Modelling of hot compaction of metal powder by homogenization

Auriault, Jean‐Louis; Bouvard, D.; Dellis, C.; Lafer, M.

doi:10.1016/0167-6636(92)90005-x

Cited by 9 publications

(11 citation statements)

References 7 publications

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“…This might more realistically model the pore structure of rocks. The equations for upscaling could also be augmented by giving the matrix a nonlinear rheology, as in the works by Auriault et al [1992] and Geindreau and Auriault [1999]. This may be particularly important for magma migration; a nonlinear matrix rheology was needed to computationally model physical experiments for shear bands in the work by Katz et al [2006], and in general, nonlinear power law rheologies are expected in the dislocation creep regime [e.g., Hirth and Kohlstedt , 1995b].…”

Section: Discussionmentioning

confidence: 99%

A multiscale model of partial melts: 1. Effective equations

Simpson¹,

Spiegelman²,

Weinstein³

2010

J. Geophys. Res.

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[1] Developing accurate and tractable mathematical models for partially molten systems is critical for understanding the dynamics of magmatic plate boundaries as well as the geochemical evolution of the planet. Because these systems include interacting fluid and solid phases, developing such models can be challenging. The composite material of melt and solid may have emergent properties, such as permeability and compressibility that are absent in each phase alone. Previous work by several authors have used multiphase flow theory to derive macroscopic equations based on conservation principles and assumptions about interphase forces and interactions. Here we present a complementary approach using homogenization, a multiple scale theory. Our point of departure is a model of the microstructure, assumed to possess an arbitrary, but periodic, microscopic geometry of interpenetrating melt and matrix. At this scale, incompressible Stokes flow is assumed to govern both phases, with appropriate interface conditions. Homogenization systematically leads to macroscopic equations for the melt and matrix velocities, as well as the bulk parameters, permeability and bulk viscosity, without requiring ad hoc closures for interphase forces. We show that homogenization can lead to a range of macroscopic models depending on the relative contrast in melt and solid properties such as viscosity or velocity. In particular, we identify a regime that is in good agreement with previous formulations, without including their attendant assumptions. Thus, this work serves as independent verification of these models. In addition, homogenization provides a consistent machinery for computing consistent macroscopic constitutive relations such as permeability and bulk viscosity that are consistent with a given microstructure. These relations are explored numerically in the companion paper.

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Section: Discussionmentioning

confidence: 99%

A multiscale model of partial melts: 1. Effective equations

Simpson¹,

Spiegelman²,

Weinstein³

2010

J. Geophys. Res.

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“…We slightly deviate from the original approach described in Auriault (2002), Auriault and Adler (1995), Auriault et al (1992Auriault et al ( , 2005, Auriault and Lewandowska (1993, 1997, and Auriault and Royer (1993a,b). We split the differentiation ∇() = ∇ b ()+∇ s () thus distinguishing between differentiation toward r s on the small scale denoted by ∇ s (c(r b , r s )) and differentiation toward r b on the larger scale denoted by ∇ b (c(r b , r s )).…”

Section: The Non-dimensional Equations At Two Scalesmentioning

confidence: 99%

“…It follows from the problem statement above that of the many articles of Auriault, we use article (Bouddour et al 1996) as a key-reference for the derivations. There are, however, more than 70 articles and a book (Auriault et al 2009) coauthored by Auriault that deal with similar aspects (Auriault 2002;Auriault and Adler 1995;Auriault et al 1992Auriault et al , 2005Auriault and Lewandowska 1993, 1997Auriault and Royer 1993a,b). Moreover, there are new developments (Allaire et al 2010a,b) that concern homogenization for reactive flow in a moving frame of reference.…”

Section: Introductionmentioning

confidence: 99%

Computation of the Longitudinal and Transverse Dispersion Coefficient in an Adsorbing Porous Medium Using Homogenization

2011

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This article compares for the first time, local longitudinal and transverse dispersion coefficients obtained by homogenization with experimental data of dispersion coefficients in porous media, using the correct porosity dependence. It is shown that the longitudinal dispersion coefficient can be reasonably represented by a simple periodic unit cell (PUC), which consists of a single sphere in a cube. We present a slightly modified and simplified approach to derive the homogenized equations, which emphasizes physical aspects of homogenization. Subsequently, we give full dimensional expressions for the dispersion tensor based on a comparison with the convective dispersion equation used for contaminant transport, inclusive the correct dependence on porosity. For the PUC of choice, the dispersion relations are identical to the relations obtained for periodic media. We show that commercial finite element software can be readily used to compute longitudinal and transverse dispersion coefficients in 2D and 3D. The 3D results are for the first time obtained at relevant Peclet numbers. There is good agreement for longitudinal dispersion. The computed transverse dispersion coefficients for a single sphere in a cube are much too low. The effect of adsorption on the dispersion coefficient is also studied. Adsorption does not affect the transverse dispersion coefficient. However, adsorption enhances the longitudinal dispersion coefficient in agreement with an analysis of homogenization applied to Taylor dispersion discussed in the literature.

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“…By contrast, the homogenization of the viscoplastic behavior requires a priori an infinite number of numerical simulations. However, this number of simulations can be reduced by taking into account some theoretical results (Auriault et al, 1992;Suquet, 1993;Orgéas et al, 2007). Indeed, the following can be shown:…”

Section: Macro-strain Paths Definition (Step 4)mentioning

confidence: 99%

“…In this upscaling process, the viscous behavior of ice is described by a power law of exponent n (secondary creep) as in Theile et al (2011). Due to the nonlinear behavior under consideration, the homogenization does not provide the complete structure of the macroscopic constitutive equation (Auriault et al, 1992(Auriault et al, , 2002Geindreau and Auriault, 1999;Orgéas et al, 2007). Nevertheless, it can be shown that the exponent n is preserved at the macroscopic scale and that the macroscopic dissipation power is the volume average of the local one (Suquet, 1993).…”

Section: Introductionmentioning

confidence: 99%

Numerical homogenization of the viscoplastic behavior of snow based on X-ray tomography images

2017

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Abstract. While the homogenization of snow elastic properties has been widely reported in the literature, homogeneous rate-dependent behavior responsible for the densification of the snowpack has hardly ever been upscaled from snow microstructure. We therefore adapt homogenization techniques developed within the framework of elasticity to the study of snow viscoplastic behavior. Based on the definition of kinematically uniform boundary conditions, homogenization problems are applied to 3-D images obtained from X-ray tomography, and the mechanical response of snow samples is explored for several densities. We propose an original postprocessing approach in terms of viscous dissipated powers in order to formulate snow macroscopic behavior. Then, we show that Abouaf models are able to capture snow viscoplastic behavior and we formulate a homogenized constitutive equation based on a density parametrization. Eventually, we demonstrate the ability of the proposed models to account for the macroscopic mechanical response of snow for classical laboratory tests.

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Modelling of hot compaction of metal powder by homogenization

Cited by 9 publications

References 7 publications

A multiscale model of partial melts: 1. Effective equations

A multiscale model of partial melts: 1. Effective equations

Computation of the Longitudinal and Transverse Dispersion Coefficient in an Adsorbing Porous Medium Using Homogenization

Numerical homogenization of the viscoplastic behavior of snow based on X-ray tomography images

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