Homogenization of random heterogeneous media with inclusions of arbitrary shape modeled by XFEM

Effective properties of multiphase composites made of elastic materials with hierarchical structure

“…It is known that finite element analysis (FEA) methods are not sufficient to solve heterogeneous problems, since heterogeneities impose restrictions on the size of elements and make too expensive the discretization of heterogeneous structures (Bensoussan et al, 1978;Guedes & Kikuchi, 1990;Hollister & Kikuchi, 1992), unless they are combined with micromechanical and/or analytical methods (Hashin, 1983;Kalamkarov & Kolpakov, 1997;Nemat-Nasser & Hori, 1999;Aboudi et al, 1999;Nemat-Nasser, 1999;Guinovart-Diaz et al, 2005;Chatzigeorgiou & Charalambakis, 2005;Love & Batra, 2006;Chatzigeorgiou et al, 2007;Kalamkarov et al, 2009;Nie et al, 2011;Wu et al, 2014;Chatzigeorgiou et al, 2014;Berrehili, 2014;Abd-Alla et al, 2014;Tu & Pindera, 2014;Savvas et al, 2014;Mahmoud et al, 2014).…”

A Stochastic Multi-scale Approach for Numerical Modeling of Complex Materials—Application to Uniaxial Cyclic Response of Concrete

“…We recall that we only consider the case of linear displacements imposed all over the boundary of the ED (see equation (3)). Following the work presented in [22,30], we can write these imposed displacements at any node q of the N bo nodes of the boundary as:…”

“…To that purpose, the cross-correlation matrix R g g g (ξ ξ ξ ) of the underlying standard Gaussian fields g j (x; θ ) has to be determined. We recall (see equations (29) and (30)) that: (48) can be analytically or numerically inverted to calculate a unique ρ ρ ρ g g g (ξ ξ ξ ). Besides, it must be verified that the matrix ρ ρ ρ g g g (ξ ξ ξ ) really is a correlation matrix, namely that the auto-correlation functions ρ g g g j j (ξ ξ ξ ), j ∈ [1, ...m], as well as the correlation matrix ρ ρ ρ g g g (ξ ξ ξ ) are positive semidefinite for every separation distance ξ ξ ξ .…”

“…In [5], an efficient numerical strategy is presented to obtain effective tensors of random materials by coupling random micro-structures to tentative effective models within the Arlequin framework for model superposition [1]. In [30], micro-structures composed of a medium with randomly distributed inclusions of random shapes are generated and their behaviors are simulated with the extended finite element method (XFEM); homogenized properties at macroscale are then derived through the computation of mean response using Monte Carlo simulations. 3 In the work presented in this chapter, we focus on the numerical representation of the homogenized one-dimensional response of a concrete specimen in cyclic compressive loading, as it can be observed in lab tests.…”

Computational homogenisation from a 3D finite element model of asphalt concrete—linear elastic computations

“…3D computations are used to determine the effective properties. Savvas et al [44] study the homogenisation of structures with arbitrarily shaped inclusions by means of the extended finite element method. They investigate values of material contrast up to 1000, and volume fractions between 0.2 and 0.4.…”