Determination of the size of the representative volume element for random composites: statistical and numerical approach

Kanit, Toufik; Forest, Samuel; Galliet, Isabelle; Mounoury, Valérie; Jeulin, Dominique

doi:10.1016/s0020-7683(03)00143-4

Cited by 1,649 publications

(1,232 citation statements)

References 33 publications

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“…For periodic structures, under the assumption of the periodic local field, the RVE can be selected as the repeated unit since this RVE contains all necessary information about the micro-structure. For random structures, the RVEs are defined in a statistical way (Kanit et al, 2003). In all generalities, the relation between the homogenized fieldf and the local field f is expressed under the form of the volume integral (so-called homogenization) over the RVE as…”

Section: Computational Micro-mechanics Methodsmentioning

confidence: 99%

Experimental and computational micro-mechanical investigations of compressive properties of polypropylene/multi-walled carbon nanotubes nanocomposite foams

Wan¹,

Tran

Leblanc

et al. 2015

Mechanics of Materials

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The compressive behavior of nanocomposite foams is studied by both experimental and computational micro-mechanics approaches with the aim of providing an efficient computational model for this kind of material.The nanocomposites based on polypropylene (PP) and different contents of multi-walled carbon nanotubes (CNTs) method. The nanocomposite samples are foamed using super-critical carbon dioxide (ScCO 2 ) as blowing agent at different soaking temperatures. The influence of this foaming parameter on the morphological characteristics of the foam micro-structure is discussed. Differential Scanning Calorimetry (DSC) measurements are used to quantify the crystallinity degree of both nanocomposites and foams showing that the crystallinity degree is reduced after the foaming process. This modification leads to mechanical properties of the foam cell walls that are different from the raw nanocomposite PP/CNTs material. Three-point bending tests are performed on the latter to measure the flexural modulus in terms of the crystallinity degree. Uniaxial compression tests are then performed on the foamed samples under quasi-static conditions in order to extract the macro-scale compressive response. Next, a two-level multi-scale approach is developed to model the behavior of the foamed nanocomposite material. On the one hand, the micromechanical properties of nanocomposite PP/CNTs cell walls are evaluated from a theoretical homogenization model accounting for the micro-structure of the semi-crystalline PP, for the degree of crystallinity, and for the CNT volume fraction. The applicability of this theoretical model is demonstrated via the comparison with experimental data from the described experimental measurements and from literature. On the other hand, the macroscopic behavior of the foamed material is evaluated using a computational micromechanics model using tetrakaidecahedron unit cells and periodic boundary conditions to estimate the homogenized properties. The unit cell is combined with several geometrical imperfections in order to capture the elastic collapse of the foamed material. The numerical results are compared to the experimental measurements and it is shown that the proposed unit cell computational micro-mechanics model can be used to estimate the homogenized behavior, including the linear and plateau regimes, of nanocomposite foams.

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Section: Computational Micro-mechanics Methodsmentioning

confidence: 99%

Experimental and computational micro-mechanical investigations of compressive properties of polypropylene/multi-walled carbon nanotubes nanocomposite foams

Wan¹,

Tran

Leblanc

et al. 2015

Mechanics of Materials

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“…Many authors, e.g., Refs. [215][216][217][218][219][220][221][222][223][224], have shown that in pure mechanical linear and nonlinear problems, the effective behavior derived under periodic boundary conditions is bounded by linear displacement boundary conditions from above and constant traction boundary conditions from below for a finite size of the RVE. Kaczmarczyk et al [225] made similar conclusions in the context of second-order computational homogenization.…”

Section: Computational Homogenizationmentioning

confidence: 99%

“…The RVE must be large enough to be statistically representative of the composite so that it effectively includes a sampling of all microstructural heterogeneities that occur in the composite [245]. On the other hand, it must remain sufficiently small to be considered as a volume element of continuum mechanics [223]. The first-order computational homogenization scheme critically relies on the principle of separation of scales, which requires that "the microscopic length scale is assumed to be much smaller than the characteristic length over which the macroscopic loading varies in space" [197].…”

Section: 22mentioning

confidence: 99%

“…Shan and Gokhale [264] utilized probability density functions of critical microstructural variables such as nearest neighbor distances and also microstress distribution to derive a sufficiently small RVE for a ceramic matrix composite possessing fiber-rich and fiber-poor regions. Kanit et al [223] proposed a quantitative definition for the RVE through statistical and numerical approaches in the case of linear elasticity and thermal conductivity. Based on their studies, the size of the RVE is a function of five parameters: the physical property, the contrast of properties, the volume fractions of components, the relative precision for the estimation of the effective property, and the number of realizations of the microstructure.…”

Section: 22mentioning

confidence: 99%

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

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

185

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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“…In the context of computational homogenization methods, the meso-scale finite element problem is solved by applying on the RVE adequate Boundary Conditions (BCs) which should satisfy the Hill-Mandel condition stating that the deformation energy at the macroscopic level should be equal to the volume average of the micro-scale stress work. To be called strictly representative a RVE should be defined so that the homogenized results are not dependent on the (energetically consistent) BCs, although the use of periodic boundary conditions (PBCs) allows using meso-scale volume elements of smaller sizes as the convergence rate of the homogenized properties with respect to the RVE size is faster than with other BCs [17,18]. A review of multi-scale computational homogenization can be found in reference [19].…”

Section: Introductionmentioning

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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Determination of the size of the representative volume element for random composites: statistical and numerical approach

Cited by 1,649 publications

References 33 publications

Experimental and computational micro-mechanical investigations of compressive properties of polypropylene/multi-walled carbon nanotubes nanocomposite foams

Experimental and computational micro-mechanical investigations of compressive properties of polypropylene/multi-walled carbon nanotubes nanocomposite foams

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

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