Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume

Hazanov, S.; Huet, C.

doi:10.1016/0022-5096(94)90022-1

Cited by 301 publications

(150 citation statements)

References 7 publications

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“…Xia et al [234] reported that the homogeneous boundary conditions, when applied on periodic microstructures, "are not only overconstrained, but may also violate the boundary traction periodicity conditions" under loading types with shear components. Hazanov and Huet [235], Hazanov and Amieur [236], and Pahr and Zysset [237] proposed uniform mixedtype boundary conditions that consider applying constant traction boundary conditions on some parts of the boundary and linear displacement boundary conditions to the other parts such that the apparent elasticity tensor for this boundary condition lies between the apparent tensors obtained with homogeneous boundary conditions. Mesarovic and Padbidri [238] argued that there is no reason to assume that an RVE with random microstructure behaves as a periodic unit cell and suggested the use of minimal kinematic boundary conditions.…”

Section: Computational Homogenizationmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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Section: Computational Homogenizationmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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“…As previously stated, if the meso-scale volume-element ω on which the averaging is performed is large enough, the so-called RVE is statistically representative and a unique material tensor C eff M can be obtained for these different 1 Although not true for general MBCs, particular MBCs such as the orthogonal uniform ones can be defined in a particular way as to satisfy the Hill-Mandel condition (17), see the discussion in [42]. Assuming a rectangular parallelepiped RVE, on every face we constrain along one direction -says x-the displacements to um x = z i=x ε Mxi x i , so that u mx = 0, and along the two other directions tm j = z k=x σ Mjk nm k , j = y, z.…”

Section: Generalities On Representative and Statistical Volume Elementsmentioning

confidence: 99%

A stochastic computational multiscale approach; Application to MEMS resonators

Lucas¹,

Golinval²,

Paquay³

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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The aim of this work is to develop a stochastic multiscale model for polycrystalline materials, which accounts for the uncertainties in the micro-structure. At the finest scale, we model the micro-structure using a random Voronoï tessellation, each grain being assigned a random orientation. Then, we apply a computational homogenization procedure on statistical volume elements to obtain a stochastic characterization of the elasticity tensor at the meso-scale. A random field of the meso-scale elasticity tensor can then be generated based on the information obtained from the SVE simulations. Finally, using a stochastic finite element method, these meso-scale uncertainties are propagated to the coarser scale. As an illustration we study the resonance frequencies of MEMS micro-beams made of poly-silicon materials, and we show that the stochastic multiscale approach predicts results in agreement with a Monte Carlo analysis applied directly on the fine finite-element model, i.e. with an explicit discretization of the grains.

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“…Consequently, we can define the corresponding static and kinematic statistical apparent elasticity matrices, denoted respectively by [L app σ ] and [L app ]. Using the same methodology as in Section 2.1.3 (that is, introducing a partitioning of Ω), it can be shown that the following inequality holds for any given realization of the random media [10]: The boundedness property (18) holds for apparent tensors defined with respect to very particular kinds of boundary conditions (namely, KUBC and SUBC) which may not be encountered in practical applications or may be tricky to reproduce experimentally (as pointed out in [8] …”

Section: Inequalities Between Apparent Tensors At Various Mesoscalesmentioning

confidence: 99%

“…Note that the above mechanical definitions do not coincide with the energetic ones, unless the retained boundary conditions satisfy the Hill condition (see [8] for a discussion). A particular family of MBC satisfying Eq.…”

Section: Inequalities Between Apparent Tensors At Various Mesoscalesmentioning

confidence: 99%

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

Guilleminot

Noshadravan

Soize

et al. 2011

Computer Methods in Applied Mechanics and Engineering

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In this paper, we address the construction of a prior stochastic model for nonGaussian deterministically-bounded positive-definite matrix-valued random fields in the context of mesoscale modeling of heterogeneous elastic microstructures. We first introduce the micromechanical framework and recall, in particular, Huet's Partition Theorem. Based on the latter, we discuss the nature of hierarchical bounds and define, under some given assumptions, deterministic bounds for the apparent elasticity tensor. Having recourse to the Maximum Entropy Principle under the constraints defined by the available information, we then introduce two random matrix models. It is shown that an alternative formulation of the boundedness constraints further allows constructing a probabilistic model for deterministically-bounded positive-definite matrix-valued random fields. Such a construction is presented and relies on a class of random fields previously defined. We finally exemplify the overall methodology considering an experimental database obtained from EBSD measurements and provide a simple numerical application.Key words: Micromechanics; Heterogeneous materials; Apparent elasticity tensor; Mesoscale modeling; Random field; Non-Gaussian. $ J. Guilleminot, A. Noshadravan, R. Ghanem and C. Soize, A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures,

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Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume

Cited by 301 publications

References 7 publications

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

A stochastic computational multiscale approach; Application to MEMS resonators

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

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