabilistic approach is based on the generalized stochastic perturbation technique allowing for large random dispersions of the input random variables and is implemented using the poly- nomial response functions recovered using the Least Squares Method. Homogenization technique employed is dual and consists of (1) stress version of the effective modules method and (2) its displace- ments counterpart based on the deformation energies of the real and homogenized composites. The cell problem is solved for the first case by the plane strain homogenization-oriented code MCCEFF and, in the 3D case, using the system ABAQUS Ò (8-node linear brick finite elements C3D8), where the uniform defor- mations are imposed on specific outer surfaces of the composite cell; probabilistic part is carried out in the symbolic computations package MAPLE Ò . We compare probabilistic coefficients of the effective elas- ticity tensor computed in this way with the corresponding coefficients for their upper and lower bounds and this is done for the composite with small and larger contrast between Young moduli of the fiber and the matrix. The main conclusion coming from the performed numerical analysis is a very good agreement of the probabilistic moments resulting from 2 and 3D computer models; this conclusion is totally inde- pendent from the contrast between elastic moduli of both composite components. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction There is a variety of homogenization techniques leading to a determination of the effective tensors characterizing the homoge- neous medium equivalent to the original composite structure [1,2,5,13,16]. They are derived as some analytical bounds or the ex- act approximations based upon the deformation criteria or the deformation energy. Analogously, there are several numerical ap- proaches to the homogenization problem, where the so-called cell problem on the Representative Volume Element (RVE) [4,14] is usually solved to predict the homogenized behavior of the entire composite structure [5–8]. It is most frequently carried out using the Finite Element Method by the straightforward spatial discreti- zation of this RVE and the relevant solution via the displacement or the stress-based formulations [7]. Different typical stress boundary conditions in-between the composite components and periodicity conditions on external edges of the RVE can be applied in this case [5,6]. Alternatively, constant deformations on the RVE external boundaries are imposed and we assure their continuity at the interface to calculate effective tensor components [9]. The first ap- proach needs additional spatial averaging of the induced stresses fields, while the second allows for a direct computation of the homogenized tensor components from the internal energy accu- mulated in the RVE as the result of the applied uniform deformation. The uncertainty in the composite parameters essentially does not change the homogenization methods in the sense that some probabilistic technique needs to be added to the deterministic apparatus to perform randomization of the model. The Monte-Car- lo simulation in its various implementations can be traditionally used in this context to get the statistical estimators of the effective tensor. Doing so, a direct probabilistic integration technique can be applied, when the analytical approximations for this tensor are ta- ken into account as well as using the perturbation or the expan- sion-based methodologies. The stochastic perturbation technique is quite sufficient to carry out all computations because the elastic 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.10.035 ⇑ Corresponding author. Tel.: +48 42 6313571. E-mail address: Marcin.Kaminski@p.lodz.pl (M. Kamin ´ ski). URL: http://www.kmk.p.lodz.pl/pracownicy/kaminski/index.htm (M. Kamin ´ ski). Composite Structures 108 (2014) 1009–1018 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct