Numerization of a memory effect for an homogenized composite material with a large contrast in the phase thermal conductivities

Dureisseix, David; Royer, Pascale; Faverjon, Béatrice

doi:10.1016/j.ijheatmasstransfer.2015.06.048

Cited by 3 publications

(3 citation statements)

References 30 publications

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“…As the heredity function is vanishing for long past times, it can be truncated as well, with a good approximation. A similar situation may arise for homogenisation of thermal problems, see [6]. For a discretised problem, the strict equivalence is obtained if the number of macroscopic internal variables is the same as the sum of the number of microscopic internal variables of the cell behind a macroscopic integration point that may still be huge.…”

Section: Reduced Order Modelling For Storage Compressionmentioning

confidence: 83%

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

Bhattacharyya

Dureisseix

Faverjon

2020

International Journal of Damage Mechanics

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This article deals with damage computation of heterogeneous structures containing locally periodic micro-structures. Such heterogeneous structure is extremely expensive to simulate using classical finite element methods, as the level of discretisation required to capture the micro-structural effects is too fine. The simulation time becomes even higher when dealing with highly non-linear material behaviour, e.g. damage, plasticity and such others. Therefore, a multi-scale strategy is proposed here that facilitates the simulation of non-linear heterogeneous material behaviour in a manner that is computationally feasible. Based on the asymptotic homogenisation theory, this multi-scale technique explores the micro–macro behaviour for elasto-(visco)plasticity coupled with damage. The theory inherently segregates the heterogeneous continua into a macroscopic homogeneous structure and an underlying heterogeneous microscopic periodic unit cell. Several heterogeneous structures have been simulated using the multi-scale method along with a one-dimensional verification with respect to a reference solution. Additionally, a reduced order modelling is used to prevent large memory requirement for storing micro-structural quantities of interest.

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Section: Reduced Order Modelling For Storage Compressionmentioning

confidence: 83%

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

Bhattacharyya

Dureisseix

Faverjon

2020

International Journal of Damage Mechanics

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“…Auriault and Lewandowska (1995) developed the parallel theory for mass transport in double-diffusivity media, leading to similar conclusions. Dureisseix et al (2015) recently proposed a computational approach to estimate the memory function by direct finite element calculation on a unit cell of the microstructure. Alternatively, Curto Idiart (2015, 2016) used the multiscale convergence approach (Allaire, 1992;Allaire and Briane, 1996) to determine the effective diffusion behaviour of ions in microstructured electrolytes driven by concentration gradients and electric fields.…”

Section: Review Of Existing Approachesmentioning

confidence: 99%

Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities

Brassart

Stainier

2019

Journal of the Mechanics and Physics of Solids

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This paper presents a homogenisation-based constitutive model to describe the effective transient diffusion behaviour in heterogeneous media in which there is a large contrast between the phase diffusivities. In this case mobile species can diffuse over long distances through the fast phase in the time scale of diffusion in the slow phase. At macroscopic scale, contrasted phase diffusivities lead to a memory effect that cannot be properly described by classical Fick's second law. Here we obtain effective governing equations through a two-scale approach for composite materials consisting of a fast matrix and slow inclusions. The micro-macro transition is similar to first-order computational homogenisation, and involves the solution of a transient diffusion boundary-value problem in a Representative Volume Element of the microstructure. Different from computational homogenisation, we propose a semi-analytical mean-field estimate of the composite response based on the exact solution for a single inclusion developed in our previous work [Brassart, L., Stainier, L., 2018. Effective transient behaviour of inclusions in diffusion problems. Z. Angew Math. Mech. 98, 981-998]. A key outcome of the model is that the macroscopic concentration is not one-to-one related to the macroscopic chemical potential, but obeys a local kinetic equation associated with diffusion in the slow phase. The history-dependent macroscopic response admits a representation based on internal variables, enabling efficient time integration. We show that the local chemical kinetics can result in non-Fickian behaviour in macroscale boundary-value problems.

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“…The analysis was later adapted to mass transfer [8]. Recently, Dureisseix et al [9] developed a finite element (FE) approach to evaluate the incremental evolution of the memory function for periodic microstructures by solving the transient diffusion problem in the slow phase, and also proposed heuristic semi-analytical estimates. An asymptotic homogenisation approach was also developed by Matine et al [10] to include edge and short time effects in transient heat conduction in laminate microstructures.…”

Section: Introductionmentioning

confidence: 99%

Effective transient behaviour of inclusions in diffusion problems

Brassart

Stainier

2018

Z Angew Math Mech

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This paper is concerned with the effective transport properties of heterogeneous media in which there is a high contrast between the phase diffusivities. In this case the transient response of the slow phase induces a memory effect at the macroscopic scale, which needs to be included in a macroscopic continuum description. This paper focuses on the slow phase, which we take as a dispersion of inclusions of arbitrary shape. We revisit the linear diffusion problem in such inclusions in order to identify the structure of the effective (average) inclusion response to a chemical load applied on the inclusion boundary. We identify a chemical creep function (similar to the creep function of viscoelasticity), from which we construct estimates with a reduced number of relaxation modes. The proposed estimates admit an equivalent representation based on a finite number of internal variables. These estimates allow us to predict the average inclusion response under arbitrary time‐varying boundary conditions at very low computational cost. A heuristic generalisation to concentration‐dependent diffusion coefficient is also presented. The proposed estimates for the effective transient response of an inclusion can serve as a building block for the formulation of multi‐inclusion homogenisation schemes.

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Numerization of a memory effect for an homogenized composite material with a large contrast in the phase thermal conductivities

Cited by 3 publications

References 30 publications

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

Numerical homogenisation based on asymptotic theory and model reduction for coupled elastic-viscoplastic damage

Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities

Effective transient behaviour of inclusions in diffusion problems

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